From 2b4d9dd60cdd3bba37591b832d5c3fb4609a9d14 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Sun, 22 Aug 2021 23:23:31 +0200 Subject: [PATCH] deleted files --- .../_build/.doctrees/Clustering.doctree | Bin 411603 -> 0 bytes .../_build/.doctrees/chapter1.doctree | Bin 432356 -> 0 bytes .../_build/.doctrees/chapter10.doctree | Bin 418991 -> 0 bytes .../_build/.doctrees/chapter11.doctree | Bin 554808 -> 0 bytes .../_build/.doctrees/chapter2.doctree | Bin 169314 -> 0 bytes .../_build/.doctrees/chapter3.doctree | Bin 144163 -> 0 bytes .../_build/.doctrees/chapter4.doctree | Bin 281174 -> 0 bytes .../_build/.doctrees/chapter5.doctree | Bin 379356 -> 0 bytes .../_build/.doctrees/chapter6.doctree | Bin 208987 -> 0 bytes .../_build/.doctrees/chapter7.doctree | Bin 148410 -> 0 bytes .../_build/.doctrees/chapter8.doctree | Bin 202140 -> 0 bytes .../_build/.doctrees/chapter9.doctree | Bin 198601 -> 0 bytes .../_build/.doctrees/content.doctree | Bin 2850 -> 0 bytes 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"cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Linear Regression, basic Elements\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureAug21.mp4?vrtx=view-as-webpage)\n", - "\n", - "\n", - "## Introduction\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "Our emphasis throughout this series of lectures \n", - "is on understanding the mathematical aspects of\n", - "different algorithms used in the fields of data analysis and machine learning. \n", - "\n", - "However, where possible we will emphasize the\n", - "importance of using available software. We start thus with a hands-on\n", - "and top-down approach to machine learning. The aim is thus to start with\n", - "relevant data or data we have produced \n", - "and use these to introduce statistical data analysis\n", - "concepts and machine learning algorithms before we delve into the\n", - "algorithms themselves. The examples we will use in the beginning, start with simple\n", - "polynomials with random noise added. We will use the Python\n", - "software package [Scikit-Learn](http://scikit-learn.org/stable/) and\n", - "introduce various machine learning algorithms to make fits of\n", - "the data and predictions. We move thereafter to more interesting\n", - "cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example).\n", - "These are examples where we can easily set up the data and\n", - "then use machine learning algorithms included in for example\n", - "**Scikit-Learn**. \n", - "\n", - "These examples will serve us the purpose of getting\n", - "started. Furthermore, they allow us to catch more than two birds with\n", - "a stone. They will allow us to bring in some programming specific\n", - "topics and tools as well as showing the power of various Python \n", - "libraries for machine learning and statistical data analysis. \n", - "\n", - "Here, we will mainly focus on two\n", - "specific Python packages for Machine Learning, Scikit-Learn and\n", - "Tensorflow (see below for links etc). Moreover, the examples we\n", - "introduce will serve as inputs to many of our discussions later, as\n", - "well as allowing you to set up models and produce your own data and\n", - "get started with programming.\n", - "\n", - "\n", - "\n", - "## What is Machine Learning?\n", - "\n", - "Statistics, data science and machine learning form important fields of\n", - "research in modern science. They describe how to learn and make\n", - "predictions from data, as well as allowing us to extract important\n", - "correlations about physical process and the underlying laws of motion\n", - "in large data sets. The latter, big data sets, appear frequently in\n", - "essentially all disciplines, from the traditional Science, Technology,\n", - "Mathematics and Engineering fields to Life Science, Law, education\n", - "research, the Humanities and the Social Sciences. \n", - "\n", - "It has become more\n", - "and more common to see research projects on big data in for example\n", - "the Social Sciences where extracting patterns from complicated survey\n", - "data is one of many research directions. Having a solid grasp of data\n", - "analysis and machine learning is thus becoming central to scientific\n", - "computing in many fields, and competences and skills within the fields\n", - "of machine learning and scientific computing are nowadays strongly\n", - "requested by many potential employers. The latter cannot be\n", - "overstated, familiarity with machine learning has almost become a\n", - "prerequisite for many of the most exciting employment opportunities,\n", - "whether they are in bioinformatics, life science, physics or finance,\n", - "in the private or the public sector. This author has had several\n", - "students or met students who have been hired recently based on their\n", - "skills and competences in scientific computing and data science, often\n", - "with marginal knowledge of machine learning.\n", - "\n", - "Machine learning is a subfield of computer science, and is closely\n", - "related to computational statistics. It evolved from the study of\n", - "pattern recognition in artificial intelligence (AI) research, and has\n", - "made contributions to AI tasks like computer vision, natural language\n", - "processing and speech recognition. Many of the methods we will study are also \n", - "strongly rooted in basic mathematics and physics research. \n", - "\n", - "Ideally, machine learning represents the science of giving computers\n", - "the ability to learn without being explicitly programmed. The idea is\n", - "that there exist generic algorithms which can be used to find patterns\n", - "in a broad class of data sets without having to write code\n", - "specifically for each problem. The algorithm will build its own logic\n", - "based on the data. You should however always keep in mind that\n", - "machines and algorithms are to a large extent developed by humans. The\n", - "insights and knowledge we have about a specific system, play a central\n", - "role when we develop a specific machine learning algorithm. \n", - "\n", - "Machine learning is an extremely rich field, in spite of its young\n", - "age. The increases we have seen during the last three decades in\n", - "computational capabilities have been followed by developments of\n", - "methods and techniques for analyzing and handling large date sets,\n", - "relying heavily on statistics, computer science and mathematics. The\n", - "field is rather new and developing rapidly. Popular software packages\n", - "written in Python for machine learning like\n", - "[Scikit-learn](http://scikit-learn.org/stable/),\n", - "[Tensorflow](https://www.tensorflow.org/),\n", - "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all\n", - "freely available at their respective GitHub sites, encompass\n", - "communities of developers in the thousands or more. And the number of\n", - "code developers and contributors keeps increasing. Not all the\n", - "algorithms and methods can be given a rigorous mathematical\n", - "justification, opening up thereby large rooms for experimenting and\n", - "trial and error and thereby exciting new developments. However, a\n", - "solid command of linear algebra, multivariate theory, probability\n", - "theory, statistical data analysis, understanding errors and Monte\n", - "Carlo methods are central elements in a proper understanding of many\n", - "of algorithms and methods we will discuss.\n", - "\n", - "\n", - "\n", - "The approaches to machine learning are many, but are often split into\n", - "two main categories. In *supervised learning* we know the answer to a\n", - "problem, and let the computer deduce the logic behind it. On the other\n", - "hand, *unsupervised learning* is a method for finding patterns and\n", - "relationship in data sets without any prior knowledge of the system.\n", - "Some authours also operate with a third category, namely\n", - "*reinforcement learning*. This is a paradigm of learning inspired by\n", - "behavioral psychology, where learning is achieved by trial-and-error,\n", - "solely from rewards and punishment.\n", - "\n", - "Another way to categorize machine learning tasks is to consider the\n", - "desired output of a system. Some of the most common tasks are:\n", - "\n", - " * Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.\n", - "\n", - " * Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.\n", - "\n", - " * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n", - "\n", - "The methods we cover have three main topics in common, irrespective of\n", - "whether we deal with supervised or unsupervised learning. The first\n", - "ingredient is normally our data set (which can be subdivided into\n", - "training and test data), the second item is a model which is normally a\n", - "function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. \n", - "\n", - "The last ingredient is a so-called **cost**\n", - "function which allows us to present an estimate on how good our model\n", - "is in reproducing the data it is supposed to train. \n", - "At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of **gradient** methods.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Software and needed installations\n", - "\n", - "We will make extensive use of Python as programming language and its\n", - "myriad of available libraries. You will find\n", - "Jupyter notebooks invaluable in your work. You can run **R**\n", - "codes in the Jupyter/IPython notebooks, with the immediate benefit of\n", - "visualizing your data. You can also use compiled languages like C++,\n", - "Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be\n", - "on Python.\n", - "\n", - "\n", - "If you have Python installed (we strongly recommend Python3) and you feel\n", - "pretty familiar with installing different packages, we recommend that\n", - "you install the following Python packages via **pip** as \n", - "\n", - "1. pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow \n", - "\n", - "For Python3, replace **pip** with **pip3**.\n", - "\n", - "For OSX users we recommend, after having installed Xcode, to\n", - "install **brew**. Brew allows for a seamless installation of additional\n", - "software via for example \n", - "\n", - "1. brew install python3\n", - "\n", - "For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,\n", - "you can use **pip** as well and simply install Python as \n", - "\n", - "1. sudo apt-get install python3 (or python for pyhton2.7)\n", - "\n", - "etc etc. \n", - "\n", - "\n", - "\n", - "## Python installers\n", - "\n", - "If you don't want to perform these operations separately and venture\n", - "into the hassle of exploring how to set up dependencies and paths, we\n", - "recommend two widely used distrubutions which set up all relevant\n", - "dependencies for Python, namely \n", - "\n", - "* [Anaconda](https://docs.anaconda.com/), \n", - "\n", - "which is an open source\n", - "distribution of the Python and R programming languages for large-scale\n", - "data processing, predictive analytics, and scientific computing, that\n", - "aims to simplify package management and deployment. Package versions\n", - "are managed by the package management system **conda**. \n", - "\n", - "* [Enthought canopy](https://www.enthought.com/product/canopy/) \n", - "\n", - "is a Python\n", - "distribution for scientific and analytic computing distribution and\n", - "analysis environment, available for free and under a commercial\n", - "license.\n", - "\n", - "Furthermore, [Google's Colab](https://colab.research.google.com/notebooks/welcome.ipynb) is a free Jupyter notebook environment that requires \n", - "no setup and runs entirely in the cloud. Try it out!\n", - "\n", - "\n", - "## Useful Python libraries\n", - "Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)\n", - "\n", - "* [NumPy](https://www.numpy.org/) is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays\n", - "\n", - "* [The pandas](https://pandas.pydata.org/) library provides high-performance, easy-to-use data structures and data analysis tools \n", - "\n", - "* [Xarray](http://xarray.pydata.org/en/stable/) is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!\n", - "\n", - "* [Scipy](https://www.scipy.org/) (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. \n", - "\n", - "* [Matplotlib](https://matplotlib.org/) is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.\n", - "\n", - "* [Autograd](https://github.com/HIPS/autograd) can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives\n", - "\n", - "* [SymPy](https://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. \n", - "\n", - "* [scikit-learn](https://scikit-learn.org/stable/) has simple and efficient tools for machine learning, data mining and data analysis\n", - "\n", - "* [TensorFlow](https://www.tensorflow.org/) is a Python library for fast numerical computing created and released by Google\n", - "\n", - "* [Keras](https://keras.io/) is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano\n", - "\n", - "* And many more such as [pytorch](https://pytorch.org/), [Theano](https://pypi.org/project/Theano/) etc \n", - "\n", - "## Installing R, C++, cython or Julia\n", - "\n", - "You will also find it convenient to utilize **R**. We will mainly\n", - "use Python during our lectures and in various projects and exercises.\n", - "Those of you\n", - "already familiar with **R** should feel free to continue using **R**, keeping\n", - "however an eye on the parallel Python set ups. Similarly, if you are a\n", - "Python afecionado, feel free to explore **R** as well. Jupyter/Ipython\n", - "notebook allows you to run **R** codes interactively in your\n", - "browser. The software library **R** is really tailored for statistical data analysis\n", - "and allows for an easy usage of the tools and algorithms we will discuss in these\n", - "lectures.\n", - "\n", - "To install **R** with Jupyter notebook \n", - "[follow the link here](https://mpacer.org/maths/r-kernel-for-ipython-notebook)\n", - "\n", - "\n", - "\n", - "\n", - "## Installing R, C++, cython, Numba etc\n", - "\n", - "\n", - "For the C++ aficionados, Jupyter/IPython notebook allows you also to\n", - "install C++ and run codes written in this language interactively in\n", - "the browser. Since we will emphasize writing many of the algorithms\n", - "yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming\n", - "languages.\n", - "\n", - "To add more entropy, **cython** can also be used when running your\n", - "notebooks. It means that Python with the jupyter notebook\n", - "setup allows you to integrate widely popular softwares and tools for\n", - "scientific computing. Similarly, the \n", - "[Numba Python package](https://numba.pydata.org/) delivers increased performance\n", - "capabilities with minimal rewrites of your codes. With its\n", - "versatility, including symbolic operations, Python offers a unique\n", - "computational environment. Your jupyter notebook can easily be\n", - "converted into a nicely rendered **PDF** file or a Latex file for\n", - "further processing. For example, convert to latex as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " pycod jupyter nbconvert filename.ipynb --to latex \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "And to add more versatility, the Python package [SymPy](http://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python. \n", - "\n", - "Finally, if you wish to use the light mark-up language \n", - "[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML \n", - "formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using **doconce**.\n", - "\n", - "\n", - "\n", - "## Numpy examples and Important Matrix and vector handling packages\n", - "\n", - "There are several central software libraries for linear algebra and eigenvalue problems. Several of the more\n", - "popular ones have been wrapped into ofter software packages like those from the widely used text **Numerical Recipes**. The original source codes in many of the available packages are often taken from the widely used\n", - "software package LAPACK, which follows two other popular packages\n", - "developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.\n", - "\n", - " * LINPACK: package for linear equations and least square problems.\n", - "\n", - " * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.\n", - "\n", - " * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from .\n", - "\n", - "## Basic Matrix Features\n", - "\n", - "Matrix properties reminder" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A} =\n", - " \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", - " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", - " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", - " a_{41} & a_{42} & a_{43} & a_{44}\n", - " \\end{bmatrix}\\qquad\n", - "\\mathbf{I} =\n", - " \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n", - " 0 & 1 & 0 & 0 \\\\\n", - " 0 & 0 & 1 & 0 \\\\\n", - " 0 & 0 & 0 & 1\n", - " \\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The inverse of a matrix is defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \\left (A^{T} \\right )^{-1}$ real orthogonal $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \\left (A^{\\dagger} \\right )^{-1}$ unitary $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$
\n", - "\n", - "\n", - "### Some famous Matrices\n", - "\n", - " * Diagonal if $a_{ij}=0$ for $i\\ne j$\n", - "\n", - " * Upper triangular if $a_{ij}=0$ for $i > j$\n", - "\n", - " * Lower triangular if $a_{ij}=0$ for $i < j$\n", - "\n", - " * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$\n", - "\n", - " * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$\n", - "\n", - " * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$\n", - "\n", - " * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$\n", - "\n", - " * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$\n", - "\n", - " * Banded, block upper triangular, block lower triangular....\n", - "\n", - "### More Basic Matrix Features\n", - "\n", - "Some Equivalent Statements\n", - "For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", - "\n", - " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", - "\n", - " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", - "\n", - " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * $\\mathbf{A}$ is a product of elementary matrices.\n", - "\n", - " * $0$ is not eigenvalue of $\\mathbf{A}$.\n", - "\n", - "## Numpy and arrays\n", - "[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution," - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "n = 10\n", - "x = np.random.normal(size=n)\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.\n", - "Another alternative is to declare a vector as follows" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.array([1, 2, 3])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++\n", - "start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \\dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4, 7, 8]))\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the last example we used Numpy's unary function $np.log$. This function is\n", - "highly tuned to compute array elements since the code is vectorized\n", - "and does not require looping. We normaly recommend that you use the\n", - "Numpy intrinsic functions instead of the corresponding **log** function\n", - "from Python's **math** module. The looping is done explicitely by the\n", - "**np.log** function. The alternative, and slower way to compute the\n", - "logarithms of a vector would be to write" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "from math import log\n", - "x = np.array([4, 7, 8])\n", - "for i in range(0, len(x)):\n", - " x[i] = log(x[i])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note that our code is much longer already and we need to import the **log** function from the **math** module. \n", - "The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4, 7, 8], dtype = np.float64))\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", - "print(x.itemsize)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Matrices in Python\n", - "\n", - "Having defined vectors, we are now ready to try out matrices. We can\n", - "define a $3 \\times 3 $ real matrix $\\hat{A}$ as (recall that we user\n", - "lowercase letters for vectors and uppercase letters for matrices)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "# print the first column, row-major order and elements start with 0\n", - "print(A[:,0])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can continue this was by printing out other columns or rows. The example here prints out the second column" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "# print the first column, row-major order and elements start with 0\n", - "print(A[1,:])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to zero\n", - "A = np.zeros( (n, n) )\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or initializing all elements to" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to one\n", - "A = np.ones( (n, n) )\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or as unitarily distributed random numbers (see the material on random number generators in the statistics part)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \\in [0, 1]\n", - "A = np.random.rand(n, n)\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n", - "As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n", - "$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", - " \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n", - " \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where for example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma_{xy} =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. \n", - "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\hat{W}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n", - " x_1 & y_1 & z_1 \\\\\n", - " x_2 & y_2 & z_2 \\\\\n", - " \\dots & \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2} & z_{n-2} \\\\\n", - " x_{n-1} & y_{n-1} & z_{n-1}\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in turn is converted into into the $3\\times 3$ covariance matrix\n", - "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", - "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", - "function **np.mean(x)**. We can also extract the eigenvalues of the\n", - "covariance matrix through the **np.linalg.eig()** function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "z = x**3+np.random.normal(size=n)\n", - "print(np.mean(z))\n", - "W = np.vstack((x, y, z))\n", - "Sigma = np.cov(W)\n", - "print(Sigma)\n", - "Eigvals, Eigvecs = np.linalg.eig(Sigma)\n", - "print(Eigvals)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from scipy import sparse\n", - "eye = np.eye(4)\n", - "print(eye)\n", - "sparse_mtx = sparse.csr_matrix(eye)\n", - "print(sparse_mtx)\n", - "x = np.linspace(-10,10,100)\n", - "y = np.sin(x)\n", - "plt.plot(x,y,marker='x')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Meet the Pandas\n", - "\n", - "\n", - "\n", - "\n", - "Another useful Python package is\n", - "[pandas](https://pandas.pydata.org/), which is an open source library\n", - "providing high-performance, easy-to-use data structures and data\n", - "analysis tools for Python. **pandas** stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.\n", - "**pandas** has two major classes, the **DataFrame** class with two-dimensional data objects and tabular data organized in columns and the class **Series** with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. \n", - "**pandas** allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. \n", - "\n", - "The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of **pandas**, in particular in connection with classification of data." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import pandas as pd\n", - "from IPython.display import display\n", - "data = {'First Name': [\"Frodo\", \"Bilbo\", \"Aragorn II\", \"Samwise\"],\n", - " 'Last Name': [\"Baggins\", \"Baggins\",\"Elessar\",\"Gamgee\"],\n", - " 'Place of birth': [\"Shire\", \"Shire\", \"Eriador\", \"Shire\"],\n", - " 'Date of Birth T.A.': [2968, 2890, 2931, 2980]\n", - " }\n", - "data_pandas = pd.DataFrame(data)\n", - "display(data_pandas)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the above we have imported **pandas** with the shorthand **pd**, the latter has become the standard way we import **pandas**. We make then a list of various variables\n", - "and reorganize the aboves lists into a **DataFrame** and then print out a neat table with specific column labels as *Name*, *place of birth* and *date of birth*.\n", - "Displaying these results, we see that the indices are given by the default numbers from zero to three.\n", - "**pandas** is extremely flexible and we can easily change the above indices by defining a new type of indexing as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])\n", - "display(data_pandas)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Thereafter we display the content of the row which begins with the index **Aragorn**" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "display(data_pandas.loc['Aragorn'])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can easily append data to this, for example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "new_hobbit = {'First Name': [\"Peregrin\"],\n", - " 'Last Name': [\"Took\"],\n", - " 'Place of birth': [\"Shire\"],\n", - " 'Date of Birth T.A.': [2990]\n", - " }\n", - "data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))\n", - "display(data_pandas)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here are other examples where we use the **DataFrame** functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix \n", - "of dimensionality $10\\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import pandas as pd\n", - "from IPython.display import display\n", - "np.random.seed(100)\n", - "# setting up a 10 x 5 matrix\n", - "rows = 10\n", - "cols = 5\n", - "a = np.random.randn(rows,cols)\n", - "df = pd.DataFrame(a)\n", - "display(df)\n", - "print(df.mean())\n", - "print(df.std())\n", - "display(df**2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Thereafter we can select specific columns only and plot final results" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']\n", - "df.index = np.arange(10)\n", - "\n", - "display(df)\n", - "print(df['Second'].mean() )\n", - "print(df.info())\n", - "print(df.describe())\n", - "\n", - "from pylab import plt, mpl\n", - "plt.style.use('seaborn')\n", - "mpl.rcParams['font.family'] = 'serif'\n", - "\n", - "df.cumsum().plot(lw=2.0, figsize=(10,6))\n", - "plt.show()\n", - "\n", - "\n", - "df.plot.bar(figsize=(10,6), rot=15)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can produce a $4\\times 4$ matrix" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "b = np.arange(16).reshape((4,4))\n", - "print(b)\n", - "df1 = pd.DataFrame(b)\n", - "print(df1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and many other operations. \n", - "\n", - "The **Series** class is another important class included in\n", - "**pandas**. You can view it as a specialization of **DataFrame** but where\n", - "we have just a single column of data. It shares many of the same features as _DataFrame. As with **DataFrame**,\n", - "most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.\n", - "As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.\n", - "For multidimensional arrays, we recommend strongly [xarray](http://xarray.pydata.org/en/stable/). **xarray** has much of the same flexibility as **pandas**, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both **pandas** and **xarray**. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "In order to study various Machine Learning algorithms, we need to\n", - "access data. Acccessing data is an essential step in all machine\n", - "learning algorithms. In particular, setting up the so-called **design\n", - "matrix** (to be defined below) is often the first element we need in\n", - "order to perform our calculations. To set up the design matrix means\n", - "reading (and later, when the calculations are done, writing) data\n", - "in various formats, The formats span from reading files from disk,\n", - "loading data from databases and interacting with online sources\n", - "like web application programming interfaces (APIs).\n", - "\n", - "In handling various input formats, as discussed above, we will mainly stay with **pandas**,\n", - "a Python package which allows us, in a seamless and painless way, to\n", - "deal with a multitude of formats, from standard **csv** (comma separated\n", - "values) files, via **excel**, **html** to **hdf5** formats. With **pandas**\n", - "and the **DataFrame** and **Series** functionalities we are able to convert text data\n", - "into the calculational formats we need for a specific algorithm. And our code is going to be \n", - "pretty close the basic mathematical expressions.\n", - "\n", - "Our first data set is going to be a classic from nuclear physics, namely all\n", - "available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set. \n", - "\n", - "We will show some of the\n", - "strengths of packages like **Scikit-Learn** in fitting nuclear binding energies to\n", - "specific functions using linear regression first. Then, as a teaser, we will show you how \n", - "you can easily implement other algorithms like decision trees and random forests and neural networks.\n", - "\n", - "But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,\n", - "(don't be offended) fitting straight lines!\n", - "\n", - "\n", - "\n", - "\n", - "## Simple linear regression model using **scikit-learn**\n", - "\n", - "We start with perhaps our simplest possible example, using **Scikit-Learn** to perform linear regression analysis on a data set produced by us. \n", - "\n", - "What follows is a simple Python code where we have defined a function\n", - "$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. \n", - "The numbers in the vector $\\hat{x}$ are given\n", - "by random numbers generated with a uniform distribution with entries\n", - "$x_i \\in [0,1]$ (more about probability distribution functions\n", - "later). These values are then used to define a function $y(x)$\n", - "(tabulated again as a vector) with a linear dependence on $x$ plus a\n", - "random noise added via the normal distribution.\n", - "\n", - "\n", - "The Numpy functions are imported used the **import numpy as np**\n", - "statement and the random number generator for the uniform distribution\n", - "is called using the function **np.random.rand()**, where we specificy\n", - "that we want $100$ random variables. Using Numpy we define\n", - "automatically an array with the specified number of elements, $100$ in\n", - "our case. With the Numpy function **randn()** we can compute random\n", - "numbers with the normal distribution (mean value $\\mu$ equal to zero and\n", - "variance $\\sigma^2$ set to one) and produce the values of $y$ assuming a linear\n", - "dependence as function of $x$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y = 2x+N(0,1),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $N(0,1)$ represents random numbers generated by the normal\n", - "distribution. From **Scikit-Learn** we import then the\n", - "**LinearRegression** functionality and make a prediction $\\tilde{y} =\n", - "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n", - "data $(\\hat{x},\\hat{y})$ for our training data. The Python package\n", - "**scikit-learn** has also a functionality which extracts the above\n", - "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n", - "distinguish between training data and test data.\n", - "\n", - "For plotting we use the Python package\n", - "[matplotlib](https://matplotlib.org/) which produces publication\n", - "quality figures. Feel free to explore the extensive\n", - "[gallery](https://matplotlib.org/gallery/index.html) of examples. In\n", - "this example we plot our original values of $x$ and $y$ as well as the\n", - "prediction **ypredict** ($\\tilde{y}$), which attempts at fitting our\n", - "data with a straight line.\n", - "\n", - "The Python code follows here." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression\n", - "\n", - "x = np.random.rand(100,1)\n", - "y = 2*x+np.random.randn(100,1)\n", - "linreg = LinearRegression()\n", - "linreg.fit(x,y)\n", - "xnew = np.array([[0],[1]])\n", - "ypredict = linreg.predict(xnew)\n", - "\n", - "plt.plot(xnew, ypredict, \"r-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0,1.0,0, 5.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Simple Linear Regression')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This example serves several aims. It allows us to demonstrate several\n", - "aspects of data analysis and later machine learning algorithms. The\n", - "immediate visualization shows that our linear fit is not\n", - "impressive. It goes through the data points, but there are many\n", - "outliers which are not reproduced by our linear regression. We could\n", - "now play around with this small program and change for example the\n", - "factor in front of $x$ and the normal distribution. Try to change the\n", - "function $y$ to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y = 10x+0.01 \\times N(0,1),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $x$ is defined as before. Does the fit look better? Indeed, by\n", - "reducing the role of the noise given by the normal distribution we see immediately that\n", - "our linear prediction seemingly reproduces better the training\n", - "set. However, this testing 'by the eye' is obviouly not satisfactory in the\n", - "long run. Here we have only defined the training data and our model, and \n", - "have not discussed a more rigorous approach to the **cost** function.\n", - "\n", - "We need more rigorous criteria in defining whether we have succeeded or\n", - "not in modeling our training data. You will be surprised to see that\n", - "many scientists seldomly venture beyond this 'by the eye' approach. A\n", - "standard approach for the *cost* function is the so-called $\\chi^2$\n", - "function (a variant of the mean-squared error (MSE))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\chi^2 = \\frac{1}{n}\n", - "\\sum_{i=0}^{n-1}\\frac{(y_i-\\tilde{y}_i)^2}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\sigma_i^2$ is the variance (to be defined later) of the entry\n", - "$y_i$. We may not know the explicit value of $\\sigma_i^2$, it serves\n", - "however the aim of scaling the equations and make the cost function\n", - "dimensionless. \n", - "\n", - "Minimizing the cost function is a central aspect of\n", - "our discussions to come. Finding its minima as function of the model\n", - "parameters ($\\alpha$ and $\\beta$ in our case) will be a recurring\n", - "theme in these series of lectures. Essentially all machine learning\n", - "algorithms we will discuss center around the minimization of the\n", - "chosen cost function. This depends in turn on our specific\n", - "model for describing the data, a typical situation in supervised\n", - "learning. Automatizing the search for the minima of the cost function is a\n", - "central ingredient in all algorithms. Typical methods which are\n", - "employed are various variants of **gradient** methods. These will be\n", - "discussed in more detail later. Again, you'll be surprised to hear that\n", - "many practitioners minimize the above function ''by the eye', popularly dubbed as \n", - "'chi by the eye'. That is, change a parameter and see (visually and numerically) that \n", - "the $\\chi^2$ function becomes smaller. \n", - "\n", - "There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define \n", - "the relative error (why would we prefer the MSE instead of the relative error?) as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\hat{y} -\\hat{\\tilde{y}}\\vert}{\\vert \\hat{y}\\vert}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The squared cost function results in an arithmetic mean-unbiased\n", - "estimator, and the absolute-value cost function results in a\n", - "median-unbiased estimator (in the one-dimensional case, and a\n", - "geometric median-unbiased estimator for the multi-dimensional\n", - "case). The squared cost function has the disadvantage that it has the tendency\n", - "to be dominated by outliers.\n", - "\n", - "We can modify easily the above Python code and plot the relative error instead" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression\n", - "\n", - "x = np.random.rand(100,1)\n", - "y = 5*x+0.01*np.random.randn(100,1)\n", - "linreg = LinearRegression()\n", - "linreg.fit(x,y)\n", - "ypredict = linreg.predict(x)\n", - "\n", - "plt.plot(x, np.abs(ypredict-y)/abs(y), \"ro\")\n", - "plt.axis([0,1.0,0.0, 0.5])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$\\epsilon_{\\mathrm{relative}}$')\n", - "plt.title(r'Relative error')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Depending on the parameter in front of the normal distribution, we may\n", - "have a small or larger relative error. Try to play around with\n", - "different training data sets and study (graphically) the value of the\n", - "relative error.\n", - "\n", - "As mentioned above, **Scikit-Learn** has an impressive functionality.\n", - "We can for example extract the values of $\\alpha$ and $\\beta$ and\n", - "their error estimates, or the variance and standard deviation and many\n", - "other properties from the statistical data analysis. \n", - "\n", - "Here we show an\n", - "example of the functionality of **Scikit-Learn**." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np \n", - "import matplotlib.pyplot as plt \n", - "from sklearn.linear_model import LinearRegression \n", - "from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error\n", - "\n", - "x = np.random.rand(100,1)\n", - "y = 2.0+ 5*x+0.5*np.random.randn(100,1)\n", - "linreg = LinearRegression()\n", - "linreg.fit(x,y)\n", - "ypredict = linreg.predict(x)\n", - "print('The intercept alpha: \\n', linreg.intercept_)\n", - "print('Coefficient beta : \\n', linreg.coef_)\n", - "# The mean squared error \n", - "print(\"Mean squared error: %.2f\" % mean_squared_error(y, ypredict))\n", - "# Explained variance score: 1 is perfect prediction \n", - "print('Variance score: %.2f' % r2_score(y, ypredict))\n", - "# Mean squared log error \n", - "print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )\n", - "# Mean absolute error \n", - "print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))\n", - "plt.plot(x, ypredict, \"r-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0.0,1.0,1.5, 7.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Linear Regression fit ')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n", - "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n", - "\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The smaller the value, the better the fit. Ideally we would like to\n", - "have an MSE equal zero. The attentive reader has probably recognized\n", - "this function as being similar to the $\\chi^2$ function defined above.\n", - "\n", - "The **r2score** function computes $R^2$, the coefficient of\n", - "determination. It provides a measure of how well future samples are\n", - "likely to be predicted by the model. Best possible score is 1.0 and it\n", - "can be negative (because the model can be arbitrarily worse). A\n", - "constant model that always predicts the expected value of $\\hat{y}$,\n", - "disregarding the input features, would get a $R^2$ score of $0.0$.\n", - "\n", - "If $\\tilde{\\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined the mean value of $\\hat{y}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Another quantity taht we will meet again in our discussions of regression analysis is \n", - " the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error.\n", - "The MAE is defined as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\text{MAE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We present the \n", - "squared logarithmic (quadratic) error" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\text{MSLE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\log_e (x)$ stands for the natural logarithm of $x$. This error\n", - "estimate is best to use when targets having exponential growth, such\n", - "as population counts, average sales of a commodity over a span of\n", - "years etc. \n", - "\n", - "\n", - "Finally, another cost function is the Huber cost function used in robust regression.\n", - "\n", - "The rationale behind this possible cost function is its reduced\n", - "sensitivity to outliers in the data set. In our discussions on\n", - "dimensionality reduction and normalization of data we will meet other\n", - "ways of dealing with outliers.\n", - "\n", - "The Huber cost function is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "H_{\\delta}(a)={\\begin{cases}{\\frac {1}{2}}{a^{2}}&{\\text{for }}|a|\\leq \\delta ,\\\\\\delta (|a|-{\\frac {1}{2}}\\delta ),&{\\text{otherwise.}}\\end{cases}}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here $a=\\boldsymbol{y} - \\boldsymbol{\\tilde{y}}$.\n", - "We will discuss in more\n", - "detail these and other functions in the various lectures. We conclude this part with another example. Instead of \n", - "a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "import random\n", - "from sklearn.linear_model import Ridge\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.linear_model import LinearRegression\n", - "\n", - "x=np.linspace(0.02,0.98,200)\n", - "noise = np.asarray(random.sample((range(200)),200))\n", - "y=x**3*noise\n", - "yn=x**3*100\n", - "poly3 = PolynomialFeatures(degree=3)\n", - "X = poly3.fit_transform(x[:,np.newaxis])\n", - "clf3 = LinearRegression()\n", - "clf3.fit(X,y)\n", - "\n", - "Xplot=poly3.fit_transform(x[:,np.newaxis])\n", - "poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')\n", - "plt.plot(x,yn, color='red', label=\"True Cubic\")\n", - "plt.scatter(x, y, label='Data', color='orange', s=15)\n", - "plt.legend()\n", - "plt.show()\n", - "\n", - "def error(a):\n", - " for i in y:\n", - " err=(y-yn)/yn\n", - " return abs(np.sum(err))/len(err)\n", - "\n", - "print (error(y))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding\n", - "energies. A basic quantity which can be measured for the ground\n", - "states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with\n", - "atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). \n", - "\n", - "Atomic masses are usually tabulated in terms of the mass excess defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\Delta M(N, Z) = M(N, Z) - uA,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $u$ is the Atomic Mass Unit" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u = M(^{12}\\mathrm{C})/12 = 931.4940954(57) \\hspace{0.1cm} \\mathrm{MeV}/c^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The nucleon masses are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "m_p = 1.00727646693(9)u,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "m_n = 939.56536(8)\\hspace{0.1cm} \\mathrm{MeV}/c^2 = 1.0086649156(6)u.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the [2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu](http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf)\n", - "there are data on masses and decays of 3437 nuclei.\n", - "\n", - "The nuclear binding energy is defined as the energy required to break\n", - "up a given nucleus into its constituent parts of $N$ neutrons and $Z$\n", - "protons. In terms of the atomic masses $M(N, Z)$ the binding energy is\n", - "defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.\n", - "In terms of the mass excess the binding energy is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "BE(N, Z) = Z\\Delta_H c^2 + N\\Delta_n c^2 -\\Delta(N, Z)c^2 ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\Delta_H c^2 = 7.2890$ MeV and $\\Delta_n c^2 = 8.0713$ MeV.\n", - "\n", - "\n", - "A popular and physically intuitive model which can be used to parametrize \n", - "the experimental binding energies as function of $A$, is the so-called \n", - "**liquid drop model**. The ansatz is based on the following expression" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "BE(N,Z) = a_1A-a_2A^{2/3}-a_3\\frac{Z^2}{A^{1/3}}-a_4\\frac{(N-Z)^2}{A},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit \n", - "to the experimental data. \n", - "\n", - "\n", - "\n", - "\n", - "To arrive at the above expression we have assumed that we can make the following assumptions:\n", - "\n", - " * There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.\n", - "\n", - " * There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.\n", - "\n", - " * There is a Coulomb energy term $a_3\\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding. \n", - "\n", - " * There is an asymmetry term $a_4\\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.\n", - "\n", - "We could also add a so-called pairing term, which is a correction term that\n", - "arises from the tendency of proton pairs and neutron pairs to\n", - "occur. An even number of particles is more stable than an odd number. \n", - "\n", - "\n", - "### Organizing our data\n", - "\n", - "Let us start with reading and organizing our data. \n", - "We start with the compilation of masses and binding energies from 2016.\n", - "After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.\n", - "\n", - "\n", - "We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of **scikit-learn**." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "import sklearn.linear_model as skl\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error\n", - "import os\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"MassEval2016.dat\"),'r')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various **matplotlib** commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from pylab import plt, mpl\n", - "plt.style.use('seaborn')\n", - "mpl.rcParams['font.family'] = 'serif'\n", - "\n", - "def MakePlot(x,y, styles, labels, axlabels):\n", - " plt.figure(figsize=(10,6))\n", - " for i in range(len(x)):\n", - " plt.plot(x[i], y[i], styles[i], label = labels[i])\n", - " plt.xlabel(axlabels[0])\n", - " plt.ylabel(axlabels[1])\n", - " plt.legend(loc=0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Our next step is to read the data on experimental binding energies and\n", - "reorganize them as functions of the mass number $A$, the number of\n", - "protons $Z$ and neutrons $N$ using **pandas**. Before we do this it is\n", - "always useful (unless you have a binary file or other types of compressed\n", - "data) to actually open the file and simply take a look at it!\n", - "\n", - "\n", - "In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with **pandas**. The file begins with some basic format information." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\"\"\" \n", - "This is taken from the data file of the mass 2016 evaluation. \n", - "All files are 3436 lines long with 124 character per line. \n", - " Headers are 39 lines long. \n", - " col 1 : Fortran character control: 1 = page feed 0 = line feed \n", - " format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 \n", - " These formats are reflected in the pandas widths variable below, see the statement \n", - " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), \n", - " Pandas has also a variable header, with length 39 in this case. \n", - "\"\"\"" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The data we are interested in are in columns 2, 3, 4 and 11, giving us\n", - "the number of neutrons, protons, mass numbers and binding energies,\n", - "respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will\n", - "covert them into the **pandas** DataFrame structure." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Read the experimental data with Pandas\n", - "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", - " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", - " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", - " header=39,\n", - " index_col=False)\n", - "\n", - "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", - "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", - "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", - "Masses = Masses.dropna()\n", - "# Convert from keV to MeV.\n", - "Masses['Ebinding'] /= 1000\n", - "\n", - "# Group the DataFrame by nucleon number, A.\n", - "Masses = Masses.groupby('A')\n", - "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", - "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We have now read in the data, grouped them according to the variables we are interested in. \n", - "We see how easy it is to reorganize the data using **pandas**. If we\n", - "were to do these operations in C/C++ or Fortran, we would have had to\n", - "write various functions/subroutines which perform the above\n", - "reorganizations for us. Having reorganized the data, we can now start\n", - "to make some simple fits using both the functionalities in **numpy** and\n", - "**Scikit-Learn** afterwards. \n", - "\n", - "Now we define five variables which contain\n", - "the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "A = Masses['A']\n", - "Z = Masses['Z']\n", - "N = Masses['N']\n", - "Element = Masses['Element']\n", - "Energies = Masses['Ebinding']\n", - "print(Masses)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The next step, and we will define this mathematically later, is to set up the so-called **design matrix**. We will throughout call this matrix $\\boldsymbol{X}$.\n", - "It has dimensionality $p\\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Now we set up the design matrix X\n", - "X = np.zeros((len(A),5))\n", - "X[:,0] = 1\n", - "X[:,1] = A\n", - "X[:,2] = A**(2.0/3.0)\n", - "X[:,3] = A**(-1.0/3.0)\n", - "X[:,4] = A**(-1.0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With **scikitlearn** we are now ready to use linear regression and fit our data." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "clf = skl.LinearRegression().fit(X, Energies)\n", - "fity = clf.predict(X)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Pretty simple! \n", - "Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# The mean squared error \n", - "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, fity))\n", - "# Explained variance score: 1 is perfect prediction \n", - "print('Variance score: %.2f' % r2_score(Energies, fity))\n", - "# Mean absolute error \n", - "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))\n", - "print(clf.coef_, clf.intercept_)\n", - "\n", - "Masses['Eapprox'] = fity\n", - "# Generate a plot comparing the experimental with the fitted values values.\n", - "fig, ax = plt.subplots()\n", - "ax.set_xlabel(r'$A = N + Z$')\n", - "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", - "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", - " label='Ame2016')\n", - "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", - " label='Fit')\n", - "ax.legend()\n", - "save_fig(\"Masses2016\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As a teaser, let us now see how we can do this with decision trees using **scikit-learn**. Later we will switch to so-called **random forests**!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "#Decision Tree Regression\n", - "from sklearn.tree import DecisionTreeRegressor\n", - "regr_1=DecisionTreeRegressor(max_depth=5)\n", - "regr_2=DecisionTreeRegressor(max_depth=7)\n", - "regr_3=DecisionTreeRegressor(max_depth=9)\n", - "regr_1.fit(X, Energies)\n", - "regr_2.fit(X, Energies)\n", - "regr_3.fit(X, Energies)\n", - "\n", - "\n", - "y_1 = regr_1.predict(X)\n", - "y_2 = regr_2.predict(X)\n", - "y_3=regr_3.predict(X)\n", - "Masses['Eapprox'] = y_3\n", - "# Plot the results\n", - "plt.figure()\n", - "plt.plot(A, Energies, color=\"blue\", label=\"Data\", linewidth=2)\n", - "plt.plot(A, y_1, color=\"red\", label=\"max_depth=5\", linewidth=2)\n", - "plt.plot(A, y_2, color=\"green\", label=\"max_depth=7\", linewidth=2)\n", - "plt.plot(A, y_3, color=\"m\", label=\"max_depth=9\", linewidth=2)\n", - "\n", - "plt.xlabel(\"$A$\")\n", - "plt.ylabel(\"$E$[MeV]\")\n", - "plt.title(\"Decision Tree Regression\")\n", - "plt.legend()\n", - "save_fig(\"Masses2016Trees\")\n", - "plt.show()\n", - "print(Masses)\n", - "print(np.mean( (Energies-y_1)**2))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The **seaborn** package allows us to visualize data in an efficient way. Note that we use **scikit-learn**'s multi-layer perceptron (or feed forward neural network) \n", - "functionality." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.neural_network import MLPRegressor\n", - "from sklearn.metrics import accuracy_score\n", - "import seaborn as sns\n", - "\n", - "X_train = X\n", - "Y_train = Energies\n", - "n_hidden_neurons = 100\n", - "epochs = 100\n", - "# store models for later use\n", - "eta_vals = np.logspace(-5, 1, 7)\n", - "lmbd_vals = np.logspace(-5, 1, 7)\n", - "# store the models for later use\n", - "DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "sns.set()\n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',\n", - " alpha=lmbd, learning_rate_init=eta, max_iter=epochs)\n", - " dnn.fit(X_train, Y_train)\n", - " DNN_scikit[i][j] = dnn\n", - " train_accuracy[i][j] = dnn.score(X_train, Y_train)\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Linear Regression, basic elements\n", - "\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureAug27.mp4?vrtx=view-as-webpage).\n", - "\n", - "\n", - "Fitting a continuous function with linear parameterization in terms of the parameters $\\boldsymbol{\\beta}$.\n", - "* Method of choice for fitting a continuous function!\n", - "\n", - "* Gives an excellent introduction to central Machine Learning features with **understandable pedagogical** links to other methods like **Neural Networks**, **Support Vector Machines** etc\n", - "\n", - "* Analytical expression for the fitting parameters $\\boldsymbol{\\beta}$\n", - "\n", - "* Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more\n", - "\n", - "* Analytical relation with probabilistic interpretations \n", - "\n", - "* Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics\n", - "\n", - "* Easy to code! And links well with classification problems and logistic regression and neural networks\n", - "\n", - "* Allows for **easy** hands-on understanding of gradient descent methods\n", - "\n", - "* and many more features\n", - "\n", - "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", - "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended.\n", - "\n", - "\n", - "\n", - "Regression modeling deals with the description of the sampling distribution of a given random variable $y$ and how it varies as function of another variable or a set of such variables $\\boldsymbol{x} =[x_0, x_1,\\dots, x_{n-1}]^T$. \n", - "The first variable is called the **dependent**, the **outcome** or the **response** variable while the set of variables $\\boldsymbol{x}$ is called the independent variable, or the predictor variable or the explanatory variable. \n", - "\n", - "A regression model aims at finding a likelihood function $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$, that is the conditional distribution for $\\boldsymbol{y}$ with a given $\\boldsymbol{x}$. The estimation of $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$ is made using a data set with \n", - "* $n$ cases $i = 0, 1, 2, \\dots, n-1$ \n", - "\n", - "* Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \\dots, n-1$ \n", - "\n", - "* $p$ so-called explanatory (independent or predictor) variables $\\boldsymbol{x}_i=[x_{i0}, x_{i1}, \\dots, x_{ip-1}]$ with $i = 0, 1, 2, \\dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples. \n", - "\n", - " The goal of the regression analysis is to extract/exploit relationship between $\\boldsymbol{y}$ and $\\boldsymbol{x}$ in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.\n", - "\n", - "\n", - "Consider an experiment in which $p$ characteristics of $n$ samples are\n", - "measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix \n", - "$\\mathbf{X}$.\n", - "\n", - "The matrix $\\mathbf{X}$ is called the *design\n", - "matrix*. Additional information of the samples is available in the\n", - "form of $\\boldsymbol{y}$ (also as above). The variable $\\boldsymbol{y}$ is\n", - "generally referred to as the *response variable*. The aim of\n", - "regression analysis is to explain $\\boldsymbol{y}$ in terms of\n", - "$\\boldsymbol{X}$ through a functional relationship like $y_i =\n", - "f(\\mathbf{X}_{i,\\ast})$. When no prior knowledge on the form of\n", - "$f(\\cdot)$ is available, it is common to assume a linear relationship\n", - "between $\\boldsymbol{X}$ and $\\boldsymbol{y}$. This assumption gives rise to\n", - "the *linear regression model* where $\\boldsymbol{\\beta} = [\\beta_0, \\ldots,\n", - "\\beta_{p-1}]^{T}$ are the *regression parameters*. \n", - "\n", - "Linear regression gives us a set of analytical equations for the parameters $\\beta_j$.\n", - "\n", - "\n", - "In order to understand the relation among the predictors $p$, the set of data $n$ and the target (outcome, output etc) $\\boldsymbol{y}$,\n", - "consider the model we discussed for describing nuclear binding energies. \n", - "\n", - "There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.\n", - "Assuming" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we have five predictors, that is the intercept, the $A$ dependent term, the $A^{2/3}$ term and the $A^{-1/3}$ and $A^{-1}$ terms.\n", - "This gives $p=0,1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n", - "$p\\times n$ matrix $\\boldsymbol{X}$.\n", - "\n", - "Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the\n", - "so-called [credit card default data from Taiwan](https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub). The data set contains data on $n=30000$ credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are $24$ such predictors or attributes leading to a design matrix of dimensionality $24 \\times 30000$. This is however a classification problem and we will come back to it when we discuss Logistic Regression. \n", - "\n", - "\n", - "Before we proceed let us study a case from linear algebra where we aim at fitting a set of data $\\boldsymbol{y}=[y_0,y_1,\\dots,y_{n-1}]$. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables $\\boldsymbol{x}=[x_0,x_1,\\dots,x_{n-1}]$, that is $y_i = y(x_i)$ with $i=0,1,2,\\dots,n-1$. The variables $x_i$ could represent physical quantities like time, temperature, position etc. We assume that $y(x)$ is a smooth function. \n", - "\n", - "Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of $y$ which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree $n-1$ with $n$ points, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y=y(x) \\rightarrow y(x_i)=\\tilde{y}_i+\\epsilon_i=\\sum_{j=0}^{n-1} \\beta_j x_i^j+\\epsilon_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\epsilon_i$ is the error in our approximation. \n", - "\n", - "\n", - "For every set of values $y_i,x_i$ we have thus the corresponding set of equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "y_0&=\\beta_0+\\beta_1x_0^1+\\beta_2x_0^2+\\dots+\\beta_{n-1}x_0^{n-1}+\\epsilon_0\\\\\n", - "y_1&=\\beta_0+\\beta_1x_1^1+\\beta_2x_1^2+\\dots+\\beta_{n-1}x_1^{n-1}+\\epsilon_1\\\\\n", - "y_2&=\\beta_0+\\beta_1x_2^1+\\beta_2x_2^2+\\dots+\\beta_{n-1}x_2^{n-1}+\\epsilon_2\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{n-1}&=\\beta_0+\\beta_1x_{n-1}^1+\\beta_2x_{n-1}^2+\\dots+\\beta_{n-1}x_{n-1}^{n-1}+\\epsilon_{n-1}.\\\\\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Defining the vectors" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y} = [y_0,y_1, y_2,\\dots, y_{n-1}]^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta} = [\\beta_0,\\beta_1, \\beta_2,\\dots, \\beta_{n-1}]^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\epsilon} = [\\epsilon_0,\\epsilon_1, \\epsilon_2,\\dots, \\epsilon_{n-1}]^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the design matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\n", - "\\begin{bmatrix} \n", - "1& x_{0}^1 &x_{0}^2& \\dots & \\dots &x_{0}^{n-1}\\\\\n", - "1& x_{1}^1 &x_{1}^2& \\dots & \\dots &x_{1}^{n-1}\\\\\n", - "1& x_{2}^1 &x_{2}^2& \\dots & \\dots &x_{2}^{n-1}\\\\ \n", - "\\dots& \\dots &\\dots& \\dots & \\dots &\\dots\\\\\n", - "1& x_{n-1}^1 &x_{n-1}^2& \\dots & \\dots &x_{n-1}^{n-1}\\\\\n", - "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we can rewrite our equations as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta}+\\boldsymbol{\\epsilon}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The above design matrix is called a [Vandermonde matrix](https://en.wikipedia.org/wiki/Vandermonde_matrix).\n", - "\n", - "We are obviously not limited to the above polynomial expansions. We\n", - "could replace the various powers of $x$ with elements of Fourier\n", - "series or instead of $x_i^j$ we could have $\\cos{(j x_i)}$ or $\\sin{(j\n", - "x_i)}$, or time series or other orthogonal functions. For every set\n", - "of values $y_i,x_i$ we can then generalize the equations to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "y_0&=\\beta_0x_{00}+\\beta_1x_{01}+\\beta_2x_{02}+\\dots+\\beta_{n-1}x_{0n-1}+\\epsilon_0\\\\\n", - "y_1&=\\beta_0x_{10}+\\beta_1x_{11}+\\beta_2x_{12}+\\dots+\\beta_{n-1}x_{1n-1}+\\epsilon_1\\\\\n", - "y_2&=\\beta_0x_{20}+\\beta_1x_{21}+\\beta_2x_{22}+\\dots+\\beta_{n-1}x_{2n-1}+\\epsilon_2\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{i}&=\\beta_0x_{i0}+\\beta_1x_{i1}+\\beta_2x_{i2}+\\dots+\\beta_{n-1}x_{in-1}+\\epsilon_i\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{n-1}&=\\beta_0x_{n-1,0}+\\beta_1x_{n-1,2}+\\beta_2x_{n-1,2}+\\dots+\\beta_{n-1}x_{n-1,n-1}+\\epsilon_{n-1}.\\\\\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!**\n", - "\n", - "We redefine in turn the matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\n", - "\\begin{bmatrix} \n", - "x_{00}& x_{01} &x_{02}& \\dots & \\dots &x_{0,n-1}\\\\\n", - "x_{10}& x_{11} &x_{12}& \\dots & \\dots &x_{1,n-1}\\\\\n", - "x_{20}& x_{21} &x_{22}& \\dots & \\dots &x_{2,n-1}\\\\ \n", - "\\dots& \\dots &\\dots& \\dots & \\dots &\\dots\\\\\n", - "x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \\dots & \\dots &x_{n-1,n-1}\\\\\n", - "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and without loss of generality we rewrite again our equations as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta}+\\boldsymbol{\\epsilon}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The left-hand side of this equation is kwown. Our error vector $\\boldsymbol{\\epsilon}$ and the parameter vector $\\boldsymbol{\\beta}$ are our unknow quantities. How can we obtain the optimal set of $\\beta_i$ values? \n", - "\n", - "We have defined the matrix $\\boldsymbol{X}$ via the equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "y_0&=\\beta_0x_{00}+\\beta_1x_{01}+\\beta_2x_{02}+\\dots+\\beta_{n-1}x_{0n-1}+\\epsilon_0\\\\\n", - "y_1&=\\beta_0x_{10}+\\beta_1x_{11}+\\beta_2x_{12}+\\dots+\\beta_{n-1}x_{1n-1}+\\epsilon_1\\\\\n", - "y_2&=\\beta_0x_{20}+\\beta_1x_{21}+\\beta_2x_{22}+\\dots+\\beta_{n-1}x_{2n-1}+\\epsilon_1\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{i}&=\\beta_0x_{i0}+\\beta_1x_{i1}+\\beta_2x_{i2}+\\dots+\\beta_{n-1}x_{in-1}+\\epsilon_1\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{n-1}&=\\beta_0x_{n-1,0}+\\beta_1x_{n-1,2}+\\beta_2x_{n-1,2}+\\dots+\\beta_{n-1}x_{n-1,n-1}+\\epsilon_{n-1}.\\\\\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As we noted above, we stayed with a system with the design matrix \n", - " $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times n}$, that is we have $p=n$. For reasons to come later (algorithmic arguments) we will hereafter define \n", - "our matrix as $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors refering to the column numbers and the entries $n$ being the row elements.\n", - "\n", - "In our [introductory notes](https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html) we looked at the so-called [liquid drop model](https://en.wikipedia.org/wiki/Semi-empirical_mass_formula). Let us remind ourselves about what we did by looking at the code.\n", - "\n", - "We restate the parts of the code we are most interested in." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from IPython.display import display\n", - "import os\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"MassEval2016.dat\"),'r')\n", - "\n", - "\n", - "# Read the experimental data with Pandas\n", - "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", - " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", - " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", - " header=39,\n", - " index_col=False)\n", - "\n", - "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", - "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", - "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", - "Masses = Masses.dropna()\n", - "# Convert from keV to MeV.\n", - "Masses['Ebinding'] /= 1000\n", - "\n", - "# Group the DataFrame by nucleon number, A.\n", - "Masses = Masses.groupby('A')\n", - "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", - "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])\n", - "A = Masses['A']\n", - "Z = Masses['Z']\n", - "N = Masses['N']\n", - "Element = Masses['Element']\n", - "Energies = Masses['Ebinding']\n", - "\n", - "# Now we set up the design matrix X\n", - "X = np.zeros((len(A),5))\n", - "X[:,0] = 1\n", - "X[:,1] = A\n", - "X[:,2] = A**(2.0/3.0)\n", - "X[:,3] = A**(-1.0/3.0)\n", - "X[:,4] = A**(-1.0)\n", - "# Then nice printout using pandas\n", - "DesignMatrix = pd.DataFrame(X)\n", - "DesignMatrix.index = A\n", - "DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']\n", - "display(DesignMatrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With $\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p\\times 1}$, it means that we will hereafter write our equations for the approximation as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "throughout these lectures. \n", - "\n", - "With the above we use the design matrix to define the approximation $\\boldsymbol{\\tilde{y}}$ via the unknown quantity $\\boldsymbol{\\beta}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and in order to find the optimal parameters $\\beta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parameterized values $\\tilde{y}_i$, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or using the matrix $\\boldsymbol{X}$ and in a more compact matrix-vector notation as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This function is one possible way to define the so-called cost function.\n", - "\n", - "\n", - "\n", - "It is also common to define\n", - "the function $C$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta})=\\frac{1}{2n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "since when taking the first derivative with respect to the unknown parameters $\\beta$, the factor of $2$ cancels out. \n", - "\n", - "The function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "can be linked to the variance of the quantity $y_i$ if we interpret the latter as the mean value. \n", - "When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret $y_i$ as a mean value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_{i}=\\langle y_i \\rangle = \\beta_0x_{i,0}+\\beta_1x_{i,1}+\\beta_2x_{i,2}+\\dots+\\beta_{n-1}x_{i,n-1}+\\epsilon_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\langle y_i \\rangle$ is the mean value. Keep in mind also that\n", - "till now we have treated $y_i$ as the exact value. Normally, the\n", - "response (dependent or outcome) variable $y_i$ the outcome of a\n", - "numerical experiment or another type of experiment and is thus only an\n", - "approximation to the true value. It is then always accompanied by an\n", - "error estimate, often limited to a statistical error estimate given by\n", - "the standard deviation discussed earlier. In the discussion here we\n", - "will treat $y_i$ as our exact value for the response variable.\n", - "\n", - "In order to find the parameters $\\beta_i$ we will then minimize the spread of $C(\\boldsymbol{\\beta})$, that is we are going to solve the problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", - "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In practical terms it means we will require" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_j} = \\frac{\\partial }{\\partial \\beta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}\\right)^2\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which results in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_{ij}\\left(y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}\\right)\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or in a matrix-vector form as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can rewrite" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{y} = \\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and if the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ is invertible we have the solution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta} =\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n", - "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n", - "{\\mathbb{R}}^{p\\times p}$. In the above case we have that $p \\ll n$,\n", - "in our case $p=5$ meaning that we end up with inverting a small\n", - "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n", - "matrices to invert. The methods discussed here and for many other\n", - "supervised learning algorithms like classification with logistic\n", - "regression or support vector machines, exhibit dimensionalities which\n", - "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n", - "$\\boldsymbol{X}^T\\boldsymbol{X}$. \n", - "\n", - "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect? \n", - "\n", - "\n", - "The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and \n", - "matrices as upper case boldfaced letters." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "4\n", - "8\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "4\n", - "9\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "5\n", - "0\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}} = (\\boldsymbol{A}^{-1})^T.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The residuals $\\boldsymbol{\\epsilon}$ are in turn given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\epsilon} = \\boldsymbol{y}-\\boldsymbol{\\tilde{y}} = \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{\\epsilon}=\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "meaning that the solution for $\\boldsymbol{\\beta}$ is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.\n", - "\n", - "\n", - "Let us now return to our nuclear binding energies and simply code the above equations. \n", - "\n", - "\n", - "It is rather straightforward to implement the matrix inversion and obtain the parameters $\\boldsymbol{\\beta}$. After having defined the matrix $\\boldsymbol{X}$ we simply need to \n", - "write" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# matrix inversion to find beta\n", - "beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)\n", - "# and then make the prediction\n", - "ytilde = X @ beta" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Alternatively, you can use the least squares functionality in **Numpy** as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "fit = np.linalg.lstsq(X, Energies, rcond =None)[0]\n", - "ytildenp = np.dot(fit,X.T)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "And finally we plot our fit with and compare with data" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "Masses['Eapprox'] = ytilde\n", - "# Generate a plot comparing the experimental with the fitted values values.\n", - "fig, ax = plt.subplots()\n", - "ax.set_xlabel(r'$A = N + Z$')\n", - "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", - "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", - " label='Ame2016')\n", - "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", - " label='Fit')\n", - "ax.legend()\n", - "save_fig(\"Masses2016OLS\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can easily test our fit by computing the $R2$ score that we discussed in connection with the functionality of **Scikit-Learn** in the introductory slides.\n", - "Since we are not using **Scikit-Learn** here we can define our own $R2$ function as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def R2(y_data, y_model):\n", - " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we would be using it as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "print(R2(Energies,ytilde))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can easily add our **MSE** score as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def MSE(y_data,y_model):\n", - " n = np.size(y_model)\n", - " return np.sum((y_data-y_model)**2)/n\n", - "\n", - "print(MSE(Energies,ytilde))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and finally the relative error as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def RelativeError(y_data,y_model):\n", - " return abs((y_data-y_model)/y_data)\n", - "print(RelativeError(Energies, ytilde))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### The $\\chi^2$ function\n", - "\n", - "Normally, the response (dependent or outcome) variable $y_i$ is the\n", - "outcome of a numerical experiment or another type of experiment and is\n", - "thus only an approximation to the true value. It is then always\n", - "accompanied by an error estimate, often limited to a statistical error\n", - "estimate given by the standard deviation discussed earlier. In the\n", - "discussion here we will treat $y_i$ as our exact value for the\n", - "response variable.\n", - "\n", - "Introducing the standard deviation $\\sigma_i$ for each measurement\n", - "$y_i$, we define now the $\\chi^2$ function (omitting the $1/n$ term)\n", - "as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\chi^2(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\frac{\\left(y_i-\\tilde{y}_i\\right)^2}{\\sigma_i^2}=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\frac{1}{\\boldsymbol{\\Sigma^2}}\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the matrix $\\boldsymbol{\\Sigma}$ is a diagonal matrix with $\\sigma_i$ as matrix elements. \n", - "\n", - "\n", - "In order to find the parameters $\\beta_i$ we will then minimize the spread of $\\chi^2(\\boldsymbol{\\beta})$ by requiring" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_j} = \\frac{\\partial }{\\partial \\beta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)^2\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which results in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}\\frac{x_{ij}}{\\sigma_i}\\left(\\frac{y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or in a matrix-vector form as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\beta}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined the matrix $\\boldsymbol{A} =\\boldsymbol{X}/\\boldsymbol{\\Sigma}$ with matrix elements $a_{ij} = x_{ij}/\\sigma_i$ and the vector $\\boldsymbol{b}$ with elements $b_i = y_i/\\sigma_i$. \n", - "\n", - "We can rewrite" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\beta}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}^T\\boldsymbol{b} = \\boldsymbol{A}^T\\boldsymbol{A}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and if the matrix $\\boldsymbol{A}^T\\boldsymbol{A}$ is invertible we have the solution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta} =\\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1}\\boldsymbol{A}^T\\boldsymbol{b}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we then introduce the matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{H} = \\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we have then the following expression for the parameters $\\beta_j$ (the matrix elements of $\\boldsymbol{H}$ are $h_{ij}$)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_j = \\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}\\frac{y_i}{\\sigma_i}\\frac{x_{ik}}{\\sigma_i} = \\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}b_ia_{ik}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We state without proof the expression for the uncertainty in the parameters $\\beta_j$ as (we leave this as an exercise)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2(\\beta_j) = \\sum_{i=0}^{n-1}\\sigma_i^2\\left( \\frac{\\partial \\beta_j}{\\partial y_i}\\right)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2(\\beta_j) = \\left(\\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}a_{ik}\\right)\\left(\\sum_{l=0}^{p-1}h_{jl}\\sum_{m=0}^{n-1}a_{ml}\\right) = h_{jj}!\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The first step here is to approximate the function $y$ with a first-order polynomial, that is we write" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y=y(x) \\rightarrow y(x_i) \\approx \\beta_0+\\beta_1 x_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "By computing the derivatives of $\\chi^2$ with respect to $\\beta_0$ and $\\beta_1$ show that these are given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_0} = -2\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\beta_0-\\beta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_1} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_i\\left(\\frac{y_i-\\beta_0-\\beta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For a linear fit (a first-order polynomial) we don't need to invert a matrix!! \n", - "Defining" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma = \\sum_{i=0}^{n-1}\\frac{1}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_x = \\sum_{i=0}^{n-1}\\frac{x_{i}}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_y = \\sum_{i=0}^{n-1}\\left(\\frac{y_i}{\\sigma_i^2}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_{xx} = \\sum_{i=0}^{n-1}\\frac{x_ix_{i}}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_{xy} = \\sum_{i=0}^{n-1}\\frac{y_ix_{i}}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_0 = \\frac{\\gamma_{xx}\\gamma_y-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_1 = \\frac{\\gamma_{xy}\\gamma-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This approach (different linear and non-linear regression) suffers\n", - "often from both being underdetermined and overdetermined in the\n", - "unknown coefficients $\\beta_i$. A better approach is to use the\n", - "Singular Value Decomposition (SVD) method discussed below. Or using\n", - "Lasso and Ridge regression. See below.\n", - "\n", - "\n", - "### Fitting an Equation of State for Dense Nuclear Matter\n", - "\n", - "Before we continue, let us introduce yet another example. We are going to fit the\n", - "nuclear equation of state using results from many-body calculations.\n", - "The equation of state we have made available here, as function of\n", - "density, has been derived using modern nucleon-nucleon potentials with\n", - "[the addition of three-body\n", - "forces](https://www.sciencedirect.com/science/article/pii/S0370157399001106). This\n", - "time the file is presented as a standard **csv** file.\n", - "\n", - "The beginning of the Python code here is similar to what you have seen\n", - "before, with the same initializations and declarations. We use also\n", - "**pandas** again, rather extensively in order to organize our data.\n", - "\n", - "The difference now is that we use **Scikit-Learn's** regression tools\n", - "instead of our own matrix inversion implementation. Furthermore, we\n", - "sneak in **Ridge** regression (to be discussed below) which includes a\n", - "hyperparameter $\\lambda$, also to be explained below." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "import matplotlib.pyplot as plt\n", - "import sklearn.linear_model as skl\n", - "from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"EoS.csv\"),'r')\n", - "\n", - "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", - "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", - "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", - "EoS = EoS.dropna()\n", - "Energies = EoS['Energy']\n", - "Density = EoS['Density']\n", - "# The design matrix now as function of various polytrops\n", - "X = np.zeros((len(Density),4))\n", - "X[:,3] = Density**(4.0/3.0)\n", - "X[:,2] = Density\n", - "X[:,1] = Density**(2.0/3.0)\n", - "X[:,0] = 1\n", - "\n", - "# We use now Scikit-Learn's linear regressor and ridge regressor\n", - "# OLS part\n", - "clf = skl.LinearRegression().fit(X, Energies)\n", - "ytilde = clf.predict(X)\n", - "EoS['Eols'] = ytilde\n", - "# The mean squared error \n", - "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, ytilde))\n", - "# Explained variance score: 1 is perfect prediction \n", - "print('Variance score: %.2f' % r2_score(Energies, ytilde))\n", - "# Mean absolute error \n", - "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde))\n", - "print(clf.coef_, clf.intercept_)\n", - "\n", - "# The Ridge regression with a hyperparameter lambda = 0.1\n", - "_lambda = 0.1\n", - "clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies)\n", - "yridge = clf_ridge.predict(X)\n", - "EoS['Eridge'] = yridge\n", - "# The mean squared error \n", - "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, yridge))\n", - "# Explained variance score: 1 is perfect prediction \n", - "print('Variance score: %.2f' % r2_score(Energies, yridge))\n", - "# Mean absolute error \n", - "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge))\n", - "print(clf_ridge.coef_, clf_ridge.intercept_)\n", - "\n", - "fig, ax = plt.subplots()\n", - "ax.set_xlabel(r'$\\rho[\\mathrm{fm}^{-3}]$')\n", - "ax.set_ylabel(r'Energy per particle')\n", - "ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2,\n", - " label='Theoretical data')\n", - "ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m',\n", - " label='OLS')\n", - "ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g',\n", - " label='Ridge $\\lambda = 0.1$')\n", - "ax.legend()\n", - "save_fig(\"EoSfitting\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The above simple polynomial in density $\\rho$ gives an excellent fit\n", - "to the data. \n", - "\n", - "We note also that there is a small deviation between the\n", - "standard OLS and the Ridge regression at higher densities. We discuss this in more detail\n", - "below.\n", - "\n", - "\n", - "## Splitting our Data in Training and Test data\n", - "\n", - "It is normal in essentially all Machine Learning studies to split the\n", - "data in a training set and a test set (sometimes also an additional\n", - "validation set). **Scikit-Learn** has an own function for this. There\n", - "is no explicit recipe for how much data should be included as training\n", - "data and say test data. An accepted rule of thumb is to use\n", - "approximately $2/3$ to $4/5$ of the data as training data. We will\n", - "postpone a discussion of this splitting to the end of these notes and\n", - "our discussion of the so-called **bias-variance** tradeoff. Here we\n", - "limit ourselves to repeat the above equation of state fitting example\n", - "but now splitting the data into a training set and a test set." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.model_selection import train_test_split\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "def R2(y_data, y_model):\n", - " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n", - "def MSE(y_data,y_model):\n", - " n = np.size(y_model)\n", - " return np.sum((y_data-y_model)**2)/n\n", - "\n", - "infile = open(data_path(\"EoS.csv\"),'r')\n", - "\n", - "# Read the EoS data as csv file and organized into two arrays with density and energies\n", - "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", - "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", - "EoS = EoS.dropna()\n", - "Energies = EoS['Energy']\n", - "Density = EoS['Density']\n", - "# The design matrix now as function of various polytrops\n", - "X = np.zeros((len(Density),5))\n", - "X[:,0] = 1\n", - "X[:,1] = Density**(2.0/3.0)\n", - "X[:,2] = Density\n", - "X[:,3] = Density**(4.0/3.0)\n", - "X[:,4] = Density**(5.0/3.0)\n", - "# We split the data in test and training data\n", - "X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)\n", - "# matrix inversion to find beta\n", - "beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)\n", - "# and then make the prediction\n", - "ytilde = X_train @ beta\n", - "print(\"Training R2\")\n", - "print(R2(y_train,ytilde))\n", - "print(\"Training MSE\")\n", - "print(MSE(y_train,ytilde))\n", - "ypredict = X_test @ beta\n", - "print(\"Test R2\")\n", - "print(R2(y_test,ypredict))\n", - "print(\"Test MSE\")\n", - "print(MSE(y_test,ypredict))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The Boston housing data example\n", - "\n", - "The Boston housing \n", - "data set was originally a part of UCI Machine Learning Repository\n", - "and has been removed now. The data set is now included in **Scikit-Learn**'s \n", - "library. There are 506 samples and 13 feature (predictor) variables\n", - "in this data set. The objective is to predict the value of prices of\n", - "the house using the features (predictors) listed here.\n", - "\n", - "The features/predictors are\n", - "1. CRIM: Per capita crime rate by town\n", - "\n", - "2. ZN: Proportion of residential land zoned for lots over 25000 square feet\n", - "\n", - "3. INDUS: Proportion of non-retail business acres per town\n", - "\n", - "4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n", - "\n", - "5. NOX: Nitric oxide concentration (parts per 10 million)\n", - "\n", - "6. RM: Average number of rooms per dwelling\n", - "\n", - "7. AGE: Proportion of owner-occupied units built prior to 1940\n", - "\n", - "8. DIS: Weighted distances to five Boston employment centers\n", - "\n", - "9. RAD: Index of accessibility to radial highways\n", - "\n", - "10. TAX: Full-value property tax rate per USD10000\n", - "\n", - "11. B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town\n", - "\n", - "12. LSTAT: Percentage of lower status of the population\n", - "\n", - "13. MEDV: Median value of owner-occupied homes in USD 1000s\n", - "\n", - "## Housing data, the code\n", - "We start by importing the libraries" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt \n", - "\n", - "import pandas as pd \n", - "import seaborn as sns" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and load the Boston Housing DataSet from **Scikit-Learn**" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.datasets import load_boston\n", - "\n", - "boston_dataset = load_boston()\n", - "\n", - "# boston_dataset is a dictionary\n", - "# let's check what it contains\n", - "boston_dataset.keys()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Then we invoke Pandas" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)\n", - "boston.head()\n", - "boston['MEDV'] = boston_dataset.target" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and preprocess the data" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# check for missing values in all the columns\n", - "boston.isnull().sum()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can then visualize the data" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# set the size of the figure\n", - "sns.set(rc={'figure.figsize':(11.7,8.27)})\n", - "\n", - "# plot a histogram showing the distribution of the target values\n", - "sns.distplot(boston['MEDV'], bins=30)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is now useful to look at the correlation matrix" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# compute the pair wise correlation for all columns \n", - "correlation_matrix = boston.corr().round(2)\n", - "# use the heatmap function from seaborn to plot the correlation matrix\n", - "# annot = True to print the values inside the square\n", - "sns.heatmap(data=correlation_matrix, annot=True)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "From the above coorelation plot we can see that **MEDV** is strongly correlated to **LSTAT** and **RM**. We see also that **RAD** and **TAX** are stronly correlated, but we don't include this in our features together to avoid multi-colinearity" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "plt.figure(figsize=(20, 5))\n", - "\n", - "features = ['LSTAT', 'RM']\n", - "target = boston['MEDV']\n", - "\n", - "for i, col in enumerate(features):\n", - " plt.subplot(1, len(features) , i+1)\n", - " x = boston[col]\n", - " y = target\n", - " plt.scatter(x, y, marker='o')\n", - " plt.title(col)\n", - " plt.xlabel(col)\n", - " plt.ylabel('MEDV')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now we start training our model" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])\n", - "Y = boston['MEDV']" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We split the data into training and test sets" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.model_selection import train_test_split\n", - "\n", - "# splits the training and test data set in 80% : 20%\n", - "# assign random_state to any value.This ensures consistency.\n", - "X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "print(Y_train.shape)\n", - "print(Y_test.shape)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Then we use the linear regression functionality from **Scikit-Learn**" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.linear_model import LinearRegression\n", - "from sklearn.metrics import mean_squared_error, r2_score\n", - "\n", - "lin_model = LinearRegression()\n", - "lin_model.fit(X_train, Y_train)\n", - "\n", - "# model evaluation for training set\n", - "\n", - "y_train_predict = lin_model.predict(X_train)\n", - "rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))\n", - "r2 = r2_score(Y_train, y_train_predict)\n", - "\n", - "print(\"The model performance for training set\")\n", - "print(\"--------------------------------------\")\n", - "print('RMSE is {}'.format(rmse))\n", - "print('R2 score is {}'.format(r2))\n", - "print(\"\\n\")\n", - "\n", - "# model evaluation for testing set\n", - "\n", - "y_test_predict = lin_model.predict(X_test)\n", - "# root mean square error of the model\n", - "rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))\n", - "\n", - "# r-squared score of the model\n", - "r2 = r2_score(Y_test, y_test_predict)\n", - "\n", - "print(\"The model performance for testing set\")\n", - "print(\"--------------------------------------\")\n", - "print('RMSE is {}'.format(rmse))\n", - "print('R2 score is {}'.format(r2))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# plotting the y_test vs y_pred\n", - "# ideally should have been a straight line\n", - "plt.scatter(Y_test, y_test_predict)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Reducing the number of degrees of freedom, overarching view\n", - "\n", - "Many Machine Learning problems involve thousands or even millions of\n", - "features for each training instance. Not only does this make training\n", - "extremely slow, it can also make it much harder to find a good\n", - "solution, as we will see. This problem is often referred to as the\n", - "curse of dimensionality. Fortunately, in real-world problems, it is\n", - "often possible to reduce the number of features considerably, turning\n", - "an intractable problem into a tractable one.\n", - "\n", - "Later we will discuss some of the most popular dimensionality reduction\n", - "techniques: the principal component analysis (PCA), Kernel PCA, and\n", - "Locally Linear Embedding (LLE). \n", - "\n", - "\n", - "Principal component analysis and its various variants deal with the\n", - "problem of fitting a low-dimensional [affine\n", - "subspace](https://en.wikipedia.org/wiki/Affine_space) to a set of of\n", - "data points in a high-dimensional space. With its family of methods it\n", - "is one of the most used tools in data modeling, compression and\n", - "visualization.\n", - "\n", - "\n", - "Before we proceed however, we will discuss how to preprocess our\n", - "data. Till now and in connection with our previous examples we have\n", - "not met so many cases where we are too sensitive to the scaling of our\n", - "data. Normally the data may need a rescaling and/or may be sensitive\n", - "to extreme values. Scaling the data renders our inputs much more\n", - "suitable for the algorithms we want to employ.\n", - "\n", - "**Scikit-Learn** has several functions which allow us to rescale the\n", - "data, normally resulting in much better results in terms of various\n", - "accuracy scores. The **StandardScaler** function in **Scikit-Learn**\n", - "ensures that for each feature/predictor we study the mean value is\n", - "zero and the variance is one (every column in the design/feature\n", - "matrix). This scaling has the drawback that it does not ensure that\n", - "we have a particular maximum or minimum in our data set. Another\n", - "function included in **Scikit-Learn** is the **MinMaxScaler** which\n", - "ensures that all features are exactly between $0$ and $1$. The\n", - "\n", - "\n", - "The **Normalizer** scales each data\n", - "point such that the feature vector has a euclidean length of one. In other words, it\n", - "projects a data point on the circle (or sphere in the case of higher dimensions) with a\n", - "radius of 1. This means every data point is scaled by a different number (by the\n", - "inverse of it’s length).\n", - "This normalization is often used when only the direction (or angle) of the data matters,\n", - "not the length of the feature vector.\n", - "\n", - "The **RobustScaler** works similarly to the StandardScaler in that it\n", - "ensures statistical properties for each feature that guarantee that\n", - "they are on the same scale. However, the RobustScaler uses the median\n", - "and quartiles, instead of mean and variance. This makes the\n", - "RobustScaler ignore data points that are very different from the rest\n", - "(like measurement errors). These odd data points are also called\n", - "outliers, and might often lead to trouble for other scaling\n", - "techniques.\n", - "\n", - "\n", - "### Simple preprocessing examples, Franke function and regression" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "import sklearn.linear_model as skl\n", - "from sklearn.metrics import mean_squared_error\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "\n", - "def FrankeFunction(x,y):\n", - "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", - "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", - "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", - "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", - "\treturn term1 + term2 + term3 + term4\n", - "\n", - "\n", - "def create_X(x, y, n ):\n", - "\tif len(x.shape) > 1:\n", - "\t\tx = np.ravel(x)\n", - "\t\ty = np.ravel(y)\n", - "\n", - "\tN = len(x)\n", - "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", - "\tX = np.ones((N,l))\n", - "\n", - "\tfor i in range(1,n+1):\n", - "\t\tq = int((i)*(i+1)/2)\n", - "\t\tfor k in range(i+1):\n", - "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", - "\n", - "\treturn X\n", - "\n", - "\n", - "# Making meshgrid of datapoints and compute Franke's function\n", - "n = 5\n", - "N = 1000\n", - "x = np.sort(np.random.uniform(0, 1, N))\n", - "y = np.sort(np.random.uniform(0, 1, N))\n", - "z = FrankeFunction(x, y)\n", - "X = create_X(x, y, n=n) \n", - "# split in training and test data\n", - "X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=0.2)\n", - "\n", - "\n", - "clf = skl.LinearRegression().fit(X_train, y_train)\n", - "\n", - "# The mean squared error and R2 score\n", - "print(\"MSE before scaling: {:.2f}\".format(mean_squared_error(clf.predict(X_test), y_test)))\n", - "print(\"R2 score before scaling {:.2f}\".format(clf.score(X_test,y_test)))\n", - "\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "print(\"Feature min values before scaling:\\n {}\".format(X_train.min(axis=0)))\n", - "print(\"Feature max values before scaling:\\n {}\".format(X_train.max(axis=0)))\n", - "\n", - "print(\"Feature min values after scaling:\\n {}\".format(X_train_scaled.min(axis=0)))\n", - "print(\"Feature max values after scaling:\\n {}\".format(X_train_scaled.max(axis=0)))\n", - "\n", - "clf = skl.LinearRegression().fit(X_train_scaled, y_train)\n", - "\n", - "\n", - "print(\"MSE after scaling: {:.2f}\".format(mean_squared_error(clf.predict(X_test_scaled), y_test)))\n", - "print(\"R2 score for scaled data: {:.2f}\".format(clf.score(X_test_scaled,y_test)))" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter10.ipynb b/doc/LectureNotes/_build/html/_sources/chapter10.ipynb deleted file mode 100644 index cc65cfe33..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter10.ipynb +++ /dev/null @@ -1,2047 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Building a Feed Forward Neural Network\n", - "\n", - "We are now gong to develop an example based on the MNIST data\n", - "base. This is a classification problem and we need to use our\n", - "cross-entropy function we discussed in connection with logistic\n", - "regression. The cross-entropy defines our cost function for the\n", - "classificaton problems with neural networks.\n", - "\n", - "In binary classification with two classes $(0, 1)$ we define the\n", - "logistic/sigmoid function as the probability that a particular input\n", - "is in class $0$ or $1$. This is possible because the logistic\n", - "function takes any input from the real numbers and inputs a number\n", - "between 0 and 1, and can therefore be interpreted as a probability. It\n", - "also has other nice properties, such as a derivative that is simple to\n", - "calculate.\n", - "\n", - "For an input $\\boldsymbol{a}$ from the hidden layer, the probability that the input $\\boldsymbol{x}$\n", - "is in class 0 or 1 is just. We let $\\theta$ represent the unknown weights and biases to be adjusted by our equations). The variable $x$\n", - "represents our activation values $z$. We have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(y = 0 \\mid \\hat{x}, \\hat{\\theta}) = \\frac{1}{1 + \\exp{(- \\hat{x}})} ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(y = 1 \\mid \\hat{x}, \\hat{\\theta}) = 1 - P(y = 0 \\mid \\hat{x}, \\hat{\\theta}) ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $y \\in \\{0, 1\\}$ and $\\hat{\\theta}$ represents the weights and biases\n", - "of our network.\n", - "\n", - "\n", - "\n", - "## Defining the cost function\n", - "\n", - "Our cost function is given as (see the Logistic regression lectures)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\theta}) = - \\ln P(\\mathcal{D} \\mid \\hat{\\theta}) = - \\sum_{i=1}^n\n", - "y_i \\ln[P(y_i = 0)] + (1 - y_i) \\ln [1 - P(y_i = 0)] = \\sum_{i=1}^n \\mathcal{L}_i(\\hat{\\theta}) .\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This last equality means that we can interpret our *cost* function as a sum over the *loss* function\n", - "for each point in the dataset $\\mathcal{L}_i(\\hat{\\theta})$. \n", - "The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather\n", - "than maximizing a negative number. \n", - "\n", - "In *multiclass* classification it is common to treat each integer label as a so called *one-hot* vector: \n", - "\n", - "$y = 5 \\quad \\rightarrow \\quad \\hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$ and\n", - "\n", - "\n", - "$y = 1 \\quad \\rightarrow \\quad \\hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$ \n", - "\n", - "\n", - "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset (numbers from $0$ to $9$).. \n", - "\n", - "If $\\hat{x}_i$ is the $i$-th input (image), $y_{ic}$ refers to the $c$-th component of the $i$-th\n", - "output vector $\\hat{y}_i$. \n", - "The probability of $\\hat{x}_i$ being in class $c$ will be given by the softmax function:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(y_{ic} = 1 \\mid \\hat{x}_i, \\hat{\\theta}) = \\frac{\\exp{((\\hat{a}_i^{hidden})^T \\hat{w}_c)}}\n", - "{\\sum_{c'=0}^{C-1} \\exp{((\\hat{a}_i^{hidden})^T \\hat{w}_{c'})}} ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which reduces to the logistic function in the binary case. \n", - "The likelihood of this $C$-class classifier\n", - "is now given as:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(\\mathcal{D} \\mid \\hat{\\theta}) = \\prod_{i=1}^n \\prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Again we take the negative log-likelihood to define our cost function:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\theta}) = - \\log{P(\\mathcal{D} \\mid \\hat{\\theta})}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "See the logistic regression lectures for a full definition of the cost function.\n", - "\n", - "The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!\n", - "\n", - "\n", - "### Example: binary classification problem\n", - "\n", - "As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters $\\beta$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\beta}) = - \\sum_{i=1}^n \\left(y_i\\log{p(y_i \\vert x_i,\\hat{\\beta})}+(1-y_i)\\log{1-p(y_i \\vert x_i,\\hat{\\beta})}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we had defined the logistic (sigmoid) function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(y_i =1\\vert x_i,\\hat{\\beta})=\\frac{\\exp{(\\beta_0+\\beta_1 x_i)}}{1+\\exp{(\\beta_0+\\beta_1 x_i)}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(y_i =0\\vert x_i,\\hat{\\beta})=1-p(y_i =1\\vert x_i,\\hat{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The parameters $\\hat{\\beta}$ were defined using a minimization method like gradient descent or Newton-Raphson's method. \n", - "\n", - "Now we replace $x_i$ with the activation $z_i^l$ for a given layer $l$ and the outputs as $y_i=a_i^l=f(z_i^l)$, with $z_i^l$ now being a function of the weights $w_{ij}^l$ and biases $b_i^l$. \n", - "We have then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_i^l = y_i = \\frac{\\exp{(z_i^l)}}{1+\\exp{(z_i^l)}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "z_i^l = \\sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the superscript $l-1$ indicates that these are the outputs from layer $l-1$.\n", - "Our cost function at the final layer $l=L$ is now" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{W}) = - \\sum_{i=1}^n \\left(t_i\\log{a_i^L}+(1-t_i)\\log{(1-a_i^L)}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined the targets $t_i$. The derivatives of the cost function with respect to the output $a_i^L$ are then easily calculated and we get" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\hat{W})}{\\partial a_i^L} = \\frac{a_i^L-t_i}{a_i^L(1-a_i^L)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In case we use another activation function than the logistic one, we need to evaluate other derivatives. \n", - "\n", - "\n", - "\n", - "### The Softmax function\n", - "\n", - "In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f(z_i^l)}{\\partial w_{jk}^l} =\n", - "\\frac{\\partial f(z_i^l)}{\\partial z_j^l} \\frac{\\partial z_j^l}{\\partial w_{jk}^l}= \\frac{\\partial f(z_i^l)}{\\partial z_j^l}a_k^{l-1}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For the Softmax function we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(z_i^l) = \\frac{\\exp{(z_i^l)}}{\\sum_{m=1}^K\\exp{(z_m^l)}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Its derivative with respect to $z_j^l$ gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f(z_i^l)}{\\partial z_j^l}= f(z_i^l)\\left(\\delta_{ij}-f(z_j^l)\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in case of the simply binary model reduces to having $i=j$. \n", - "\n", - "\n", - "## Developing a code for doing neural networks with back propagation\n", - "\n", - "\n", - "One can identify a set of key steps when using neural networks to solve supervised learning problems: \n", - "\n", - "1. Collect and pre-process data \n", - "\n", - "2. Define model and architecture \n", - "\n", - "3. Choose cost function and optimizer \n", - "\n", - "4. Train the model \n", - "\n", - "5. Evaluate model performance on test data \n", - "\n", - "6. Adjust hyperparameters (if necessary, network architecture)\n", - "\n", - "### Collect and pre-process data\n", - "\n", - "Here we will be using the MNIST dataset, which is readily available through the **scikit-learn**\n", - "package. You may also find it for example [here](http://yann.lecun.com/exdb/mnist/). \n", - "The *MNIST* (Modified National Institute of Standards and Technology) database is a large database\n", - "of handwritten digits that is commonly used for training various image processing systems. \n", - "The MNIST dataset consists of 70 000 images of size $28\\times 28$ pixels, each labeled from 0 to 9. \n", - "The scikit-learn dataset we will use consists of a selection of 1797 images of size $8\\times 8$ collected and processed from this database. \n", - "\n", - "To feed data into a feed-forward neural network we need to represent\n", - "the inputs as a design/feature matrix $X = (n_{inputs}, n_{features})$. Each\n", - "row represents an *input*, in this case a handwritten digit, and\n", - "each column represents a *feature*, in this case a pixel. The\n", - "correct answers, also known as *labels* or *targets* are\n", - "represented as a 1D array of integers \n", - "$Y = (n_{inputs}) = (5, 3, 1, 8,...)$.\n", - "\n", - "As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from\n", - "measurements of height (in m) \n", - "and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example: \n", - "\n", - "$$ X = \\begin{bmatrix}\n", - "1.85 & 81\\\\\n", - "1.71 & 65\\\\\n", - "1.95 & 103\\\\\n", - "1.55 & 42\\\\\n", - "1.63 & 56\n", - "\\end{bmatrix} ,$$ \n", - "\n", - "and the targets would be: \n", - "\n", - "$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$ \n", - "\n", - "Since each input image is a 2D matrix, we need to flatten the image\n", - "(i.e. \"unravel\" the 2D matrix into a 1D array) to turn the data into a\n", - "design/feature matrix. This means we lose all spatial information in the\n", - "image, such as locality and translational invariance. More complicated\n", - "architectures such as Convolutional Neural Networks can take advantage\n", - "of such information, and are most commonly applied when analyzing\n", - "images." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "# import necessary packages\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn import datasets\n", - "\n", - "\n", - "# ensure the same random numbers appear every time\n", - "np.random.seed(0)\n", - "\n", - "# display images in notebook\n", - "%matplotlib inline\n", - "plt.rcParams['figure.figsize'] = (12,12)\n", - "\n", - "\n", - "# download MNIST dataset\n", - "digits = datasets.load_digits()\n", - "\n", - "# define inputs and labels\n", - "inputs = digits.images\n", - "labels = digits.target\n", - "\n", - "print(\"inputs = (n_inputs, pixel_width, pixel_height) = \" + str(inputs.shape))\n", - "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", - "\n", - "\n", - "# flatten the image\n", - "# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64\n", - "n_inputs = len(inputs)\n", - "inputs = inputs.reshape(n_inputs, -1)\n", - "print(\"X = (n_inputs, n_features) = \" + str(inputs.shape))\n", - "\n", - "\n", - "# choose some random images to display\n", - "indices = np.arange(n_inputs)\n", - "random_indices = np.random.choice(indices, size=5)\n", - "\n", - "for i, image in enumerate(digits.images[random_indices]):\n", - " plt.subplot(1, 5, i+1)\n", - " plt.axis('off')\n", - " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", - " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Train and test datasets\n", - "\n", - "Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions. \n", - "\n", - "We will reserve $80 \\%$ of our dataset for training and $20 \\%$ for testing. \n", - "\n", - "It is important that the train and test datasets are drawn randomly from our dataset, to ensure\n", - "no bias in the sampling. \n", - "Say you are taking measurements of weather data to predict the weather in the coming 5 days.\n", - "You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data\n", - "collected from 12.00 to 24.00." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.model_selection import train_test_split\n", - "\n", - "# one-liner from scikit-learn library\n", - "train_size = 0.8\n", - "test_size = 1 - train_size\n", - "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", - " test_size=test_size)\n", - "\n", - "# equivalently in numpy\n", - "def train_test_split_numpy(inputs, labels, train_size, test_size):\n", - " n_inputs = len(inputs)\n", - " inputs_shuffled = inputs.copy()\n", - " labels_shuffled = labels.copy()\n", - " \n", - " np.random.shuffle(inputs_shuffled)\n", - " np.random.shuffle(labels_shuffled)\n", - " \n", - " train_end = int(n_inputs*train_size)\n", - " X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]\n", - " Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]\n", - " \n", - " return X_train, X_test, Y_train, Y_test\n", - "\n", - "#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)\n", - "\n", - "print(\"Number of training images: \" + str(len(X_train)))\n", - "print(\"Number of test images: \" + str(len(X_test)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Define model and architecture\n", - "\n", - "Our simple feed-forward neural network will consist of an *input* layer, a single *hidden* layer and an *output* layer. The activation $y$ of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have \n", - "\n", - "$$ z = \\sum_{i=1}^n w_i a_i ,$$\n", - "\n", - "$$ y = f(z) ,$$\n", - "\n", - "where $f$ is the activation function, $a_i$ represents input from neuron $i$ in the preceding layer\n", - "and $w_i$ is the weight to input $i$. \n", - "The activation of the neurons in the input layer is just the features (e.g. a pixel value). \n", - "\n", - "The simplest activation function for a neuron is the *Heaviside* function:\n", - "\n", - "$$ f(z) = \n", - "\\begin{cases}\n", - "1, & z > 0\\\\\n", - "0, & \\text{otherwise}\n", - "\\end{cases}\n", - "$$\n", - "\n", - "A feed-forward neural network with this activation is known as a *perceptron*. \n", - "For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer. \n", - "This activation can be generalized to $k$ classes (using e.g. the *one-against-all* strategy), \n", - "and we call these architectures *multiclass perceptrons*. \n", - "\n", - "However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and \n", - "Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function. \n", - "\n", - "Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU). \n", - "We will be using the sigmoid function $\\sigma(x)$: \n", - "\n", - "$$ f(x) = \\sigma(x) = \\frac{1}{1 + e^{-x}} ,$$\n", - "\n", - "which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.\n", - "\n", - "### Layers\n", - "\n", - "* Input \n", - "\n", - "Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons. \n", - "\n", - "* Hidden layer\n", - "\n", - "We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer. \n", - "Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer. \n", - "\n", - "* Output\n", - "\n", - "If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,\n", - "which could output 0 or 1 according to the Heaviside function. This would be an example of a *hard* classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a *soft* classifier, which outputs the probability of being in class 0 or 1. \n", - "\n", - "For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class. \n", - "\n", - "Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons $j = 0,1,...,9$. The activation of each output neuron $j$ will be according to the *softmax* function: \n", - "\n", - "$$ P(\\text{class $j$} \\mid \\text{input $\\hat{a}$}) = \\frac{\\exp{(\\hat{a}^T \\hat{w}_j)}}\n", - "{\\sum_{c=0}^{9} \\exp{(\\hat{a}^T \\hat{w}_c)}} ,$$ \n", - "\n", - "i.e. each neuron $j$ outputs the probability of being in class $j$ given an input from the hidden layer $\\hat{a}$, with $\\hat{w}_j$ the weights of neuron $j$ to the inputs. \n", - "The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1. \n", - "The exponent is just the weighted sum of inputs as before: \n", - "\n", - "$$ z_j = \\sum_{i=1}^n w_ {ij} a_i+b_j.$$ \n", - "\n", - "Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500\n", - "weights to the output layer.\n", - "\n", - "\n", - "Typically weights are initialized with small values distributed around zero, drawn from a uniform\n", - "or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. \n", - "\n", - "Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range\n", - "of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron $j$, $b_j$: \n", - "\n", - "$$ z_j = \\sum_{i=1}^n w_ {ij} a_i + b_j.$$ \n", - "\n", - "The bias weights $\\hat{b}$ are often initialized to zero, but a small value like $0.01$ ensures all neurons have some output which can be backpropagated in the first training cycle." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# building our neural network\n", - "\n", - "n_inputs, n_features = X_train.shape\n", - "n_hidden_neurons = 50\n", - "n_categories = 10\n", - "\n", - "# we make the weights normally distributed using numpy.random.randn\n", - "\n", - "# weights and bias in the hidden layer\n", - "hidden_weights = np.random.randn(n_features, n_hidden_neurons)\n", - "hidden_bias = np.zeros(n_hidden_neurons) + 0.01\n", - "\n", - "# weights and bias in the output layer\n", - "output_weights = np.random.randn(n_hidden_neurons, n_categories)\n", - "output_bias = np.zeros(n_categories) + 0.01" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Feed-forward pass\n", - "\n", - "Denote $F$ the number of features, $H$ the number of hidden neurons and $C$ the number of categories. \n", - "For each input image we calculate a weighted sum of input features (pixel values) to each neuron $j$ in the hidden layer $l$: \n", - "\n", - "$$ z_{j}^{l} = \\sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$\n", - "\n", - "this is then passed through our activation function \n", - "\n", - "$$ a_{j}^{l} = f(z_{j}^{l}) .$$ \n", - "\n", - "We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron $j$ in the output layer: \n", - "\n", - "$$ z_{j}^{L} = \\sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$ \n", - "\n", - "Finally we calculate the output of neuron $j$ in the output layer using the softmax function: \n", - "\n", - "$$ a_{j}^{L} = \\frac{\\exp{(z_j^{L})}}\n", - "{\\sum_{c=0}^{C-1} \\exp{(z_c^{L})}} .$$ \n", - "\n", - "\n", - "Since our data has the dimensions $X = (n_{inputs}, n_{features})$ and our weights to the hidden\n", - "layer have the dimensions \n", - "$W_{hidden} = (n_{features}, n_{hidden})$,\n", - "we can easily feed the network all our training data in one go by taking the matrix product \n", - "\n", - "$$ X W^{h} = (n_{inputs}, n_{hidden}),$$ \n", - "\n", - "and obtain a matrix that holds the weighted sum of inputs to the hidden layer\n", - "for each input image and each hidden neuron. \n", - "We also add the bias to obtain a matrix of weighted sums to the hidden layer $Z^{h}$: \n", - "\n", - "$$ \\hat{z}^{l} = \\hat{X} \\hat{W}^{l} + \\hat{b}^{l} ,$$\n", - "\n", - "meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image. \n", - "This is then passed through the activation: \n", - "\n", - "$$ \\hat{a}^{l} = f(\\hat{z}^l) .$$ \n", - "\n", - "This is fed to the output layer: \n", - "\n", - "$$ \\hat{z}^{L} = \\hat{a}^{L} \\hat{W}^{L} + \\hat{b}^{L} .$$\n", - "\n", - "Finally we receive our output values for each image and each category by passing it through the softmax function: \n", - "\n", - "$$ output = softmax (\\hat{z}^{L}) = (n_{inputs}, n_{categories}) .$$" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# setup the feed-forward pass, subscript h = hidden layer\n", - "\n", - "def sigmoid(x):\n", - " return 1/(1 + np.exp(-x))\n", - "\n", - "def feed_forward(X):\n", - " # weighted sum of inputs to the hidden layer\n", - " z_h = np.matmul(X, hidden_weights) + hidden_bias\n", - " # activation in the hidden layer\n", - " a_h = sigmoid(z_h)\n", - " \n", - " # weighted sum of inputs to the output layer\n", - " z_o = np.matmul(a_h, output_weights) + output_bias\n", - " # softmax output\n", - " # axis 0 holds each input and axis 1 the probabilities of each category\n", - " exp_term = np.exp(z_o)\n", - " probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n", - " \n", - " return probabilities\n", - "\n", - "probabilities = feed_forward(X_train)\n", - "print(\"probabilities = (n_inputs, n_categories) = \" + str(probabilities.shape))\n", - "print(\"probability that image 0 is in category 0,1,2,...,9 = \\n\" + str(probabilities[0]))\n", - "print(\"probabilities sum up to: \" + str(probabilities[0].sum()))\n", - "print()\n", - "\n", - "# we obtain a prediction by taking the class with the highest likelihood\n", - "def predict(X):\n", - " probabilities = feed_forward(X)\n", - " return np.argmax(probabilities, axis=1)\n", - "\n", - "predictions = predict(X_train)\n", - "print(\"predictions = (n_inputs) = \" + str(predictions.shape))\n", - "print(\"prediction for image 0: \" + str(predictions[0]))\n", - "print(\"correct label for image 0: \" + str(Y_train[0]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Choose cost function and optimizer\n", - "\n", - "To measure how well our neural network is doing we need to introduce a cost function. \n", - "We will call the function that gives the error of a single sample output the *loss* function, and the function\n", - "that gives the total error of our network across all samples the *cost* function.\n", - "A typical choice for multiclass classification is the *cross-entropy* loss, also known as the negative log likelihood. \n", - "\n", - "In *multiclass* classification it is common to treat each integer label as a so called *one-hot* vector: \n", - "\n", - "$$ y = 5 \\quad \\rightarrow \\quad \\hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$ \n", - "\n", - "\n", - "$$ y = 1 \\quad \\rightarrow \\quad \\hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$ \n", - "\n", - "\n", - "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset. \n", - "\n", - "Let $y_{ic}$ denote the $c$-th component of the $i$-th one-hot vector. \n", - "We define the cost function $\\mathcal{C}$ as a sum over the cross-entropy loss for each point $\\hat{x}_i$ in the dataset.\n", - "\n", - "In the one-hot representation only one of the terms in the loss function is non-zero, namely the\n", - "probability of the correct category $c'$ \n", - "(i.e. the category $c'$ such that $y_{ic'} = 1$). This means that the cross entropy loss only punishes you for how wrong\n", - "you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\\hat{\\theta}$ represents the parameters of our network, i.e. all the weights and biases. \n", - "\n", - "\n", - "\n", - "### Optimizing the cost function\n", - "\n", - "The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is *gradient descent* and its generalizations. The idea behind gradient descent\n", - "is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a *local* minimum of the cost function. \n", - "Each parameter $\\theta$ is iteratively adjusted according to the rule \n", - "\n", - "$$ \\theta_{i+1} = \\theta_i - \\eta \\nabla \\mathcal{C}(\\theta_i) ,$$\n", - "\n", - "where $\\eta$ is known as the *learning rate*, which controls how big a step we take towards the minimum. \n", - "This update can be repeated for any number of iterations, or until we are satisfied with the result. \n", - "\n", - "A simple and effective improvement is a variant called *Batch Gradient Descent*. \n", - "Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient\n", - "on a subset of the data called a *minibatch*. \n", - "If there are $N$ data points and we have a minibatch size of $M$, the total number of batches\n", - "is $N/M$. \n", - "We denote each minibatch $B_k$, with $k = 1, 2,...,N/M$. The gradient then becomes: \n", - "\n", - "$$ \\nabla \\mathcal{C}(\\theta) = \\frac{1}{N} \\sum_{i=1}^N \\nabla \\mathcal{L}_i(\\theta) \\quad \\rightarrow \\quad\n", - "\\frac{1}{M} \\sum_{i \\in B_k} \\nabla \\mathcal{L}_i(\\theta) ,$$\n", - "\n", - "i.e. instead of averaging the loss over the entire dataset, we average over a minibatch. \n", - "\n", - "This has two important benefits: \n", - "1. Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima. \n", - "\n", - "2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient. \n", - "\n", - "The various optmization methods, with codes and algorithms, are discussed in our lectures on [Gradient descent approaches](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", - "\n", - "\n", - "### Regularization\n", - "\n", - "It is common to add an extra term to the cost function, proportional\n", - "to the size of the weights. This is equivalent to constraining the\n", - "size of the weights, so that they do not grow out of control.\n", - "Constraining the size of the weights means that the weights cannot\n", - "grow arbitrarily large to fit the training data, and in this way\n", - "reduces *overfitting*.\n", - "\n", - "We will measure the size of the weights using the so called *L2-norm*, meaning our cost function becomes: \n", - "\n", - "$$ \\mathcal{C}(\\theta) = \\frac{1}{N} \\sum_{i=1}^N \\mathcal{L}_i(\\theta) \\quad \\rightarrow \\quad\n", - "\\frac{1}{N} \\sum_{i=1}^N \\mathcal{L}_i(\\theta) + \\lambda \\lvert \\lvert \\hat{w} \\rvert \\rvert_2^2 \n", - "= \\frac{1}{N} \\sum_{i=1}^N \\mathcal{L}(\\theta) + \\lambda \\sum_{ij} w_{ij}^2,$$ \n", - "\n", - "i.e. we sum up all the weights squared. The factor $\\lambda$ is known as a regularization parameter.\n", - "\n", - "\n", - "In order to train the model, we need to calculate the derivative of\n", - "the cost function with respect to every bias and weight in the\n", - "network. In total our network has $(64 + 1)\\times 50=3250$ weights in\n", - "the hidden layer and $(50 + 1)\\times 10=510$ weights to the output\n", - "layer ($+1$ for the bias), and the gradient must be calculated for\n", - "every parameter. We use the *backpropagation* algorithm discussed\n", - "above. This is a clever use of the chain rule that allows us to\n", - "calculate the gradient efficently. \n", - "\n", - "\n", - "### Matrix multiplication\n", - "\n", - "To more efficently train our network these equations are implemented using matrix operations. \n", - "The error in the output layer is calculated simply as, with $\\hat{t}$ being our targets, \n", - "\n", - "$$ \\delta_L = \\hat{t} - \\hat{y} = (n_{inputs}, n_{categories}) .$$ \n", - "\n", - "The gradient for the output weights is calculated as \n", - "\n", - "$$ \\nabla W_{L} = \\hat{a}^T \\delta_L = (n_{hidden}, n_{categories}) ,$$\n", - "\n", - "where $\\hat{a} = (n_{inputs}, n_{hidden})$. This simply means that we are summing up the gradients for each input. \n", - "Since we are going backwards we have to transpose the activation matrix. \n", - "\n", - "The gradient with respect to the output bias is then \n", - "\n", - "$$ \\nabla \\hat{b}_{L} = \\sum_{i=1}^{n_{inputs}} \\delta_L = (n_{categories}) .$$ \n", - "\n", - "The error in the hidden layer is \n", - "\n", - "$$ \\Delta_h = \\delta_L W_{L}^T \\circ f'(z_{h}) = \\delta_L W_{L}^T \\circ a_{h} \\circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$ \n", - "\n", - "where $f'(a_{h})$ is the derivative of the activation in the hidden layer. The matrix products mean\n", - "that we are summing up the products for each neuron in the output layer. The symbol $\\circ$ denotes\n", - "the *Hadamard product*, meaning element-wise multiplication. \n", - "\n", - "This again gives us the gradients in the hidden layer: \n", - "\n", - "$$ \\nabla W_{h} = X^T \\delta_h = (n_{features}, n_{hidden}) ,$$ \n", - "\n", - "$$ \\nabla b_{h} = \\sum_{i=1}^{n_{inputs}} \\delta_h = (n_{hidden}) .$$" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# to categorical turns our integer vector into a onehot representation\n", - "from sklearn.metrics import accuracy_score\n", - "\n", - "# one-hot in numpy\n", - "def to_categorical_numpy(integer_vector):\n", - " n_inputs = len(integer_vector)\n", - " n_categories = np.max(integer_vector) + 1\n", - " onehot_vector = np.zeros((n_inputs, n_categories))\n", - " onehot_vector[range(n_inputs), integer_vector] = 1\n", - " \n", - " return onehot_vector\n", - "\n", - "#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)\n", - "Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)\n", - "\n", - "def feed_forward_train(X):\n", - " # weighted sum of inputs to the hidden layer\n", - " z_h = np.matmul(X, hidden_weights) + hidden_bias\n", - " # activation in the hidden layer\n", - " a_h = sigmoid(z_h)\n", - " \n", - " # weighted sum of inputs to the output layer\n", - " z_o = np.matmul(a_h, output_weights) + output_bias\n", - " # softmax output\n", - " # axis 0 holds each input and axis 1 the probabilities of each category\n", - " exp_term = np.exp(z_o)\n", - " probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n", - " \n", - " # for backpropagation need activations in hidden and output layers\n", - " return a_h, probabilities\n", - "\n", - "def backpropagation(X, Y):\n", - " a_h, probabilities = feed_forward_train(X)\n", - " \n", - " # error in the output layer\n", - " error_output = probabilities - Y\n", - " # error in the hidden layer\n", - " error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)\n", - " \n", - " # gradients for the output layer\n", - " output_weights_gradient = np.matmul(a_h.T, error_output)\n", - " output_bias_gradient = np.sum(error_output, axis=0)\n", - " \n", - " # gradient for the hidden layer\n", - " hidden_weights_gradient = np.matmul(X.T, error_hidden)\n", - " hidden_bias_gradient = np.sum(error_hidden, axis=0)\n", - "\n", - " return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient\n", - "\n", - "print(\"Old accuracy on training data: \" + str(accuracy_score(predict(X_train), Y_train)))\n", - "\n", - "eta = 0.01\n", - "lmbd = 0.01\n", - "for i in range(1000):\n", - " # calculate gradients\n", - " dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)\n", - " \n", - " # regularization term gradients\n", - " dWo += lmbd * output_weights\n", - " dWh += lmbd * hidden_weights\n", - " \n", - " # update weights and biases\n", - " output_weights -= eta * dWo\n", - " output_bias -= eta * dBo\n", - " hidden_weights -= eta * dWh\n", - " hidden_bias -= eta * dBh\n", - "\n", - "print(\"New accuracy on training data: \" + str(accuracy_score(predict(X_train), Y_train)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Improving performance\n", - "\n", - "As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image. \n", - "In order to obtain a network that does something useful, we will have to do a bit more work. \n", - "\n", - "The choice of *hyperparameters* such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a *grid-search* is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates $\\eta = 10^{-6}, 10^{-5},...,10^{-1}$ with different regularization parameters $\\lambda = 10^{-6},...,10^{-0}$. \n", - "\n", - "Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an *iteration*, and a full training period\n", - "going through the entire dataset ($n/M$ batches) an *epoch*.\n", - "\n", - "If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers. \n", - "Andrew Ng goes through some of these considerations in this [video](https://youtu.be/F1ka6a13S9I). You can find a summary of the video [here](https://kevinzakka.github.io/2016/09/26/applying-deep-learning/). \n", - "\n", - "\n", - "It is very natural to think of the network as an object, with specific instances of the network\n", - "being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "class NeuralNetwork:\n", - " def __init__(\n", - " self,\n", - " X_data,\n", - " Y_data,\n", - " n_hidden_neurons=50,\n", - " n_categories=10,\n", - " epochs=10,\n", - " batch_size=100,\n", - " eta=0.1,\n", - " lmbd=0.0):\n", - "\n", - " self.X_data_full = X_data\n", - " self.Y_data_full = Y_data\n", - "\n", - " self.n_inputs = X_data.shape[0]\n", - " self.n_features = X_data.shape[1]\n", - " self.n_hidden_neurons = n_hidden_neurons\n", - " self.n_categories = n_categories\n", - "\n", - " self.epochs = epochs\n", - " self.batch_size = batch_size\n", - " self.iterations = self.n_inputs // self.batch_size\n", - " self.eta = eta\n", - " self.lmbd = lmbd\n", - "\n", - " self.create_biases_and_weights()\n", - "\n", - " def create_biases_and_weights(self):\n", - " self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)\n", - " self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01\n", - "\n", - " self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)\n", - " self.output_bias = np.zeros(self.n_categories) + 0.01\n", - "\n", - " def feed_forward(self):\n", - " # feed-forward for training\n", - " self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias\n", - " self.a_h = sigmoid(self.z_h)\n", - "\n", - " self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias\n", - "\n", - " exp_term = np.exp(self.z_o)\n", - " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n", - "\n", - " def feed_forward_out(self, X):\n", - " # feed-forward for output\n", - " z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias\n", - " a_h = sigmoid(z_h)\n", - "\n", - " z_o = np.matmul(a_h, self.output_weights) + self.output_bias\n", - " \n", - " exp_term = np.exp(z_o)\n", - " probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n", - " return probabilities\n", - "\n", - " def backpropagation(self):\n", - " error_output = self.probabilities - self.Y_data\n", - " error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)\n", - "\n", - " self.output_weights_gradient = np.matmul(self.a_h.T, error_output)\n", - " self.output_bias_gradient = np.sum(error_output, axis=0)\n", - "\n", - " self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)\n", - " self.hidden_bias_gradient = np.sum(error_hidden, axis=0)\n", - "\n", - " if self.lmbd > 0.0:\n", - " self.output_weights_gradient += self.lmbd * self.output_weights\n", - " self.hidden_weights_gradient += self.lmbd * self.hidden_weights\n", - "\n", - " self.output_weights -= self.eta * self.output_weights_gradient\n", - " self.output_bias -= self.eta * self.output_bias_gradient\n", - " self.hidden_weights -= self.eta * self.hidden_weights_gradient\n", - " self.hidden_bias -= self.eta * self.hidden_bias_gradient\n", - "\n", - " def predict(self, X):\n", - " probabilities = self.feed_forward_out(X)\n", - " return np.argmax(probabilities, axis=1)\n", - "\n", - " def predict_probabilities(self, X):\n", - " probabilities = self.feed_forward_out(X)\n", - " return probabilities\n", - "\n", - " def train(self):\n", - " data_indices = np.arange(self.n_inputs)\n", - "\n", - " for i in range(self.epochs):\n", - " for j in range(self.iterations):\n", - " # pick datapoints with replacement\n", - " chosen_datapoints = np.random.choice(\n", - " data_indices, size=self.batch_size, replace=False\n", - " )\n", - "\n", - " # minibatch training data\n", - " self.X_data = self.X_data_full[chosen_datapoints]\n", - " self.Y_data = self.Y_data_full[chosen_datapoints]\n", - "\n", - " self.feed_forward()\n", - " self.backpropagation()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Evaluate model performance on test data\n", - "\n", - "To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data. \n", - "We measure the performance of the network using the *accuracy* score. \n", - "The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of $1$. \n", - "\n", - "$$ \\text{Accuracy} = \\frac{\\sum_{i=1}^n I(\\hat{y}_i = y_i)}{n} ,$$ \n", - "\n", - "where $I$ is the indicator function, $1$ if $\\hat{y}_i = y_i$ and $0$ otherwise." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "epochs = 100\n", - "batch_size = 100\n", - "\n", - "dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,\n", - " n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)\n", - "dnn.train()\n", - "test_predict = dnn.predict(X_test)\n", - "\n", - "# accuracy score from scikit library\n", - "print(\"Accuracy score on test set: \", accuracy_score(Y_test, test_predict))\n", - "\n", - "# equivalent in numpy\n", - "def accuracy_score_numpy(Y_test, Y_pred):\n", - " return np.sum(Y_test == Y_pred) / len(Y_test)\n", - "\n", - "#print(\"Accuracy score on test set: \", accuracy_score_numpy(Y_test, test_predict))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Adjust hyperparameters\n", - "\n", - "We now perform a grid search to find the optimal hyperparameters for the network. \n", - "Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around $98\\%$ ($2\\%$ error rate)." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "eta_vals = np.logspace(-5, 1, 7)\n", - "lmbd_vals = np.logspace(-5, 1, 7)\n", - "# store the models for later use\n", - "DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - "\n", - "# grid search\n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,\n", - " n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)\n", - " dnn.train()\n", - " \n", - " DNN_numpy[i][j] = dnn\n", - " \n", - " test_predict = dnn.predict(X_test)\n", - " \n", - " print(\"Learning rate = \", eta)\n", - " print(\"Lambda = \", lmbd)\n", - " print(\"Accuracy score on test set: \", accuracy_score(Y_test, test_predict))\n", - " print()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Visualization" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# visual representation of grid search\n", - "# uses seaborn heatmap, you can also do this with matplotlib imshow\n", - "import seaborn as sns\n", - "\n", - "sns.set()\n", - "\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "\n", - "for i in range(len(eta_vals)):\n", - " for j in range(len(lmbd_vals)):\n", - " dnn = DNN_numpy[i][j]\n", - " \n", - " train_pred = dnn.predict(X_train) \n", - " test_pred = dnn.predict(X_test)\n", - "\n", - " train_accuracy[i][j] = accuracy_score(Y_train, train_pred)\n", - " test_accuracy[i][j] = accuracy_score(Y_test, test_pred)\n", - "\n", - " \n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Test Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## scikit-learn implementation\n", - "\n", - "**scikit-learn** focuses more\n", - "on traditional machine learning methods, such as regression,\n", - "clustering, decision trees, etc. As such, it has only two types of\n", - "neural networks: Multi Layer Perceptron outputting continuous values,\n", - "*MPLRegressor*, and Multi Layer Perceptron outputting labels,\n", - "*MLPClassifier*. We will see how simple it is to use these classes.\n", - "\n", - "**scikit-learn** implements a few improvements from our neural network,\n", - "such as early stopping, a varying learning rate, different\n", - "optimization methods, etc. We would therefore expect a better\n", - "performance overall." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.neural_network import MLPClassifier\n", - "# store models for later use\n", - "DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - "\n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',\n", - " alpha=lmbd, learning_rate_init=eta, max_iter=epochs)\n", - " dnn.fit(X_train, Y_train)\n", - " \n", - " DNN_scikit[i][j] = dnn\n", - " \n", - " print(\"Learning rate = \", eta)\n", - " print(\"Lambda = \", lmbd)\n", - " print(\"Accuracy score on test set: \", dnn.score(X_test, Y_test))\n", - " print()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Visualization" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# optional\n", - "# visual representation of grid search\n", - "# uses seaborn heatmap, could probably do this in matplotlib\n", - "import seaborn as sns\n", - "\n", - "sns.set()\n", - "\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "\n", - "for i in range(len(eta_vals)):\n", - " for j in range(len(lmbd_vals)):\n", - " dnn = DNN_scikit[i][j]\n", - " \n", - " train_pred = dnn.predict(X_train) \n", - " test_pred = dnn.predict(X_test)\n", - "\n", - " train_accuracy[i][j] = accuracy_score(Y_train, train_pred)\n", - " test_accuracy[i][j] = accuracy_score(Y_test, test_pred)\n", - "\n", - " \n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Test Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Building neural networks in Tensorflow and Keras\n", - "\n", - "Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn\n", - "and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy\n", - "and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer. \n", - "\n", - "In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite\n", - "clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or\n", - "NumPy arrays.\n", - "\n", - "\n", - "Tensorflow is an open source library machine learning library\n", - "developed by the Google Brain team for internal use. It was released\n", - "under the Apache 2.0 open source license in November 9, 2015.\n", - "\n", - "Tensorflow is a computational framework that allows you to construct\n", - "machine learning models at different levels of abstraction, from\n", - "high-level, object-oriented APIs like Keras, down to the C++ kernels\n", - "that Tensorflow is built upon. The higher levels of abstraction are\n", - "simpler to use, but less flexible, and our choice of implementation\n", - "should reflect the problems we are trying to solve.\n", - "\n", - "[Tensorflow uses](https://www.tensorflow.org/guide/graphs) so-called graphs to represent your computation\n", - "in terms of the dependencies between individual operations, such that you first build a Tensorflow *graph*\n", - "to represent your model, and then create a Tensorflow *session* to run the graph.\n", - "\n", - "In this guide we will analyze the same data as we did in our NumPy and\n", - "scikit-learn tutorial, gathered from the MNIST database of images. We\n", - "will give an introduction to the lower level Python Application\n", - "Program Interfaces (APIs), and see how we use them to build our graph.\n", - "Then we will build (effectively) the same graph in Keras, to see just\n", - "how simple solving a machine learning problem can be.\n", - "\n", - "To install tensorflow on Unix/Linux systems, use pip as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pip3 install tensorflow" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and/or if you use **anaconda**, just write (or install from the graphical user interface)\n", - "(current release of CPU-only TensorFlow)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "conda create -n tf tensorflow\n", - "conda activate tf" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To install the current release of GPU TensorFlow" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "conda create -n tf-gpu tensorflow-gpu\n", - "conda activate tf-gpu" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Keras is a high level [neural network](https://en.wikipedia.org/wiki/Application_programming_interface)\n", - "that supports Tensorflow, CTNK and Theano as backends. \n", - "If you have Anaconda installed you may run the following command" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "conda install keras" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "You can look up the [instructions here](https://keras.io/) for more information.\n", - "\n", - "We will to a large extent use **keras** in this course. \n", - "\n", - "\n", - "Let us look again at the MINST data set." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# import necessary packages\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "import tensorflow as tf\n", - "from sklearn import datasets\n", - "\n", - "\n", - "# ensure the same random numbers appear every time\n", - "np.random.seed(0)\n", - "\n", - "# display images in notebook\n", - "%matplotlib inline\n", - "plt.rcParams['figure.figsize'] = (12,12)\n", - "\n", - "\n", - "# download MNIST dataset\n", - "digits = datasets.load_digits()\n", - "\n", - "# define inputs and labels\n", - "inputs = digits.images\n", - "labels = digits.target\n", - "\n", - "print(\"inputs = (n_inputs, pixel_width, pixel_height) = \" + str(inputs.shape))\n", - "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", - "\n", - "\n", - "# flatten the image\n", - "# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64\n", - "n_inputs = len(inputs)\n", - "inputs = inputs.reshape(n_inputs, -1)\n", - "print(\"X = (n_inputs, n_features) = \" + str(inputs.shape))\n", - "\n", - "\n", - "# choose some random images to display\n", - "indices = np.arange(n_inputs)\n", - "random_indices = np.random.choice(indices, size=5)\n", - "\n", - "for i, image in enumerate(digits.images[random_indices]):\n", - " plt.subplot(1, 5, i+1)\n", - " plt.axis('off')\n", - " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", - " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from tensorflow.keras.layers import Input\n", - "from tensorflow.keras.models import Sequential #This allows appending layers to existing models\n", - "from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer\n", - "from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)\n", - "from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)\n", - "from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function\n", - "\n", - "from sklearn.model_selection import train_test_split\n", - "\n", - "# one-hot representation of labels\n", - "labels = to_categorical(labels)\n", - "\n", - "# split into train and test data\n", - "train_size = 0.8\n", - "test_size = 1 - train_size\n", - "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", - " test_size=test_size)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "epochs = 100\n", - "batch_size = 100\n", - "n_neurons_layer1 = 100\n", - "n_neurons_layer2 = 50\n", - "n_categories = 10\n", - "eta_vals = np.logspace(-5, 1, 7)\n", - "lmbd_vals = np.logspace(-5, 1, 7)\n", - "def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):\n", - " model = Sequential()\n", - " model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))\n", - " model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))\n", - " model.add(Dense(n_categories, activation='softmax'))\n", - " \n", - " sgd = optimizers.SGD(lr=eta)\n", - " model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])\n", - " \n", - " return model" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - " \n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,\n", - " eta=eta, lmbd=lmbd)\n", - " DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)\n", - " scores = DNN.evaluate(X_test, Y_test)\n", - " \n", - " DNN_keras[i][j] = DNN\n", - " \n", - " print(\"Learning rate = \", eta)\n", - " print(\"Lambda = \", lmbd)\n", - " print(\"Test accuracy: %.3f\" % scores[1])\n", - " print()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# optional\n", - "# visual representation of grid search\n", - "# uses seaborn heatmap, could probably do this in matplotlib\n", - "import seaborn as sns\n", - "\n", - "sns.set()\n", - "\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "\n", - "for i in range(len(eta_vals)):\n", - " for j in range(len(lmbd_vals)):\n", - " DNN = DNN_keras[i][j]\n", - "\n", - " train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]\n", - " test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]\n", - "\n", - " \n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Test Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The Breast Cancer Data, now with Keras" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "import tensorflow as tf\n", - "from tensorflow.keras.layers import Input\n", - "from tensorflow.keras.models import Sequential #This allows appending layers to existing models\n", - "from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer\n", - "from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)\n", - "from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)\n", - "from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "import seaborn as sns\n", - "from sklearn.model_selection import train_test_split as splitter\n", - "from sklearn.datasets import load_breast_cancer\n", - "import pickle\n", - "import os \n", - "\n", - "\n", - "\"\"\"Load breast cancer dataset\"\"\"\n", - "\n", - "np.random.seed(0) #create same seed for random number every time\n", - "\n", - "cancer=load_breast_cancer() #Download breast cancer dataset\n", - "\n", - "inputs=cancer.data #Feature matrix of 569 rows (samples) and 30 columns (parameters)\n", - "outputs=cancer.target #Label array of 569 rows (0 for benign and 1 for malignant)\n", - "labels=cancer.feature_names[0:30]\n", - "\n", - "print('The content of the breast cancer dataset is:') #Print information about the datasets\n", - "print(labels)\n", - "print('-------------------------')\n", - "print(\"inputs = \" + str(inputs.shape))\n", - "print(\"outputs = \" + str(outputs.shape))\n", - "print(\"labels = \"+ str(labels.shape))\n", - "\n", - "x=inputs #Reassign the Feature and Label matrices to other variables\n", - "y=outputs\n", - "\n", - "#%% \n", - "\n", - "# Visualisation of dataset (for correlation analysis)\n", - "\n", - "plt.figure()\n", - "plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral)\n", - "plt.xlabel('Mean radius',fontweight='bold')\n", - "plt.ylabel('Mean perimeter',fontweight='bold')\n", - "plt.show()\n", - "\n", - "plt.figure()\n", - "plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral)\n", - "plt.xlabel('Mean compactness',fontweight='bold')\n", - "plt.ylabel('Mean concavity',fontweight='bold')\n", - "plt.show()\n", - "\n", - "\n", - "plt.figure()\n", - "plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)\n", - "plt.xlabel('Mean radius',fontweight='bold')\n", - "plt.ylabel('Mean texture',fontweight='bold')\n", - "plt.show()\n", - "\n", - "plt.figure()\n", - "plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)\n", - "plt.xlabel('Mean perimeter',fontweight='bold')\n", - "plt.ylabel('Mean compactness',fontweight='bold')\n", - "plt.show()\n", - "\n", - "\n", - "# Generate training and testing datasets\n", - "\n", - "#Select features relevant to classification (texture,perimeter,compactness and symmetery) \n", - "#and add to input matrix\n", - "\n", - "temp1=np.reshape(x[:,1],(len(x[:,1]),1))\n", - "temp2=np.reshape(x[:,2],(len(x[:,2]),1))\n", - "X=np.hstack((temp1,temp2)) \n", - "temp=np.reshape(x[:,5],(len(x[:,5]),1))\n", - "X=np.hstack((X,temp)) \n", - "temp=np.reshape(x[:,8],(len(x[:,8]),1))\n", - "X=np.hstack((X,temp)) \n", - "\n", - "X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1) #Split datasets into training and testing\n", - "\n", - "y_train=to_categorical(y_train) #Convert labels to categorical when using categorical cross entropy\n", - "y_test=to_categorical(y_test)\n", - "\n", - "del temp1,temp2,temp\n", - "\n", - "# %%\n", - "\n", - "# Define tunable parameters\"\n", - "\n", - "eta=np.logspace(-3,-1,3) #Define vector of learning rates (parameter to SGD optimiser)\n", - "lamda=0.01 #Define hyperparameter\n", - "n_layers=2 #Define number of hidden layers in the model\n", - "n_neuron=np.logspace(0,3,4,dtype=int) #Define number of neurons per layer\n", - "epochs=100 #Number of reiterations over the input data\n", - "batch_size=100 #Number of samples per gradient update\n", - "\n", - "# %%\n", - "\n", - "\"\"\"Define function to return Deep Neural Network model\"\"\"\n", - "\n", - "def NN_model(inputsize,n_layers,n_neuron,eta,lamda):\n", - " model=Sequential() \n", - " for i in range(n_layers): #Run loop to add hidden layers to the model\n", - " if (i==0): #First layer requires input dimensions\n", - " model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize))\n", - " else: #Subsequent layers are capable of automatic shape inferencing\n", - " model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda)))\n", - " model.add(Dense(2,activation='softmax')) #2 outputs - ordered and disordered (softmax for prob)\n", - " sgd=optimizers.SGD(lr=eta)\n", - " model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy'])\n", - " return model\n", - "\n", - " \n", - "Train_accuracy=np.zeros((len(n_neuron),len(eta))) #Define matrices to store accuracy scores as a function\n", - "Test_accuracy=np.zeros((len(n_neuron),len(eta))) #of learning rate and number of hidden neurons for \n", - "\n", - "for i in range(len(n_neuron)): #run loops over hidden neurons and learning rates to calculate \n", - " for j in range(len(eta)): #accuracy scores \n", - " DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda)\n", - " DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1)\n", - " Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1]\n", - " Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1]\n", - " \n", - "\n", - "def plot_data(x,y,data,title=None):\n", - "\n", - " # plot results\n", - " fontsize=16\n", - "\n", - "\n", - " fig = plt.figure()\n", - " ax = fig.add_subplot(111)\n", - " cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1)\n", - " \n", - " cbar=fig.colorbar(cax)\n", - " cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize)\n", - " cbar.set_ticks([0,.2,.4,0.6,0.8,1.0])\n", - " cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%'])\n", - "\n", - " # put text on matrix elements\n", - " for i, x_val in enumerate(np.arange(len(x))):\n", - " for j, y_val in enumerate(np.arange(len(y))):\n", - " c = \"${0:.1f}\\\\%$\".format( 100*data[j,i]) \n", - " ax.text(x_val, y_val, c, va='center', ha='center')\n", - "\n", - " # convert axis vaues to to string labels\n", - " x=[str(i) for i in x]\n", - " y=[str(i) for i in y]\n", - "\n", - "\n", - " ax.set_xticklabels(['']+x)\n", - " ax.set_yticklabels(['']+y)\n", - "\n", - " ax.set_xlabel('$\\\\mathrm{learning\\\\ rate}$',fontsize=fontsize)\n", - " ax.set_ylabel('$\\\\mathrm{hidden\\\\ neurons}$',fontsize=fontsize)\n", - " if title is not None:\n", - " ax.set_title(title)\n", - "\n", - " plt.tight_layout()\n", - "\n", - " plt.show()\n", - " \n", - "plot_data(eta,n_neuron,Train_accuracy, 'training')\n", - "plot_data(eta,n_neuron,Test_accuracy, 'testing')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Fine-tuning neural network hyperparameters\n", - "\n", - "The flexibility of neural networks is also one of their main\n", - "drawbacks: there are many hyperparameters to tweak. Not only can you\n", - "use any imaginable network topology (how neurons/nodes are interconnected),\n", - "but even in a simple FFNN you can change the number of layers, the\n", - "number of neurons per layer, the type of activation function to use in\n", - "each layer, the weight initialization logic, the stochastic gradient optmized and much more. How do you\n", - "know what combination of hyperparameters is the best for your task?\n", - "\n", - "* You can use grid search with cross-validation to find the right hyperparameters.\n", - "\n", - "However,since there are many hyperparameters to tune, and since\n", - "training a neural network on a large dataset takes a lot of time, you\n", - "will only be able to explore a tiny part of the hyperparameter space.\n", - "\n", - "\n", - "* You can use randomized search.\n", - "\n", - "* Or use tools like [Oscar](http://oscar.calldesk.ai/), which implements more complex algorithms to help you find a good set of hyperparameters quickly. \n", - "\n", - "For many problems you can start with just one or two hidden layers and it will work just fine.\n", - "For the MNIST data set you ca easily get a high accuracy using just one hidden layer with a\n", - "few hundred neurons.\n", - "You can reach for this data set above 98% accuracy using two hidden layers with the same total amount of\n", - "neurons, in roughly the same amount of training time. \n", - "\n", - "For more complex problems, you can gradually\n", - "ramp up the number of hidden layers, until you start overfitting the training set. Very complex tasks, such\n", - "as large image classification or speech recognition, typically require networks with dozens of layers\n", - "and they need a huge amount\n", - "of training data. However, you will rarely have to train such networks from scratch: it is much more\n", - "common to reuse parts of a pretrained state-of-the-art network that performs a similar task.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Which activation function should I use?\n", - "\n", - "The Back propagation algorithm we derived above works by going from\n", - "the output layer to the input layer, propagating the error gradient on\n", - "the way. Once the algorithm has computed the gradient of the cost\n", - "function with regards to each parameter in the network, it uses these\n", - "gradients to update each parameter with a Gradient Descent (GD) step.\n", - "\n", - "\n", - "Unfortunately for us, the gradients often get smaller and smaller as the\n", - "algorithm progresses down to the first hidden layers. As a result, the\n", - "GD update leaves the lower layer connection weights\n", - "virtually unchanged, and training never converges to a good\n", - "solution. This is known in the literature as \n", - "**the vanishing gradients problem**. \n", - "\n", - "In other cases, the opposite can happen, namely the the gradients can grow bigger and\n", - "bigger. The result is that many of the layers get large updates of the \n", - "weights the\n", - "algorithm diverges. This is the **exploding gradients problem**, which is\n", - "mostly encountered in recurrent neural networks. More generally, deep\n", - "neural networks suffer from unstable gradients, different layers may\n", - "learn at widely different speeds\n", - "\n", - "\n", - "\n", - "\n", - "Although this unfortunate behavior has been empirically observed for\n", - "quite a while (it was one of the reasons why deep neural networks were\n", - "mostly abandoned for a long time), it is only around 2010 that\n", - "significant progress was made in understanding it.\n", - "\n", - "A paper titled [Understanding the Difficulty of Training Deep\n", - "Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio](http://proceedings.mlr.press/v9/glorot10a.html) found that\n", - "the problems with the popular logistic\n", - "sigmoid activation function and the weight initialization technique\n", - "that was most popular at the time, namely random initialization using\n", - "a normal distribution with a mean of 0 and a standard deviation of\n", - "1. \n", - "\n", - "They showed that with this activation function and this\n", - "initialization scheme, the variance of the outputs of each layer is\n", - "much greater than the variance of its inputs. Going forward in the\n", - "network, the variance keeps increasing after each layer until the\n", - "activation function saturates at the top layers. This is actually made\n", - "worse by the fact that the logistic function has a mean of 0.5, not 0\n", - "(the hyperbolic tangent function has a mean of 0 and behaves slightly\n", - "better than the logistic function in deep networks).\n", - "\n", - "\n", - "\n", - "Looking at the logistic activation function, when inputs become large\n", - "(negative or positive), the function saturates at 0 or 1, with a\n", - "derivative extremely close to 0. Thus when backpropagation kicks in,\n", - "it has virtually no gradient to propagate back through the network,\n", - "and what little gradient exists keeps getting diluted as\n", - "backpropagation progresses down through the top layers, so there is\n", - "really nothing left for the lower layers.\n", - "\n", - "In their paper, Glorot and Bengio propose a way to significantly\n", - "alleviate this problem. We need the signal to flow properly in both\n", - "directions: in the forward direction when making predictions, and in\n", - "the reverse direction when backpropagating gradients. We don’t want\n", - "the signal to die out, nor do we want it to explode and saturate. For\n", - "the signal to flow properly, the authors argue that we need the\n", - "variance of the outputs of each layer to be equal to the variance of\n", - "its inputs, and we also need the gradients to have equal variance\n", - "before and after flowing through a layer in the reverse direction.\n", - "\n", - "\n", - "\n", - "One of the insights in the 2010 paper by Glorot and Bengio was that\n", - "the vanishing/exploding gradients problems were in part due to a poor\n", - "choice of activation function. Until then most people had assumed that\n", - "if Nature had chosen to use roughly sigmoid activation functions in\n", - "biological neurons, they must be an excellent choice. But it turns out\n", - "that other activation functions behave much better in deep neural\n", - "networks, in particular the ReLU activation function, mostly because\n", - "it does not saturate for positive values (and also because it is quite\n", - "fast to compute).\n", - "\n", - "\n", - "## The RELU function family\n", - "\n", - "The ReLU activation function suffers from a problem known as the dying\n", - "ReLUs: during training, some neurons effectively die, meaning they\n", - "stop outputting anything other than 0.\n", - "\n", - "In some cases, you may find that half of your network’s neurons are\n", - "dead, especially if you used a large learning rate. During training,\n", - "if a neuron’s weights get updated such that the weighted sum of the\n", - "neuron’s inputs is negative, it will start outputting 0. When this\n", - "happen, the neuron is unlikely to come back to life since the gradient\n", - "of the ReLU function is 0 when its input is negative.\n", - "\n", - "To solve this problem, nowadays practitioners use a variant of the ReLU\n", - "function, such as the leaky ReLU discussed above or the so-called\n", - "exponential linear unit (ELU) function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "ELU(z) = \\left\\{\\begin{array}{cc} \\alpha\\left( \\exp{(z)}-1\\right) & z < 0,\\\\ z & z \\ge 0.\\end{array}\\right.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In general it seems that the ELU activation function is better than\n", - "the leaky ReLU function (and its variants), which is better than\n", - "ReLU. ReLU performs better than $\\tanh$ which in turn performs better\n", - "than the logistic function. \n", - "\n", - "If runtime\n", - "performance is an issue, then you may opt for the leaky ReLU function over the \n", - "ELU function If you don’t\n", - "want to tweak yet another hyperparameter, you may just use the default\n", - "$\\alpha$ of $0.01$ for the leaky ReLU, and $1$ for ELU. If you have\n", - "spare time and computing power, you can use cross-validation or\n", - "bootstrap to evaluate other activation functions.\n", - "\n", - "\n", - "\n", - "In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).\n", - "\n", - "It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.\n", - "\n", - "**For the output layer:**\n", - "\n", - "* For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).\n", - "\n", - "* For regression tasks, you can simply use no activation function at all.\n", - "\n", - "## Batch Normalization\n", - "\n", - "Batch Normalization\n", - "aims to address the vanishing/exploding gradients problems, and more generally the problem that the\n", - "distribution of each layer’s inputs changes during training, as the parameters of the previous layers change.\n", - "\n", - "The technique consists of adding an operation in the model just before the activation function of each\n", - "layer, simply zero-centering and normalizing the inputs, then scaling and shifting the result using two new\n", - "parameters per layer (one for scaling, the other for shifting). In other words, this operation lets the model\n", - "learn the optimal scale and mean of the inputs for each layer.\n", - "In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and\n", - "standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current\n", - "mini-batch, from this the name batch normalization.\n", - "\n", - "## Dropout\n", - "\n", - "It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but\n", - "excluding the output neurons) has a probability $p$ of being temporarily dropped out, meaning it will be\n", - "entirely ignored during this training step, but it may be active during the next step.\n", - "\n", - "The\n", - "hyperparameter $p$ is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore.\n", - " It is viewed as one of the most popular regularization techniques.\n", - "\n", - "## Gradient Clipping\n", - "\n", - "A popular technique to lessen the exploding gradients problem is to simply clip the gradients during\n", - "backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural\n", - "networks).\n", - "\n", - "This technique is called Gradient Clipping.\n", - "\n", - "In general however, Batch\n", - "Normalization is preferred.\n", - "\n", - "\n", - "## A top-down perspective on Neural networks\n", - "\n", - "\n", - "The first thing we would like to do is divide the data into two or three\n", - "parts. A training set, a validation or dev (development) set, and a\n", - "test set. The test set is the data on which we want to make\n", - "predictions. The dev set is a subset of the training data we use to\n", - "check how well we are doing out-of-sample, after training the model on\n", - "the training dataset. We use the validation error as a proxy for the\n", - "test error in order to make tweaks to our model. It is crucial that we\n", - "do not use any of the test data to train the algorithm. This is a\n", - "cardinal sin in ML. Then:\n", - "\n", - "\n", - "* Estimate optimal error rate\n", - "\n", - "* Minimize underfitting (bias) on training data set.\n", - "\n", - "* Make sure you are not overfitting.\n", - "\n", - "If the validation and test sets are drawn from the same distributions,\n", - "then a good performance on the validation set should lead to similarly\n", - "good performance on the test set. \n", - "\n", - "However, sometimes\n", - "the training data and test data differ in subtle ways because, for\n", - "example, they are collected using slightly different methods, or\n", - "because it is cheaper to collect data in one way versus another. In\n", - "this case, there can be a mismatch between the training and test\n", - "data. This can lead to the neural network overfitting these small\n", - "differences between the test and training sets, and a poor performance\n", - "on the test set despite having a good performance on the validation\n", - "set. To rectify this, Andrew Ng suggests making two validation or dev\n", - "sets, one constructed from the training data and one constructed from\n", - "the test data. The difference between the performance of the algorithm\n", - "on these two validation sets quantifies the train-test mismatch. This\n", - "can serve as another important diagnostic when using DNNs for\n", - "supervised learning.\n", - "\n", - "\n", - "## Limitations of supervised learning with deep networks\n", - "\n", - "Like all statistical methods, supervised learning using neural\n", - "networks has important limitations. This is especially important when\n", - "one seeks to apply these methods, especially to physics problems. Like\n", - "all tools, DNNs are not a universal solution. Often, the same or\n", - "better performance on a task can be achieved by using a few\n", - "hand-engineered features (or even a collection of random\n", - "features). \n", - "\n", - "Here we list some of the important limitations of supervised neural network based models. \n", - "\n", - "\n", - "\n", - "* **Need labeled data**. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).\n", - "\n", - "* **Supervised neural networks are extremely data intensive.** DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.\n", - "\n", - "* **Homogeneous data.** Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types.\n", - "\n", - "* **Many problems are not about prediction.** In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a *wrong* model. The model might or might not be useful for understanding the underlying science.\n", - "\n", - "Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems." - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter11.ipynb b/doc/LectureNotes/_build/html/_sources/chapter11.ipynb deleted file mode 100644 index c03e4a368..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter11.ipynb +++ /dev/null @@ -1,3023 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Solving Differential Equations with Deep Learning\n", - "\n", - "The Universal Approximation Theorem states that a neural network can\n", - "approximate any function at a single hidden layer along with one input\n", - "and output layer to any given precision. \n", - "\n", - "\n", - "An ordinary differential equation (ODE) is an equation involving functions having one variable.\n", - "\n", - "In general, an ordinary differential equation looks like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "

\n", - "\n", - "$$\n", - "\\begin{equation} \\label{ode} \\tag{1}\n", - "f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right) = 0\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(x)$ is the function to find, and $g^{(n)}(x)$ is the $n$-th derivative of $g(x)$.\n", - "\n", - "The $f\\left(x, g(x), g'(x), g''(x), \\, \\dots \\, , g^{(n)}(x)\\right)$ is just a way to write that there is an expression involving $x$ and $g(x), \\ g'(x), \\ g''(x), \\, \\dots \\, , \\text{ and } g^{(n)}(x)$ on the left side of the equality sign in ([1](#ode)).\n", - "The highest order of derivative, that is the value of $n$, determines to the order of the equation.\n", - "The equation is referred to as a $n$-th order ODE.\n", - "Along with ([1](#ode)), some additional conditions of the function $g(x)$ are typically given\n", - "for the solution to be unique.\n", - "\n", - "\n", - "\n", - "Let the trial solution $g_t(x)$ be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\tg_t(x) = h_1(x) + h_2(x,N(x,P))\n", - "\\label{_auto1} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $h_1(x)$ is a function that makes $g_t(x)$ satisfy a given set\n", - "of conditions, $N(x,P)$ a neural network with weights and biases\n", - "described by $P$ and $h_2(x, N(x,P))$ some expression involving the\n", - "neural network. The role of the function $h_2(x, N(x,P))$, is to\n", - "ensure that the output from $N(x,P)$ is zero when $g_t(x)$ is\n", - "evaluated at the values of $x$ where the given conditions must be\n", - "satisfied. The function $h_1(x)$ should alone make $g_t(x)$ satisfy\n", - "the conditions.\n", - "\n", - "But what about the network $N(x,P)$?\n", - "\n", - "\n", - "As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.\n", - "\n", - "\n", - "\n", - "For the minimization to be defined, we need to have a cost function at hand to minimize.\n", - "\n", - "It is given that $f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)$ should be equal to zero in ([1](#ode)).\n", - "We can choose to consider the mean squared error as the cost function for an input $x$.\n", - "Since we are looking at one input, the cost function is just $f$ squared.\n", - "The cost function $c\\left(x, P \\right)$ can therefore be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(x, P\\right) = \\big(f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)\\big)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If $N$ inputs are given as a vector $\\boldsymbol{x}$ with elements $x_i$ for $i = 1,\\dots,N$,\n", - "the cost function becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{cost} \\tag{3}\n", - "\tC\\left(\\boldsymbol{x}, P\\right) = \\frac{1}{N} \\sum_{i=1}^N \\big(f\\left(x_i, \\, g(x_i), \\, g'(x_i), \\, g''(x_i), \\, \\dots \\, , \\, g^{(n)}(x_i)\\right)\\big)^2\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The neural net should then find the parameters $P$ that minimizes the cost function in\n", - "([3](#cost)) for a set of $N$ training samples $x_i$.\n", - "\n", - "\n", - "\n", - "To perform the minimization using gradient descent, the gradient of $C\\left(\\boldsymbol{x}, P\\right)$ is needed.\n", - "It might happen so that finding an analytical expression of the gradient of $C(\\boldsymbol{x}, P)$ from ([3](#cost)) gets too messy, depending on which cost function one desires to use.\n", - "\n", - "Luckily, there exists libraries that makes the job for us through automatic differentiation.\n", - "Automatic differentiation is a method of finding the derivatives numerically with very high precision.\n", - "\n", - "\n", - "### Example: Exponential decay\n", - "\n", - "An exponential decay of a quantity $g(x)$ is described by the equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solve_expdec} \\tag{4}\n", - " g'(x) = -\\gamma g(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $g(0) = g_0$ for some chosen initial value $g_0$.\n", - "\n", - "The analytical solution of ([4](#solve_expdec)) is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " g(x) = g_0 \\exp\\left(-\\gamma x\\right)\n", - "\\label{_auto2} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of ([4](#solve_expdec)).\n", - "\n", - "\n", - "\n", - "The program will use a neural network to solve" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solveode} \\tag{6}\n", - "g'(x) = -\\gamma g(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(0) = g_0$ with $\\gamma$ and $g_0$ being some chosen values.\n", - "\n", - "In this example, $\\gamma = 2$ and $g_0 = 10$.\n", - "\n", - "\n", - "To begin with, a trial solution $g_t(t)$ must be chosen. A general trial solution for ordinary differential equations could be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x, P) = h_1(x) + h_2(x, N(x, P))\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and $h_2(x,N(x, P))$ an expression involving $x$ and the output from the neural network $N(x,P)$ with $P $ being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.\n", - "\n", - "\n", - "\n", - "In this network, there are no weights and bias at the input layer, so $P = \\{ P_{\\text{hidden}}, P_{\\text{output}} \\}$.\n", - "If there are $N_{\\text{hidden} }$ neurons in the hidden layer, then $P_{\\text{hidden}}$ is a $N_{\\text{hidden} } \\times (1 + N_{\\text{input}})$ matrix, given that there are $N_{\\text{input}}$ neurons in the input layer.\n", - "\n", - "The first column in $P_{\\text{hidden} }$ represents the bias for each neuron in the hidden layer and the second column represents the weights for each neuron in the hidden layer from the input layer.\n", - "If there are $N_{\\text{output} }$ neurons in the output layer, then $P_{\\text{output}} $ is a $N_{\\text{output} } \\times (1 + N_{\\text{hidden} })$ matrix.\n", - "\n", - "Its first column represents the bias of each neuron and the remaining columns represents the weights to each neuron.\n", - "\n", - "It is given that $g(0) = g_0$. The trial solution must fulfill this condition to be a proper solution of ([6](#solveode)). A possible way to ensure that $g_t(0, P) = g_0$, is to let $F(N(x,P)) = x \\cdot N(x,P)$ and $A(x) = g_0$. This gives the following trial solution:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{trial} \\tag{7}\n", - "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Reformulating the problem\n", - "\n", - "We wish that our neural network manages to minimize a given cost function.\n", - "\n", - "A reformulation of out equation, ([6](#solveode)), must therefore be done,\n", - "such that it describes the problem a neural network can solve for.\n", - "\n", - "The neural network must find the set of weights and biases $P$ such that the trial solution in ([7](#trial)) satisfies ([6](#solveode)).\n", - "\n", - "The trial solution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "has been chosen such that it already solves the condition $g(0) = g_0$. What remains, is to find $P$ such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{nnmin} \\tag{8}\n", - "g_t'(x, P) = - \\gamma g_t(x, P)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "is fulfilled as *best as possible*.\n", - "\n", - "\n", - "The left hand side and right hand side of ([8](#nnmin)) must be computed separately, and then the neural network must choose weights and biases, contained in $P$, such that the sides are equal as best as possible.\n", - "This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.\n", - "In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to $P$ of the neural network.\n", - "\n", - "This gives the following cost function our neural network must solve for:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P}\\Big\\{ \\big(g_t'(x, P) - ( -\\gamma g_t(x, P) \\big)^2 \\Big\\}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "(the notation $\\min_{P}\\{ f(x, P) \\}$ means that we desire to find $P$ that yields the minimum of $f(x, P)$)\n", - "\n", - "or, in terms of weights and biases for the hidden and output layer in our network:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }}\\Big\\{ \\big(g_t'(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) - ( -\\gamma g_t(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) \\big)^2 \\Big\\}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for an input value $x$.\n", - "\n", - "\n", - "\n", - "If the neural network evaluates $g_t(x, P)$ at more values for $x$, say $N$ values $x_i$ for $i = 1, \\dots, N$, then the *total* error to minimize becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{min} \\tag{9}\n", - "\\min_{P}\\Big\\{\\frac{1}{N} \\sum_{i=1}^N \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2 \\Big\\}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Letting $\\boldsymbol{x}$ be a vector with elements $x_i$ and $C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2$ denote the cost function, the minimization problem that our network must solve, becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P} C(\\boldsymbol{x}, P)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In terms of $P_{\\text{hidden} }$ and $P_{\\text{output} }$, this could also be expressed as\n", - "\n", - "$$\n", - "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }} C(\\boldsymbol{x}, \\{P_{\\text{hidden} }, P_{\\text{output} }\\})\n", - "$$\n", - "\n", - "\n", - "For simplicity, it is assumed that the input is an array $\\boldsymbol{x} = (x_1, \\dots, x_N)$ with $N$ elements. It is at these points the neural network should find $P$ such that it fulfills ([9](#min)).\n", - "\n", - "First, the neural network must feed forward the inputs.\n", - "This means that $\\boldsymbol{x}s$ must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further.\n", - "The input layer will consist of $N_{\\text{input} }$ neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be $N_{\\text{hidden} }$.\n", - "\n", - "\n", - "For the $i$-th in the hidden layer with weight $w_i^{\\text{hidden} }$ and bias $b_i^{\\text{hidden} }$, the weighting from the $j$-th neuron at the input layer is:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "z_{i,j}^{\\text{hidden}} &= b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_j \\\\\n", - "&=\n", - "\\begin{pmatrix}\n", - "b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 \\\\\n", - "x_j\n", - "\\end{pmatrix}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The result after weighting the inputs at the $i$-th hidden neuron can be written as a vector:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "\\boldsymbol{z}_{i}^{\\text{hidden}} &= \\Big( b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_1 , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_2, \\ \\dots \\, , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_N\\Big) \\\\\n", - "&=\n", - "\\begin{pmatrix}\n", - " b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 & 1 & \\dots & 1 \\\\\n", - "x_1 & x_2 & \\dots & x_N\n", - "\\end{pmatrix} \\\\\n", - "&= \\boldsymbol{p}_{i, \\text{hidden}}^T X\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The vector $\\boldsymbol{p}_{i, \\text{hidden}}^T$ constitutes each row in $P_{\\text{hidden} }$, which contains the weights for the neural network to minimize according to ([9](#min)).\n", - "\n", - "After having found $\\boldsymbol{z}_{i}^{\\text{hidden}} $ for every $i$-th neuron within the hidden layer, the vector will be sent to an activation function $a_i(\\boldsymbol{z})$.\n", - "\n", - "In this example, the sigmoid function has been chosen to be the activation function for each hidden neuron:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(z) = \\frac{1}{1 + \\exp{(-z)}}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is possible to use other activations functions for the hidden layer also.\n", - "\n", - "The output $\\boldsymbol{x}_i^{\\text{hidden}}$ from each $i$-th hidden neuron is:\n", - "\n", - "$$\n", - "\\boldsymbol{x}_i^{\\text{hidden} } = f\\big( \\boldsymbol{z}_{i}^{\\text{hidden}} \\big)\n", - "$$\n", - "\n", - "The outputs $\\boldsymbol{x}_i^{\\text{hidden} } $ are then sent to the output layer.\n", - "\n", - "The output layer consists of one neuron in this case, and combines the\n", - "output from each of the neurons in the hidden layers. The output layer\n", - "combines the results from the hidden layer using some weights $w_i^{\\text{output}}$\n", - "and biases $b_i^{\\text{output}}$. In this case,\n", - "it is assumes that the number of neurons in the output layer is one.\n", - "\n", - "\n", - "\n", - "The procedure of weighting the output neuron $j$ in the hidden layer to the $i$-th neuron in the output layer is similar as for the hidden layer described previously." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "z_{1,j}^{\\text{output}} & =\n", - "\\begin{pmatrix}\n", - "b_1^{\\text{output}} & \\boldsymbol{w}_1^{\\text{output}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 \\\\\n", - "\\boldsymbol{x}_j^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Expressing $z_{1,j}^{\\text{output}}$ as a vector gives the following way of weighting the inputs from the hidden layer:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{z}_{1}^{\\text{output}} =\n", - "\\begin{pmatrix}\n", - "b_1^{\\text{output}} & \\boldsymbol{w}_1^{\\text{output}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 & 1 & \\dots & 1 \\\\\n", - "\\boldsymbol{x}_1^{\\text{hidden}} & \\boldsymbol{x}_2^{\\text{hidden}} & \\dots & \\boldsymbol{x}_N^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\\boldsymbol{z}_{1}^{\\text{output}}$ the neural network has finished its feed forward step, and $\\boldsymbol{z}_{1}^{\\text{output}}$ is the final output of the network.\n", - "\n", - "\n", - "The next step is to decide how the parameters should be changed such that they minimize the cost function.\n", - "\n", - "The chosen cost function for this problem is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to minimize the cost function, an optimization method must be chosen.\n", - "\n", - "Here, gradient descent with a constant step size has been chosen.\n", - "\n", - "### Gradient descent\n", - "\n", - "The idea of the gradient descent algorithm is to update parameters in\n", - "a direction where the cost function decreases goes to a minimum.\n", - "\n", - "In general, the update of some parameters $\\boldsymbol{\\omega}$ given a cost\n", - "function defined by some weights $\\boldsymbol{\\omega}$, $C(\\boldsymbol{x},\n", - "\\boldsymbol{\\omega})$, goes as follows:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\omega}_{\\text{new} } = \\boldsymbol{\\omega} - \\lambda \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for a number of iterations or until $ \\big|\\big| \\boldsymbol{\\omega}_{\\text{new} } - \\boldsymbol{\\omega} \\big|\\big|$ becomes smaller than some given tolerance.\n", - "\n", - "The value of $\\lambda$ decides how large steps the algorithm must take\n", - "in the direction of $ \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})$.\n", - "The notation $\\nabla_{\\boldsymbol{\\omega}}$ express the gradient with respect\n", - "to the elements in $\\boldsymbol{\\omega}$.\n", - "\n", - "In our case, we have to minimize the cost function $C(\\boldsymbol{x}, P)$ with\n", - "respect to the two sets of weights and biases, that is for the hidden\n", - "layer $P_{\\text{hidden} }$ and for the output layer $P_{\\text{output}\n", - "}$ .\n", - "\n", - "This means that $P_{\\text{hidden} }$ and $P_{\\text{output} }$ is updated by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "P_{\\text{hidden},\\text{new}} &= P_{\\text{hidden}} - \\lambda \\nabla_{P_{\\text{hidden}}} C(\\boldsymbol{x}, P) \\\\\n", - "P_{\\text{output},\\text{new}} &= P_{\\text{output}} - \\lambda \\nabla_{P_{\\text{output}}} C(\\boldsymbol{x}, P)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### The code for solving the ODE" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# Assuming one input, hidden, and output layer\n", - "def neural_network(params, x):\n", - "\n", - " # Find the weights (including and biases) for the hidden and output layer.\n", - " # Assume that params is a list of parameters for each layer.\n", - " # The biases are the first element for each array in params,\n", - " # and the weights are the remaning elements in each array in params.\n", - "\n", - " w_hidden = params[0]\n", - " w_output = params[1]\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " ## Hidden layer:\n", - "\n", - " # Add a row of ones to include bias\n", - " x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_input)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " ## Output layer:\n", - "\n", - " # Include bias:\n", - " x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_hidden)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial(x,params, g0 = 10):\n", - " return g0 + x*neural_network(params,x)\n", - "\n", - "# The right side of the ODE:\n", - "def g(x, g_trial, gamma = 2):\n", - " return -gamma*g_trial\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the neural network\n", - " d_net_out = elementwise_grad(neural_network,1)(P,x)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = g(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# Solve the exponential decay ODE using neural network with one input, hidden, and output layer\n", - "def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):\n", - " ## Set up initial weights and biases\n", - "\n", - " # For the hidden layer\n", - " p0 = npr.randn(num_neurons_hidden, 2 )\n", - "\n", - " # For the output layer\n", - " p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included\n", - "\n", - " P = [p0, p1]\n", - "\n", - " print('Initial cost: %g'%cost_function(P, x))\n", - "\n", - " ## Start finding the optimal weights using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of two arrays;\n", - " # one for the gradient w.r.t P_hidden and\n", - " # one for the gradient w.r.t P_output\n", - " cost_grad = cost_function_grad(P, x)\n", - "\n", - " P[0] = P[0] - lmb * cost_grad[0]\n", - " P[1] = P[1] - lmb * cost_grad[1]\n", - "\n", - " print('Final cost: %g'%cost_function(P, x))\n", - "\n", - " return P\n", - "\n", - "def g_analytic(x, gamma = 2, g0 = 10):\n", - " return g0*np.exp(-gamma*x)\n", - "\n", - "# Solve the given problem\n", - "if __name__ == '__main__':\n", - " # Set seed such that the weight are initialized\n", - " # with same weights and biases for every run.\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " N = 10\n", - " x = np.linspace(0, 1, N)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = 10\n", - " num_iter = 10000\n", - " lmb = 0.001\n", - "\n", - " # Use the network\n", - " P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " # Print the deviation from the trial solution and true solution\n", - " res = g_trial(x,P)\n", - " res_analytical = g_analytic(x)\n", - "\n", - " print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))\n", - "\n", - " # Plot the results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, res_analytical)\n", - " plt.plot(x, res[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The network with one input layer, specified number of hidden layers, and one output layer\n", - "\n", - "It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.\n", - "\n", - "The number of neurons within each hidden layer are given as a list of integers in the program below." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# The neural network with one input layer and one output layer,\n", - "# but with number of hidden layers specified by the user.\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - "\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consists of\n", - " # parameters to all the hidden\n", - " # layers AND the output layer.\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial_deep(x,params, g0 = 10):\n", - " return g0 + x*deep_neural_network(params, x)\n", - "\n", - "# The right side of the ODE:\n", - "def g(x, g_trial, gamma = 2):\n", - " return -gamma*g_trial\n", - "\n", - "# The same cost function as before, but calls deep_neural_network instead.\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the neural network\n", - " d_net_out = elementwise_grad(deep_neural_network,1)(P,x)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial_deep,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = g(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# Solve the exponential decay ODE using neural network with one input and one output layer,\n", - "# but with specified number of hidden layers from the user.\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # The number of elements in the list num_hidden_neurons thus represents\n", - " # the number of hidden layers.\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weights and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weights using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "def g_analytic(x, gamma = 2, g0 = 10):\n", - " return g0*np.exp(-gamma*x)\n", - "\n", - "# Solve the given problem\n", - "if __name__ == '__main__':\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " N = 10\n", - " x = np.linspace(0, 1, N)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = np.array([10,10])\n", - " num_iter = 10000\n", - " lmb = 0.001\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " res = g_trial_deep(x,P)\n", - " res_analytical = g_analytic(x)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, res_analytical)\n", - " plt.plot(x, res[0,:])\n", - " plt.legend(['analytical','dnn'])\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Example: Population growth\n", - "\n", - "A logistic model of population growth assumes that a population converges toward an equilibrium.\n", - "The population growth can be modeled by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{log} \\tag{10}\n", - "\tg'(t) = \\alpha g(t)(A - g(t))\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(t)$ is the population density at time $t$, $\\alpha > 0$ the growth rate and $A > 0$ is the maximum population number in the environment.\n", - "Also, at $t = 0$ the population has the size $g(0) = g_0$, where $g_0$ is some chosen constant.\n", - "\n", - "In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability\n", - "and high execution time (this might be more apparent in the examples solving PDEs),\n", - "using a library like TensorFlow is recommended.\n", - "Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.\n", - "\n", - "\n", - "\n", - "Here, we will model a population $g(t)$ in an environment having carrying capacity $A$.\n", - "The population follows the model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solveode_population} \\tag{11}\n", - "g'(t) = \\alpha g(t)(A - g(t))\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(0) = g_0$.\n", - "\n", - "In this example, we let $\\alpha = 2$, $A = 1$, and $g_0 = 1.2$.\n", - "\n", - "\n", - "We will get a slightly different trial solution, as the boundary conditions are different\n", - "compared to the case for exponential decay.\n", - "\n", - "A possible trial solution satisfying the condition $g(0) = g_0$ could be\n", - "\n", - "$$\n", - "h_1(t) = g_0 + t \\cdot N(t,P)\n", - "$$\n", - "\n", - "with $N(t,P)$ being the output from the neural network with weights and biases for each layer collected in the set $P$.\n", - "\n", - "The analytical solution is\n", - "\n", - "$$\n", - "g(t) = \\frac{Ag_0}{g_0 + (A - g_0)\\exp(-\\alpha A t)}\n", - "$$\n", - "\n", - "\n", - "\n", - "The network will be the similar as for the exponential decay example, but with some small modifications for our problem." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# Function to get the parameters.\n", - "# Done such that one can easily change the paramaters after one's liking.\n", - "def get_parameters():\n", - " alpha = 2\n", - " A = 1\n", - " g0 = 1.2\n", - " return alpha, A, g0\n", - "\n", - "def deep_neural_network(P, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(P) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = P[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = P[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial_deep,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = f(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# The right side of the ODE:\n", - "def f(x, g_trial):\n", - " alpha,A, g0 = get_parameters()\n", - " return alpha*g_trial*(A - g_trial)\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial_deep(x, params):\n", - " alpha,A, g0 = get_parameters()\n", - " return g0 + x*deep_neural_network(params,x)\n", - "\n", - "# The analytical solution:\n", - "def g_analytic(t):\n", - " alpha,A, g0 = get_parameters()\n", - " return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nt = 10\n", - " T = 1\n", - " t = np.linspace(0,T, Nt)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [100, 50, 25]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(t,P)\n", - " g_analytical = g_analytic(t)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%diff_ag)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(t, g_analytical)\n", - " plt.plot(t, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Using forward Euler to solve the ODE\n", - "\n", - "A straightforward way of solving an ODE numerically, is to use Euler's method.\n", - "\n", - "Euler's method uses Taylor series to approximate the value at a function $f$ at a step $\\Delta x$ from $x$:\n", - "\n", - "$$\n", - "f(x + \\Delta x) \\approx f(x) + \\Delta x f'(x)\n", - "$$\n", - "\n", - "In our case, using Euler's method to approximate the value of $g$ at a step $\\Delta t$ from $t$ yields" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - " g(t + \\Delta t) &\\approx g(t) + \\Delta t g'(t) \\\\\n", - " &= g(t) + \\Delta t \\big(\\alpha g(t)(A - g(t))\\big)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "along with the condition that $g(0) = g_0$.\n", - "\n", - "Let $t_i = i \\cdot \\Delta t$ where $\\Delta t = \\frac{T}{N_t-1}$ where $T$ is the final time our solver must solve for and $N_t$ the number of values for $t \\in [0, T]$ for $i = 0, \\dots, N_t-1$.\n", - "\n", - "For $i \\geq 1$, we have that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "t_i &= i\\Delta t \\\\\n", - "&= (i - 1)\\Delta t + \\Delta t \\\\\n", - "&= t_{i-1} + \\Delta t\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now, if $g_i = g(t_i)$ then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\begin{aligned}\n", - " g_i &= g(t_i) \\\\\n", - " &= g(t_{i-1} + \\Delta t) \\\\\n", - " &\\approx g(t_{i-1}) + \\Delta t \\big(\\alpha g(t_{i-1})(A - g(t_{i-1}))\\big) \\\\\n", - " &= g_{i-1} + \\Delta t \\big(\\alpha g_{i-1}(A - g_{i-1})\\big)\n", - " \\end{aligned}\n", - "\\end{equation} \\label{odenum} \\tag{12}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for $i \\geq 1$ and $g_0 = g(t_0) = g(0) = g_0$.\n", - "\n", - "Equation ([12](#odenum)) could be implemented in the following way,\n", - "extending the program that uses the network using Autograd:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Assume that all function definitions from the example program using Autograd\n", - "# are located here.\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nt = 10\n", - " T = 1\n", - " t = np.linspace(0,T, Nt)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [100,50,25]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(t,P)\n", - " g_analytical = g_analytic(t)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%diff_ag)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(t, g_analytical)\n", - " plt.plot(t, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " ## Find an approximation to the funtion using forward Euler\n", - "\n", - " alpha, A, g0 = get_parameters()\n", - " dt = T/(Nt - 1)\n", - "\n", - " # Perform forward Euler to solve the ODE\n", - " g_euler = np.zeros(Nt)\n", - " g_euler[0] = g0\n", - "\n", - " for i in range(1,Nt):\n", - " g_euler[i] = g_euler[i-1] + dt*(alpha*g_euler[i-1]*(A - g_euler[i-1]))\n", - "\n", - " # Print the errors done by each method\n", - " diff1 = np.max(np.abs(g_euler - g_analytical))\n", - " diff2 = np.max(np.abs(g_dnn_ag[0,:] - g_analytical))\n", - "\n", - " print('Max absolute difference between Euler method and analytical: %g'%diff1)\n", - " print('Max absolute difference between deep neural network and analytical: %g'%diff2)\n", - "\n", - " # Plot results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.plot(t,g_euler)\n", - " plt.plot(t,g_analytical)\n", - " plt.plot(t,g_dnn_ag[0,:])\n", - "\n", - " plt.legend(['euler','analytical','dnn'])\n", - " plt.xlabel('Time t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Solving the one dimensional Poisson equation\n", - "\n", - "The Poisson equation for $g(x)$ in one dimension is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{poisson} \\tag{13}\n", - " -g''(x) = f(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f(x)$ is a given function for $x \\in (0,1)$.\n", - "\n", - "The conditions that $g(x)$ is chosen to fulfill, are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " g(0) &= 0 \\\\\n", - " g(1) &= 0\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used.\n", - "The results from the networks can then be compared to the analytical solution.\n", - "In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.\n", - "\n", - "\n", - "Here, the function $g(x)$ to solve for follows the equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "-g''(x) = f(x),\\qquad x \\in (0,1)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f(x)$ is a given function, along with the chosen conditions" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{aligned}\n", - "g(0) = g(1) = 0\n", - "\\end{aligned}\\label{cond} \\tag{14}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this example, we consider the case when $f(x) = (3x + x^2)\\exp(x)$.\n", - "\n", - "For this case, a possible trial solution satisfying the conditions could be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x) = x \\cdot (1-x) \\cdot N(P,x)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The analytical solution for this problem is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g(x) = x(1 - x)\\exp(x)\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "## Set up the cost function specified for this Poisson equation:\n", - "\n", - "# The right side of the ODE\n", - "def f(x):\n", - " return (3*x + x**2)*np.exp(x)\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)\n", - "\n", - " right_side = f(x)\n", - "\n", - " err_sqr = (-d2_g_t - right_side)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum/np.size(err_sqr)\n", - "\n", - "# The trial solution:\n", - "def g_trial_deep(x,P):\n", - " return x*(1-x)*deep_neural_network(P,x)\n", - "\n", - "# The analytic solution;\n", - "def g_analytic(x):\n", - " return x*(1-x)*np.exp(x)\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10\n", - " x = np.linspace(0,1, Nx)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [200,100]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(x,P)\n", - " g_analytical = g_analytic(x)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " max_diff = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%max_diff)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, g_analytical)\n", - " plt.plot(x, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Comparing with a numerical scheme\n", - "\n", - "The Poisson equation is possible to solve using Taylor series to approximate the second derivative.\n", - "\n", - "Using Taylor series, the second derivative can be expressed as\n", - "\n", - "$$\n", - "g''(x) = \\frac{g(x + \\Delta x) - 2g(x) + g(x-\\Delta x)}{\\Delta x^2} + E_{\\Delta x}(x)\n", - "$$\n", - "\n", - "where $\\Delta x$ is a small step size and $E_{\\Delta x}(x)$ being the error term.\n", - "\n", - "Looking away from the error terms gives an approximation to the second derivative:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{approx} \\tag{15}\n", - "g''(x) \\approx \\frac{g(x + \\Delta x) - 2g(x) + g(x-\\Delta x)}{\\Delta x^2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If $x_i = i \\Delta x = x_{i-1} + \\Delta x$ and $g_i = g(x_i)$ for $i = 1,\\dots N_x - 2$ with $N_x$ being the number of values for $x$, ([15](#approx)) becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "g''(x_i) &\\approx \\frac{g(x_i + \\Delta x) - 2g(x_i) + g(x_i -\\Delta x)}{\\Delta x^2} \\\\\n", - "&= \\frac{g_{i+1} - 2g_i + g_{i-1}}{\\Delta x^2}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Since we know from our problem that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "-g''(x) &= f(x) \\\\\n", - "&= (3x + x^2)\\exp(x)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "along with the conditions $g(0) = g(1) = 0$,\n", - "the following scheme can be used to find an approximate solution for $g(x)$ numerically:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\begin{aligned}\n", - " -\\Big( \\frac{g_{i+1} - 2g_i + g_{i-1}}{\\Delta x^2} \\Big) &= f(x_i) \\\\\n", - " -g_{i+1} + 2g_i - g_{i-1} &= \\Delta x^2 f(x_i)\n", - " \\end{aligned}\n", - "\\end{equation} \\label{odesys} \\tag{16}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for $i = 1, \\dots, N_x - 2$ where $g_0 = g_{N_x - 1} = 0$ and $f(x_i) = (3x_i + x_i^2)\\exp(x_i)$, which is given for our specific problem.\n", - "\n", - "The equation can be rewritten into a matrix equation:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "\\begin{pmatrix}\n", - "2 & -1 & 0 & \\dots & 0 \\\\\n", - "-1 & 2 & -1 & \\dots & 0 \\\\\n", - "\\vdots & & \\ddots & & \\vdots \\\\\n", - "0 & \\dots & -1 & 2 & -1 \\\\\n", - "0 & \\dots & 0 & -1 & 2\\\\\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "g_1 \\\\\n", - "g_2 \\\\\n", - "\\vdots \\\\\n", - "g_{N_x - 3} \\\\\n", - "g_{N_x - 2}\n", - "\\end{pmatrix}\n", - "&=\n", - "\\Delta x^2\n", - "\\begin{pmatrix}\n", - "f(x_1) \\\\\n", - "f(x_2) \\\\\n", - "\\vdots \\\\\n", - "f(x_{N_x - 3}) \\\\\n", - "f(x_{N_x - 2})\n", - "\\end{pmatrix} \\\\\n", - "\\boldsymbol{A}\\boldsymbol{g} &= \\boldsymbol{f},\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which makes it possible to solve for the vector $\\boldsymbol{g}$.\n", - "\n", - "\n", - "We can then compare the result from this numerical scheme with the output from our network using Autograd:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "## Set up the cost function specified for this Poisson equation:\n", - "\n", - "# The right side of the ODE\n", - "def f(x):\n", - " return (3*x + x**2)*np.exp(x)\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)\n", - "\n", - " right_side = f(x)\n", - "\n", - " err_sqr = (-d2_g_t - right_side)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum/np.size(err_sqr)\n", - "\n", - "# The trial solution:\n", - "def g_trial_deep(x,P):\n", - " return x*(1-x)*deep_neural_network(P,x)\n", - "\n", - "# The analytic solution;\n", - "def g_analytic(x):\n", - " return x*(1-x)*np.exp(x)\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10\n", - " x = np.linspace(0,1, Nx)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [200,100]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(x,P)\n", - " g_analytical = g_analytic(x)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, g_analytical)\n", - " plt.plot(x, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - "\n", - " ## Perform the computation using the numerical scheme\n", - "\n", - " dx = 1/(Nx - 1)\n", - "\n", - " # Set up the matrix A\n", - " A = np.zeros((Nx-2,Nx-2))\n", - "\n", - " A[0,0] = 2\n", - " A[0,1] = -1\n", - "\n", - " for i in range(1,Nx-3):\n", - " A[i,i-1] = -1\n", - " A[i,i] = 2\n", - " A[i,i+1] = -1\n", - "\n", - " A[Nx - 3, Nx - 4] = -1\n", - " A[Nx - 3, Nx - 3] = 2\n", - "\n", - " # Set up the vector f\n", - " f_vec = dx**2 * f(x[1:-1])\n", - "\n", - " # Solve the equation\n", - " g_res = np.linalg.solve(A,f_vec)\n", - "\n", - " g_vec = np.zeros(Nx)\n", - " g_vec[1:-1] = g_res\n", - "\n", - " # Print the differences between each method\n", - " max_diff1 = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " max_diff2 = np.max(np.abs(g_vec - g_analytical))\n", - " print(\"The max absolute difference between the analytical solution and DNN Autograd: %g\"%max_diff1)\n", - " print(\"The max absolute difference between the analytical solution and numerical scheme: %g\"%max_diff2)\n", - "\n", - " # Plot the results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.plot(x,g_vec)\n", - " plt.plot(x,g_analytical)\n", - " plt.plot(x,g_dnn_ag[0,:])\n", - "\n", - " plt.legend(['numerical scheme','analytical','dnn'])\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Partial Differential Equations\n", - "\n", - "A partial differential equation (PDE) has a solution here the function\n", - "is defined by multiple variables. The equation may involve all kinds\n", - "of combinations of which variables the function is differentiated with\n", - "respect to.\n", - "\n", - "In general, a partial differential equation for a function $g(x_1,\\dots,x_N)$ with $N$ variables may be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{PDE} \\tag{17}\n", - " f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) = 0\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given.\n", - "\n", - "### Type of problem\n", - "\n", - "The problem our network must solve for, is similar to the ODE case.\n", - "We must have a trial solution $g_t$ at hand.\n", - "\n", - "For instance, the trial solution could be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " g_t(x_1,\\dots,x_N) = h_1(x_1,\\dots,x_N) + h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $h_1(x_1,\\dots,x_N)$ is a function that ensures $g_t(x_1,\\dots,x_N)$ satisfies some given conditions.\n", - "The neural network $N(x_1,\\dots,x_N,P)$ has weights and biases described by $P$ and $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$ is an expression using the output from the neural network in some way.\n", - "\n", - "The role of the function $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$, is to ensure that the output of $N(x_1,\\dots,x_N,P)$ is zero when $g_t(x_1,\\dots,x_N)$ is evaluated at the values of $x_1,\\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\\dots,x_N)$ should alone make $g_t(x_1,\\dots,x_N)$ satisfy the conditions.\n", - "\n", - "\n", - "\n", - "### Network requirements\n", - "\n", - "The network tries then the minimize the cost function following the\n", - "same ideas as described for the ODE case, but now with more than one\n", - "variables to consider. The concept still remains the same; find a set\n", - "of parameters $P$ such that the expression $f$ in ([17](#PDE)) is as\n", - "close to zero as possible.\n", - "\n", - "As for the ODE case, the cost function is the mean squared error that\n", - "the network must try to minimize. The cost function for the network to\n", - "minimize is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(x_1, \\dots, x_N, P\\right) = \\left( f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) \\right)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we let $\\boldsymbol{x} = \\big( x_1, \\dots, x_N \\big)$ be an array containing the values for $x_1, \\dots, x_N$ respectively, the cost function can be reformulated into the following:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(\\boldsymbol{x}, P\\right) = f\\left( \\left( \\boldsymbol{x}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}) }{\\partial x_N^n} \\right) \\right)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we also have $M$ different sets of values for $x_1, \\dots, x_N$, that is $\\boldsymbol{x}_i = \\big(x_1^{(i)}, \\dots, x_N^{(i)}\\big)$ for $i = 1,\\dots,M$ being the rows in matrix $X$, the cost function can be generalized into" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(X, P \\right) = \\sum_{i=1}^M f\\left( \\left( \\boldsymbol{x}_i, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}_i) }{\\partial x_N^n} \\right) \\right)^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Example: The diffusion equation\n", - "\n", - "In one spatial dimension, the equation reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where a possible choice of conditions are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "g(0,t) &= 0 ,\\qquad t \\geq 0 \\\\\n", - "g(1,t) &= 0, \\qquad t \\geq 0 \\\\\n", - "g(x,0) &= u(x),\\qquad x\\in [0,1]\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $u(x)$ being some given function.\n", - "\n", - "\n", - "\n", - "For this case, we want to find $g(x,t)$ such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "\\end{equation} \\label{diffonedim} \\tag{18}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "g(0,t) &= 0 ,\\qquad t \\geq 0 \\\\\n", - "g(1,t) &= 0, \\qquad t \\geq 0 \\\\\n", - "g(x,0) &= u(x),\\qquad x\\in [0,1]\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $u(x) = \\sin(\\pi x)$.\n", - "\n", - "First, let us set up the deep neural network.\n", - "The deep neural network will follow the same structure as discussed in the examples solving the ODEs.\n", - "First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions.\n", - "\n", - "\n", - "\n", - "\n", - "The only change to do here, is to extend our network such that\n", - "functions of multiple parameters are correctly handled. In this case\n", - "we have two variables in our function to solve for, that is time $t$\n", - "and position $x$. The variables will be represented by a\n", - "one-dimensional array in the program. The program will evaluate the\n", - "network at each possible pair $(x,t)$, given an array for the desired\n", - "$x$-values and $t$-values to approximate the solution at." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The cost function must then iterate through the given arrays\n", - "containing values for $x$ and $t$, defines a point $(x,t)$ the deep\n", - "neural network and the trial solution is evaluated at, and then finds\n", - "the Jacobian of the trial solution.\n", - "\n", - "A possible trial solution for this PDE is\n", - "\n", - "$$\n", - "g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P)\n", - "$$\n", - "\n", - "with $A(x,t)$ being a function ensuring that $g_t(x,t)$ satisfies our given conditions, and $N(x,t,P)$ being the output from the deep neural network using weights and biases for each layer from $P$.\n", - "\n", - "To fulfill the conditions, $A(x,t)$ could be:\n", - "\n", - "$$\n", - "h_1(x,t) = (1-t)\\Big(u(x) - \\big((1-x)u(0) + x u(1)\\big)\\Big) = (1-t)u(x) = (1-t)\\sin(\\pi x)\n", - "$$\n", - "since $(0) = u(1) = 0$ and $u(x) = \\sin(\\pi x)$.\n", - "\n", - "\n", - "\n", - "The Jacobian is used because the program must find the derivative of\n", - "the trial solution with respect to $x$ and $t$.\n", - "\n", - "This gives the necessity of computing the Jacobian matrix, as we want\n", - "to evaluate the gradient with respect to $x$ and $t$ (note that the\n", - "Jacobian of a scalar-valued multivariate function is simply its\n", - "gradient).\n", - "\n", - "In Autograd, the differentiation is by default done with respect to\n", - "the first input argument of your Python function. Since the points is\n", - "an array representing $x$ and $t$, the Jacobian is calculated using\n", - "the values of $x$ and $t$.\n", - "\n", - "To find the second derivative with respect to $x$ and $t$, the\n", - "Jacobian can be found for the second time. The result is a Hessian\n", - "matrix, which is the matrix containing all the possible second order\n", - "mixed derivatives of $g(x,t)$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Set up the trial function:\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)\n", - "\n", - "# The right side of the ODE:\n", - "def f(point):\n", - " return 0.\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_jacobian_func = jacobian(g_trial)\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t = g_trial(point,P)\n", - " g_t_jacobian = g_t_jacobian_func(point,P)\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_dt = g_t_jacobian[1]\n", - " g_t_d2x = g_t_hessian[0][0]\n", - "\n", - " func = f(point)\n", - "\n", - " err_sqr = ( (g_t_dt - g_t_d2x) - func)**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Setting up the network using Autograd; The full program\n", - "\n", - "Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.\n", - "\n", - "The analytical solution of our problem is\n", - "\n", - "$$\n", - "g(x,t) = \\exp(-\\pi^2 t)\\sin(\\pi x)\n", - "$$\n", - "\n", - "A possible way to implement a neural network solving the PDE, is given below.\n", - "Be aware, though, that it is fairly slow for the parameters used.\n", - "A better result is possible, but requires more iterations, and thus longer time to complete.\n", - "\n", - "\n", - "Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.\n", - "Using TensorFlow results in a much better execution time. Try it!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import jacobian,hessian,grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import cm\n", - "from matplotlib import pyplot as plt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "## Set up the network\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]\n", - "\n", - "## Define the trial solution and cost function\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)\n", - "\n", - "# The right side of the ODE:\n", - "def f(point):\n", - " return 0.\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_jacobian_func = jacobian(g_trial)\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t = g_trial(point,P)\n", - " g_t_jacobian = g_t_jacobian_func(point,P)\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_dt = g_t_jacobian[1]\n", - " g_t_d2x = g_t_hessian[0][0]\n", - "\n", - " func = f(point)\n", - "\n", - " err_sqr = ( (g_t_dt - g_t_d2x) - func)**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum /( np.size(x)*np.size(t) )\n", - "\n", - "## For comparison, define the analytical solution\n", - "def g_analytic(point):\n", - " x,t = point\n", - " return np.exp(-np.pi**2*t)*np.sin(np.pi*x)\n", - "\n", - "## Set up a function for training the network to solve for the equation\n", - "def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):\n", - " ## Set up initial weigths and biases\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: ',cost_function(P, x, t))\n", - "\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " cost_grad = cost_function_grad(P, x , t)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_grad[l]\n", - "\n", - " print('Final cost: ',cost_function(P, x, t))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " ### Use the neural network:\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10; Nt = 10\n", - " x = np.linspace(0, 1, Nx)\n", - " t = np.linspace(0,1,Nt)\n", - "\n", - " ## Set up the parameters for the network\n", - " num_hidden_neurons = [100, 25]\n", - " num_iter = 250\n", - " lmb = 0.01\n", - "\n", - " P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " ## Store the results\n", - " g_dnn_ag = np.zeros((Nx, Nt))\n", - " G_analytical = np.zeros((Nx, Nt))\n", - " for i,x_ in enumerate(x):\n", - " for j, t_ in enumerate(t):\n", - " point = np.array([x_, t_])\n", - " g_dnn_ag[i,j] = g_trial(point,P)\n", - "\n", - " G_analytical[i,j] = g_analytic(point)\n", - "\n", - " # Find the map difference between the analytical and the computed solution\n", - " diff_ag = np.abs(g_dnn_ag - G_analytical)\n", - " print('Max absolute difference between the analytical solution and the network: %g'%np.max(diff_ag))\n", - "\n", - " ## Plot the solutions in two dimensions, that being in position and time\n", - "\n", - " T,X = np.meshgrid(t,x)\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", - " s = ax.plot_surface(T,X,g_dnn_ag,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Analytical solution')\n", - " s = ax.plot_surface(T,X,G_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Difference')\n", - " s = ax.plot_surface(T,X,diff_ag,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " ## Take some slices of the 3D plots just to see the solutions at particular times\n", - " indx1 = 0\n", - " indx2 = int(Nt/2)\n", - " indx3 = Nt-1\n", - "\n", - " t1 = t[indx1]\n", - " t2 = t[indx2]\n", - " t3 = t[indx3]\n", - "\n", - " # Slice the results from the DNN\n", - " res1 = g_dnn_ag[:,indx1]\n", - " res2 = g_dnn_ag[:,indx2]\n", - " res3 = g_dnn_ag[:,indx3]\n", - "\n", - " # Slice the analytical results\n", - " res_analytical1 = G_analytical[:,indx1]\n", - " res_analytical2 = G_analytical[:,indx2]\n", - " res_analytical3 = G_analytical[:,indx3]\n", - "\n", - " # Plot the slices\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t1)\n", - " plt.plot(x, res1)\n", - " plt.plot(x,res_analytical1)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t2)\n", - " plt.plot(x, res2)\n", - " plt.plot(x,res_analytical2)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t3)\n", - " plt.plot(x, res3)\n", - " plt.plot(x,res_analytical3)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Solving the wave equation with Neural Networks\n", - "\n", - "The wave equation is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2\\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $c$ being the specified wave speed.\n", - "\n", - "Here, the chosen conditions are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\tg(0,t) &= 0 \\\\\n", - "\tg(1,t) &= 0 \\\\\n", - "\tg(x,0) &= u(x) \\\\\n", - "\t\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0} &= v(x)\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0}$ means the derivative of $g(x,t)$ with respect to $t$ is evaluated at $t = 0$, and $u(x)$ and $v(x)$ being given functions.\n", - "\n", - "\n", - "The wave equation to solve for, is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{wave} \\tag{19}\n", - "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2 \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $c$ is the given wave speed.\n", - "The chosen conditions for this equation are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{aligned}\n", - "g(0,t) &= 0, &t \\geq 0 \\\\\n", - "g(1,t) &= 0, &t \\geq 0 \\\\\n", - "g(x,0) &= u(x), &x\\in[0,1] \\\\\n", - "\\frac{\\partial g(x,t)}{\\partial t}\\Big |_{t = 0} &= v(x), &x \\in [0,1]\n", - "\\end{aligned} \\label{condwave} \\tag{20}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this example, let $c = 1$ and $u(x) = \\sin(\\pi x)$ and $v(x) = -\\pi\\sin(\\pi x)$.\n", - "\n", - "\n", - "\n", - "Setting up the network is done in similar matter as for the example of solving the diffusion equation.\n", - "The only things we have to change, is the trial solution such that it satisfies the conditions from ([20](#condwave)) and the cost function.\n", - "\n", - "The trial solution becomes slightly different since we have other conditions than in the example of solving the diffusion equation. Here, a possible trial solution $g_t(x,t)$ is\n", - "\n", - "$$\n", - "g_t(x,t) = h_1(x,t) + x(1-x)t^2N(x,t,P)\n", - "$$\n", - "\n", - "where\n", - "\n", - "$$\n", - "h_1(x,t) = (1-t^2)u(x) + tv(x)\n", - "$$\n", - "\n", - "Note that this trial solution satisfies the conditions only if $u(0) = v(0) = u(1) = v(1) = 0$, which is the case in this example.\n", - "\n", - "\n", - "The analytical solution for our specific problem, is\n", - "\n", - "$$\n", - "g(x,t) = \\sin(\\pi x)\\cos(\\pi t) - \\sin(\\pi x)\\sin(\\pi t)\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import hessian,grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import cm\n", - "from matplotlib import pyplot as plt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "## Set up the trial function:\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def v(x):\n", - " return -np.pi*np.sin(np.pi*x)\n", - "\n", - "def h1(point):\n", - " x,t = point\n", - " return (1 - t**2)*u(x) + t*v(x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return h1(point) + x*(1-x)*t**2*deep_neural_network(P,point)\n", - "\n", - "## Define the cost function\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_d2x = g_t_hessian[0][0]\n", - " g_t_d2t = g_t_hessian[1][1]\n", - "\n", - " err_sqr = ( (g_t_d2t - g_t_d2x) )**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum / (np.size(t) * np.size(x))\n", - "\n", - "## The neural network\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]\n", - "\n", - "## The analytical solution\n", - "def g_analytic(point):\n", - " x,t = point\n", - " return np.sin(np.pi*x)*np.cos(np.pi*t) - np.sin(np.pi*x)*np.sin(np.pi*t)\n", - "\n", - "def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):\n", - " ## Set up initial weigths and biases\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: ',cost_function(P, x, t))\n", - "\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " cost_grad = cost_function_grad(P, x , t)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_grad[l]\n", - "\n", - "\n", - " print('Final cost: ',cost_function(P, x, t))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " ### Use the neural network:\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10; Nt = 10\n", - " x = np.linspace(0, 1, Nx)\n", - " t = np.linspace(0,1,Nt)\n", - "\n", - " ## Set up the parameters for the network\n", - " num_hidden_neurons = [50,20]\n", - " num_iter = 1000\n", - " lmb = 0.01\n", - "\n", - " P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " ## Store the results\n", - " res = np.zeros((Nx, Nt))\n", - " res_analytical = np.zeros((Nx, Nt))\n", - " for i,x_ in enumerate(x):\n", - " for j, t_ in enumerate(t):\n", - " point = np.array([x_, t_])\n", - " res[i,j] = g_trial(point,P)\n", - "\n", - " res_analytical[i,j] = g_analytic(point)\n", - "\n", - " diff = np.abs(res - res_analytical)\n", - " print(\"Max difference between analytical and solution from nn: %g\"%np.max(diff))\n", - "\n", - " ## Plot the solutions in two dimensions, that being in position and time\n", - "\n", - " T,X = np.meshgrid(t,x)\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", - " s = ax.plot_surface(T,X,res,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Analytical solution')\n", - " s = ax.plot_surface(T,X,res_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Difference')\n", - " s = ax.plot_surface(T,X,diff,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " ## Take some slices of the 3D plots just to see the solutions at particular times\n", - " indx1 = 0\n", - " indx2 = int(Nt/2)\n", - " indx3 = Nt-1\n", - "\n", - " t1 = t[indx1]\n", - " t2 = t[indx2]\n", - " t3 = t[indx3]\n", - "\n", - " # Slice the results from the DNN\n", - " res1 = res[:,indx1]\n", - " res2 = res[:,indx2]\n", - " res3 = res[:,indx3]\n", - "\n", - " # Slice the analytical results\n", - " res_analytical1 = res_analytical[:,indx1]\n", - " res_analytical2 = res_analytical[:,indx2]\n", - " res_analytical3 = res_analytical[:,indx3]\n", - "\n", - " # Plot the slices\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t1)\n", - " plt.plot(x, res1)\n", - " plt.plot(x,res_analytical1)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t2)\n", - " plt.plot(x, res2)\n", - " plt.plot(x,res_analytical2)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t3)\n", - " plt.plot(x, res3)\n", - " plt.plot(x,res_analytical3)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Resources on differential equations and deep learning\n", - "\n", - "1. [Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al](https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf)\n", - "\n", - "2. [Neural networks for solving differential equations by A. Honchar](https://becominghuman.ai/neural-networks-for-solving-differential-equations-fa230ac5e04c)\n", - "\n", - "3. [Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener](http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf)\n", - "\n", - "4. [Introduction to Partial Differential Equations by A. Tveito, R. Winther](https://www.springer.com/us/book/9783540225515)" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter2.ipynb b/doc/LectureNotes/_build/html/_sources/chapter2.ipynb deleted file mode 100644 index 721b43cde..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter2.ipynb +++ /dev/null @@ -1,1355 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Resampling Methods\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSept3.mp4?vrtx=view-as-webpage)\n", - "\n", - "\n", - "## Introduction\n", - "\n", - "Resampling methods are an indispensable tool in modern\n", - "statistics. They involve repeatedly drawing samples from a training\n", - "set and refitting a model of interest on each sample in order to\n", - "obtain additional information about the fitted model. For example, in\n", - "order to estimate the variability of a linear regression fit, we can\n", - "repeatedly draw different samples from the training data, fit a linear\n", - "regression to each new sample, and then examine the extent to which\n", - "the resulting fits differ. Such an approach may allow us to obtain\n", - "information that would not be available from fitting the model only\n", - "once using the original training sample.\n", - "\n", - "Two resampling methods are often used in Machine Learning analyses,\n", - "1. The **bootstrap method**\n", - "\n", - "2. and **Cross-Validation**\n", - "\n", - "In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular\n", - "cross-validation and the bootstrap method. \n", - "\n", - "\n", - "Resampling approaches can be computationally expensive, because they\n", - "involve fitting the same statistical method multiple times using\n", - "different subsets of the training data. However, due to recent\n", - "advances in computing power, the computational requirements of\n", - "resampling methods generally are not prohibitive. In this chapter, we\n", - "discuss two of the most commonly used resampling methods,\n", - "cross-validation and the bootstrap. Both methods are important tools\n", - "in the practical application of many statistical learning\n", - "procedures. For example, cross-validation can be used to estimate the\n", - "test error associated with a given statistical learning method in\n", - "order to evaluate its performance, or to select the appropriate level\n", - "of flexibility. The process of evaluating a model’s performance is\n", - "known as model assessment, whereas the process of selecting the proper\n", - "level of flexibility for a model is known as model selection. The\n", - "bootstrap is widely used.\n", - "\n", - "\n", - "* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n", - "\n", - "* The results can be analysed with the same statistical tools as we would use analysing experimental data.\n", - "\n", - "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n", - "\n", - "## Reminder on Statistics\n", - "\n", - "\n", - "* As in other experiments, many numerical experiments have two classes of errors:\n", - "\n", - " * Statistical errors\n", - "\n", - " * Systematical errors\n", - "\n", - "\n", - "* Statistical errors can be estimated using standard tools from statistics\n", - "\n", - "* Systematical errors are method specific and must be treated differently from case to case. \n", - "\n", - "The\n", - "advantage of doing linear regression is that we actually end up with\n", - "analytical expressions for several statistical quantities. \n", - "Standard least squares and Ridge regression allow us to\n", - "derive quantities like the variance and other expectation values in a\n", - "rather straightforward way.\n", - "\n", - "\n", - "It is assumed that $\\varepsilon_i\n", - "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n", - "independent, i.e.:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*} \n", - "\\mbox{Cov}(\\varepsilon_{i_1},\n", - "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n", - "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2. \\end{array} \\right.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The randomness of $\\varepsilon_i$ implies that\n", - "$\\mathbf{y}_i$ is also a random variable. In particular,\n", - "$\\mathbf{y}_i$ is normally distributed, because $\\varepsilon_i \\sim\n", - "\\mathcal{N}(0, \\sigma^2)$ and $\\mathbf{X}_{i,\\ast} \\, \\boldsymbol{\\beta}$ is a\n", - "non-random scalar. To specify the parameters of the distribution of\n", - "$\\mathbf{y}_i$ we need to calculate its first two moments. \n", - "\n", - "Recall that $\\boldsymbol{X}$ is a matrix of dimensionality $n\\times p$. The\n", - "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n", - "row number $i$ and perform a sum over all values $p$.\n", - "\n", - "\n", - "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n", - "that there exists a function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n", - "which describe our data" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We approximate this function with our model from the solution of the linear regression equations, that is our\n", - "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*} \n", - "\\mathbb{E}(y_i) & =\n", - "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n", - "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "while\n", - "its variance is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n", - "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n", - "[\\mathbb{E}(y_i)]^2 \\\\ & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n", - "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n", - "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n", - "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n", - "\\ast} \\, \\beta)^2 \\\\ & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n", - "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n", - "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n", - "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n", - "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2. \n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n", - "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD). \n", - "\n", - "\n", - "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we can evaluate the expectation value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}(\\boldsymbol{\\beta}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This means that the estimator of the regression parameters is unbiased.\n", - "\n", - "We can also calculate the variance\n", - "\n", - "The variance of $\\boldsymbol{\\beta}$ is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{eqnarray*}\n", - "\\mbox{Var}(\\boldsymbol{\\beta}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n", - "\\\\\n", - "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}]^{T} \\}\n", - "\\\\\n", - "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} \\, \\mathbf{Y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{Y} \\, \\mathbf{Y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "\\\\\n", - "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "% & = & (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^T \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T % \\mathbf{X})^{-1}\n", - "% \\\\\n", - "% & & + \\, \\, \\sigma^2 \\, (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\boldsymbol{\\beta}^T\n", - "\\\\\n", - "& = & \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} + \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "\\, \\, \\, = \\, \\, \\, \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1},\n", - "\\end{eqnarray*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have used that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n", - "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n", - "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n", - "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n", - "variance of the estimate of the $j$-th regression coefficient:\n", - "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 \\sqrt{\n", - "[(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} }$. This may be used to\n", - "construct a confidence interval for the estimates.\n", - "\n", - "\n", - "In a similar way, we can obtain analytical expressions for say the\n", - "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n", - "when we employ Ridge regression, allowing us again to define a confidence interval. \n", - "\n", - "It is rather straightforward to show that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see clearly that \n", - "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n", - "\n", - "We can also compute the variance as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n", - "\n", - "With this, we can compute the difference" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The difference is non-negative definite since each component of the\n", - "matrix product is non-negative definite. \n", - "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. \n", - "\n", - "\n", - "\n", - "## Resampling methods\n", - "\n", - "With all these analytical equations for both the OLS and Ridge\n", - "regression, we will now outline how to assess a given model. This will\n", - "lead us to a discussion of the so-called bias-variance tradeoff (see\n", - "below) and so-called resampling methods.\n", - "\n", - "One of the quantities we have discussed as a way to measure errors is\n", - "the mean-squared error (MSE), mainly used for fitting of continuous\n", - "functions. Another choice is the absolute error.\n", - "\n", - "In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data,\n", - "we discuss the\n", - "1. prediction error or simply the **test error** $\\mathrm{Err_{Test}}$, where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the \n", - "\n", - "2. training error $\\mathrm{Err_{Train}}$, which is the average loss over the training data.\n", - "\n", - "As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n", - "For a certain level of complexity the test error will reach minimum, before starting to increase again. The\n", - "training error reaches a saturation.\n", - "\n", - "\n", - "\n", - "Two famous\n", - "resampling methods are the **independent bootstrap** and **the jackknife**. \n", - "\n", - "The jackknife is a special case of the independent bootstrap. Still, the jackknife was made\n", - "popular prior to the independent bootstrap. And as the popularity of\n", - "the independent bootstrap soared, new variants, such as **the dependent bootstrap**.\n", - "\n", - "The Jackknife and independent bootstrap work for\n", - "independent, identically distributed random variables.\n", - "If these conditions are not\n", - "satisfied, the methods will fail. Yet, it should be said that if the data are\n", - "independent, identically distributed, and we only want to estimate the\n", - "variance of $\\overline{X}$ (which often is the case), then there is no\n", - "need for bootstrapping. \n", - "\n", - "\n", - "The Jackknife works by making many replicas of the estimator $\\widehat{\\theta}$. \n", - "The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\\boldsymbol{x} = (x_1,x_2,\\cdots,X_n)$. \n", - "Let $\\boldsymbol{x}_i$ denote the vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i = (x_1,x_2,\\cdots,x_{i-1},x_{i+1},\\cdots,x_n),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which equals the vector $\\boldsymbol{x}$ with the exception that observation\n", - "number $i$ is left out. Using this notation, define\n", - "$\\widehat{\\theta}_i$ to be the estimator\n", - "$\\widehat{\\theta}$ computed using $\\vec{X}_i$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from numpy import *\n", - "from numpy.random import randint, randn\n", - "from time import time\n", - "\n", - "def jackknife(data, stat):\n", - " n = len(data);t = zeros(n); inds = arange(n); t0 = time()\n", - " ## 'jackknifing' by leaving out an observation for each i \n", - " for i in range(n):\n", - " t[i] = stat(delete(data,i) )\n", - "\n", - " # analysis \n", - " print(\"Runtime: %g sec\" % (time()-t0)); print(\"Jackknife Statistics :\")\n", - " print(\"original bias std. error\")\n", - " print(\"%8g %14g %15g\" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))\n", - "\n", - " return t\n", - "\n", - "\n", - "# Returns mean of data samples \n", - "def stat(data):\n", - " return mean(data)\n", - "\n", - "\n", - "mu, sigma = 100, 15\n", - "datapoints = 10000\n", - "x = mu + sigma*random.randn(datapoints)\n", - "# jackknife returns the data sample \n", - "t = jackknife(x, stat)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Bootstrap\n", - "\n", - "Bootstrapping is a nonparametric approach to statistical inference\n", - "that substitutes computation for more traditional distributional\n", - "assumptions and asymptotic results. Bootstrapping offers a number of\n", - "advantages: \n", - "1. The bootstrap is quite general, although there are some cases in which it fails. \n", - "\n", - "2. Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small. \n", - "\n", - "3. It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically. \n", - "\n", - "4. It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).\n", - "\n", - "Since $\\widehat{\\theta} = \\widehat{\\theta}(\\boldsymbol{X})$ is a function of random variables,\n", - "$\\widehat{\\theta}$ itself must be a random variable. Thus it has\n", - "a pdf, call this function $p(\\boldsymbol{t})$. The aim of the bootstrap is to\n", - "estimate $p(\\boldsymbol{t})$ by the relative frequency of\n", - "$\\widehat{\\theta}$. You can think of this as using a histogram\n", - "in the place of $p(\\boldsymbol{t})$. If the relative frequency closely\n", - "resembles $p(\\vec{t})$, then using numerics, it is straight forward to\n", - "estimate all the interesting parameters of $p(\\boldsymbol{t})$ using point\n", - "estimators. \n", - "\n", - "\n", - "\n", - "In the case that $\\widehat{\\theta}$ has\n", - "more than one component, and the components are independent, we use the\n", - "same estimator on each component separately. If the probability\n", - "density function of $X_i$, $p(x)$, had been known, then it would have\n", - "been straight forward to do this by: \n", - "1. Drawing lots of numbers from $p(x)$, suppose we call one such set of numbers $(X_1^*, X_2^*, \\cdots, X_n^*)$. \n", - "\n", - "2. Then using these numbers, we could compute a replica of $\\widehat{\\theta}$ called $\\widehat{\\theta}^*$. \n", - "\n", - "By repeated use of (1) and (2), many\n", - "estimates of $\\widehat{\\theta}$ could have been obtained. The\n", - "idea is to use the relative frequency of $\\widehat{\\theta}^*$\n", - "(think of a histogram) as an estimate of $p(\\boldsymbol{t})$.\n", - "\n", - "\n", - "But\n", - "unless there is enough information available about the process that\n", - "generated $X_1,X_2,\\cdots,X_n$, $p(x)$ is in general\n", - "unknown. Therefore, [Efron in 1979](https://projecteuclid.org/euclid.aos/1176344552) asked the\n", - "question: What if we replace $p(x)$ by the relative frequency\n", - "of the observation $X_i$; if we draw observations in accordance with\n", - "the relative frequency of the observations, will we obtain the same\n", - "result in some asymptotic sense? The answer is yes.\n", - "\n", - "\n", - "Instead of generating the histogram for the relative\n", - "frequency of the observation $X_i$, just draw the values\n", - "$(X_1^*,X_2^*,\\cdots,X_n^*)$ with replacement from the vector\n", - "$\\boldsymbol{X}$. \n", - "\n", - "\n", - "The independent bootstrap works like this: \n", - "\n", - "1. Draw with replacement $n$ numbers for the observed variables $\\boldsymbol{x} = (x_1,x_2,\\cdots,x_n)$. \n", - "\n", - "2. Define a vector $\\boldsymbol{x}^*$ containing the values which were drawn from $\\boldsymbol{x}$. \n", - "\n", - "3. Using the vector $\\boldsymbol{x}^*$ compute $\\widehat{\\theta}^*$ by evaluating $\\widehat \\theta$ under the observations $\\boldsymbol{x}^*$. \n", - "\n", - "4. Repeat this process $k$ times. \n", - "\n", - "When you are done, you can draw a histogram of the relative frequency\n", - "of $\\widehat \\theta^*$. This is your estimate of the probability\n", - "distribution $p(t)$. Using this probability distribution you can\n", - "estimate any statistics thereof. In principle you never draw the\n", - "histogram of the relative frequency of $\\widehat{\\theta}^*$. Instead\n", - "you use the estimators corresponding to the statistic of interest. For\n", - "example, if you are interested in estimating the variance of $\\widehat\n", - "\\theta$, apply the etsimator $\\widehat \\sigma^2$ to the values\n", - "$\\widehat \\theta ^*$.\n", - "\n", - "\n", - "\n", - "The following code starts with a Gaussian distribution with mean value\n", - "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n", - "used in the bootstrap analysis. The bootstrap analysis returns a data\n", - "set after a given number of bootstrap operations (as many as we have\n", - "data points). This data set consists of estimated mean values for each\n", - "bootstrap operation. The histogram generated by the bootstrap method\n", - "shows that the distribution for these mean values is also a Gaussian,\n", - "centered around the mean value $\\mu=100$ but with standard deviation\n", - "$\\sigma/\\sqrt{n}$, where $n$ is the number of bootstrap samples (in\n", - "this case the same as the number of original data points). The value\n", - "of the standard deviation is what we expect from the central limit\n", - "theorem." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "from numpy import *\n", - "from numpy.random import randint, randn\n", - "from time import time\n", - "import matplotlib.mlab as mlab\n", - "import matplotlib.pyplot as plt\n", - "\n", - "# Returns mean of bootstrap samples \n", - "def stat(data):\n", - " return mean(data)\n", - "\n", - "# Bootstrap algorithm\n", - "def bootstrap(data, statistic, R):\n", - " t = zeros(R); n = len(data); inds = arange(n); t0 = time()\n", - " # non-parametric bootstrap \n", - " for i in range(R):\n", - " t[i] = statistic(data[randint(0,n,n)])\n", - "\n", - " # analysis \n", - " print(\"Runtime: %g sec\" % (time()-t0)); print(\"Bootstrap Statistics :\")\n", - " print(\"original bias std. error\")\n", - " print(\"%8g %8g %14g %15g\" % (statistic(data), std(data),mean(t),std(t)))\n", - " return t\n", - "\n", - "\n", - "mu, sigma = 100, 15\n", - "datapoints = 10000\n", - "x = mu + sigma*random.randn(datapoints)\n", - "# bootstrap returns the data sample \n", - "t = bootstrap(x, stat, datapoints)\n", - "# the histogram of the bootstrapped data \n", - "n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)\n", - "\n", - "# add a 'best fit' line \n", - "y = mlab.normpdf( binsboot, mean(t), std(t))\n", - "lt = plt.plot(binsboot, y, 'r--', linewidth=1)\n", - "plt.xlabel('Smarts')\n", - "plt.ylabel('Probability')\n", - "plt.axis([99.5, 100.6, 0, 3.0])\n", - "plt.grid(True)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Various steps in cross-validation\n", - "\n", - "When the repetitive splitting of the data set is done randomly,\n", - "samples may accidently end up in a fast majority of the splits in\n", - "either training or test set. Such samples may have an unbalanced\n", - "influence on either model building or prediction evaluation. To avoid\n", - "this $k$-fold cross-validation structures the data splitting. The\n", - "samples are divided into $k$ more or less equally sized exhaustive and\n", - "mutually exclusive subsets. In turn (at each split) one of these\n", - "subsets plays the role of the test set while the union of the\n", - "remaining subsets constitutes the training set. Such a splitting\n", - "warrants a balanced representation of each sample in both training and\n", - "test set over the splits. Still the division into the $k$ subsets\n", - "involves a degree of randomness. This may be fully excluded when\n", - "choosing $k=n$. This particular case is referred to as leave-one-out\n", - "cross-validation (LOOCV). \n", - "\n", - "\n", - "* Define a range of interest for the penalty parameter.\n", - "\n", - "* Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n", - "\n", - "* Fit the linear regression model by means of ridge estimation for each $\\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\\boldsymbol{\\sigma}_{-i}^2(\\lambda)$, as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\boldsymbol{\\beta}_{-i}(\\lambda) & = ( \\boldsymbol{X}_{-i, \\ast}^{T}\n", - "\\boldsymbol{X}_{-i, \\ast} + \\lambda \\boldsymbol{I}_{pp})^{-1}\n", - "\\boldsymbol{X}_{-i, \\ast}^{T} \\boldsymbol{y}_{-i}\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "* Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n", - "\n", - "* Repeat the first three steps such that each sample plays the role of the test set once.\n", - "\n", - "* Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\frac{1}{n} \\sum_{i = 1}^n \\log\\{L[y_i, \\mathbf{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For the various values of $k$\n", - "\n", - "1. shuffle the dataset randomly.\n", - "\n", - "2. Split the dataset into $k$ groups.\n", - "\n", - "3. For each unique group:\n", - "\n", - "a. Decide which group to use as set for test data\n", - "\n", - "b. Take the remaining groups as a training data set\n", - "\n", - "c. Fit a model on the training set and evaluate it on the test set\n", - "\n", - "d. Retain the evaluation score and discard the model\n", - "\n", - "\n", - "5. Summarize the model using the sample of model evaluation scores\n", - "\n", - "The code here uses Ridge regression with cross-validation (CV) resampling and $k$-fold CV in order to fit a specific polynomial." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.model_selection import KFold\n", - "from sklearn.linear_model import Ridge\n", - "from sklearn.model_selection import cross_val_score\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "\n", - "# A seed just to ensure that the random numbers are the same for every run.\n", - "# Useful for eventual debugging.\n", - "np.random.seed(3155)\n", - "\n", - "# Generate the data.\n", - "nsamples = 100\n", - "x = np.random.randn(nsamples)\n", - "y = 3*x**2 + np.random.randn(nsamples)\n", - "\n", - "## Cross-validation on Ridge regression using KFold only\n", - "\n", - "# Decide degree on polynomial to fit\n", - "poly = PolynomialFeatures(degree = 6)\n", - "\n", - "# Decide which values of lambda to use\n", - "nlambdas = 500\n", - "lambdas = np.logspace(-3, 5, nlambdas)\n", - "\n", - "# Initialize a KFold instance\n", - "k = 5\n", - "kfold = KFold(n_splits = k)\n", - "\n", - "# Perform the cross-validation to estimate MSE\n", - "scores_KFold = np.zeros((nlambdas, k))\n", - "\n", - "i = 0\n", - "for lmb in lambdas:\n", - " ridge = Ridge(alpha = lmb)\n", - " j = 0\n", - " for train_inds, test_inds in kfold.split(x):\n", - " xtrain = x[train_inds]\n", - " ytrain = y[train_inds]\n", - "\n", - " xtest = x[test_inds]\n", - " ytest = y[test_inds]\n", - "\n", - " Xtrain = poly.fit_transform(xtrain[:, np.newaxis])\n", - " ridge.fit(Xtrain, ytrain[:, np.newaxis])\n", - "\n", - " Xtest = poly.fit_transform(xtest[:, np.newaxis])\n", - " ypred = ridge.predict(Xtest)\n", - "\n", - " scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)\n", - "\n", - " j += 1\n", - " i += 1\n", - "\n", - "\n", - "estimated_mse_KFold = np.mean(scores_KFold, axis = 1)\n", - "\n", - "## Cross-validation using cross_val_score from sklearn along with KFold\n", - "\n", - "# kfold is an instance initialized above as:\n", - "# kfold = KFold(n_splits = k)\n", - "\n", - "estimated_mse_sklearn = np.zeros(nlambdas)\n", - "i = 0\n", - "for lmb in lambdas:\n", - " ridge = Ridge(alpha = lmb)\n", - "\n", - " X = poly.fit_transform(x[:, np.newaxis])\n", - " estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)\n", - "\n", - " # cross_val_score return an array containing the estimated negative mse for every fold.\n", - " # we have to the the mean of every array in order to get an estimate of the mse of the model\n", - " estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)\n", - "\n", - " i += 1\n", - "\n", - "## Plot and compare the slightly different ways to perform cross-validation\n", - "\n", - "plt.figure()\n", - "\n", - "plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')\n", - "plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')\n", - "\n", - "plt.xlabel('log10(lambda)')\n", - "plt.ylabel('mse')\n", - "\n", - "plt.legend()\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The bias-variance tradeoff\n", - "\n", - "\n", - "We will discuss the bias-variance tradeoff in the context of\n", - "continuous predictions such as regression. However, many of the\n", - "intuitions and ideas discussed here also carry over to classification\n", - "tasks. Consider a dataset $\\mathcal{L}$ consisting of the data\n", - "$\\mathbf{X}_\\mathcal{L}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$. \n", - "\n", - "Let us assume that the true data is generated from a noisy model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\epsilon$ is normally distributed with mean zero and standard deviation $\\sigma^2$.\n", - "\n", - "In our derivation of the ordinary least squares method we defined then\n", - "an approximation to the function $f$ in terms of the parameters\n", - "$\\boldsymbol{\\beta}$ and the design matrix $\\boldsymbol{X}$ which embody our model,\n", - "that is $\\boldsymbol{\\tilde{y}}=\\boldsymbol{X}\\boldsymbol{\\beta}$. \n", - "\n", - "Thereafter we found the parameters $\\boldsymbol{\\beta}$ by optimizing the means squared error via the so-called cost function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can rewrite this as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The three terms represent the square of the bias of the learning\n", - "method, which can be thought of as the error caused by the simplifying\n", - "assumptions built into the method. The second term represents the\n", - "variance of the chosen model and finally the last terms is variance of\n", - "the error $\\boldsymbol{\\epsilon}$.\n", - "\n", - "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\\boldsymbol{\\tilde{y}}$.\n", - "We use a more compact notation in terms of the expectation value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}})^2\\right],\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and adding and subtracting $\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]$ we get" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}}+\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right],\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which, using the abovementioned expectation values can be rewritten as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right]+\\mathrm{Var}\\left[\\boldsymbol{\\tilde{y}}\\right]+\\sigma^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "that is the rewriting in terms of the so-called bias, the variance of the model $\\boldsymbol{\\tilde{y}}$ and the variance of $\\boldsymbol{\\epsilon}$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.utils import resample\n", - "\n", - "np.random.seed(2018)\n", - "\n", - "n = 500\n", - "n_boostraps = 100\n", - "degree = 18 # A quite high value, just to show.\n", - "noise = 0.1\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-1, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape)\n", - "\n", - "# Hold out some test data that is never used in training.\n", - "x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "\n", - "# Combine x transformation and model into one operation.\n", - "# Not neccesary, but convenient.\n", - "model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))\n", - "\n", - "# The following (m x n_bootstraps) matrix holds the column vectors y_pred\n", - "# for each bootstrap iteration.\n", - "y_pred = np.empty((y_test.shape[0], n_boostraps))\n", - "for i in range(n_boostraps):\n", - " x_, y_ = resample(x_train, y_train)\n", - "\n", - " # Evaluate the new model on the same test data each time.\n", - " y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()\n", - "\n", - "# Note: Expectations and variances taken w.r.t. different training\n", - "# data sets, hence the axis=1. Subsequent means are taken across the test data\n", - "# set in order to obtain a total value, but before this we have error/bias/variance\n", - "# calculated per data point in the test set.\n", - "# Note 2: The use of keepdims=True is important in the calculation of bias as this \n", - "# maintains the column vector form. Dropping this yields very unexpected results.\n", - "error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n", - "bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n", - "variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n", - "print('Error:', error)\n", - "print('Bias^2:', bias)\n", - "print('Var:', variance)\n", - "print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))\n", - "\n", - "plt.plot(x[::5, :], y[::5, :], label='f(x)')\n", - "plt.scatter(x_test, y_test, label='Data points')\n", - "plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.utils import resample\n", - "\n", - "np.random.seed(2018)\n", - "\n", - "n = 40\n", - "n_boostraps = 100\n", - "maxdegree = 14\n", - "\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "error = np.zeros(maxdegree)\n", - "bias = np.zeros(maxdegree)\n", - "variance = np.zeros(maxdegree)\n", - "polydegree = np.zeros(maxdegree)\n", - "x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "\n", - "for degree in range(maxdegree):\n", - " model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))\n", - " y_pred = np.empty((y_test.shape[0], n_boostraps))\n", - " for i in range(n_boostraps):\n", - " x_, y_ = resample(x_train, y_train)\n", - " y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()\n", - "\n", - " polydegree[degree] = degree\n", - " error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n", - " bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n", - " variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n", - " print('Polynomial degree:', degree)\n", - " print('Error:', error[degree])\n", - " print('Bias^2:', bias[degree])\n", - " print('Var:', variance[degree])\n", - " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", - "\n", - "plt.plot(polydegree, error, label='Error')\n", - "plt.plot(polydegree, bias, label='bias')\n", - "plt.plot(polydegree, variance, label='Variance')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The bias-variance tradeoff summarizes the fundamental tension in\n", - "machine learning, particularly supervised learning, between the\n", - "complexity of a model and the amount of training data needed to train\n", - "it. Since data is often limited, in practice it is often useful to\n", - "use a less-complex model with higher bias, that is a model whose asymptotic\n", - "performance is worse than another model because it is easier to\n", - "train and less sensitive to sampling noise arising from having a\n", - "finite-sized training dataset (smaller variance). \n", - "\n", - "\n", - "\n", - "The above equations tell us that in\n", - "order to minimize the expected test error, we need to select a\n", - "statistical learning method that simultaneously achieves low variance\n", - "and low bias. Note that variance is inherently a nonnegative quantity,\n", - "and squared bias is also nonnegative. Hence, we see that the expected\n", - "test MSE can never lie below $Var(\\epsilon)$, the irreducible error.\n", - "\n", - "\n", - "What do we mean by the variance and bias of a statistical learning\n", - "method? The variance refers to the amount by which our model would change if we\n", - "estimated it using a different training data set. Since the training\n", - "data are used to fit the statistical learning method, different\n", - "training data sets will result in a different estimate. But ideally the\n", - "estimate for our model should not vary too much between training\n", - "sets. However, if a method has high variance then small changes in\n", - "the training data can result in large changes in the model. In general, more\n", - "flexible statistical methods have higher variance.\n", - "\n", - "\n", - "You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\"\"\"\n", - "============================\n", - "Underfitting vs. Overfitting\n", - "============================\n", - "\n", - "This example demonstrates the problems of underfitting and overfitting and\n", - "how we can use linear regression with polynomial features to approximate\n", - "nonlinear functions. The plot shows the function that we want to approximate,\n", - "which is a part of the cosine function. In addition, the samples from the\n", - "real function and the approximations of different models are displayed. The\n", - "models have polynomial features of different degrees. We can see that a\n", - "linear function (polynomial with degree 1) is not sufficient to fit the\n", - "training samples. This is called **underfitting**. A polynomial of degree 4\n", - "approximates the true function almost perfectly. However, for higher degrees\n", - "the model will **overfit** the training data, i.e. it learns the noise of the\n", - "training data.\n", - "We evaluate quantitatively **overfitting** / **underfitting** by using\n", - "cross-validation. We calculate the mean squared error (MSE) on the validation\n", - "set, the higher, the less likely the model generalizes correctly from the\n", - "training data.\n", - "\"\"\"\n", - "\n", - "print(__doc__)\n", - "\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.pipeline import Pipeline\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.linear_model import LinearRegression\n", - "from sklearn.model_selection import cross_val_score\n", - "\n", - "\n", - "def true_fun(X):\n", - " return np.cos(1.5 * np.pi * X)\n", - "\n", - "np.random.seed(0)\n", - "\n", - "n_samples = 30\n", - "degrees = [1, 4, 15]\n", - "\n", - "X = np.sort(np.random.rand(n_samples))\n", - "y = true_fun(X) + np.random.randn(n_samples) * 0.1\n", - "\n", - "plt.figure(figsize=(14, 5))\n", - "for i in range(len(degrees)):\n", - " ax = plt.subplot(1, len(degrees), i + 1)\n", - " plt.setp(ax, xticks=(), yticks=())\n", - "\n", - " polynomial_features = PolynomialFeatures(degree=degrees[i],\n", - " include_bias=False)\n", - " linear_regression = LinearRegression()\n", - " pipeline = Pipeline([(\"polynomial_features\", polynomial_features),\n", - " (\"linear_regression\", linear_regression)])\n", - " pipeline.fit(X[:, np.newaxis], y)\n", - "\n", - " # Evaluate the models using crossvalidation\n", - " scores = cross_val_score(pipeline, X[:, np.newaxis], y,\n", - " scoring=\"neg_mean_squared_error\", cv=10)\n", - "\n", - " X_test = np.linspace(0, 1, 100)\n", - " plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label=\"Model\")\n", - " plt.plot(X_test, true_fun(X_test), label=\"True function\")\n", - " plt.scatter(X, y, edgecolor='b', s=20, label=\"Samples\")\n", - " plt.xlabel(\"x\")\n", - " plt.ylabel(\"y\")\n", - " plt.xlim((0, 1))\n", - " plt.ylim((-2, 2))\n", - " plt.legend(loc=\"best\")\n", - " plt.title(\"Degree {}\\nMSE = {:.2e}(+/- {:.2e})\".format(\n", - " degrees[i], -scores.mean(), scores.std()))\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.utils import resample\n", - "from sklearn.metrics import mean_squared_error\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"EoS.csv\"),'r')\n", - "\n", - "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", - "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", - "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", - "EoS = EoS.dropna()\n", - "Energies = EoS['Energy']\n", - "Density = EoS['Density']\n", - "# The design matrix now as function of various polytrops\n", - "\n", - "Maxpolydegree = 30\n", - "X = np.zeros((len(Density),Maxpolydegree))\n", - "X[:,0] = 1.0\n", - "testerror = np.zeros(Maxpolydegree)\n", - "trainingerror = np.zeros(Maxpolydegree)\n", - "polynomial = np.zeros(Maxpolydegree)\n", - "\n", - "trials = 100\n", - "for polydegree in range(1, Maxpolydegree):\n", - " polynomial[polydegree] = polydegree\n", - " for degree in range(polydegree):\n", - " X[:,degree] = Density**(degree/3.0)\n", - "\n", - "# loop over trials in order to estimate the expectation value of the MSE\n", - " testerror[polydegree] = 0.0\n", - " trainingerror[polydegree] = 0.0\n", - " for samples in range(trials):\n", - " x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)\n", - " model = LinearRegression(fit_intercept=True).fit(x_train, y_train)\n", - " ypred = model.predict(x_train)\n", - " ytilde = model.predict(x_test)\n", - " testerror[polydegree] += mean_squared_error(y_test, ytilde)\n", - " trainingerror[polydegree] += mean_squared_error(y_train, ypred) \n", - "\n", - " testerror[polydegree] /= trials\n", - " trainingerror[polydegree] /= trials\n", - " print(\"Degree of polynomial: %3d\"% polynomial[polydegree])\n", - " print(\"Mean squared error on training data: %.8f\" % trainingerror[polydegree])\n", - " print(\"Mean squared error on test data: %.8f\" % testerror[polydegree])\n", - "\n", - "plt.plot(polynomial, np.log10(trainingerror), label='Training Error')\n", - "plt.plot(polynomial, np.log10(testerror), label='Test Error')\n", - "plt.xlabel('Polynomial degree')\n", - "plt.ylabel('log10[MSE]')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.metrics import mean_squared_error\n", - "from sklearn.model_selection import KFold\n", - "from sklearn.model_selection import cross_val_score\n", - "\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"EoS.csv\"),'r')\n", - "\n", - "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", - "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", - "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", - "EoS = EoS.dropna()\n", - "Energies = EoS['Energy']\n", - "Density = EoS['Density']\n", - "# The design matrix now as function of various polytrops\n", - "\n", - "Maxpolydegree = 30\n", - "X = np.zeros((len(Density),Maxpolydegree))\n", - "X[:,0] = 1.0\n", - "estimated_mse_sklearn = np.zeros(Maxpolydegree)\n", - "polynomial = np.zeros(Maxpolydegree)\n", - "k =5\n", - "kfold = KFold(n_splits = k)\n", - "\n", - "for polydegree in range(1, Maxpolydegree):\n", - " polynomial[polydegree] = polydegree\n", - " for degree in range(polydegree):\n", - " X[:,degree] = Density**(degree/3.0)\n", - " OLS = LinearRegression()\n", - "# loop over trials in order to estimate the expectation value of the MSE\n", - " estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)\n", - "#[:, np.newaxis]\n", - " estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)\n", - "\n", - "plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')\n", - "plt.xlabel('Polynomial degree')\n", - "plt.ylabel('log10[MSE]')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.model_selection import KFold\n", - "from sklearn.linear_model import Ridge\n", - "from sklearn.model_selection import cross_val_score\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "\n", - "# A seed just to ensure that the random numbers are the same for every run.\n", - "np.random.seed(3155)\n", - "# Generate the data.\n", - "n = 100\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "# Decide degree on polynomial to fit\n", - "poly = PolynomialFeatures(degree = 10)\n", - "\n", - "# Decide which values of lambda to use\n", - "nlambdas = 500\n", - "lambdas = np.logspace(-3, 5, nlambdas)\n", - "# Initialize a KFold instance\n", - "k = 5\n", - "kfold = KFold(n_splits = k)\n", - "estimated_mse_sklearn = np.zeros(nlambdas)\n", - "i = 0\n", - "for lmb in lambdas:\n", - " ridge = Ridge(alpha = lmb)\n", - " estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)\n", - " estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)\n", - " i += 1\n", - "plt.figure()\n", - "plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')\n", - "plt.xlabel('log10(lambda)')\n", - "plt.ylabel('MSE')\n", - "plt.legend()\n", - "plt.show()" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter3.ipynb b/doc/LectureNotes/_build/html/_sources/chapter3.ipynb deleted file mode 100644 index 5ede9f687..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter3.ipynb +++ /dev/null @@ -1,1378 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Ridge and Lasso Regression\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember10.mp4?vrtx=view-as-webpage)\n", - "\n", - "\n", - "## The singular value decomposition\n", - "\n", - "The examples we have looked at so far are cases where we normally can\n", - "invert the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$. Using a polynomial expansion as we\n", - "did both for the masses and the fitting of the equation of state,\n", - "leads to row vectors of the design matrix which are essentially\n", - "orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. \n", - "\n", - "\n", - "\n", - "This may\n", - "however not the be case in general and a standard matrix inversion\n", - "algorithm based on say LU, QR or Cholesky decomposition may lead to singularities. We will see examples of this below.\n", - "\n", - "There is however a way to partially circumvent this problem and also gain some insights about the ordinary least squares approach, and later shrinkage methods like Ridge and Lasso regressions. \n", - "\n", - "This is given by the **Singular Value Decomposition** algorithm, perhaps\n", - "the most powerful linear algebra algorithm. Let us look at a\n", - "different example where we may have problems with the standard matrix\n", - "inversion algorithm. Thereafter we dive into the math of the SVD.\n", - "\n", - "\n", - "\n", - "One of the typical problems we encounter with linear regression, in particular \n", - "when the matrix $\\boldsymbol{X}$ (our so-called design matrix) is high-dimensional, \n", - "are problems with near singular or singular matrices. The column vectors of $\\boldsymbol{X}$ \n", - "may be linearly dependent, normally referred to as super-collinearity. \n", - "This means that the matrix may be rank deficient and it is basically impossible to \n", - "to model the data using linear regression. As an example, consider the matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\mathbf{X} & = \\left[\n", - "\\begin{array}{rrr}\n", - "1 & -1 & 2\n", - "\\\\\n", - "1 & 0 & 1\n", - "\\\\\n", - "1 & 2 & -1\n", - "\\\\\n", - "1 & 1 & 0\n", - "\\end{array} \\right]\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The columns of $\\boldsymbol{X}$ are linearly dependent. We see this easily since the \n", - "the first column is the row-wise sum of the other two columns. The rank (more correct,\n", - "the column rank) of a matrix is the dimension of the space spanned by the\n", - "column vectors. Hence, the rank of $\\mathbf{X}$ is equal to the number\n", - "of linearly independent columns. In this particular case the matrix has rank 2.\n", - "\n", - "Super-collinearity of an $(n \\times p)$-dimensional design matrix $\\mathbf{X}$ implies\n", - "that the inverse of the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\boldsymbol{X} & = \\left[\n", - "\\begin{array}{rr}\n", - "1 & -1\n", - "\\\\\n", - "1 & -1\n", - "\\end{array} \\right].\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see easily that $\\mbox{det}(\\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \\times (-1) - 1 \\times (-1) = 0$. Hence, $\\mathbf{X}$ is singular and its inverse is undefined.\n", - "This is equivalent to saying that the matrix $\\boldsymbol{X}$ has at least an eigenvalue which is zero.\n", - "\n", - "\n", - "If our design matrix $\\boldsymbol{X}$ which enters the linear regression problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\boldsymbol{\\beta} = (\\boldsymbol{X}^{T} \\boldsymbol{X})^{-1} \\boldsymbol{X}^{T} \\boldsymbol{y},\n", - "\\label{_auto1} \\tag{1}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "has linearly dependent column vectors, we will not be able to compute the inverse\n", - "of $\\boldsymbol{X}^T\\boldsymbol{X}$ and we cannot find the parameters (estimators) $\\beta_i$. \n", - "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exits. \n", - "This is more likely to happen when the matrix $\\boldsymbol{X}$ is high-dimensional. In this case it is likely to encounter a situation where \n", - "the regression parameters $\\beta_i$ cannot be estimated.\n", - "\n", - "A cheap *ad hoc* approach is simply to add a small diagonal component to the matrix to invert, that is we change" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^{T} \\boldsymbol{X} \\rightarrow \\boldsymbol{X}^{T} \\boldsymbol{X}+\\lambda \\boldsymbol{I},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\boldsymbol{I}$ is the identity matrix. When we discuss **Ridge** regression this is actually what we end up evaluating. The parameter $\\lambda$ is called a hyperparameter. More about this later. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "From standard linear algebra we know that a square matrix $\\boldsymbol{X}$ can be diagonalized if and only it is \n", - "a so-called [normal matrix](https://en.wikipedia.org/wiki/Normal_matrix), that is if $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times n}$\n", - "we have $\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{X}^T\\boldsymbol{X}$ or if $\\boldsymbol{X}\\in {\\mathbb{C}}^{n\\times n}$ we have $\\boldsymbol{X}\\boldsymbol{X}^{\\dagger}=\\boldsymbol{X}^{\\dagger}\\boldsymbol{X}$.\n", - "The matrix has then a set of eigenpairs" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\lambda_1,\\boldsymbol{u}_1),\\dots, (\\lambda_n,\\boldsymbol{u}_n),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the eigenvalues are given by the diagonal matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\Sigma}=\\mathrm{Diag}(\\lambda_1, \\dots,\\lambda_n).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The matrix $\\boldsymbol{X}$ can be written in terms of an orthogonal/unitary transformation $\\boldsymbol{U}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{U}\\boldsymbol{U}^T=\\boldsymbol{I}$ or $\\boldsymbol{U}\\boldsymbol{U}^{\\dagger}=\\boldsymbol{I}$.\n", - "\n", - "Not all square matrices are diagonalizable. A matrix like the one discussed above" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\begin{bmatrix} \n", - "1& -1 \\\\\n", - "1& -1\\\\\n", - "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "is not diagonalizable, it is a so-called [defective matrix](https://en.wikipedia.org/wiki/Defective_matrix). It is easy to see that the condition\n", - "$\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{X}^T\\boldsymbol{X}$ is not fulfilled. \n", - "\n", - "\n", - "\n", - "## The SVD, a Fantastic Algorithm\n", - "\n", - "\n", - "However, and this is the strength of the SVD algorithm, any general\n", - "matrix $\\boldsymbol{X}$ can be decomposed in terms of a diagonal matrix and\n", - "two orthogonal/unitary matrices. The [Singular Value Decompostion\n", - "(SVD) theorem](https://en.wikipedia.org/wiki/Singular_value_decomposition)\n", - "states that a general $m\\times n$ matrix $\\boldsymbol{X}$ can be written in\n", - "terms of a diagonal matrix $\\boldsymbol{\\Sigma}$ of dimensionality $m\\times n$\n", - "and two orthognal matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$, where the first has\n", - "dimensionality $m \\times m$ and the last dimensionality $n\\times n$.\n", - "We have then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As an example, the above defective matrix can be decomposed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& 1 \\\\ 1& -1\\\\ \\end{bmatrix} \\begin{bmatrix} 2& 0 \\\\ 0& 0\\\\ \\end{bmatrix} \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& -1 \\\\ 1& 1\\\\ \\end{bmatrix}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with eigenvalues $\\sigma_1=2$ and $\\sigma_2=0$. \n", - "The SVD exits always! \n", - "\n", - "The SVD\n", - "decomposition (singular values) gives eigenvalues \n", - "$\\sigma_i\\geq\\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=p$, the\n", - "eigenvalues (singular values) are zero.\n", - "\n", - "In the general case, where our design matrix $\\boldsymbol{X}$ has dimension\n", - "$n\\times p$, the matrix is thus decomposed into an $n\\times n$\n", - "orthogonal matrix $\\boldsymbol{U}$, a $p\\times p$ orthogonal matrix $\\boldsymbol{V}$\n", - "and a diagonal matrix $\\boldsymbol{\\Sigma}$ with $r=\\mathrm{min}(n,p)$\n", - "singular values $\\sigma_i\\geq 0$ on the main diagonal and zeros filling\n", - "the rest of the matrix. There are at most $p$ singular values\n", - "assuming that $n > p$. In our regression examples for the nuclear\n", - "masses and the equation of state this is indeed the case, while for\n", - "the Ising model we have $p > n$. These are often cases that lead to\n", - "near singular or singular matrices.\n", - "\n", - "The columns of $\\boldsymbol{U}$ are called the left singular vectors while the columns of $\\boldsymbol{V}$ are the right singular vectors.\n", - "\n", - "## Economy-size SVD\n", - "\n", - "If we assume that $n > p$, then our matrix $\\boldsymbol{U}$ has dimension $n\n", - "\\times n$. The last $n-p$ columns of $\\boldsymbol{U}$ become however\n", - "irrelevant in our calculations since they are multiplied with the\n", - "zeros in $\\boldsymbol{\\Sigma}$.\n", - "\n", - "The economy-size decomposition removes extra rows or columns of zeros\n", - "from the diagonal matrix of singular values, $\\boldsymbol{\\Sigma}$, along with the columns\n", - "in either $\\boldsymbol{U}$ or $\\boldsymbol{V}$ that multiply those zeros in the expression. \n", - "Removing these zeros and columns can improve execution time\n", - "and reduce storage requirements without compromising the accuracy of\n", - "the decomposition.\n", - "\n", - "If $n > p$, we keep only the first $p$ columns of $\\boldsymbol{U}$ and $\\boldsymbol{\\Sigma}$ has dimension $p\\times p$. \n", - "If $p > n$, then only the first $n$ columns of $\\boldsymbol{V}$ are computed and $\\boldsymbol{\\Sigma}$ has dimension $n\\times n$.\n", - "The $n=p$ case is obvious, we retain the full SVD. \n", - "In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "# SVD inversion\n", - "def SVDinv(A):\n", - " ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).\n", - " SVD is numerically more stable than the inversion algorithms provided by\n", - " numpy and scipy.linalg at the cost of being slower.\n", - " '''\n", - " U, s, VT = np.linalg.svd(A)\n", - "# print('test U')\n", - "# print( (np.transpose(U) @ U - U @np.transpose(U)))\n", - "# print('test VT')\n", - "# print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))\n", - " print(U)\n", - " print(s)\n", - " print(VT)\n", - "\n", - " D = np.zeros((len(U),len(VT)))\n", - " for i in range(0,len(VT)):\n", - " D[i,i]=s[i]\n", - " UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)\n", - " return np.matmul(V,np.matmul(invD,UT))\n", - "\n", - "\n", - "X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])\n", - "print(X)\n", - "A = np.transpose(X) @ X\n", - "print(A)\n", - "# Brute force inversion of super-collinear matrix\n", - "#B = np.linalg.inv(A)\n", - "#print(B)\n", - "C = SVDinv(A)\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The matrix $\\boldsymbol{X}$ has columns that are linearly dependent. The first\n", - "column is the row-wise sum of the other two columns. The rank of a\n", - "matrix (the column rank) is the dimension of space spanned by the\n", - "column vectors. The rank of the matrix is the number of linearly\n", - "independent columns, in this case just $2$. We see this from the\n", - "singular values when running the above code. Running the standard\n", - "inversion algorithm for matrix inversion with $\\boldsymbol{X}^T\\boldsymbol{X}$ results\n", - "in the program terminating due to a singular matrix.\n", - "\n", - "\n", - "\n", - "\n", - "There are several interesting mathematical properties which will be\n", - "relevant when we are going to discuss the differences between say\n", - "ordinary least squares (OLS) and **Ridge** regression.\n", - "\n", - "We have from OLS that the parameters of the linear approximation are given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The matrix to invert can be rewritten in terms of our SVD decomposition as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{X} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Using the orthogonality properties of $\\boldsymbol{U}$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{X} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{V}^T = \\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{D}$ being a diagonal matrix with values along the diagonal given by the singular values squared. \n", - "\n", - "This means that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{X}^T\\boldsymbol{X})\\boldsymbol{V} = \\boldsymbol{V}\\boldsymbol{D},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "that is the eigenvectors of $(\\boldsymbol{X}^T\\boldsymbol{X})$ are given by the columns of the right singular matrix of $\\boldsymbol{X}$ and the eigenvalues are the squared singular values. It is easy to show (show this) that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U} = \\boldsymbol{U}\\boldsymbol{D},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "that is, the eigenvectors of $(\\boldsymbol{X}\\boldsymbol{X})^T$ are the columns of the left singular matrix and the eigenvalues are the same. \n", - "\n", - "Going back to our OLS equation we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We will come back to this expression when we discuss Ridge regression. \n", - "\n", - "\n", - "$$ \\tilde{y}^{OLS}=\\boldsymbol{X}\\hat{\\beta}^{OLS}=\\sum_{j=1}^p \\boldsymbol{u}_j\\boldsymbol{u}_j^T\\boldsymbol{y}$$ and for Ridge we have \n", - "\n", - "$$ \\tilde{y}^{Ridge}=\\boldsymbol{X}\\hat{\\beta}^{Ridge}=\\sum_{j=1}^p \\boldsymbol{u}_j\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{u}_j^T\\boldsymbol{y}$$ . \n", - "\n", - "It is indeed the economy-sized SVD, note the summation runs up tp $$p$$ only and not $$n$$. \n", - "\n", - "Here we have that $$\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T$$, with $$\\Sigma$$ being an $$ n\\times p$$ matrix and $$\\boldsymbol{V}$$ being a $$ p\\times p$$ matrix. We also have assumed here that $$ n > p$$. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Ridge and LASSO Regression\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember11.mp4?vrtx=view-as-webpage)\n", - "\n", - "Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is \n", - "our optimization problem is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or we can state it as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", - "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have used the definition of a norm-2 vector, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "By minimizing the above equation with respect to the parameters\n", - "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n", - "parameters $\\boldsymbol{\\beta}$. We can add a regularization parameter $\\lambda$ by\n", - "defining a new cost function to be optimized, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", - "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to the Ridge regression minimization problem where we\n", - "require that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n", - "a finite number larger than zero. By defining" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we have a new optimization equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", - "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. \n", - "\n", - "Here we have defined the norm-1 as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\vert\\vert \\boldsymbol{x}\\vert\\vert_1 = \\sum_i \\vert x_i\\vert.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Using the matrix-vector expression for Ridge regression," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\boldsymbol{\\beta}^T\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "by taking the derivatives with respect to $\\boldsymbol{\\beta}$ we obtain then\n", - "a slightly modified matrix inversion problem which for finite values\n", - "of $\\lambda$ does not suffer from singularity problems. We obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $t$ a finite positive number. \n", - "\n", - "We see that Ridge regression is nothing but the standard\n", - "OLS with a modified diagonal term added to $\\boldsymbol{X}^T\\boldsymbol{X}$. The\n", - "consequences, in particular for our discussion of the bias-variance tradeoff \n", - "are rather interesting.\n", - "\n", - "Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U} = \\boldsymbol{U}\\boldsymbol{D}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\\boldsymbol{U}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For Ridge regression this becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$. \n", - "\n", - "\n", - "Since $\\lambda \\geq 0$, it means that compared to OLS, we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda} \\leq 1.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Ridge regression finds the coordinates of $\\boldsymbol{y}$ with respect to the\n", - "orthonormal basis $\\boldsymbol{U}$, it then shrinks the coordinates by\n", - "$\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}$. Recall that the SVD has\n", - "eigenvalues ordered in a descending way, that is $\\sigma_i \\geq\n", - "\\sigma_{i+1}$.\n", - "\n", - "For small eigenvalues $\\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.\n", - "Actually, calculating the variance of $\\boldsymbol{X}\\boldsymbol{v}_j$ shows that this quantity is equal to $\\sigma_j^2/n$.\n", - "With a parameter $\\lambda$ we can thus shrink the role of specific parameters. \n", - "\n", - "\n", - "\n", - "For the sake of simplicity, let us assume that the design matrix is orthonormal, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{X}=(\\boldsymbol{X}^T\\boldsymbol{X})^{-1} =\\boldsymbol{I}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case the standard OLS results in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{OLS}} = \\boldsymbol{X}^T\\boldsymbol{y}=\\sum_{i=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\boldsymbol{y},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{I}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\left(1+\\lambda\\right)^{-1}\\boldsymbol{\\beta}^{\\mathrm{OLS}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\\lambda$, and\n", - "the Ridge estimator converges to zero when the hyperparameter goes to\n", - "infinity.\n", - "\n", - "We will come back to more interpreations after we have gone through some of the statistical analysis part. \n", - "\n", - "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", - "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended.\n", - "\n", - "\n", - "\n", - "## A better understanding of regularization\n", - "\n", - "The parameter $\\lambda$ that we have introduced in the Ridge (and\n", - "Lasso as well) regression is often called a regularization parameter\n", - "or shrinkage parameter. It is common to call it a hyperparameter. What does it mean mathemtically?\n", - "\n", - "Here we will first look at how to analyze the difference between the\n", - "standard OLS equations and the Ridge expressions in terms of a linear\n", - "algebra analysis using the SVD algorithm. Thereafter, we will link\n", - "(see the material on the bias-variance tradeoff below) these\n", - "observation to the statisical analysis of the results. In particular\n", - "we consider how the variance of the parameters $\\boldsymbol{\\beta}$ is\n", - "affected by changing the parameter $\\lambda$.\n", - "\n", - "\n", - "We have our design matrix\n", - " $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$. With the SVD we decompose it as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\boldsymbol{U\\Sigma V^T},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{U}\\in {\\mathbb{R}}^{n\\times n}$, $\\boldsymbol{\\Sigma}\\in {\\mathbb{R}}^{n\\times p}$\n", - "and $\\boldsymbol{V}\\in {\\mathbb{R}}^{p\\times p}$.\n", - "\n", - "The matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\\boldsymbol{U}^T\\boldsymbol{U}=\\boldsymbol{U}\\boldsymbol{U}^T=\\boldsymbol{I}$ and $\\boldsymbol{V}^T\\boldsymbol{V}=\\boldsymbol{V}\\boldsymbol{V}^T=\\boldsymbol{I}$.\n", - "\n", - "\n", - "\n", - "## Introducing the Covariance and Correlation functions\n", - "\n", - "Before we discuss the link between for example Ridge regression and the singular value decomposition, we need to remind ourselves about\n", - "the definition of the covariance and the correlation function. These are quantities \n", - "\n", - "Suppose we have defined two vectors\n", - "$\\hat{x}$ and $\\hat{y}$ with $n$ elements each. The covariance matrix $\\boldsymbol{C}$ is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{y}] \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where for example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With this definition and recalling that the variance is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{var}[\\boldsymbol{x}]=\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we can rewrite the covariance matrix as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] & \\mathrm{var}[\\boldsymbol{y}] \\\\\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The covariance takes values between zero and infinity and may thus\n", - "lead to problems with loss of numerical precision for particularly\n", - "large values. It is common to scale the covariance matrix by\n", - "introducing instead the correlation matrix defined via the so-called\n", - "correlation function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]=\\frac{\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}]}{\\sqrt{\\mathrm{var}[\\boldsymbol{x}] \\mathrm{var}[\\boldsymbol{y}]}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The correlation function is then given by values $\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]\n", - "\\in [-1,1]$. This avoids eventual problems with too large values. We\n", - "can then define the correlation matrix for the two vectors $\\boldsymbol{x}$\n", - "and $\\boldsymbol{y}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{K}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} 1 & \\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{corr}[\\boldsymbol{y},\\boldsymbol{x}] & 1 \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the above example this is the function we constructed using **pandas**.\n", - "\n", - "\n", - "\n", - "In our derivation of the various regression algorithms like **Ordinary Least Squares** or **Ridge regression**\n", - "we defined the design/feature matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n", - "x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n", - "x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n", - "\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n", - "x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n", - "x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$ refering to the column numbers and the\n", - "entries $n$ being the row elements.\n", - "We can rewrite the design/feature matrix in terms of its column vectors as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix} \\boldsymbol{x}_0 & \\boldsymbol{x}_1 & \\boldsymbol{x}_2 & \\dots & \\dots & \\boldsymbol{x}_{p-1}\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with a given vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i^T = \\begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \\dots & \\dots x_{n-1,i}\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With these definitions, we can now rewrite our $2\\times 2$\n", - "correaltion/covariance matrix in terms of a moe general design/feature\n", - "matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$. This leads to a $p\\times p$\n", - "covariance matrix for the vectors $\\boldsymbol{x}_i$ with $i=0,1,\\dots,p-1$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\begin{bmatrix}\n", - "\\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & \\mathrm{var}[\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & \\mathrm{var}[\\boldsymbol{x}_{p-1}]\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the correlation matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{K}[\\boldsymbol{x}] = \\begin{bmatrix}\n", - "1 & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & 1 & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & 1 & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & 1\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The Numpy function **np.cov** calculates the covariance elements using\n", - "the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have\n", - "the exact mean values. The following simple function uses the\n", - "**np.vstack** function which takes each vector of dimension $1\\times n$\n", - "and produces a $2\\times n$ matrix $\\boldsymbol{W}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{W} = \\begin{bmatrix} x_0 & y_0 \\\\\n", - " x_1 & y_1 \\\\\n", - " x_2 & y_2\\\\\n", - " \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2}\\\\\n", - " x_{n-1} & y_{n-1} & \n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in turn is converted into into the $2\\times 2$ covariance matrix\n", - "$\\boldsymbol{C}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", - "the mean value of each set of samples $\\boldsymbol{x}$ etc using the Numpy\n", - "function **np.mean(x)**. We can also extract the eigenvalues of the\n", - "covariance matrix through the **np.linalg.eig()** function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "W = np.vstack((x, y))\n", - "C = np.cov(W)\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The previous example can be converted into the correlation matrix by\n", - "simply scaling the matrix elements with the variances. We should also\n", - "subtract the mean values for each column. This leads to the following\n", - "code which sets up the correlations matrix for the previous example in\n", - "a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\\times 2$ correlation matrix (since we have only two vectors)." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 100\n", - "# define two vectors \n", - "x = np.random.random(size=n)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "#scaling the x and y vectors \n", - "x = x - np.mean(x)\n", - "y = y - np.mean(y)\n", - "variance_x = np.sum(x@x)/n\n", - "variance_y = np.sum(y@y)/n\n", - "print(variance_x)\n", - "print(variance_y)\n", - "cov_xy = np.sum(x@y)/n\n", - "cov_xx = np.sum(x@x)/n\n", - "cov_yy = np.sum(y@y)/n\n", - "C = np.zeros((2,2))\n", - "C[0,0]= cov_xx/variance_x\n", - "C[1,1]= cov_yy/variance_y\n", - "C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)\n", - "C[1,0]= C[0,1]\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see that the matrix elements along the diagonal are one as they\n", - "should be and that the matrix is symmetric. Furthermore, diagonalizing\n", - "this matrix we easily see that it is a positive definite matrix.\n", - "\n", - "The above procedure with **numpy** can be made more compact if we use **pandas**.\n", - "\n", - "\n", - "We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import pandas as pd\n", - "n = 10\n", - "x = np.random.normal(size=n)\n", - "x = x - np.mean(x)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "y = y - np.mean(y)\n", - "X = (np.vstack((x, y))).T\n", - "print(X)\n", - "Xpd = pd.DataFrame(X)\n", - "print(Xpd)\n", - "correlation_matrix = Xpd.corr()\n", - "print(correlation_matrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We expand this model to the Franke function discussed above." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "\n", - "\n", - "def FrankeFunction(x,y):\n", - "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", - "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", - "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", - "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", - "\treturn term1 + term2 + term3 + term4\n", - "\n", - "\n", - "def create_X(x, y, n ):\n", - "\tif len(x.shape) > 1:\n", - "\t\tx = np.ravel(x)\n", - "\t\ty = np.ravel(y)\n", - "\n", - "\tN = len(x)\n", - "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", - "\tX = np.ones((N,l))\n", - "\n", - "\tfor i in range(1,n+1):\n", - "\t\tq = int((i)*(i+1)/2)\n", - "\t\tfor k in range(i+1):\n", - "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", - "\n", - "\treturn X\n", - "\n", - "\n", - "# Making meshgrid of datapoints and compute Franke's function\n", - "n = 4\n", - "N = 100\n", - "x = np.sort(np.random.uniform(0, 1, N))\n", - "y = np.sort(np.random.uniform(0, 1, N))\n", - "z = FrankeFunction(x, y)\n", - "X = create_X(x, y, n=n) \n", - "\n", - "Xpd = pd.DataFrame(X)\n", - "# subtract the mean values and set up the covariance matrix\n", - "Xpd = Xpd - Xpd.mean()\n", - "covariance_matrix = Xpd.cov()\n", - "print(covariance_matrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note here that the covariance is zero for the first rows and\n", - "columns since all matrix elements in the design matrix were set to one\n", - "(we are fitting the function in terms of a polynomial of degree $n$).\n", - "\n", - "This means that the variance for these elements will be zero and will\n", - "cause problems when we set up the correlation matrix. We can simply\n", - "drop these elements and construct a correlation\n", - "matrix without these elements. \n", - "\n", - "\n", - "\n", - "\n", - "We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}= \\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}].\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To see this let us simply look at a design matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{2\\times 2}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{00} & x_{01}\\\\\n", - "x_{10} & x_{11}\\\\\n", - "\\end{bmatrix}=\\begin{bmatrix}\n", - "\\boldsymbol{x}_{0} & \\boldsymbol{x}_{1}\\\\\n", - "\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we then compute the expectation value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\\\\n", - "x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is just" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]=\\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we wrote $$\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]$$ to indicate that this the covariance of the vectors $\\boldsymbol{x}$ of the design/feature matrix $\\boldsymbol{X}$.\n", - "\n", - "It is easy to generalize this to a matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$.\n", - "\n", - "\n", - "## Linking with SVD" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter4.ipynb b/doc/LectureNotes/_build/html/_sources/chapter4.ipynb deleted file mode 100644 index 80fb450d4..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter4.ipynb +++ /dev/null @@ -1,2875 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Logistic Regression\n", - "\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureSeptember18.mp4?vrtx=view-as-webpage)\n", - "\n", - "\n", - "## Logistic Regression\n", - "\n", - "In linear regression our main interest was centered on learning the\n", - "coefficients of a functional fit (say a polynomial) in order to be\n", - "able to predict the response of a continuous variable on some unseen\n", - "data. The fit to the continuous variable $y_i$ is based on some\n", - "independent variables $\\hat{x}_i$. Linear regression resulted in\n", - "analytical expressions for standard ordinary Least Squares or Ridge\n", - "regression (in terms of matrices to invert) for several quantities,\n", - "ranging from the variance and thereby the confidence intervals of the\n", - "parameters $\\hat{\\beta}$ to the mean squared error. If we can invert\n", - "the product of the design matrices, linear regression gives then a\n", - "simple recipe for fitting our data.\n", - "\n", - "\n", - "Classification problems, however, are concerned with outcomes taking\n", - "the form of discrete variables (i.e. categories). We may for example,\n", - "on the basis of DNA sequencing for a number of patients, like to find\n", - "out which mutations are important for a certain disease; or based on\n", - "scans of various patients' brains, figure out if there is a tumor or\n", - "not; or given a specific physical system, we'd like to identify its\n", - "state, say whether it is an ordered or disordered system (typical\n", - "situation in solid state physics); or classify the status of a\n", - "patient, whether she/he has a stroke or not and many other similar\n", - "situations.\n", - "\n", - "The most common situation we encounter when we apply logistic\n", - "regression is that of two possible outcomes, normally denoted as a\n", - "binary outcome, true or false, positive or negative, success or\n", - "failure etc.\n", - "\n", - "\n", - "Logistic regression will also serve as our stepping stone towards\n", - "neural network algorithms and supervised deep learning. For logistic\n", - "learning, the minimization of the cost function leads to a non-linear\n", - "equation in the parameters $\\hat{\\beta}$. The optimization of the\n", - "problem calls therefore for minimization algorithms. This forms the\n", - "bottle neck of all machine learning algorithms, namely how to find\n", - "reliable minima of a multi-variable function. This leads us to the\n", - "family of gradient descent methods. The latter are the working horses\n", - "of basically all modern machine learning algorithms.\n", - "\n", - "We note also that many of the topics discussed here on logistic \n", - "regression are also commonly used in modern supervised Deep Learning\n", - "models, as we will see later.\n", - "\n", - "\n", - "\n", - "## Basics\n", - "\n", - "We consider the case where the dependent variables, also called the\n", - "responses or the outcomes, $y_i$ are discrete and only take values\n", - "from $k=0,\\dots,K-1$ (i.e. $K$ classes).\n", - "\n", - "The goal is to predict the\n", - "output classes from the design matrix $\\hat{X}\\in\\mathbb{R}^{n\\times p}$\n", - "made of $n$ samples, each of which carries $p$ features or predictors. The\n", - "primary goal is to identify the classes to which new unseen samples\n", - "belong.\n", - "\n", - "Let us specialize to the case of two classes only, with outputs\n", - "$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a\n", - "credit card user that could default or not on her/his credit card\n", - "debt. That is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\ 1 & \\mathrm{yes} \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Before moving to the logistic model, let us try to use our linear\n", - "regression model to classify these two outcomes. We could for example\n", - "fit a linear model to the default case if $y_i > 0.5$ and the no\n", - "default case $y_i \\leq 0.5$.\n", - "\n", - "We would then have our \n", - "weighted linear combination, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\hat{y} = \\hat{X}^T\\hat{\\beta} + \\hat{\\epsilon},\n", - "\\label{_auto1} \\tag{1}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\hat{y}$ is a vector representing the possible outcomes, $\\hat{X}$ is our\n", - "$n\\times p$ design matrix and $\\hat{\\beta}$ represents our estimators/predictors.\n", - "\n", - "\n", - "The main problem with our function is that it takes values on the\n", - "entire real axis. In the case of logistic regression, however, the\n", - "labels $y_i$ are discrete variables. A typical example is the credit\n", - "card data discussed below here, where we can set the state of\n", - "defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons\n", - "in the data set (see the full example below).\n", - "\n", - "One simple way to get a discrete output is to have sign\n", - "functions that map the output of a linear regressor to values $\\{0,1\\}$,\n", - "$f(s_i)=sign(s_i)=1$ if $s_i\\ge 0$ and 0 if otherwise. \n", - "We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron\" model in the machine learning\n", - "literature. This model is extremely simple. However, in many cases it is more\n", - "favorable to use a ``soft\" classifier that outputs\n", - "the probability of a given category. This leads us to the logistic function.\n", - "\n", - "\n", - "The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.utils import resample\n", - "from sklearn.metrics import mean_squared_error\n", - "from IPython.display import display\n", - "from pylab import plt, mpl\n", - "plt.style.use('seaborn')\n", - "mpl.rcParams['font.family'] = 'serif'\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"chddata.csv\"),'r')\n", - "\n", - "# Read the chd data as csv file and organize the data into arrays with age group, age, and chd\n", - "chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))\n", - "chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']\n", - "output = chd['CHD']\n", - "age = chd['Age']\n", - "agegroup = chd['Agegroup']\n", - "numberID = chd['ID'] \n", - "display(chd)\n", - "\n", - "plt.scatter(age, output, marker='o')\n", - "plt.axis([18,70.0,-0.1, 1.2])\n", - "plt.xlabel(r'Age')\n", - "plt.ylabel(r'CHD')\n", - "plt.title(r'Age distribution and Coronary heart disease')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "What we could attempt however is to plot the mean value for each group." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])\n", - "group = np.array([1, 2, 3, 4, 5, 6, 7, 8])\n", - "plt.plot(group, agegroupmean, \"r-\")\n", - "plt.axis([0,9,0, 1.0])\n", - "plt.xlabel(r'Age group')\n", - "plt.ylabel(r'CHD mean values')\n", - "plt.title(r'Mean values for each age group')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We are now trying to find a function $f(y\\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$.\n", - "In standard linear regression with a linear dependence on $x$, we would write this in terms of our model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(y_i\\vert x_i)=\\beta_0+\\beta_1 x_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This expression implies however that $f(y_i\\vert x_i)$ could take any\n", - "value from minus infinity to plus infinity. If we however let\n", - "$f(y\\vert y)$ be represented by the mean value, the above example\n", - "shows us that we can constrain the function to take values between\n", - "zero and one, that is we have $0 \\le f(y_i\\vert x_i) \\le 1$. Looking\n", - "at our last curve we see also that it has an S-shaped form. This leads\n", - "us to a very popular model for the function $f$, namely the so-called\n", - "Sigmoid function or logistic model. We will consider this function as\n", - "representing the probability for finding a value of $y_i$ with a given\n", - "$x_i$.\n", - "\n", - "\n", - "## The logistic function\n", - "\n", - "Another widely studied model, is the so-called \n", - "perceptron model, which is an example of a \"hard classification\" model. We\n", - "will encounter this model when we discuss neural networks as\n", - "well. Each datapoint is deterministically assigned to a category (i.e\n", - "$y_i=0$ or $y_i=1$). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a \"soft\"\n", - "classifier that outputs the probability of a given category rather\n", - "than a single value. For example, given $x_i$, the classifier\n", - "outputs the probability of being in a category $k$. Logistic regression\n", - "is the most common example of a so-called soft classifier. In logistic\n", - "regression, the probability that a data point $x_i$\n", - "belongs to a category $y_i=\\{0,1\\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that $1-p(t)= p(-t)$.\n", - "\n", - "## Examples of likelihood functions used in logistic regression and nueral networks\n", - "\n", - "\n", - "The following code plots the logistic function, the step function and other functions we will encounter from here and on." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\"\"\"The sigmoid function (or the logistic curve) is a\n", - "function that takes any real number, z, and outputs a number (0,1).\n", - "It is useful in neural networks for assigning weights on a relative scale.\n", - "The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n", - "\n", - "import numpy\n", - "import matplotlib.pyplot as plt\n", - "import math as mt\n", - "\n", - "z = numpy.arange(-5, 5, .1)\n", - "sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n", - "sigma = sigma_fn(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, sigma)\n", - "ax.set_ylim([-0.1, 1.1])\n", - "ax.set_xlim([-5,5])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('sigmoid function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"Step Function\"\"\"\n", - "z = numpy.arange(-5, 5, .02)\n", - "step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n", - "step = step_fn(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, step)\n", - "ax.set_ylim([-0.5, 1.5])\n", - "ax.set_xlim([-5,5])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('step function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"tanh Function\"\"\"\n", - "z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n", - "t = numpy.tanh(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, t)\n", - "ax.set_ylim([-1.0, 1.0])\n", - "ax.set_xlim([-2*mt.pi,2*mt.pi])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('tanh function')\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\\beta$ in our fitting of the Sigmoid function, that is we define probabilities" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "p(y_i=1|x_i,\\hat{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n", - "p(y_i=0|x_i,\\hat{\\beta}) &= 1 - p(y_i=1|x_i,\\hat{\\beta}),\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\hat{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", - "\n", - "Note that we used" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(y_i=0\\vert x_i, \\hat{\\beta}) = 1-p(y_i=1\\vert x_i, \\hat{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to define the total likelihood for all possible outcomes from a \n", - "dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n", - "$y_i\\in\\{0,1\\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. \n", - "We aim thus at maximizing \n", - "the probability of seeing the observed data. We can then approximate the \n", - "likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "P(\\mathcal{D}|\\hat{\\beta})& = \\prod_{i=1}^n \\left[p(y_i=1|x_i,\\hat{\\beta})\\right]^{y_i}\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]^{1-y_i}\\nonumber \\\\\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "from which we obtain the log-likelihood and our **cost/loss** function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\hat{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Reordering the logarithms, we can rewrite the **cost/loss** function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\\beta$.\n", - "Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n", - "in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n", - "\n", - "\n", - "The cross entropy is a convex function of the weights $\\hat{\\beta}$ and,\n", - "therefore, any local minimizer is a global minimizer. \n", - "\n", - "\n", - "Minimizing this\n", - "cost function with respect to the two parameters $\\beta_0$ and $\\beta_1$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_1} = -\\sum_{i=1}^n \\left(y_ix_i -x_i\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us now define a vector $\\hat{y}$ with $n$ elements $y_i$, an\n", - "$n\\times p$ matrix $\\hat{X}$ which contains the $x_i$ values and a\n", - "vector $\\hat{p}$ of fitted probabilities $p(y_i\\vert x_i,\\hat{\\beta})$. We can rewrite in a more compact form the first\n", - "derivative of cost function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}} = -\\hat{X}^T\\left(\\hat{y}-\\hat{p}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we in addition define a diagonal matrix $\\hat{W}$ with elements \n", - "$p(y_i\\vert x_i,\\hat{\\beta})(1-p(y_i\\vert x_i,\\hat{\\beta})$, we can obtain a compact expression of the second derivative as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T} = \\hat{X}^T\\hat{W}\\hat{X}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\log{ \\frac{p(\\hat{\\beta}\\hat{x})}{1-p(\\hat{\\beta}\\hat{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we defined $\\hat{x}=[1,x_1,x_2,\\dots,x_p]$ and $\\hat{\\beta}=[\\beta_0, \\beta_1, \\dots, \\beta_p]$ leading to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(\\hat{\\beta}\\hat{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Till now we have mainly focused on two classes, the so-called binary\n", - "system. Suppose we wish to extend to $K$ classes. Let us for the sake\n", - "of simplicity assume we have only two predictors. We have then following model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\log{\\frac{p(C=1\\vert x)}{p(K\\vert x)}} = \\beta_{10}+\\beta_{11}x_1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and so on till the class $C=K-1$ class" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the model is specified in term of $K-1$ so-called log-odds or\n", - "**logit** transformations.\n", - "\n", - "\n", - "\n", - "In our discussion of neural networks we will encounter the above again\n", - "in terms of a slightly modified function, the so-called **Softmax** function.\n", - "\n", - "The softmax function is used in various multiclass classification\n", - "methods, such as multinomial logistic regression (also known as\n", - "softmax regression), multiclass linear discriminant analysis, naive\n", - "Bayes classifiers, and artificial neural networks. Specifically, in\n", - "multinomial logistic regression and linear discriminant analysis, the\n", - "input to the function is the result of $K$ distinct linear functions,\n", - "and the predicted probability for the $k$-th class given a sample\n", - "vector $\\hat{x}$ and a weighting vector $\\hat{\\beta}$ is (with two\n", - "predictors):" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is easy to extend to more predictors. The final class is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and they sum to one. Our earlier discussions were all specialized to\n", - "the case with two classes only. It is easy to see from the above that\n", - "what we derived earlier is compatible with these equations.\n", - "\n", - "To find the optimal parameters we would typically use a gradient\n", - "descent method. Newton's method and gradient descent methods are\n", - "discussed in the material on [optimization\n", - "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", - "\n", - "## Wisconsin Cancer Data\n", - "\n", - "We show here how we can use a simple regression case on the breast\n", - "cancer data using Logistic regression as our algorithm for\n", - "classification." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.linear_model import LogisticRegression\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "# Logistic Regression\n", - "logreg = LogisticRegression(solver='lbfgs')\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Logistic Regression\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In addition to the above scores, we could also study the covariance (and the correlation matrix).\n", - "We use **Pandas** to compute the correlation matrix." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.linear_model import LogisticRegression\n", - "cancer = load_breast_cancer()\n", - "import pandas as pd\n", - "# Making a data frame\n", - "cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)\n", - "\n", - "fig, axes = plt.subplots(15,2,figsize=(10,20))\n", - "malignant = cancer.data[cancer.target == 0]\n", - "benign = cancer.data[cancer.target == 1]\n", - "ax = axes.ravel()\n", - "\n", - "for i in range(30):\n", - " _, bins = np.histogram(cancer.data[:,i], bins =50)\n", - " ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5)\n", - " ax[i].hist(benign[:,i], bins = bins, alpha = 0.5)\n", - " ax[i].set_title(cancer.feature_names[i])\n", - " ax[i].set_yticks(())\n", - "ax[0].set_xlabel(\"Feature magnitude\")\n", - "ax[0].set_ylabel(\"Frequency\")\n", - "ax[0].legend([\"Malignant\", \"Benign\"], loc =\"best\")\n", - "fig.tight_layout()\n", - "plt.show()\n", - "\n", - "import seaborn as sns\n", - "correlation_matrix = cancerpd.corr().round(1)\n", - "# use the heatmap function from seaborn to plot the correlation matrix\n", - "# annot = True to print the values inside the square\n", - "plt.figure(figsize=(15,8))\n", - "sns.heatmap(data=correlation_matrix, annot=True)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the above example we note two things. In the first plot we display\n", - "the overlap of benign and malignant tumors as functions of the various\n", - "features in the Wisconsing breast cancer data set. We see that for\n", - "some of the features we can distinguish clearly the benign and\n", - "malignant cases while for other features we cannot. This can point to\n", - "us which features may be of greater interest when we wish to classify\n", - "a benign or not benign tumour.\n", - "\n", - "In the second figure we have computed the so-called correlation\n", - "matrix, which in our case with thirty features becomes a $30\\times 30$\n", - "matrix.\n", - "\n", - "We constructed this matrix using **pandas** via the statements" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and then" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "correlation_matrix = cancerpd.corr().round(1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Diagonalizing this matrix we can in turn say something about which\n", - "features are of relevance and which are not. This leads us to\n", - "the classical Principal Component Analysis (PCA) theorem with\n", - "applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html))." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.linear_model import LogisticRegression\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "# Logistic Regression\n", - "logreg = LogisticRegression(solver='lbfgs')\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Logistic Regression\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "\n", - "\n", - "from sklearn.preprocessing import LabelEncoder\n", - "from sklearn.model_selection import cross_validate\n", - "#Cross validation\n", - "accuracy = cross_validate(logreg,X_test_scaled,y_test,cv=10)['test_score']\n", - "print(accuracy)\n", - "print(\"Test set accuracy with Logistic Regression and scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "\n", - "\n", - "import scikitplot as skplt\n", - "y_pred = logreg.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "plt.show()\n", - "y_probas = logreg.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Optimization, the central part of any Machine Learning algortithm\n", - "\n", - "Almost every problem in machine learning and data science starts with\n", - "a dataset $X$, a model $g(\\beta)$, which is a function of the\n", - "parameters $\\beta$ and a cost function $C(X, g(\\beta))$ that allows\n", - "us to judge how well the model $g(\\beta)$ explains the observations\n", - "$X$. The model is fit by finding the values of $\\beta$ that minimize\n", - "the cost function. Ideally we would be able to solve for $\\beta$\n", - "analytically, however this is not possible in general and we must use\n", - "some approximative/numerical method to compute the minimum.\n", - "\n", - "\n", - "\n", - "## Revisiting our Logistic Regression case\n", - "\n", - "In our discussion on Logistic Regression we studied the \n", - "case of\n", - "two classes, with $y_i$ either\n", - "$0$ or $1$. Furthermore we assumed also that we have only two\n", - "parameters $\\beta$ in our fitting, that is we\n", - "defined probabilities" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "p(y_i=1|x_i,\\boldsymbol{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n", - "p(y_i=0|x_i,\\boldsymbol{\\beta}) &= 1 - p(y_i=1|x_i,\\boldsymbol{\\beta}),\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\boldsymbol{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", - "\n", - "\n", - "## The equations to solve\n", - "\n", - "Our compact equations used a definition of a vector $\\boldsymbol{y}$ with $n$\n", - "elements $y_i$, an $n\\times p$ matrix $\\boldsymbol{X}$ which contains the\n", - "$x_i$ values and a vector $\\boldsymbol{p}$ of fitted probabilities\n", - "$p(y_i\\vert x_i,\\boldsymbol{\\beta})$. We rewrote in a more compact form\n", - "the first derivative of the cost function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = -\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{p}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we in addition define a diagonal matrix $\\boldsymbol{W}$ with elements \n", - "$p(y_i\\vert x_i,\\boldsymbol{\\beta})(1-p(y_i\\vert x_i,\\boldsymbol{\\beta})$, we can obtain a compact expression of the second derivative as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T} = \\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This defines what is called the Hessian matrix.\n", - "\n", - "\n", - "## Solving using Newton-Raphson's method\n", - "\n", - "If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. \n", - "\n", - "Our iterative scheme is then given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{new}} = \\boldsymbol{\\beta}^{\\mathrm{old}}-\\left(\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T}\\right)^{-1}_{\\boldsymbol{\\beta}^{\\mathrm{old}}}\\times \\left(\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}\\right)_{\\boldsymbol{\\beta}^{\\mathrm{old}}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or in matrix form as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{new}} = \\boldsymbol{\\beta}^{\\mathrm{old}}-\\left(\\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X} \\right)^{-1}\\times \\left(-\\boldsymbol{X}^T(\\boldsymbol{y}-\\boldsymbol{p}) \\right)_{\\boldsymbol{\\beta}^{\\mathrm{old}}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The right-hand side is computed with the old values of $\\beta$. \n", - "\n", - "If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. \n", - "\n", - "\n", - "\n", - "## Brief reminder on Newton-Raphson's method\n", - "\n", - "Let us quickly remind ourselves how we derive the above method.\n", - "\n", - "Perhaps the most celebrated of all one-dimensional root-finding\n", - "routines is Newton's method, also called the Newton-Raphson\n", - "method. This method requires the evaluation of both the\n", - "function $f$ and its derivative $f'$ at arbitrary points. \n", - "If you can only calculate the derivative\n", - "numerically and/or your function is not of the smooth type, we\n", - "normally discourage the use of this method.\n", - "\n", - "\n", - "## The equations\n", - "\n", - "The Newton-Raphson formula consists geometrically of extending the\n", - "tangent line at a current point until it crosses zero, then setting\n", - "the next guess to the abscissa of that zero-crossing. The mathematics\n", - "behind this method is rather simple. Employing a Taylor expansion for\n", - "$x$ sufficiently close to the solution $s$, we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "f(s)=0=f(x)+(s-x)f'(x)+\\frac{(s-x)^2}{2}f''(x) +\\dots.\n", - " \\label{eq:taylornr} \\tag{2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For small enough values of the function and for well-behaved\n", - "functions, the terms beyond linear are unimportant, hence we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(x)+(s-x)f'(x)\\approx 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "yielding" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "s\\approx x-\\frac{f(x)}{f'(x)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Having in mind an iterative procedure, it is natural to start iterating with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Simple geometric interpretation\n", - "\n", - "The above is Newton-Raphson's method. It has a simple geometric\n", - "interpretation, namely $x_{n+1}$ is the point where the tangent from\n", - "$(x_n,f(x_n))$ crosses the $x$-axis. Close to the solution,\n", - "Newton-Raphson converges fast to the desired result. However, if we\n", - "are far from a root, where the higher-order terms in the series are\n", - "important, the Newton-Raphson formula can give grossly inaccurate\n", - "results. For instance, the initial guess for the root might be so far\n", - "from the true root as to let the search interval include a local\n", - "maximum or minimum of the function. If an iteration places a trial\n", - "guess near such a local extremum, so that the first derivative nearly\n", - "vanishes, then Newton-Raphson may fail totally\n", - "\n", - "\n", - "\n", - "## Extending to more than one variable\n", - "\n", - "Newton's method can be generalized to systems of several non-linear equations\n", - "and variables. Consider the case with two equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{array}{cc} f_1(x_1,x_2) &=0\\\\\n", - " f_2(x_1,x_2) &=0,\\end{array}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which we Taylor expand to obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1\n", - " \\partial f_1/\\partial x_1+h_2\n", - " \\partial f_1/\\partial x_2+\\dots\\\\\n", - " 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1\n", - " \\partial f_2/\\partial x_1+h_2\n", - " \\partial f_2/\\partial x_2+\\dots\n", - " \\end{array}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Defining the Jacobian matrix ${\\bf \\boldsymbol{J}}$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\bf \\boldsymbol{J}}=\\left( \\begin{array}{cc}\n", - " \\partial f_1/\\partial x_1 & \\partial f_1/\\partial x_2 \\\\\n", - " \\partial f_2/\\partial x_1 &\\partial f_2/\\partial x_2\n", - " \\end{array} \\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we can rephrase Newton's method as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\left(\\begin{array}{c} x_1^{n+1} \\\\ x_2^{n+1} \\end{array} \\right)=\n", - "\\left(\\begin{array}{c} x_1^{n} \\\\ x_2^{n} \\end{array} \\right)+\n", - "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right)=\n", - " -{\\bf \\boldsymbol{J}}^{-1}\n", - " \\left(\\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\\\ f_2(x_1^{n},x_2^{n}) \\end{array} \\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We need thus to compute the inverse of the Jacobian matrix and it\n", - "is to understand that difficulties may\n", - "arise in case ${\\bf \\boldsymbol{J}}$ is nearly singular.\n", - "\n", - "It is rather straightforward to extend the above scheme to systems of\n", - "more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. \n", - "\n", - "\n", - "\n", - "\n", - "## Steepest descent\n", - "\n", - "The basic idea of gradient descent is\n", - "that a function $F(\\mathbf{x})$, \n", - "$\\mathbf{x} \\equiv (x_1,\\cdots,x_n)$, decreases fastest if one goes from $\\bf {x}$ in the\n", - "direction of the negative gradient $-\\nabla F(\\mathbf{x})$.\n", - "\n", - "It can be shown that if" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\gamma_k > 0$.\n", - "\n", - "For $\\gamma_k$ small enough, then $F(\\mathbf{x}_{k+1}) \\leq\n", - "F(\\mathbf{x}_k)$. This means that for a sufficiently small $\\gamma_k$\n", - "we are always moving towards smaller function values, i.e a minimum.\n", - "\n", - "\n", - "## More on Steepest descent\n", - "\n", - "The previous observation is the basis of the method of steepest\n", - "descent, which is also referred to as just gradient descent (GD). One\n", - "starts with an initial guess $\\mathbf{x}_0$ for a minimum of $F$ and\n", - "computes new approximations according to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k), \\ \\ k \\geq 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The parameter $\\gamma_k$ is often referred to as the step length or\n", - "the learning rate within the context of Machine Learning.\n", - "\n", - "\n", - "## The ideal\n", - "\n", - "Ideally the sequence $\\{\\mathbf{x}_k \\}_{k=0}$ converges to a global\n", - "minimum of the function $F$. In general we do not know if we are in a\n", - "global or local minimum. In the special case when $F$ is a convex\n", - "function, all local minima are also global minima, so in this case\n", - "gradient descent can converge to the global solution. The advantage of\n", - "this scheme is that it is conceptually simple and straightforward to\n", - "implement. However the method in this form has some severe\n", - "limitations:\n", - "\n", - "In machine learing we are often faced with non-convex high dimensional\n", - "cost functions with many local minima. Since GD is deterministic we\n", - "will get stuck in a local minimum, if the method converges, unless we\n", - "have a very good intial guess. This also implies that the scheme is\n", - "sensitive to the chosen initial condition.\n", - "\n", - "Note that the gradient is a function of $\\mathbf{x} =\n", - "(x_1,\\cdots,x_n)$ which makes it expensive to compute numerically.\n", - "\n", - "\n", - "\n", - "## The sensitiveness of the gradient descent\n", - "\n", - "The gradient descent method \n", - "is sensitive to the choice of learning rate $\\gamma_k$. This is due\n", - "to the fact that we are only guaranteed that $F(\\mathbf{x}_{k+1}) \\leq\n", - "F(\\mathbf{x}_k)$ for sufficiently small $\\gamma_k$. The problem is to\n", - "determine an optimal learning rate. If the learning rate is chosen too\n", - "small the method will take a long time to converge and if it is too\n", - "large we can experience erratic behavior.\n", - "\n", - "Many of these shortcomings can be alleviated by introducing\n", - "randomness. One such method is that of Stochastic Gradient Descent\n", - "(SGD), see below.\n", - "\n", - "\n", - "\n", - "## Convex functions\n", - "\n", - "Ideally we want our cost/loss function to be convex(concave).\n", - "\n", - "First we give the definition of a convex set: A set $C$ in\n", - "$\\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and\n", - "all $t \\in (0,1)$ , the point $(1 − t)x + ty$ also belongs to\n", - "C. Geometrically this means that every point on the line segment\n", - "connecting $x$ and $y$ is in $C$ as discussed below.\n", - "\n", - "The convex subsets of $\\mathbb{R}$ are the intervals of\n", - "$\\mathbb{R}$. Examples of convex sets of $\\mathbb{R}^2$ are the\n", - "regular polygons (triangles, rectangles, pentagons, etc...).\n", - "\n", - "\n", - "## Convex function\n", - "\n", - "**Convex function**: Let $X \\subset \\mathbb{R}^n$ be a convex set. Assume that the function $f: X \\rightarrow \\mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \\leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \\in X$ and for all $t \\in [0,1]$. If $\\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \\neq x_2$ and $t\\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below.\n", - "\n", - "\n", - "## Conditions on convex functions\n", - "\n", - "In the following we state first and second-order conditions which\n", - "ensures convexity of a function $f$. We write $D_f$ to denote the\n", - "domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more\n", - "details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/, 2004).\n", - "\n", - "**First order condition.**\n", - "\n", - "Suppose $f$ is differentiable (i.e $\\nabla f(x)$ is well defined for\n", - "all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$\n", - "is a convex set and $$f(y) \\geq f(x) + \\nabla f(x)^T (y-x) $$ holds\n", - "for all $x,y \\in D_f$. This condition means that for a convex function\n", - "the first order Taylor expansion (right hand side above) at any point\n", - "a global under estimator of the function. To convince yourself you can\n", - "make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and\n", - "note that it is always below the graph.\n", - "\n", - "\n", - "\n", - "**Second order condition.**\n", - "\n", - "Assume that $f$ is twice\n", - "differentiable, i.e the Hessian matrix exists at each point in\n", - "$D_f$. Then $f$ is convex if and only if $D_f$ is a convex set and its\n", - "Hessian is positive semi-definite for all $x\\in D_f$. For a\n", - "single-variable function this reduces to $f''(x) \\geq 0$. Geometrically this means that $f$ has nonnegative curvature\n", - "everywhere.\n", - "\n", - "\n", - "\n", - "This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.\n", - "\n", - "\n", - "## More on convex functions\n", - "\n", - "The next result is of great importance to us and the reason why we are\n", - "going on about convex functions. In machine learning we frequently\n", - "have to minimize a loss/cost function in order to find the best\n", - "parameters for the model we are considering. \n", - "\n", - "Ideally we want the\n", - "global minimum (for high-dimensional models it is hard to know\n", - "if we have local or global minimum). However, if the cost/loss function\n", - "is convex the following result provides invaluable information:\n", - "\n", - "**Any minimum is global for convex functions.**\n", - "\n", - "Consider the problem of finding $x \\in \\mathbb{R}^n$ such that $f(x)$\n", - "is minimal, where $f$ is convex and differentiable. Then, any point\n", - "$x^*$ that satisfies $\\nabla f(x^*) = 0$ is a global minimum.\n", - "\n", - "\n", - "\n", - "This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.\n", - "\n", - "\n", - "## Some simple problems\n", - "\n", - "1. Show that $f(x)=x^2$ is convex for $x \\in \\mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \\in D_f$ and any $\\lambda \\in [0,1]$ $\\lambda f(x)+(1-\\lambda)f(y)-f(\\lambda x + (1-\\lambda) y ) \\geq 0$.\n", - "\n", - "2. Using the second order condition show that the following functions are convex on the specified domain.\n", - "\n", - " * $f(x) = e^x$ is convex for $x \\in \\mathbb{R}$.\n", - "\n", - " * $g(x) = -\\ln(x)$ is convex for $x \\in (0,\\infty)$.\n", - "\n", - "\n", - "3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \\in \\mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex.\n", - "\n", - "4. A norm is any function that satisfy the following properties\n", - "\n", - " * $f(\\alpha x) = |\\alpha| f(x)$ for all $\\alpha \\in \\mathbb{R}$.\n", - "\n", - " * $f(x+y) \\leq f(x) + f(y)$\n", - "\n", - " * $f(x) \\leq 0$ for all $x \\in \\mathbb{R}^n$ with equality if and only if $x = 0$\n", - "\n", - "\n", - "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).\n", - "\n", - "\n", - "\n", - "## Friday September 25\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember25.mp4?vrtx=view-as-webpage) and [link to handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember25.pdf).\n", - "\n", - "\n", - "\n", - "## Standard steepest descent\n", - "\n", - "\n", - "Before we proceed, we would like to discuss the approach called the\n", - "**standard Steepest descent** (different from the above steepest descent discussion), which again leads to us having to be able\n", - "to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).\n", - "\n", - "[The success of the CG method](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)\n", - "for finding solutions of non-linear problems is based on the theory\n", - "of conjugate gradients for linear systems of equations. It belongs to\n", - "the class of iterative methods for solving problems from linear\n", - "algebra of the type" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{x} = \\boldsymbol{b}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the iterative process we end up with a problem like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}= \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\boldsymbol{r}$ is the so-called residual or error in the iterative process.\n", - "\n", - "When we have found the exact solution, $\\boldsymbol{r}=0$.\n", - "\n", - "\n", - "## Gradient method\n", - "\n", - "The residual is zero when we reach the minimum of the quadratic equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T\\boldsymbol{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the constraint that the matrix $\\boldsymbol{A}$ is positive definite and\n", - "symmetric. This defines also the Hessian and we want it to be positive definite. \n", - "\n", - "\n", - "\n", - "## Steepest descent method\n", - "\n", - "We denote the initial guess for $\\boldsymbol{x}$ as $\\boldsymbol{x}_0$. \n", - "We can assume without loss of generality that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_0=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or consider the system" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{z} = \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "instead.\n", - "\n", - "\n", - "\n", - "## Steepest descent method\n", - "One can show that the solution $\\boldsymbol{x}$ is also the unique minimizer of the quadratic form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(\\boldsymbol{x}) = \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T \\boldsymbol{x} , \\quad \\boldsymbol{x}\\in\\mathbf{R}^n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This suggests taking the first basis vector $\\boldsymbol{r}_1$ (see below for definition) \n", - "to be the gradient of $f$ at $\\boldsymbol{x}=\\boldsymbol{x}_0$, \n", - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{x}_0-\\boldsymbol{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and \n", - "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", - "\n", - "\n", - "\n", - "\n", - "## Final expressions\n", - "We can compute the residual iteratively as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}_{k+1}=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_{k+1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{b}-\\boldsymbol{A}(\\boldsymbol{x}_k+\\alpha_k\\boldsymbol{r}_k),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k)-\\alpha_k\\boldsymbol{A}\\boldsymbol{r}_k,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\alpha_k = \\frac{\\boldsymbol{r}_k^T\\boldsymbol{r}_k}{\\boldsymbol{r}_k^T\\boldsymbol{A}\\boldsymbol{r}_k}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "leading to the iterative scheme" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_{k+1}=\\boldsymbol{x}_k-\\alpha_k\\boldsymbol{r}_{k},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Steepest descent example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import numpy.linalg as la\n", - "\n", - "import scipy.optimize as sopt\n", - "\n", - "import matplotlib.pyplot as pt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "def f(x):\n", - " return 0.5*x[0]**2 + 2.5*x[1]**2\n", - "\n", - "def df(x):\n", - " return np.array([x[0], 5*x[1]])\n", - "\n", - "fig = pt.figure()\n", - "ax = fig.gca(projection=\"3d\")\n", - "\n", - "xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]\n", - "fmesh = f(np.array([xmesh, ymesh]))\n", - "ax.plot_surface(xmesh, ymesh, fmesh)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "And then as countor plot" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pt.axis(\"equal\")\n", - "pt.contour(xmesh, ymesh, fmesh)\n", - "guesses = [np.array([2, 2./5])]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Find guesses" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "x = guesses[-1]\n", - "s = -df(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Run it!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def f1d(alpha):\n", - " return f(x + alpha*s)\n", - "\n", - "alpha_opt = sopt.golden(f1d)\n", - "next_guess = x + alpha_opt * s\n", - "guesses.append(next_guess)\n", - "print(next_guess)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "What happened?" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pt.axis(\"equal\")\n", - "pt.contour(xmesh, ymesh, fmesh, 50)\n", - "it_array = np.array(guesses)\n", - "pt.plot(it_array.T[0], it_array.T[1], \"x-\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "In the CG method we define so-called conjugate directions and two vectors \n", - "$\\boldsymbol{s}$ and $\\boldsymbol{t}$\n", - "are said to be\n", - "conjugate if" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{s}^T\\boldsymbol{A}\\boldsymbol{t}= 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The philosophy of the CG method is to perform searches in various conjugate directions\n", - "of our vectors $\\boldsymbol{x}_i$ obeying the above criterion, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i^T\\boldsymbol{A}\\boldsymbol{x}_j= 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Two vectors are conjugate if they are orthogonal with respect to \n", - "this inner product. Being conjugate is a symmetric relation: if $\\boldsymbol{s}$ is conjugate to $\\boldsymbol{t}$, then $\\boldsymbol{t}$ is conjugate to $\\boldsymbol{s}$.\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "An example is given by the eigenvectors of the matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{v}_i^T\\boldsymbol{A}\\boldsymbol{v}_j= \\lambda\\boldsymbol{v}_i^T\\boldsymbol{v}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is zero unless $i=j$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "Assume now that we have a symmetric positive-definite matrix $\\boldsymbol{A}$ of size\n", - "$n\\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_{i+1}=\\boldsymbol{x}_{i}+\\alpha_i\\boldsymbol{p}_{i}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We assume that $\\boldsymbol{p}_{i}$ is a sequence of $n$ mutually conjugate directions. \n", - "Then the $\\boldsymbol{p}_{i}$ form a basis of $R^n$ and we can expand the solution \n", - "$ \\boldsymbol{A}\\boldsymbol{x} = \\boldsymbol{b}$ in this basis, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x} = \\sum^{n}_{i=1} \\alpha_i \\boldsymbol{p}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "The coefficients are given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}\\mathbf{x} = \\sum^{n}_{i=1} \\alpha_i \\mathbf{A} \\mathbf{p}_i = \\mathbf{b}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Multiplying with $\\boldsymbol{p}_k^T$ from the left gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{x} = \\sum^{n}_{i=1} \\alpha_i\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{p}_i= \\boldsymbol{p}_k^T \\boldsymbol{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we can define the coefficients $\\alpha_k$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\alpha_k = \\frac{\\boldsymbol{p}_k^T \\boldsymbol{b}}{\\boldsymbol{p}_k^T \\boldsymbol{A} \\boldsymbol{p}_k}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method and iterations\n", - "\n", - "If we choose the conjugate vectors $\\boldsymbol{p}_k$ carefully, \n", - "then we may not need all of them to obtain a good approximation to the solution \n", - "$\\boldsymbol{x}$. \n", - "We want to regard the conjugate gradient method as an iterative method. \n", - "This will us to solve systems where $n$ is so large that the direct \n", - "method would take too much time.\n", - "\n", - "We denote the initial guess for $\\boldsymbol{x}$ as $\\boldsymbol{x}_0$. \n", - "We can assume without loss of generality that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_0=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or consider the system" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{z} = \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "instead.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "One can show that the solution $\\boldsymbol{x}$ is also the unique minimizer of the quadratic form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(\\boldsymbol{x}) = \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T \\boldsymbol{x} , \\quad \\boldsymbol{x}\\in\\mathbf{R}^n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This suggests taking the first basis vector $\\boldsymbol{p}_1$ \n", - "to be the gradient of $f$ at $\\boldsymbol{x}=\\boldsymbol{x}_0$, \n", - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{x}_0-\\boldsymbol{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and \n", - "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", - "The other vectors in the basis will be conjugate to the gradient, \n", - "hence the name conjugate gradient method.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "Let $\\boldsymbol{r}_k$ be the residual at the $k$-th step:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}_k=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that $\\boldsymbol{r}_k$ is the negative gradient of $f$ at \n", - "$\\boldsymbol{x}=\\boldsymbol{x}_k$, \n", - "so the gradient descent method would be to move in the direction $\\boldsymbol{r}_k$. \n", - "Here, we insist that the directions $\\boldsymbol{p}_k$ are conjugate to each other, \n", - "so we take the direction closest to the gradient $\\boldsymbol{r}_k$ \n", - "under the conjugacy constraint. \n", - "This gives the following expression" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{p}_{k+1}=\\boldsymbol{r}_k-\\frac{\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{r}_k}{\\boldsymbol{p}_k^T\\boldsymbol{A}\\boldsymbol{p}_k} \\boldsymbol{p}_k.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "We can also compute the residual iteratively as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}_{k+1}=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_{k+1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{b}-\\boldsymbol{A}(\\boldsymbol{x}_k+\\alpha_k\\boldsymbol{p}_k),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k)-\\alpha_k\\boldsymbol{A}\\boldsymbol{p}_k,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}_{k+1}=\\boldsymbol{r}_k-\\boldsymbol{A}\\boldsymbol{p}_{k},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Revisiting our first homework\n", - "\n", - "We will use linear regression as a case study for the gradient descent\n", - "methods. Linear regression is a great test case for the gradient\n", - "descent methods discussed in the lectures since it has several\n", - "desirable properties such as:\n", - "\n", - "1. An analytical solution (recall homework set 1).\n", - "\n", - "2. The gradient can be computed analytically.\n", - "\n", - "3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates\n", - "\n", - "We revisit an example similar to what we had in the first homework set. We had a function of the type" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "x = 2*np.random.rand(m,1)\n", - "y = 4+3*x+np.random.randn(m,1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $x_i \\in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\\cal {N}(0,1)$. \n", - "The linear regression model is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "h_\\beta(x) = \\boldsymbol{y} = \\beta_0 + \\beta_1 x,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y}_i = \\beta_0 + \\beta_1 x_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gradient descent example\n", - "\n", - "Let $\\mathbf{y} = (y_1,\\cdots,y_n)^T$, $\\mathbf{\\boldsymbol{y}} = (\\boldsymbol{y}_1,\\cdots,\\boldsymbol{y}_n)^T$ and $\\beta = (\\beta_0, \\beta_1)^T$\n", - "\n", - "It is convenient to write $\\mathbf{\\boldsymbol{y}} = X\\beta$ where $X \\in \\mathbb{R}^{100 \\times 2} $ is the design matrix given by (we keep the intercept here)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "X \\equiv \\begin{bmatrix}\n", - "1 & x_1 \\\\\n", - "\\vdots & \\vdots \\\\\n", - "1 & x_{100} & \\\\\n", - "\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The cost/loss/risk function is given by (" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\beta) = \\frac{1}{n}||X\\beta-\\mathbf{y}||_{2}^{2} = \\frac{1}{n}\\sum_{i=1}^{100}\\left[ (\\beta_0 + \\beta_1 x_i)^2 - 2 y_i (\\beta_0 + \\beta_1 x_i) + y_i^2\\right]\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we want to find $\\beta$ such that $C(\\beta)$ is minimized.\n", - "\n", - "\n", - "## The derivative of the cost/loss function\n", - "\n", - "Computing $\\partial C(\\beta) / \\partial \\beta_0$ and $\\partial C(\\beta) / \\partial \\beta_1$ we can show that the gradient can be written as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\nabla_{\\beta} C(\\beta) = \\frac{2}{n}\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n", - "\\sum_{i=1}^{100}\\left( x_i (\\beta_0+\\beta_1x_i)-y_ix_i\\right) \\\\\n", - "\\end{bmatrix} = \\frac{2}{n}X^T(X\\beta - \\mathbf{y}),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $X$ is the design matrix defined above.\n", - "\n", - "\n", - "## The Hessian matrix\n", - "The Hessian matrix of $C(\\beta)$ is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{H} \\equiv \\begin{bmatrix}\n", - "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0^2} & \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} \\\\\n", - "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} & \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_1^2} & \\\\\n", - "\\end{bmatrix} = \\frac{2}{n}X^T X.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This result implies that $C(\\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Simple program\n", - "\n", - "We can now write a program that minimizes $C(\\beta)$ using the gradient descent method with a constant learning rate $\\gamma$ according to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_{k+1} = \\beta_k - \\gamma \\nabla_\\beta C(\\beta_k), \\ k=0,1,\\cdots\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can use the expression we computed for the gradient and let use a\n", - "$\\beta_0$ be chosen randomly and let $\\gamma = 0.001$. Stop iterating\n", - "when $||\\nabla_\\beta C(\\beta_k) || \\leq \\epsilon = 10^{-8}$. **Note that the code below does not include the latter stop criterion**.\n", - "\n", - "And finally we can compare our solution for $\\beta$ with the analytic result given by \n", - "$\\beta= (X^TX)^{-1} X^T \\mathbf{y}$.\n", - "\n", - "\n", - "## Gradient Descent Example\n", - "\n", - "Here our simple example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "# Importing various packages\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from mpl_toolkits.mplot3d import Axes3D\n", - "from matplotlib import cm\n", - "from matplotlib.ticker import LinearLocator, FormatStrFormatter\n", - "import sys\n", - "\n", - "# the number of datapoints\n", - "n = 100\n", - "x = 2*np.random.rand(n,1)\n", - "y = 4+3*x+np.random.randn(n,1)\n", - "\n", - "X = np.c_[np.ones((n,1)), x]\n", - "# Hessian matrix\n", - "H = (2.0/n)* X.T @ X\n", - "# Get the eigenvalues\n", - "EigValues, EigVectors = np.linalg.eig(H)\n", - "print(EigValues)\n", - "\n", - "beta_linreg = np.linalg.inv(X.T @ X) @ X.T @ y\n", - "print(beta_linreg)\n", - "beta = np.random.randn(2,1)\n", - "\n", - "eta = 1.0/np.max(EigValues)\n", - "Niterations = 1000\n", - "\n", - "for iter in range(Niterations):\n", - " gradient = (2.0/n)*X.T @ (X @ beta-y)\n", - " beta -= eta*gradient\n", - "\n", - "print(beta)\n", - "xnew = np.array([[0],[2]])\n", - "xbnew = np.c_[np.ones((2,1)), xnew]\n", - "ypredict = xbnew.dot(beta)\n", - "ypredict2 = xbnew.dot(beta_linreg)\n", - "plt.plot(xnew, ypredict, \"r-\")\n", - "plt.plot(xnew, ypredict2, \"b-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0,2.0,0, 15.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Gradient descent example')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## And a corresponding example using **scikit-learn**" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import SGDRegressor\n", - "\n", - "n = 100\n", - "x = 2*np.random.rand(n,1)\n", - "y = 4+3*x+np.random.randn(n,1)\n", - "\n", - "X = np.c_[np.ones((n,1)), x]\n", - "beta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n", - "print(beta_linreg)\n", - "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n", - "sgdreg.fit(x,y.ravel())\n", - "print(sgdreg.intercept_, sgdreg.coef_)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gradient descent and Ridge\n", - "\n", - "We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\\beta$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C_{\\text{ridge}}(\\beta) = \\frac{1}{n}||X\\beta -\\mathbf{y}||^2 + \\lambda ||\\beta||^2, \\ \\lambda \\geq 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to minimize $C_{\\text{ridge}}(\\beta)$ using GD we only have adjust the gradient as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\nabla_\\beta C_{\\text{ridge}}(\\beta) = \\frac{2}{n}\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n", - "\\sum_{i=1}^{100}\\left( x_i (\\beta_0+\\beta_1x_i)-y_ix_i\\right) \\\\\n", - "\\end{bmatrix} + 2\\lambda\\begin{bmatrix} \\beta_0 \\\\ \\beta_1\\end{bmatrix} = 2 (X^T(X\\beta - \\mathbf{y})+\\lambda \\beta).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can easily extend our program to minimize $C_{\\text{ridge}}(\\beta)$ using gradient descent and compare with the analytical solution given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_{\\text{ridge}} = \\left(X^T X + \\lambda I_{2 \\times 2} \\right)^{-1} X^T \\mathbf{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Program example for gradient descent with Ridge Regression" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from mpl_toolkits.mplot3d import Axes3D\n", - "from matplotlib import cm\n", - "from matplotlib.ticker import LinearLocator, FormatStrFormatter\n", - "import sys\n", - "\n", - "# the number of datapoints\n", - "n = 100\n", - "x = 2*np.random.rand(n,1)\n", - "y = 4+3*x+np.random.randn(n,1)\n", - "\n", - "X = np.c_[np.ones((n,1)), x]\n", - "XT_X = X.T @ X\n", - "\n", - "#Ridge parameter lambda\n", - "lmbda = 0.001\n", - "Id = lmbda* np.eye(XT_X.shape[0])\n", - "\n", - "beta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y\n", - "print(beta_linreg)\n", - "# Start plain gradient descent\n", - "beta = np.random.randn(2,1)\n", - "\n", - "eta = 0.1\n", - "Niterations = 100\n", - "\n", - "for iter in range(Niterations):\n", - " gradients = 2.0/n*X.T @ (X @ (beta)-y)+2*lmbda*beta\n", - " beta -= eta*gradients\n", - "\n", - "print(beta)\n", - "ypredict = X @ beta\n", - "ypredict2 = X @ beta_linreg\n", - "plt.plot(x, ypredict, \"r-\")\n", - "plt.plot(x, ypredict2, \"b-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0,2.0,0, 15.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Gradient descent example for Ridge')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Using gradient descent methods, limitations\n", - "\n", - "* **Gradient descent (GD) finds local minima of our function**. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.\n", - "\n", - "* **GD is sensitive to initial conditions**. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.\n", - "\n", - "* **Gradients are computationally expensive to calculate for large datasets**. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \\propto \\sum_{i=1}^n (y_i - \\mathbf{w}^T\\cdot\\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called \"mini batches\". This has the added benefit of introducing stochasticity into our algorithm.\n", - "\n", - "* **GD is very sensitive to choices of learning rates**. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape.\n", - "\n", - "* **GD treats all directions in parameter space uniformly.** Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. \n", - "\n", - "* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.\n", - "\n", - "## Stochastic Gradient Descent\n", - "\n", - "Stochastic gradient descent (SGD) and variants thereof address some of\n", - "the shortcomings of the Gradient descent method discussed above.\n", - "\n", - "The underlying idea of SGD comes from the observation that the cost\n", - "function, which we want to minimize, can almost always be written as a\n", - "sum over $n$ data points $\\{\\mathbf{x}_i\\}_{i=1}^n$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\mathbf{\\beta}) = \\sum_{i=1}^n c_i(\\mathbf{x}_i,\n", - "\\mathbf{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Computation of gradients\n", - "\n", - "This in turn means that the gradient can be\n", - "computed as a sum over $i$-gradients" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\nabla_\\beta C(\\mathbf{\\beta}) = \\sum_i^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", - "\\mathbf{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Stochasticity/randomness is introduced by only taking the\n", - "gradient on a subset of the data called minibatches. If there are $n$\n", - "data points and the size of each minibatch is $M$, there will be $n/M$\n", - "minibatches. We denote these minibatches by $B_k$ where\n", - "$k=1,\\cdots,n/M$.\n", - "\n", - "\n", - "## SGD example\n", - "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n", - "and we choose to have $M=5$ minibathces,\n", - "then each minibatch contains two data points. In particular we have\n", - "$B_1 = (\\mathbf{x}_1,\\mathbf{x}_2), \\cdots, B_5 =\n", - "(\\mathbf{x}_9,\\mathbf{x}_{10})$. Note that if you choose $M=1$ you\n", - "have only a single batch with all data points and on the other extreme,\n", - "you may choose $M=n$ resulting in a minibatch for each datapoint, i.e\n", - "$B_k = \\mathbf{x}_k$.\n", - "\n", - "The idea is now to approximate the gradient by replacing the sum over\n", - "all data points with a sum over the data points in one the minibatches\n", - "picked at random in each gradient descent step" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\nabla_{\\beta}\n", - "C(\\mathbf{\\beta}) = \\sum_{i=1}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", - "\\mathbf{\\beta}) \\rightarrow \\sum_{i \\in B_k}^n \\nabla_\\beta\n", - "c_i(\\mathbf{x}_i, \\mathbf{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The gradient step\n", - "\n", - "Thus a gradient descent step now looks like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_{j+1} = \\beta_j - \\gamma_j \\sum_{i \\in B_k}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", - "\\mathbf{\\beta})\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $k$ is picked at random with equal\n", - "probability from $[1,n/M]$. An iteration over the number of\n", - "minibathces (n/M) is commonly referred to as an epoch. Thus it is\n", - "typical to choose a number of epochs and for each epoch iterate over\n", - "the number of minibatches, as exemplified in the code below.\n", - "\n", - "\n", - "## Simple example code" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np \n", - "\n", - "n = 100 #100 datapoints \n", - "M = 5 #size of each minibatch\n", - "m = int(n/M) #number of minibatches\n", - "n_epochs = 10 #number of epochs\n", - "\n", - "j = 0\n", - "for epoch in range(1,n_epochs+1):\n", - " for i in range(m):\n", - " k = np.random.randint(m) #Pick the k-th minibatch at random\n", - " #Compute the gradient using the data in minibatch Bk\n", - " #Compute new suggestion for \n", - " j += 1" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Taking the gradient only on a subset of the data has two important\n", - "benefits. First, it introduces randomness which decreases the chance\n", - "that our opmization scheme gets stuck in a local minima. Second, if\n", - "the size of the minibatches are small relative to the number of\n", - "datapoints ($M < n$), the computation of the gradient is much\n", - "cheaper since we sum over the datapoints in the $k-th$ minibatch and not\n", - "all $n$ datapoints.\n", - "\n", - "\n", - "## When do we stop?\n", - "\n", - "A natural question is when do we stop the search for a new minimum?\n", - "One possibility is to compute the full gradient after a given number\n", - "of epochs and check if the norm of the gradient is smaller than some\n", - "threshold and stop if true. However, the condition that the gradient\n", - "is zero is valid also for local minima, so this would only tell us\n", - "that we are close to a local/global minimum. However, we could also\n", - "evaluate the cost function at this point, store the result and\n", - "continue the search. If the test kicks in at a later stage we can\n", - "compare the values of the cost function and keep the $\\beta$ that\n", - "gave the lowest value.\n", - "\n", - "\n", - "## Slightly different approach\n", - "\n", - "Another approach is to let the step length $\\gamma_j$ depend on the\n", - "number of epochs in such a way that it becomes very small after a\n", - "reasonable time such that we do not move at all.\n", - "\n", - "As an example, let $e = 0,1,2,3,\\cdots$ denote the current epoch and let $t_0, t_1 > 0$ be two fixed numbers. Furthermore, let $t = e \\cdot m + i$ where $m$ is the number of minibatches and $i=0,\\cdots,m-1$. Then the function $$\\gamma_j(t; t_0, t_1) = \\frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length $\\gamma_j (0; t_0, t_1) = t_0/t_1$ which decays in *time* $t$.\n", - "\n", - "In this way we can fix the number of epochs, compute $\\beta$ and\n", - "evaluate the cost function at the end. Repeating the computation will\n", - "give a different result since the scheme is random by design. Then we\n", - "pick the final $\\beta$ that gives the lowest value of the cost\n", - "function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np \n", - "\n", - "def step_length(t,t0,t1):\n", - " return t0/(t+t1)\n", - "\n", - "n = 100 #100 datapoints \n", - "M = 5 #size of each minibatch\n", - "m = int(n/M) #number of minibatches\n", - "n_epochs = 500 #number of epochs\n", - "t0 = 1.0\n", - "t1 = 10\n", - "\n", - "gamma_j = t0/t1\n", - "j = 0\n", - "for epoch in range(1,n_epochs+1):\n", - " for i in range(m):\n", - " k = np.random.randint(m) #Pick the k-th minibatch at random\n", - " #Compute the gradient using the data in minibatch Bk\n", - " #Compute new suggestion for beta\n", - " t = epoch*m+i\n", - " gamma_j = step_length(t,t0,t1)\n", - " j += 1\n", - "\n", - "print(\"gamma_j after %d epochs: %g\" % (n_epochs,gamma_j))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Program for stochastic gradient" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "from math import exp, sqrt\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import SGDRegressor\n", - "\n", - "m = 100\n", - "x = 2*np.random.rand(m,1)\n", - "y = 4+3*x+np.random.randn(m,1)\n", - "\n", - "X = np.c_[np.ones((m,1)), x]\n", - "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n", - "print(\"Own inversion\")\n", - "print(theta_linreg)\n", - "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n", - "sgdreg.fit(x,y.ravel())\n", - "print(\"sgdreg from scikit\")\n", - "print(sgdreg.intercept_, sgdreg.coef_)\n", - "\n", - "\n", - "theta = np.random.randn(2,1)\n", - "eta = 0.1\n", - "Niterations = 1000\n", - "\n", - "\n", - "for iter in range(Niterations):\n", - " gradients = 2.0/m*X.T @ ((X @ theta)-y)\n", - " theta -= eta*gradients\n", - "print(\"theta from own gd\")\n", - "print(theta)\n", - "\n", - "xnew = np.array([[0],[2]])\n", - "Xnew = np.c_[np.ones((2,1)), xnew]\n", - "ypredict = Xnew.dot(theta)\n", - "ypredict2 = Xnew.dot(theta_linreg)\n", - "\n", - "\n", - "n_epochs = 50\n", - "t0, t1 = 5, 50\n", - "def learning_schedule(t):\n", - " return t0/(t+t1)\n", - "\n", - "theta = np.random.randn(2,1)\n", - "\n", - "for epoch in range(n_epochs):\n", - " for i in range(m):\n", - " random_index = np.random.randint(m)\n", - " xi = X[random_index:random_index+1]\n", - " yi = y[random_index:random_index+1]\n", - " gradients = 2 * xi.T @ ((xi @ theta)-yi)\n", - " eta = learning_schedule(epoch*m+i)\n", - " theta = theta - eta*gradients\n", - "print(\"theta from own sdg\")\n", - "print(theta)\n", - "\n", - "plt.plot(xnew, ypredict, \"r-\")\n", - "plt.plot(xnew, ypredict2, \"b-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0,2.0,0, 15.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Random numbers ')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Challenge**: try to write a similar code for a Logistic Regression case." - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter5.ipynb b/doc/LectureNotes/_build/html/_sources/chapter5.ipynb deleted file mode 100644 index 0e24adf4e..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter5.ipynb +++ /dev/null @@ -1,1867 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Support Vector Machines, overarching aims\n", - "\n", - "A Support Vector Machine (SVM) is a very powerful and versatile\n", - "Machine Learning method, capable of performing linear or nonlinear\n", - "classification, regression, and even outlier detection. It is one of\n", - "the most popular models in Machine Learning, and anyone interested in\n", - "Machine Learning should have it in their toolbox. SVMs are\n", - "particularly well suited for classification of complex but small-sized or\n", - "medium-sized datasets. \n", - "\n", - "The case with two well-separated classes only can be understood in an\n", - "intuitive way in terms of lines in a two-dimensional space separating\n", - "the two classes (see figure below).\n", - "\n", - "The basic mathematics behind the SVM is however less familiar to most of us. \n", - "It relies on the definition of hyperplanes and the\n", - "definition of a **margin** which separates classes (in case of\n", - "classification problems) of variables. It is also used for regression\n", - "problems.\n", - "\n", - "With SVMs we distinguish between hard margin and soft margins. The\n", - "latter introduces a so-called softening parameter to be discussed\n", - "below. We distinguish also between linear and non-linear\n", - "approaches. The latter are the most frequent ones since it is rather\n", - "unlikely that we can separate classes easily by say straight lines.\n", - "\n", - "\n", - "## Hyperplanes and all that\n", - "\n", - "The theory behind support vector machines (SVM hereafter) is based on\n", - "the mathematical description of so-called hyperplanes. Let us start\n", - "with a two-dimensional case. This will also allow us to introduce our\n", - "first SVM examples. These will be tailored to the case of two specific\n", - "classes, as displayed in the figure here based on the usage of the petal data.\n", - "\n", - "We assume here that our data set can be well separated into two\n", - "domains, where a straight line does the job in the separating the two\n", - "classes. Here the two classes are represented by either squares or\n", - "circles." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "from sklearn import datasets\n", - "from sklearn.svm import SVC, LinearSVC\n", - "from sklearn.linear_model import SGDClassifier\n", - "from sklearn.preprocessing import StandardScaler\n", - "import matplotlib\n", - "import matplotlib.pyplot as plt\n", - "plt.rcParams['axes.labelsize'] = 14\n", - "plt.rcParams['xtick.labelsize'] = 12\n", - "plt.rcParams['ytick.labelsize'] = 12\n", - "\n", - "\n", - "iris = datasets.load_iris()\n", - "X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n", - "y = iris[\"target\"]\n", - "\n", - "setosa_or_versicolor = (y == 0) | (y == 1)\n", - "X = X[setosa_or_versicolor]\n", - "y = y[setosa_or_versicolor]\n", - "\n", - "\n", - "\n", - "C = 5\n", - "alpha = 1 / (C * len(X))\n", - "\n", - "lin_clf = LinearSVC(loss=\"hinge\", C=C, random_state=42)\n", - "svm_clf = SVC(kernel=\"linear\", C=C)\n", - "sgd_clf = SGDClassifier(loss=\"hinge\", learning_rate=\"constant\", eta0=0.001, alpha=alpha,\n", - " max_iter=100000, random_state=42)\n", - "\n", - "scaler = StandardScaler()\n", - "X_scaled = scaler.fit_transform(X)\n", - "\n", - "lin_clf.fit(X_scaled, y)\n", - "svm_clf.fit(X_scaled, y)\n", - "sgd_clf.fit(X_scaled, y)\n", - "\n", - "print(\"LinearSVC: \", lin_clf.intercept_, lin_clf.coef_)\n", - "print(\"SVC: \", svm_clf.intercept_, svm_clf.coef_)\n", - "print(\"SGDClassifier(alpha={:.5f}):\".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)\n", - "\n", - "# Compute the slope and bias of each decision boundary\n", - "w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]\n", - "b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]\n", - "w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]\n", - "b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]\n", - "w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]\n", - "b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]\n", - "\n", - "# Transform the decision boundary lines back to the original scale\n", - "line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])\n", - "line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])\n", - "line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])\n", - "\n", - "# Plot all three decision boundaries\n", - "plt.figure(figsize=(11, 4))\n", - "plt.plot(line1[:, 0], line1[:, 1], \"k:\", label=\"LinearSVC\")\n", - "plt.plot(line2[:, 0], line2[:, 1], \"b--\", linewidth=2, label=\"SVC\")\n", - "plt.plot(line3[:, 0], line3[:, 1], \"r-\", label=\"SGDClassifier\")\n", - "plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\") # label=\"Iris-Versicolor\"\n", - "plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\") # label=\"Iris-Setosa\"\n", - "plt.xlabel(\"Petal length\", fontsize=14)\n", - "plt.ylabel(\"Petal width\", fontsize=14)\n", - "plt.legend(loc=\"upper center\", fontsize=14)\n", - "plt.axis([0, 5.5, 0, 2])\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The aim of the SVM algorithm is to find a hyperplane in a\n", - "$p$-dimensional space, where $p$ is the number of features that\n", - "distinctly classifies the data points.\n", - "\n", - "In a $p$-dimensional space, a hyperplane is what we call an affine subspace of dimension of $p-1$.\n", - "As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is \n", - "a two-dimensional subspace, or stated simply, a plane. \n", - "\n", - "In two dimensions, with the variables $x_1$ and $x_2$, the hyperplane is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b+w_1x_1+w_2x_2=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $b$ is the intercept and $w_1$ and $w_2$ define the elements of a vector orthogonal to the line \n", - "$b+w_1x_1+w_2x_2=0$. \n", - "In two dimensions we define the vectors $\\boldsymbol{x} =[x1,x2]$ and $\\boldsymbol{w}=[w1,w2]$. \n", - "We can then rewrite the above equation as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}^T\\boldsymbol{w}+b=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We limit ourselves to two classes of outputs $y_i$ and assign these classes the values $y_i = \\pm 1$. \n", - "In a $p$-dimensional space of say $p$ features we have a hyperplane defines as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b+wx_1+w_2x_2+\\dots +w_px_p=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we define a \n", - "matrix $\\boldsymbol{X}=\\left[\\boldsymbol{x}_1,\\boldsymbol{x}_2,\\dots, \\boldsymbol{x}_p\\right]$\n", - "of dimension $n\\times p$, where $n$ represents the observations for each feature and each vector $x_i$ is a column vector of the matrix $\\boldsymbol{X}$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i = \\begin{bmatrix} x_{i1} \\\\ x_{i2} \\\\ \\dots \\\\ \\dots \\\\ x_{ip} \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If the above condition is not met for a given vector $\\boldsymbol{x}_i$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip} >0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "if our output $y_i=1$.\n", - "In this case we say that $\\boldsymbol{x}_i$ lies on one of the sides of the hyperplane and if" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip} < 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for the class of observations $y_i=-1$, \n", - "then $\\boldsymbol{x}_i$ lies on the other side. \n", - "\n", - "Equivalently, for the two classes of observations we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i\\left(b+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip}\\right) > 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.\n", - "\n", - "\n", - "### The two-dimensional case\n", - "\n", - "Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional\n", - "plane. To separate the two classes of data points, there are many\n", - "possible lines (hyperplanes if you prefer a more strict naming) \n", - "that could be chosen. Our objective is to find a\n", - "plane that has the maximum margin, i.e the maximum distance between\n", - "data points of both classes. Maximizing the margin distance provides\n", - "some reinforcement so that future data points can be classified with\n", - "more confidence.\n", - "\n", - "What a linear classifier attempts to accomplish is to split the\n", - "feature space into two half spaces by placing a hyperplane between the\n", - "data points. This hyperplane will be our decision boundary. All\n", - "points on one side of the plane will belong to class one and all points\n", - "on the other side of the plane will belong to the second class two.\n", - "\n", - "Unfortunately there are many ways in which we can place a hyperplane\n", - "to divide the data. Below is an example of two candidate hyperplanes\n", - "for our data sample.\n", - "\n", - "\n", - "Let us define the function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(x) = \\boldsymbol{w}^T\\boldsymbol{x}+b = 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "as the function that determines the line $L$ that separates two classes (our two features), see the figure here. \n", - "\n", - "\n", - "Any point defined by $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_2$ on the line $L$ will satisfy $\\boldsymbol{w}^T(\\boldsymbol{x}_1-\\boldsymbol{x}_2)=0$. \n", - "\n", - "The signed distance $\\delta$ from any point defined by a vector $\\boldsymbol{x}$ and a point $\\boldsymbol{x}_0$ on the line $L$ is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta = \\frac{1}{\\vert\\vert \\boldsymbol{w}\\vert\\vert}(\\boldsymbol{w}^T\\boldsymbol{x}+b).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "How do we find the parameter $b$ and the vector $\\boldsymbol{w}$? What we could\n", - "do is to define a cost function which now contains the set of all\n", - "misclassified points $M$ and attempt to minimize this function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{w},b) = -\\sum_{i\\in M} y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We could now for example define all values $y_i =1$ as misclassified in case we have $\\boldsymbol{w}^T\\boldsymbol{x}_i+b < 0$ and the opposite if we have $y_i=-1$. Taking the derivatives gives us" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C}{\\partial b} = -\\sum_{i\\in M} y_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C}{\\partial \\boldsymbol{w}} = -\\sum_{i\\in M} y_ix_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b \\leftarrow b +\\eta \\frac{\\partial C}{\\partial b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{w} \\leftarrow \\boldsymbol{w} +\\eta \\frac{\\partial C}{\\partial \\boldsymbol{w}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\eta$ is our by now well-known learning rate. \n", - "\n", - "\n", - "\n", - "The equations we discussed above can be coded rather easily (the\n", - "framework is similar to what we developed for logistic\n", - "regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There are however problems with this approach, although it looks\n", - "pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.\n", - "\n", - "\n", - "For small\n", - "gaps between the entries, we may also end up needing many iterations\n", - "before the solutions converge and if the data cannot be separated\n", - "properly into two distinct classes, we may not experience a converge\n", - "at all.\n", - "\n", - "\n", - "### A better approach\n", - "\n", - "A better approach is rather to try to define a large margin between\n", - "the two classes (if they are well separated from the beginning).\n", - "\n", - "Thus, we wish to find a margin $M$ with $\\boldsymbol{w}$ normalized to\n", - "$\\vert\\vert \\boldsymbol{w}\\vert\\vert =1$ subject to the condition" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M \\hspace{0.1cm}\\forall i=1,2,\\dots, p.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "All points are thus at a signed distance from the decision boundary defined by the line $L$. The parameters $b$ and $w_1$ and $w_2$ define this line. \n", - "\n", - "We seek thus the largest value $M$ defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{\\vert \\vert \\boldsymbol{w}\\vert\\vert}y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M \\hspace{0.1cm}\\forall i=1,2,\\dots, n,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or just" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M\\vert \\vert \\boldsymbol{w}\\vert\\vert \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we scale the equation so that $\\vert \\vert \\boldsymbol{w}\\vert\\vert = 1/M$, we have to find the minimum of \n", - "$\\boldsymbol{w}^T\\boldsymbol{w}=\\vert \\vert \\boldsymbol{w}\\vert\\vert$ (the norm) subject to the condition" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq 1 \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We have thus defined our margin as the invers of the norm of\n", - "$\\boldsymbol{w}$. We want to minimize the norm in order to have a as large as\n", - "possible margin $M$. Before we proceed, we need to remind ourselves\n", - "about Lagrangian multipliers.\n", - "\n", - "\n", - "## A quick Reminder on Lagrangian Multipliers\n", - "\n", - "Consider a function of three independent variables $f(x,y,z)$ . For the function $f$ to be an\n", - "extreme we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "df=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "A necessary and sufficient condition is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial x} =\\frac{\\partial f}{\\partial y}=\\frac{\\partial f}{\\partial z}=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "due to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "df = \\frac{\\partial f}{\\partial x}dx+\\frac{\\partial f}{\\partial y}dy+\\frac{\\partial f}{\\partial z}dz.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In many problems the variables $x,y,z$ are often subject to constraints (such as those above for the margin)\n", - "so that they are no longer all independent. It is possible at least in principle to use each \n", - "constraint to eliminate one variable\n", - "and to proceed with a new and smaller set of independent varables.\n", - "\n", - "The use of so-called Lagrangian multipliers is an alternative technique when the elimination\n", - "of variables is incovenient or undesirable. Assume that we have an equation of constraint on \n", - "the variables $x,y,z$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\phi(x,y,z) = 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "d\\phi = \\frac{\\partial \\phi}{\\partial x}dx+\\frac{\\partial \\phi}{\\partial y}dy+\\frac{\\partial \\phi}{\\partial z}dz =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now we cannot set anymore" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial x} =\\frac{\\partial f}{\\partial y}=\\frac{\\partial f}{\\partial z}=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "if $df=0$ is wanted\n", - "because there are now only two independent variables! Assume $x$ and $y$ are the independent \n", - "variables.\n", - "Then $dz$ is no longer arbitrary.\n", - "\n", - "\n", - "However, we can add to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "df = \\frac{\\partial f}{\\partial x}dx+\\frac{\\partial f}{\\partial y}dy+\\frac{\\partial f}{\\partial z}dz,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "a multiplum of $d\\phi$, viz. $\\lambda d\\phi$, resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "df+\\lambda d\\phi = (\\frac{\\partial f}{\\partial z}+\\lambda\n", - "\\frac{\\partial \\phi}{\\partial x})dx+(\\frac{\\partial f}{\\partial y}+\\lambda\\frac{\\partial \\phi}{\\partial y})dy+\n", - "(\\frac{\\partial f}{\\partial z}+\\lambda\\frac{\\partial \\phi}{\\partial z})dz =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Our multiplier is chosen so that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial z}+\\lambda\\frac{\\partial \\phi}{\\partial z} =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We need to remember that we took $dx$ and $dy$ to be arbitrary and thus we must have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial x}+\\lambda\\frac{\\partial \\phi}{\\partial x} =0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial y}+\\lambda\\frac{\\partial \\phi}{\\partial y} =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "When all these equations are satisfied, $df=0$. We have four unknowns, $x,y,z$ and\n", - "$\\lambda$. Actually we want only $x,y,z$, $\\lambda$ needs not to be determined, \n", - "it is therefore often called\n", - "Lagrange's undetermined multiplier.\n", - "If we have a set of constraints $\\phi_k$ we have the equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial x_i}+\\sum_k\\lambda_k\\frac{\\partial \\phi_k}{\\partial x_i} =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to solve the above problem, we define the following Lagrangian function to be minimized" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}(\\lambda,b,\\boldsymbol{w})=\\frac{1}{2}\\boldsymbol{w}^T\\boldsymbol{w}-\\sum_{i=1}^n\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)-1\\right],\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\lambda_i$ is a so-called Lagrange multiplier subject to the condition $\\lambda_i \\geq 0$.\n", - "\n", - "Taking the derivatives with respect to $b$ and $\\boldsymbol{w}$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{L}}{\\partial b} = -\\sum_{i} \\lambda_iy_i=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{L}}{\\partial \\boldsymbol{w}} = 0 = \\boldsymbol{w}-\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Inserting these constraints into the equation for $\\cal{L}$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to the constraints $\\lambda_i\\geq 0$ and $\\sum_i\\lambda_iy_i=0$. \n", - "We must in addition satisfy the [Karush-Kuhn-Tucker](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions) (KKT) condition" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) -1\\right] \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "1. If $\\lambda_i > 0$, then $y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1$ and we say that $x_i$ is on the boundary.\n", - "\n", - "2. If $y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)> 1$, we say $x_i$ is not on the boundary and we set $\\lambda_i=0$. \n", - "\n", - "When $\\lambda_i > 0$, the vectors $\\boldsymbol{x}_i$ are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin $M$. \n", - "\n", - "\n", - "We can rewrite" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and its constraints in terms of a matrix-vector problem where we minimize w.r.t. $\\lambda$ the following problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{2} \\boldsymbol{\\lambda}^T\\begin{bmatrix} y_1y_1\\boldsymbol{x}_1^T\\boldsymbol{x}_1 & y_1y_2\\boldsymbol{x}_1^T\\boldsymbol{x}_2 & \\dots & \\dots & y_1y_n\\boldsymbol{x}_1^T\\boldsymbol{x}_n \\\\\n", - "y_2y_1\\boldsymbol{x}_2^T\\boldsymbol{x}_1 & y_2y_2\\boldsymbol{x}_2^T\\boldsymbol{x}_2 & \\dots & \\dots & y_1y_n\\boldsymbol{x}_2^T\\boldsymbol{x}_n \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "y_ny_1\\boldsymbol{x}_n^T\\boldsymbol{x}_1 & y_ny_2\\boldsymbol{x}_n^T\\boldsymbol{x}_2 & \\dots & \\dots & y_ny_n\\boldsymbol{x}_n^T\\boldsymbol{x}_n \\\\\n", - "\\end{bmatrix}\\boldsymbol{\\lambda}-\\mathbb{1}\\boldsymbol{\\lambda},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$. Here we defined the vectors $\\boldsymbol{\\lambda} =[\\lambda_1,\\lambda_2,\\dots,\\lambda_n]$ and \n", - "$\\boldsymbol{y}=[y_1,y_2,\\dots,y_n]$. \n", - "\n", - "\n", - "\n", - "Solving the above problem, yields the values of $\\lambda_i$.\n", - "To find the coefficients of your hyperplane we need simply to compute" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{w}=\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With our vector $\\boldsymbol{w}$ we can in turn find the value of the intercept $b$ (here in two dimensions) via" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b = \\frac{1}{y_i}-\\boldsymbol{w}^T\\boldsymbol{x}_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or if we write it out in terms of the support vectors only, with $N_s$ being their number, we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b = \\frac{1}{N_s}\\sum_{j\\in N_s}\\left(y_j-\\sum_{i=1}^n\\lambda_iy_i\\boldsymbol{x}_i^T\\boldsymbol{x}_j\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With our hyperplane coefficients we can use our classifier to assign any observation by simply using" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i = \\mathrm{sign}(\\boldsymbol{w}^T\\boldsymbol{x}_i+b).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Below we discuss how to find the optimal values of $\\lambda_i$. Before we proceed however, we discuss now the so-called soft classifier. \n", - "\n", - "\n", - "## A soft classifier\n", - "\n", - "Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.\n", - "\n", - "Suppose now that classes overlap in feature space, as shown in the\n", - "figure here. One way to deal with this problem before we define the\n", - "so-called **kernel approach**, is to allow a kind of slack in the sense\n", - "that we allow some points to be on the wrong side of the margin.\n", - "\n", - "We introduce thus the so-called **slack** variables $\\boldsymbol{\\xi} =[\\xi_1,x_2,\\dots,x_n]$ and \n", - "modify our previous equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1-\\xi_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the requirement $\\xi_i\\geq 0$. The total violation is now $\\sum_i\\xi$. \n", - "The value $\\xi_i$ in the constraint the last constraint corresponds to the amount by which the prediction\n", - "$y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1$ is on the wrong side of its margin. Hence by bounding the sum $\\sum_i \\xi_i$,\n", - "we bound the total amount by which predictions fall on the wrong side of their margins.\n", - "\n", - "Misclassifications occur when $\\xi_i > 1$. Thus bounding the total sum by some value $C$ bounds in turn the total number of\n", - "misclassifications.\n", - "\n", - "\n", - "This has in turn the consequences that we change our optmization problem to finding the minimum of" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\frac{1}{2}\\boldsymbol{w}^T\\boldsymbol{w}-\\sum_{i=1}^n\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)-(1-\\xi_)\\right]+C\\sum_{i=1}^n\\xi_i-\\sum_{i=1}^n\\gamma_i\\xi_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1-\\xi_i \\hspace{0.1cm}\\forall i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the requirement $\\xi_i\\geq 0$.\n", - "\n", - "Taking the derivatives with respect to $b$ and $\\boldsymbol{w}$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{L}}{\\partial b} = -\\sum_{i} \\lambda_iy_i=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{L}}{\\partial \\boldsymbol{w}} = 0 = \\boldsymbol{w}-\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\lambda_i = C-\\gamma_i \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Inserting these constraints into the equation for $\\cal{L}$ we obtain the same equation as before" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "but now subject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$ and $0\\leq\\lambda_i \\leq C$. \n", - "We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "5\n", - "0\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_i\\xi_i = 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) -(1-\\xi_) \\geq 0 \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Kernels and non-linearity\n", - "\n", - "The cases we have studied till now, were all characterized by two classes\n", - "with a close to linear separability. The classifiers we have described\n", - "so far find linear boundaries in our input feature space. It is\n", - "possible to make our procedure more flexible by exploring the feature\n", - "space using other basis expansions such as higher-order polynomials,\n", - "wavelets, splines etc.\n", - "\n", - "If our feature space is not easy to separate, as shown in the figure\n", - "here, we can achieve a better separation by introducing more complex\n", - "basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to \n", - "obtain a separation between the classes which is almost linear. \n", - "\n", - "The change of basis, from $x\\rightarrow z=\\phi(x)$ leads to the same type of equations to be solved, except that\n", - "we need to introduce for example a polynomial transformation to a two-dimensional training set." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import os\n", - "\n", - "np.random.seed(42)\n", - "\n", - "# To plot pretty figures\n", - "import matplotlib\n", - "import matplotlib.pyplot as plt\n", - "plt.rcParams['axes.labelsize'] = 14\n", - "plt.rcParams['xtick.labelsize'] = 12\n", - "plt.rcParams['ytick.labelsize'] = 12\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "from sklearn import datasets\n", - "\n", - "\n", - "\n", - "X1D = np.linspace(-4, 4, 9).reshape(-1, 1)\n", - "X2D = np.c_[X1D, X1D**2]\n", - "y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "\n", - "plt.subplot(121)\n", - "plt.grid(True, which='both')\n", - "plt.axhline(y=0, color='k')\n", - "plt.plot(X1D[:, 0][y==0], np.zeros(4), \"bs\")\n", - "plt.plot(X1D[:, 0][y==1], np.zeros(5), \"g^\")\n", - "plt.gca().get_yaxis().set_ticks([])\n", - "plt.xlabel(r\"$x_1$\", fontsize=20)\n", - "plt.axis([-4.5, 4.5, -0.2, 0.2])\n", - "\n", - "plt.subplot(122)\n", - "plt.grid(True, which='both')\n", - "plt.axhline(y=0, color='k')\n", - "plt.axvline(x=0, color='k')\n", - "plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], \"bs\")\n", - "plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], \"g^\")\n", - "plt.xlabel(r\"$x_1$\", fontsize=20)\n", - "plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n", - "plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])\n", - "plt.plot([-4.5, 4.5], [6.5, 6.5], \"r--\", linewidth=3)\n", - "plt.axis([-4.5, 4.5, -1, 17])\n", - "plt.subplots_adjust(right=1)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with $x_i$ and $y_i$ as variables)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "z = \\phi(x_i) =\\left(x_i^2, y_i^2, \\sqrt{2}x_iy_i\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{z}_i^T\\boldsymbol{z}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$, and for the support vectors" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{z}_i+b)= 1 \\hspace{0.1cm}\\forall i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "from which we also find $b$.\n", - "To compute $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we define the kernel $K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=\\boldsymbol{z}_i^T\\boldsymbol{z}_j= \\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For the above example, the kernel reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=[x_i^2, y_i^2, \\sqrt{2}x_iy_i]^T\\begin{bmatrix} x_j^2 \\\\ y_j^2 \\\\ \\sqrt{2}x_jy_j \\end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note that this is nothing but the dot product of the two original\n", - "vectors $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$. Instead of thus computing the\n", - "product in the Lagrangian of $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we simply compute\n", - "the dot product $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$.\n", - "\n", - "\n", - "This leads to the so-called\n", - "kernel trick and the result leads to the same as if we went through\n", - "the trouble of performing the transformation\n", - "$\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j)$ during the SVM calculations.\n", - "\n", - "\n", - "\n", - "Using our definition of the kernel We can rewrite again the Lagrangian" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{z}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$ in terms of a convex optimization problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{2} \\boldsymbol{\\lambda}^T\\begin{bmatrix} y_1y_1K(\\boldsymbol{x}_1,\\boldsymbol{x}_1) & y_1y_2K(\\boldsymbol{x}_1,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_1,\\boldsymbol{x}_n) \\\\\n", - "y_2y_1K(\\boldsymbol{x}_2,\\boldsymbol{x}_1) & y_2y_2(\\boldsymbol{x}_2,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_2,\\boldsymbol{x}_n) \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "y_ny_1K(\\boldsymbol{x}_n,\\boldsymbol{x}_1) & y_ny_2K(\\boldsymbol{x}_n\\boldsymbol{x}_2) & \\dots & \\dots & y_ny_nK(\\boldsymbol{x}_n,\\boldsymbol{x}_n) \\\\\n", - "\\end{bmatrix}\\boldsymbol{\\lambda}-\\mathbb{1}\\boldsymbol{\\lambda},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$. Here we defined the vectors $\\boldsymbol{\\lambda} =[\\lambda_1,\\lambda_2,\\dots,\\lambda_n]$ and \n", - "$\\boldsymbol{y}=[y_1,y_2,\\dots,y_n]$. \n", - "If we add the slack constants this leads to the additional constraint $0\\leq \\lambda_i \\leq C$.\n", - "\n", - "We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " &\\mathrm{min}_{\\lambda}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{\\lambda}^T\\boldsymbol{P}\\boldsymbol{\\lambda}+\\boldsymbol{q}^T\\boldsymbol{\\lambda},\\\\ \\nonumber\n", - " &\\mathrm{subject\\hspace{0.1cm}to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{\\lambda} \\preceq \\boldsymbol{h} \\hspace{0.2cm} \\wedge \\boldsymbol{A}\\boldsymbol{\\lambda}=f.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Below we discuss how to solve these equations. Here we note that the matrix $\\boldsymbol{P}$ has matrix elements $p_{ij}=y_iy_jK(\\boldsymbol{x}_i,\\boldsymbol{x}_j)$.\n", - "Given a kernel $K$ and the targets $y_i$ this matrix is easy to set up. The constraint $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$ leads to $f=0$ and $\\boldsymbol{A}=\\boldsymbol{y}$. How to set up the matrix $\\boldsymbol{G}$ is discussed later. Here note that the inequalities $0\\leq \\lambda_i \\leq C$ can be split up into\n", - "$0\\leq \\lambda_i$ and $\\lambda_i \\leq C$. These two inequalities define then the matrix $\\boldsymbol{G}$ and the vector $\\boldsymbol{h}$.\n", - "\n", - "\n", - "\n", - "## Different kernels and Mercer's theorem\n", - "\n", - "There are several popular kernels being used. These are\n", - "1. Linear: $K(\\boldsymbol{x},\\boldsymbol{y})=\\boldsymbol{x}^T\\boldsymbol{y}$,\n", - "\n", - "2. Polynomial: $K(\\boldsymbol{x},\\boldsymbol{y})=(\\boldsymbol{x}^T\\boldsymbol{y}+\\gamma)^d$,\n", - "\n", - "3. Gaussian Radial Basis Function: $K(\\boldsymbol{x},\\boldsymbol{y})=\\exp{\\left(-\\gamma\\vert\\vert\\boldsymbol{x}-\\boldsymbol{y}\\vert\\vert^2\\right)}$,\n", - "\n", - "4. Tanh: $K(\\boldsymbol{x},\\boldsymbol{y})=\\tanh{(\\boldsymbol{x}^T\\boldsymbol{y}+\\gamma)}$,\n", - "\n", - "and many other ones.\n", - "\n", - "An important theorem for us is [Mercer's\n", - "theorem](https://en.wikipedia.org/wiki/Mercer%27s_theorem). The\n", - "theorem states that if a kernel function $K$ is symmetric, continuous\n", - "and leads to a positive semi-definite matrix $\\boldsymbol{P}$ then there\n", - "exists a function $\\phi$ that maps $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_j$ into\n", - "another space (possibly with much higher dimensions) such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "So you can use $K$ as a kernel since you know $\\phi$ exists, even if\n", - "you don’t know what $\\phi$ is. \n", - "\n", - "Note that some frequently used kernels (such as the Sigmoid kernel)\n", - "don’t respect all of Mercer’s conditions, yet they generally work well\n", - "in practice.\n", - "\n", - "\n", - "## The moons example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from __future__ import division, print_function, unicode_literals\n", - "\n", - "import numpy as np\n", - "np.random.seed(42)\n", - "\n", - "import matplotlib\n", - "import matplotlib.pyplot as plt\n", - "plt.rcParams['axes.labelsize'] = 14\n", - "plt.rcParams['xtick.labelsize'] = 12\n", - "plt.rcParams['ytick.labelsize'] = 12\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "from sklearn import datasets\n", - "\n", - "\n", - "\n", - "from sklearn.pipeline import Pipeline\n", - "from sklearn.preprocessing import StandardScaler\n", - "from sklearn.svm import LinearSVC\n", - "\n", - "\n", - "from sklearn.datasets import make_moons\n", - "X, y = make_moons(n_samples=100, noise=0.15, random_state=42)\n", - "\n", - "def plot_dataset(X, y, axes):\n", - " plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"bs\")\n", - " plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"g^\")\n", - " plt.axis(axes)\n", - " plt.grid(True, which='both')\n", - " plt.xlabel(r\"$x_1$\", fontsize=20)\n", - " plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n", - "\n", - "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - "plt.show()\n", - "\n", - "from sklearn.datasets import make_moons\n", - "from sklearn.pipeline import Pipeline\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "\n", - "polynomial_svm_clf = Pipeline([\n", - " (\"poly_features\", PolynomialFeatures(degree=3)),\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", LinearSVC(C=10, loss=\"hinge\", random_state=42))\n", - " ])\n", - "\n", - "polynomial_svm_clf.fit(X, y)\n", - "\n", - "def plot_predictions(clf, axes):\n", - " x0s = np.linspace(axes[0], axes[1], 100)\n", - " x1s = np.linspace(axes[2], axes[3], 100)\n", - " x0, x1 = np.meshgrid(x0s, x1s)\n", - " X = np.c_[x0.ravel(), x1.ravel()]\n", - " y_pred = clf.predict(X).reshape(x0.shape)\n", - " y_decision = clf.decision_function(X).reshape(x0.shape)\n", - " plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)\n", - " plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)\n", - "\n", - "plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])\n", - "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - "\n", - "plt.show()\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "\n", - "poly_kernel_svm_clf = Pipeline([\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", SVC(kernel=\"poly\", degree=3, coef0=1, C=5))\n", - " ])\n", - "poly_kernel_svm_clf.fit(X, y)\n", - "\n", - "poly100_kernel_svm_clf = Pipeline([\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", SVC(kernel=\"poly\", degree=10, coef0=100, C=5))\n", - " ])\n", - "poly100_kernel_svm_clf.fit(X, y)\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "\n", - "plt.subplot(121)\n", - "plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\n", - "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - "plt.title(r\"$d=3, r=1, C=5$\", fontsize=18)\n", - "\n", - "plt.subplot(122)\n", - "plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\n", - "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - "plt.title(r\"$d=10, r=100, C=5$\", fontsize=18)\n", - "\n", - "plt.show()\n", - "\n", - "def gaussian_rbf(x, landmark, gamma):\n", - " return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)\n", - "\n", - "gamma = 0.3\n", - "\n", - "x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)\n", - "x2s = gaussian_rbf(x1s, -2, gamma)\n", - "x3s = gaussian_rbf(x1s, 1, gamma)\n", - "\n", - "XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]\n", - "yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "\n", - "plt.subplot(121)\n", - "plt.grid(True, which='both')\n", - "plt.axhline(y=0, color='k')\n", - "plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c=\"red\")\n", - "plt.plot(X1D[:, 0][yk==0], np.zeros(4), \"bs\")\n", - "plt.plot(X1D[:, 0][yk==1], np.zeros(5), \"g^\")\n", - "plt.plot(x1s, x2s, \"g--\")\n", - "plt.plot(x1s, x3s, \"b:\")\n", - "plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])\n", - "plt.xlabel(r\"$x_1$\", fontsize=20)\n", - "plt.ylabel(r\"Similarity\", fontsize=14)\n", - "plt.annotate(r'$\\mathbf{x}$',\n", - " xy=(X1D[3, 0], 0),\n", - " xytext=(-0.5, 0.20),\n", - " ha=\"center\",\n", - " arrowprops=dict(facecolor='black', shrink=0.1),\n", - " fontsize=18,\n", - " )\n", - "plt.text(-2, 0.9, \"$x_2$\", ha=\"center\", fontsize=20)\n", - "plt.text(1, 0.9, \"$x_3$\", ha=\"center\", fontsize=20)\n", - "plt.axis([-4.5, 4.5, -0.1, 1.1])\n", - "\n", - "plt.subplot(122)\n", - "plt.grid(True, which='both')\n", - "plt.axhline(y=0, color='k')\n", - "plt.axvline(x=0, color='k')\n", - "plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], \"bs\")\n", - "plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], \"g^\")\n", - "plt.xlabel(r\"$x_2$\", fontsize=20)\n", - "plt.ylabel(r\"$x_3$ \", fontsize=20, rotation=0)\n", - "plt.annotate(r'$\\phi\\left(\\mathbf{x}\\right)$',\n", - " xy=(XK[3, 0], XK[3, 1]),\n", - " xytext=(0.65, 0.50),\n", - " ha=\"center\",\n", - " arrowprops=dict(facecolor='black', shrink=0.1),\n", - " fontsize=18,\n", - " )\n", - "plt.plot([-0.1, 1.1], [0.57, -0.1], \"r--\", linewidth=3)\n", - "plt.axis([-0.1, 1.1, -0.1, 1.1])\n", - " \n", - "plt.subplots_adjust(right=1)\n", - "\n", - "plt.show()\n", - "\n", - "\n", - "x1_example = X1D[3, 0]\n", - "for landmark in (-2, 1):\n", - " k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)\n", - " print(\"Phi({}, {}) = {}\".format(x1_example, landmark, k))\n", - "\n", - "rbf_kernel_svm_clf = Pipeline([\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", SVC(kernel=\"rbf\", gamma=5, C=0.001))\n", - " ])\n", - "rbf_kernel_svm_clf.fit(X, y)\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "\n", - "gamma1, gamma2 = 0.1, 5\n", - "C1, C2 = 0.001, 1000\n", - "hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)\n", - "\n", - "svm_clfs = []\n", - "for gamma, C in hyperparams:\n", - " rbf_kernel_svm_clf = Pipeline([\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", SVC(kernel=\"rbf\", gamma=gamma, C=C))\n", - " ])\n", - " rbf_kernel_svm_clf.fit(X, y)\n", - " svm_clfs.append(rbf_kernel_svm_clf)\n", - "\n", - "plt.figure(figsize=(11, 7))\n", - "\n", - "for i, svm_clf in enumerate(svm_clfs):\n", - " plt.subplot(221 + i)\n", - " plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])\n", - " plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - " gamma, C = hyperparams[i]\n", - " plt.title(r\"$\\gamma = {}, C = {}$\".format(gamma, C), fontsize=16)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Mathematical optimization of convex functions\n", - "\n", - "A mathematical (quadratic) optimization problem, or just optimization problem, has the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " &\\mathrm{min}_{\\lambda}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{\\lambda}^T\\boldsymbol{P}\\boldsymbol{\\lambda}+\\boldsymbol{q}^T\\boldsymbol{\\lambda},\\\\ \\nonumber\n", - " &\\mathrm{subject\\hspace{0.1cm}to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{\\lambda} \\preceq \\boldsymbol{h} \\wedge \\boldsymbol{A}\\boldsymbol{\\lambda}=f.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to some constraints for say a selected set $i=1,2,\\dots, n$.\n", - "In our case we are optimizing with respect to the Lagrangian multipliers $\\lambda_i$, and the\n", - "vector $\\boldsymbol{\\lambda}=[\\lambda_1, \\lambda_2,\\dots, \\lambda_n]$ is the optimization variable we are dealing with.\n", - "\n", - "In our case we are particularly interested in a class of optimization problems called convex optmization problems. \n", - "In our discussion on gradient descent methods we discussed at length the definition of a convex function. \n", - "\n", - "Convex optimization problems play a central role in applied mathematics and we recommend strongly [Boyd and Vandenberghe's text on the topics](http://web.stanford.edu/~boyd/cvxbook/).\n", - "\n", - "\n", - "\n", - "\n", - "If we use Python as programming language and wish to venture beyond\n", - "**scikit-learn**, **tensorflow** and similar software which makes our\n", - "lives so much easier, we need to dive into the wonderful world of\n", - "quadratic programming. We can, if we wish, solve the minimization\n", - "problem using say standard gradient methods or conjugate gradient\n", - "methods. However, these methods tend to exhibit a rather slow\n", - "converge. So, welcome to the promised land of quadratic programming.\n", - "\n", - "The functions we need are contained in the quadratic programming package **CVXOPT** and we need to import it together with **numpy** as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy\n", - "import cvxopt" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This will make our life much easier. You don't need t write your own optimizer.\n", - "\n", - "\n", - "\n", - "We remind ourselves about the general problem we want to solve" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " &\\mathrm{min}_{x}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{P}\\boldsymbol{x}+\\boldsymbol{q}^T\\boldsymbol{x},\\\\ \\nonumber\n", - " &\\mathrm{subject\\hspace{0.1cm} to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{x} \\preceq \\boldsymbol{h} \\wedge \\boldsymbol{A}\\boldsymbol{x}=f.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " &\\mathrm{min}_{x}\\hspace{0.2cm} \\frac{1}{2}x^2+5x+3y \\\\ \\nonumber\n", - " &\\mathrm{subject to} \\\\ \\nonumber\n", - " &x, y \\geq 0 \\\\ \\nonumber\n", - " &x+3y \\geq 15 \\\\ \\nonumber\n", - " &2x+5y \\leq 100 \\\\ \\nonumber\n", - " &3x+4y \\leq 80. \\\\ \\nonumber\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The minimization problem can be rewritten in terms of vectors and matrices as (with $x$ and $y$ being the unknowns)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{2}\\begin{bmatrix} x\\\\ y \\end{bmatrix}^T \\begin{bmatrix} 1 & 0\\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} + \\begin{bmatrix}3\\\\ 4 \\end{bmatrix}^T \\begin{bmatrix}x \\\\ y \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Similarly, we can now set up the inequalities (we need to change $\\geq$ to $\\leq$ by multiplying with $-1$ on bot sides) as the following matrix-vector equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\\\ -1 & -3 \\\\ 2 & 5 \\\\ 3 & 4\\end{bmatrix}\\begin{bmatrix} x \\\\ y\\end{bmatrix} \\preceq \\begin{bmatrix}0 \\\\ 0\\\\ -15 \\\\ 100 \\\\ 80\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We have collapsed all the inequalities into a single matrix $\\boldsymbol{G}$. We see also that our matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{P} =\\begin{bmatrix} 1 & 0\\\\ 0 & 0 \\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "is clearly positive semi-definite (all eigenvalues larger or equal zero). \n", - "Finally, the vector $\\boldsymbol{h}$ is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{h} = \\begin{bmatrix}0 \\\\ 0\\\\ -15 \\\\ 100 \\\\ 80\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Since we don't have any equalities the matrix $\\boldsymbol{A}$ is set to zero\n", - "The following code solves the equations for us" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Import the necessary packages\n", - "import numpy\n", - "from cvxopt import matrix\n", - "from cvxopt import solvers\n", - "P = matrix(numpy.diag([1,0]), tc=’d’)\n", - "q = matrix(numpy.array([3,4]), tc=’d’)\n", - "G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)\n", - "h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)\n", - "# Construct the QP, invoke solver\n", - "sol = solvers.qp(P,q,G,h)\n", - "# Extract optimal value and solution\n", - "sol[’x’] \n", - "sol[’primal objective’]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the **slack** parameter $C$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{2} \\boldsymbol{\\lambda}^T\\begin{bmatrix} y_1y_1K(\\boldsymbol{x}_1,\\boldsymbol{x}_1) & y_1y_2K(\\boldsymbol{x}_1,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_1,\\boldsymbol{x}_n) \\\\\n", - "y_2y_1K(\\boldsymbol{x}_2,\\boldsymbol{x}_1) & y_2y_2K(\\boldsymbol{x}_2,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_2,\\boldsymbol{x}_n) \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "y_ny_1K(\\boldsymbol{x}_n,\\boldsymbol{x}_1) & y_ny_2K(\\boldsymbol{x}_n\\boldsymbol{x}_2) & \\dots & \\dots & y_ny_nK(\\boldsymbol{x}_n,\\boldsymbol{x}_n) \\\\\n", - "\\end{bmatrix}\\boldsymbol{\\lambda}-\\mathbb{I}\\boldsymbol{\\lambda},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$. Here we defined the vectors $\\boldsymbol{\\lambda} =[\\lambda_1,\\lambda_2,\\dots,\\lambda_n]$ and \n", - "$\\boldsymbol{y}=[y_1,y_2,\\dots,y_n]$. \n", - "With the slack constants this leads to the additional constraint $0\\leq \\lambda_i \\leq C$.\n", - "\n", - "**code will be added**" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter6.ipynb b/doc/LectureNotes/_build/html/_sources/chapter6.ipynb deleted file mode 100644 index 01884f54a..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter6.ipynb +++ /dev/null @@ -1,1301 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Decision trees, overarching aims\n", - "\n", - "\n", - "We start here with the most basic algorithm, the so-called decision\n", - "tree. With this basic algorithm we can in turn build more complex\n", - "networks, spanning from homogeneous and heterogenous forests (bagging,\n", - "random forests and more) to one of the most popular supervised\n", - "algorithms nowadays, the extreme gradient boosting, or just\n", - "XGBoost. But let us start with the simplest possible ingredient.\n", - "\n", - "Decision trees are supervised learning algorithms used for both,\n", - "classification and regression tasks.\n", - "\n", - "\n", - "The main idea of decision trees\n", - "is to find those descriptive features which contain the most\n", - "**information** regarding the target feature and then split the dataset\n", - "along the values of these features such that the target feature values\n", - "for the resulting underlying datasets are as pure as possible.\n", - "\n", - "The descriptive features which reproduce best the target/output features are normally said\n", - "to be the most informative ones. The process of finding the **most\n", - "informative** feature is done until we accomplish a stopping criteria\n", - "where we then finally end up in so called **leaf nodes**. \n", - "\n", - "## Basics of a tree\n", - "\n", - "A decision tree is typically divided into a **root node**, the **interior nodes**,\n", - "and the final **leaf nodes** or just **leaves**. These entities are then connected by so-called **branches**.\n", - "\n", - "The leaf nodes\n", - "contain the predictions we will make for new query instances presented\n", - "to our trained model. This is possible since the model has \n", - "learned the underlying structure of the training data and hence can,\n", - "given some assumptions, make predictions about the target feature value\n", - "(class) of unseen query instances.\n", - "\n", - "\n", - "## General Features\n", - "\n", - "The overarching approach to decision trees is a top-down approach.\n", - "\n", - "* A leaf provides the classification of a given instance.\n", - "\n", - "* A node specifies a test of some attribute of the instance.\n", - "\n", - "* A branch corresponds to a possible values of an attribute.\n", - "\n", - "* An instance is classified by starting at the root node of the tree, testing the attribute specified by this node, then moving down the tree branch corresponding to the value of the attribute in the given example.\n", - "\n", - "This process is then repeated for the subtree rooted at the new\n", - "node.\n", - "\n", - "\n", - "\n", - "In simplified terms, the process of training a decision tree and\n", - "predicting the target features of query instances is as follows:\n", - "\n", - "1. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature\n", - "\n", - "2. Train the decision tree model by continuously splitting the target feature along the values of the descriptive features using a measure of information gain during the training process\n", - "\n", - "3. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the *predictions* we want to make for new query instances\n", - "\n", - "4. Show query instances to the tree and run down the tree until we arrive at leaf nodes\n", - "\n", - "Then we are essentially done!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.linear_model import LinearRegression\n", - "\n", - "steps=250\n", - "\n", - "distance=0\n", - "x=0\n", - "distance_list=[]\n", - "steps_list=[]\n", - "while x\n", - "\n", - "Day Outlook Temperature Humidity Wind Ride \n", - "\n", - "\n", - " 1 Sunny Hot High Weak 0 \n", - " 2 Sunny Hot High Strong 1 \n", - " 3 Overcast Hot High Weak 1 \n", - " 4 Rain Mild High Weak 1 \n", - " 5 Rain Cool Normal Weak 1 \n", - " 6 Rain Cool Normal Strong 0 \n", - " 7 Overcast Cool Normal Strong 1 \n", - " 8 Sunny Mild High Weak 0 \n", - " 9 Sunny Cool Normal Weak 1 \n", - " 10 Rain Mild Normal Weak 1 \n", - " 11 Sunny Mild Normal Strong 1 \n", - " 12 Overcast Mild High Strong 1 \n", - " 13 Overcast Hot Normal Weak 1 \n", - " 14 Rain Mild High Strong 0 \n", - "\n", - "\n", - "\n", - "### Simple Python Code to read in Data and perform Classification" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.tree import export_graphviz\n", - "from sklearn.preprocessing import StandardScaler, OneHotEncoder\n", - "from sklearn.compose import ColumnTransformer\n", - "from IPython.display import Image \n", - "from pydot import graph_from_dot_data\n", - "import os\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"rideclass.csv\"),'r')\n", - "\n", - "# Read the experimental data with Pandas\n", - "from IPython.display import display\n", - "ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))\n", - "ridedata = pd.DataFrame(ridedata)\n", - "\n", - "# Features and targets\n", - "X = ridedata.loc[:, ridedata.columns != 'Ride'].values\n", - "y = ridedata.loc[:, ridedata.columns == 'Ride'].values\n", - "\n", - "# Create the encoder.\n", - "encoder = OneHotEncoder(handle_unknown=\"ignore\")\n", - "# Assume for simplicity all features are categorical.\n", - "encoder.fit(X) \n", - "# Apply the encoder.\n", - "X = encoder.transform(X)\n", - "print(X)\n", - "# Then do a Classification tree\n", - "tree_clf = DecisionTreeClassifier(max_depth=2)\n", - "tree_clf.fit(X, y)\n", - "print(\"Train set accuracy with Decision Tree: {:.2f}\".format(tree_clf.score(X,y)))\n", - "#transfer to a decision tree graph\n", - "export_graphviz(\n", - " tree_clf,\n", - " out_file=\"DataFiles/ride.dot\",\n", - " rounded=True,\n", - " filled=True\n", - ")\n", - "cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'\n", - "os.system(cmd)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The above functions (gini, entropy and misclassification error) are\n", - "important components of the so-called CART algorithm. We will discuss\n", - "this algorithm below after we have discussed the information gain\n", - "algorithm ID3.\n", - "\n", - "In the example here we have converted all our attributes into numerical values $0,1,2$ etc." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Split a dataset based on an attribute and an attribute value\n", - "def test_split(index, value, dataset):\n", - "\tleft, right = list(), list()\n", - "\tfor row in dataset:\n", - "\t\tif row[index] < value:\n", - "\t\t\tleft.append(row)\n", - "\t\telse:\n", - "\t\t\tright.append(row)\n", - "\treturn left, right\n", - " \n", - "# Calculate the Gini index for a split dataset\n", - "def gini_index(groups, classes):\n", - "\t# count all samples at split point\n", - "\tn_instances = float(sum([len(group) for group in groups]))\n", - "\t# sum weighted Gini index for each group\n", - "\tgini = 0.0\n", - "\tfor group in groups:\n", - "\t\tsize = float(len(group))\n", - "\t\t# avoid divide by zero\n", - "\t\tif size == 0:\n", - "\t\t\tcontinue\n", - "\t\tscore = 0.0\n", - "\t\t# score the group based on the score for each class\n", - "\t\tfor class_val in classes:\n", - "\t\t\tp = [row[-1] for row in group].count(class_val) / size\n", - "\t\t\tscore += p * p\n", - "\t\t# weight the group score by its relative size\n", - "\t\tgini += (1.0 - score) * (size / n_instances)\n", - "\treturn gini\n", - "\n", - "# Select the best split point for a dataset\n", - "def get_split(dataset):\n", - "\tclass_values = list(set(row[-1] for row in dataset))\n", - "\tb_index, b_value, b_score, b_groups = 999, 999, 999, None\n", - "\tfor index in range(len(dataset[0])-1):\n", - "\t\tfor row in dataset:\n", - "\t\t\tgroups = test_split(index, row[index], dataset)\n", - "\t\t\tgini = gini_index(groups, class_values)\n", - "\t\t\tprint('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))\n", - "\t\t\tif gini < b_score:\n", - "\t\t\t\tb_index, b_value, b_score, b_groups = index, row[index], gini, groups\n", - "\treturn {'index':b_index, 'value':b_value, 'groups':b_groups}\n", - " \n", - "dataset = [[0,0,0,0,0],\n", - " [0,0,0,1,1],\n", - " [1,0,0,0,1],\n", - " [2,1,0,0,1],\n", - " [2,2,1,0,1],\n", - " [2,2,1,1,0],\n", - " [1,2,1,1,1],\n", - " [0,1,0,0,0],\n", - " [0,2,1,0,1],\n", - " [2,1,1,0,1],\n", - " [0,1,1,1,1],\n", - " [1,1,0,1,1],\n", - " [1,0,1,0,1],\n", - " [2,1,0,1,0]]\n", - "\n", - "split = get_split(dataset)\n", - "print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Entropy and the ID3 algorithm\n", - "\n", - "The ID3 algorithm learns decision trees by constructing\n", - "them in a top down way, beginning with the question **which attribute should be tested at the root of the tree**?\n", - "\n", - "1. Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.\n", - "\n", - "2. The best attribute is selected and used as the test at the root node of the tree.\n", - "\n", - "3. A descendant of the root node is then created for each possible value of this attribute.\n", - "\n", - "4. Training examples are sorted to the appropriate descendant node.\n", - "\n", - "5. The entire process is then repeated using the training examples associated with each descendant node to select the best attribute to test at that point in the tree.\n", - "\n", - "6. This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices. \n", - "\n", - "The ID3 algorithm selects which attribute to test at each node in the\n", - "tree.\n", - "\n", - "We would like to select the attribute that is most useful for classifying\n", - "examples.\n", - "\n", - "What is a good quantitative measure of the worth of an attribute?\n", - "\n", - "Information gain measures how well a given attribute separates the\n", - "training examples according to their target classification.\n", - "\n", - "The ID3 algorithm uses this information gain measure to select among the candidate\n", - "attributes at each step while growing the tree.\n", - "\n", - "\n", - "### Cancer Data again now with Decision Trees and other Methods" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.svm import SVC\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "# Logistic Regression\n", - "logreg = LogisticRegression(solver='lbfgs')\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n", - "# Support vector machine\n", - "svm = SVC(gamma='auto', C=100)\n", - "svm.fit(X_train, y_train)\n", - "print(\"Test set accuracy with SVM: {:.2f}\".format(svm.score(X_test,y_test)))\n", - "# Decision Trees\n", - "deep_tree_clf = DecisionTreeClassifier(max_depth=None)\n", - "deep_tree_clf.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Decision Trees: {:.2f}\".format(deep_tree_clf.score(X_test,y_test)))\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Logistic Regression\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "# Support Vector Machine\n", - "svm.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy SVM with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "# Decision Trees\n", - "deep_tree_clf.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy with Decision Trees and scaled data: {:.2f}\".format(deep_tree_clf.score(X_test_scaled,y_test)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Another example, the moons again" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from __future__ import division, print_function, unicode_literals\n", - "\n", - "# Common imports\n", - "import numpy as np\n", - "import os\n", - "\n", - "# to make this notebook's output stable across runs\n", - "np.random.seed(42)\n", - "\n", - "# To plot pretty figures\n", - "import matplotlib\n", - "import matplotlib.pyplot as plt\n", - "from matplotlib.colors import ListedColormap\n", - "plt.rcParams['axes.labelsize'] = 14\n", - "plt.rcParams['xtick.labelsize'] = 12\n", - "plt.rcParams['ytick.labelsize'] = 12\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "from sklearn import datasets\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.datasets import make_moons\n", - "from sklearn.tree import export_graphviz\n", - "\n", - "Xm, ym = make_moons(n_samples=100, noise=0.25, random_state=53)\n", - "\n", - "deep_tree_clf1 = DecisionTreeClassifier(random_state=42)\n", - "deep_tree_clf2 = DecisionTreeClassifier(min_samples_leaf=4, random_state=42)\n", - "deep_tree_clf1.fit(Xm, ym)\n", - "deep_tree_clf2.fit(Xm, ym)\n", - "\n", - "\n", - "def plot_decision_boundary(clf, X, y, axes=[0, 7.5, 0, 3], iris=True, legend=False, plot_training=True):\n", - " x1s = np.linspace(axes[0], axes[1], 100)\n", - " x2s = np.linspace(axes[2], axes[3], 100)\n", - " x1, x2 = np.meshgrid(x1s, x2s)\n", - " X_new = np.c_[x1.ravel(), x2.ravel()]\n", - " y_pred = clf.predict(X_new).reshape(x1.shape)\n", - " custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])\n", - " plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)\n", - " if not iris:\n", - " custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])\n", - " plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)\n", - " if plot_training:\n", - " plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\", label=\"Iris-Setosa\")\n", - " plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\", label=\"Iris-Versicolor\")\n", - " plt.plot(X[:, 0][y==2], X[:, 1][y==2], \"g^\", label=\"Iris-Virginica\")\n", - " plt.axis(axes)\n", - " if iris:\n", - " plt.xlabel(\"Petal length\", fontsize=14)\n", - " plt.ylabel(\"Petal width\", fontsize=14)\n", - " else:\n", - " plt.xlabel(r\"$x_1$\", fontsize=18)\n", - " plt.ylabel(r\"$x_2$\", fontsize=18, rotation=0)\n", - " if legend:\n", - " plt.legend(loc=\"lower right\", fontsize=14)\n", - "plt.figure(figsize=(11, 4))\n", - "plt.subplot(121)\n", - "plot_decision_boundary(deep_tree_clf1, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)\n", - "plt.title(\"No restrictions\", fontsize=16)\n", - "plt.subplot(122)\n", - "plot_decision_boundary(deep_tree_clf2, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)\n", - "plt.title(\"min_samples_leaf = {}\".format(deep_tree_clf2.min_samples_leaf), fontsize=14)\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "np.random.seed(6)\n", - "Xs = np.random.rand(100, 2) - 0.5\n", - "ys = (Xs[:, 0] > 0).astype(np.float32) * 2\n", - "\n", - "angle = np.pi/4\n", - "rotation_matrix = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])\n", - "Xsr = Xs.dot(rotation_matrix)\n", - "\n", - "tree_clf_s = DecisionTreeClassifier(random_state=42)\n", - "tree_clf_s.fit(Xs, ys)\n", - "tree_clf_sr = DecisionTreeClassifier(random_state=42)\n", - "tree_clf_sr.fit(Xsr, ys)\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "plt.subplot(121)\n", - "plot_decision_boundary(tree_clf_s, Xs, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)\n", - "plt.subplot(122)\n", - "plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Quadratic training set + noise\n", - "np.random.seed(42)\n", - "m = 200\n", - "X = np.random.rand(m, 1)\n", - "y = 4 * (X - 0.5) ** 2\n", - "y = y + np.random.randn(m, 1) / 10" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.tree import DecisionTreeRegressor\n", - "\n", - "tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)\n", - "tree_reg.fit(X, y)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.tree import DecisionTreeRegressor\n", - "\n", - "tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)\n", - "tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)\n", - "tree_reg1.fit(X, y)\n", - "tree_reg2.fit(X, y)\n", - "\n", - "def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel=\"$y$\"):\n", - " x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)\n", - " y_pred = tree_reg.predict(x1)\n", - " plt.axis(axes)\n", - " plt.xlabel(\"$x_1$\", fontsize=18)\n", - " if ylabel:\n", - " plt.ylabel(ylabel, fontsize=18, rotation=0)\n", - " plt.plot(X, y, \"b.\")\n", - " plt.plot(x1, y_pred, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "plt.subplot(121)\n", - "plot_regression_predictions(tree_reg1, X, y)\n", - "for split, style in ((0.1973, \"k-\"), (0.0917, \"k--\"), (0.7718, \"k--\")):\n", - " plt.plot([split, split], [-0.2, 1], style, linewidth=2)\n", - "plt.text(0.21, 0.65, \"Depth=0\", fontsize=15)\n", - "plt.text(0.01, 0.2, \"Depth=1\", fontsize=13)\n", - "plt.text(0.65, 0.8, \"Depth=1\", fontsize=13)\n", - "plt.legend(loc=\"upper center\", fontsize=18)\n", - "plt.title(\"max_depth=2\", fontsize=14)\n", - "\n", - "plt.subplot(122)\n", - "plot_regression_predictions(tree_reg2, X, y, ylabel=None)\n", - "for split, style in ((0.1973, \"k-\"), (0.0917, \"k--\"), (0.7718, \"k--\")):\n", - " plt.plot([split, split], [-0.2, 1], style, linewidth=2)\n", - "for split in (0.0458, 0.1298, 0.2873, 0.9040):\n", - " plt.plot([split, split], [-0.2, 1], \"k:\", linewidth=1)\n", - "plt.text(0.3, 0.5, \"Depth=2\", fontsize=13)\n", - "plt.title(\"max_depth=3\", fontsize=14)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "tree_reg1 = DecisionTreeRegressor(random_state=42)\n", - "tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)\n", - "tree_reg1.fit(X, y)\n", - "tree_reg2.fit(X, y)\n", - "\n", - "x1 = np.linspace(0, 1, 500).reshape(-1, 1)\n", - "y_pred1 = tree_reg1.predict(x1)\n", - "y_pred2 = tree_reg2.predict(x1)\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "\n", - "plt.subplot(121)\n", - "plt.plot(X, y, \"b.\")\n", - "plt.plot(x1, y_pred1, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n", - "plt.axis([0, 1, -0.2, 1.1])\n", - "plt.xlabel(\"$x_1$\", fontsize=18)\n", - "plt.ylabel(\"$y$\", fontsize=18, rotation=0)\n", - "plt.legend(loc=\"upper center\", fontsize=18)\n", - "plt.title(\"No restrictions\", fontsize=14)\n", - "\n", - "plt.subplot(122)\n", - "plt.plot(X, y, \"b.\")\n", - "plt.plot(x1, y_pred2, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n", - "plt.axis([0, 1, -0.2, 1.1])\n", - "plt.xlabel(\"$x_1$\", fontsize=18)\n", - "plt.title(\"min_samples_leaf={}\".format(tree_reg2.min_samples_leaf), fontsize=14)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Pros and cons of trees, pros\n", - "\n", - "* White box, easy to interpret model. Some people believe that decision trees more closely mirror human decision-making than do the regression and classification approaches discussed earlier (think of support vector machines)\n", - "\n", - "* Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!\n", - "\n", - "* No feature normalization needed\n", - "\n", - "* Tree models can handle both continuous and categorical data (Classification and Regression Trees)\n", - "\n", - "* Can model nonlinear relationships\n", - "\n", - "* Can model interactions between the different descriptive features\n", - "\n", - "* Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)\n", - "\n", - "### Disadvantages\n", - "\n", - "* Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches\n", - "\n", - "* If continuous features are used the tree may become quite large and hence less interpretable\n", - "\n", - "* Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented\n", - "\n", - "* Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests\n", - "\n", - "* Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones. \n", - "\n", - "* If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data\n", - "\n", - "* Features with many levels may be preferred over features with less levels since for them it is *more easy* to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain\n", - "\n", - "However, by aggregating many decision trees, using methods like\n", - "bagging, random forests, and boosting, the predictive performance of\n", - "trees can be substantially improved." - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter7.ipynb b/doc/LectureNotes/_build/html/_sources/chapter7.ipynb deleted file mode 100644 index 49dd8591c..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter7.ipynb +++ /dev/null @@ -1,1580 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods\n", - "\n", - "As stated previously and seen in many of the examples discussed in the previous chapter about\n", - "a single decision tree, we often end up overfitting our training\n", - "data. This normally means that we have a high variance. Can we reduce\n", - "the variance of a statistical learning method?\n", - "\n", - "This leads us to a set of different methods that can combine different\n", - "machine learning algorithms or just use one of them to construct\n", - "forests and jungles of trees, homogeneous ones or heterogenous\n", - "ones. These methods are recognized by different names which we will\n", - "try to explain here. These are\n", - "\n", - "1. Voting classifiers\n", - "\n", - "2. Bagging and Pasting\n", - "\n", - "3. Random forests\n", - "\n", - "4. Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)\n", - "\n", - "We discuss these methods here.\n", - "\n", - "### An Overview of Ensemble Methods\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Bagging\n", - "\n", - "The **plain** decision trees suffer from high\n", - "variance. This means that if we split the training data into two parts\n", - "at random, and fit a decision tree to both halves, the results that we\n", - "get could be quite different. In contrast, a procedure with low\n", - "variance will yield similar results if applied repeatedly to distinct\n", - "data sets; linear regression tends to have low variance, if the ratio\n", - "of $n$ to $p$ is moderately large. \n", - "\n", - "**Bootstrap aggregation**, or just **bagging**, is a\n", - "general-purpose procedure for reducing the variance of a statistical\n", - "learning method. \n", - "\n", - "\n", - "Bagging typically results in improved accuracy\n", - "over prediction using a single tree. Unfortunately, however, it can be\n", - "difficult to interpret the resulting model. Recall that one of the\n", - "advantages of decision trees is the attractive and easily interpreted\n", - "diagram that results.\n", - "\n", - "However, when we bag a large number of trees, it is no longer\n", - "possible to represent the resulting statistical learning procedure\n", - "using a single tree, and it is no longer clear which variables are\n", - "most important to the procedure. Thus, bagging improves prediction\n", - "accuracy at the expense of interpretability. Although the collection\n", - "of bagged trees is much more difficult to interpret than a single\n", - "tree, one can obtain an overall summary of the importance of each\n", - "predictor using the MSE (for bagging regression trees) or the Gini\n", - "index (for bagging classification trees). In the case of bagging\n", - "regression trees, we can record the total amount that the MSE is\n", - "decreased due to splits over a given predictor, averaged over all $B$ possible\n", - "trees. A large value indicates an important predictor. Similarly, in\n", - "the context of bagging classification trees, we can add up the total\n", - "amount that the Gini index is decreased by splits over a given\n", - "predictor, averaged over all $B$ trees." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "heads_proba = 0.51\n", - "coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)\n", - "cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)\n", - "plt.figure(figsize=(8,3.5))\n", - "plt.plot(cumulative_heads_ratio)\n", - "plt.plot([0, 10000], [0.51, 0.51], \"k--\", linewidth=2, label=\"51%\")\n", - "plt.plot([0, 10000], [0.5, 0.5], \"k-\", label=\"50%\")\n", - "plt.xlabel(\"Number of coin tosses\")\n", - "plt.ylabel(\"Heads ratio\")\n", - "plt.legend(loc=\"lower right\")\n", - "plt.axis([0, 10000, 0.42, 0.58])\n", - "save_fig(\"votingsimple\")\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.model_selection import train_test_split\n", - "from sklearn.datasets import make_moons\n", - "\n", - "X, y = make_moons(n_samples=500, noise=0.30, random_state=42)\n", - "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)\n", - "\n", - "from sklearn.ensemble import RandomForestClassifier\n", - "from sklearn.ensemble import VotingClassifier\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.svm import SVC\n", - "\n", - "log_clf = LogisticRegression(solver=\"liblinear\", random_state=42)\n", - "rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)\n", - "svm_clf = SVC(gamma=\"auto\", random_state=42)\n", - "\n", - "voting_clf = VotingClassifier(\n", - " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n", - " voting='hard')\n", - "\n", - "voting_clf.fit(X_train, y_train)\n", - "\n", - "from sklearn.metrics import accuracy_score\n", - "\n", - "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n", - " clf.fit(X_train, y_train)\n", - " y_pred = clf.predict(X_test)\n", - " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))\n", - "\n", - "log_clf = LogisticRegression(solver=\"liblinear\", random_state=42)\n", - "rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)\n", - "svm_clf = SVC(gamma=\"auto\", probability=True, random_state=42)\n", - "\n", - "voting_clf = VotingClassifier(\n", - " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n", - " voting='soft')\n", - "voting_clf.fit(X_train, y_train)\n", - "\n", - "from sklearn.metrics import accuracy_score\n", - "\n", - "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n", - " clf.fit(X_train, y_train)\n", - " y_pred = clf.predict(X_test)\n", - " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.model_selection import train_test_split\n", - "from sklearn.datasets import make_moons\n", - "\n", - "X, y = make_moons(n_samples=500, noise=0.30, random_state=42)\n", - "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)\n", - "from sklearn.ensemble import RandomForestClassifier\n", - "from sklearn.ensemble import VotingClassifier\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.svm import SVC\n", - "\n", - "log_clf = LogisticRegression(random_state=42)\n", - "rnd_clf = RandomForestClassifier(random_state=42)\n", - "svm_clf = SVC(random_state=42)\n", - "\n", - "voting_clf = VotingClassifier(\n", - " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n", - " voting='hard')\n", - "voting_clf.fit(X_train, y_train)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.metrics import accuracy_score\n", - "\n", - "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n", - " clf.fit(X_train, y_train)\n", - " y_pred = clf.predict(X_test)\n", - " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "log_clf = LogisticRegression(random_state=42)\n", - "rnd_clf = RandomForestClassifier(random_state=42)\n", - "svm_clf = SVC(probability=True, random_state=42)\n", - "\n", - "voting_clf = VotingClassifier(\n", - " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n", - " voting='soft')\n", - "voting_clf.fit(X_train, y_train)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.metrics import accuracy_score\n", - "\n", - "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n", - " clf.fit(X_train, y_train)\n", - " y_pred = clf.predict(X_test)\n", - " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Bagging Examples" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.ensemble import BaggingClassifier\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "\n", - "bag_clf = BaggingClassifier(\n", - " DecisionTreeClassifier(random_state=42), n_estimators=500,\n", - " max_samples=100, bootstrap=True, n_jobs=-1, random_state=42)\n", - "bag_clf.fit(X_train, y_train)\n", - "y_pred = bag_clf.predict(X_test)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.metrics import accuracy_score\n", - "print(accuracy_score(y_test, y_pred))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "tree_clf = DecisionTreeClassifier(random_state=42)\n", - "tree_clf.fit(X_train, y_train)\n", - "y_pred_tree = tree_clf.predict(X_test)\n", - "print(accuracy_score(y_test, y_pred_tree))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "from matplotlib.colors import ListedColormap\n", - "\n", - "def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True):\n", - " x1s = np.linspace(axes[0], axes[1], 100)\n", - " x2s = np.linspace(axes[2], axes[3], 100)\n", - " x1, x2 = np.meshgrid(x1s, x2s)\n", - " X_new = np.c_[x1.ravel(), x2.ravel()]\n", - " y_pred = clf.predict(X_new).reshape(x1.shape)\n", - " custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])\n", - " plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)\n", - " if contour:\n", - " custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])\n", - " plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)\n", - " plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\", alpha=alpha)\n", - " plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\", alpha=alpha)\n", - " plt.axis(axes)\n", - " plt.xlabel(r\"$x_1$\", fontsize=18)\n", - " plt.ylabel(r\"$x_2$\", fontsize=18, rotation=0)\n", - "plt.figure(figsize=(11,4))\n", - "plt.subplot(121)\n", - "plot_decision_boundary(tree_clf, X, y)\n", - "plt.title(\"Decision Tree\", fontsize=14)\n", - "plt.subplot(122)\n", - "plot_decision_boundary(bag_clf, X, y)\n", - "plt.title(\"Decision Trees with Bagging\", fontsize=14)\n", - "save_fig(\"baggingtree\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Making your own Bootstrap: Changing the Level of the Decision Tree\n", - "\n", - "Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with\n", - "a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points $n$)." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.utils import resample\n", - "from sklearn.tree import DecisionTreeRegressor\n", - "\n", - "n = 100\n", - "n_boostraps = 100\n", - "maxdepth = 8\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "error = np.zeros(maxdepth)\n", - "bias = np.zeros(maxdepth)\n", - "variance = np.zeros(maxdepth)\n", - "polydegree = np.zeros(maxdepth)\n", - "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "# we produce a simple tree first as benchmark\n", - "simpletree = DecisionTreeRegressor(max_depth=3) \n", - "simpletree.fit(X_train_scaled, y_train)\n", - "simpleprediction = simpletree.predict(X_test_scaled)\n", - "for degree in range(1,maxdepth):\n", - " model = DecisionTreeRegressor(max_depth=degree) \n", - " y_pred = np.empty((y_test.shape[0], n_boostraps))\n", - " for i in range(n_boostraps):\n", - " x_, y_ = resample(X_train_scaled, y_train)\n", - " model.fit(x_, y_)\n", - " y_pred[:, i] = model.predict(X_test_scaled)#.ravel()\n", - "\n", - " polydegree[degree] = degree\n", - " error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n", - " bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n", - " variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n", - " print('Polynomial degree:', degree)\n", - " print('Error:', error[degree])\n", - " print('Bias^2:', bias[degree])\n", - " print('Var:', variance[degree])\n", - " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", - " \n", - "mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2)\n", - "print(mse_simpletree)\n", - "plt.xlim(1,maxdepth)\n", - "plt.plot(polydegree, error, label='MSE')\n", - "plt.plot(polydegree, bias, label='bias')\n", - "plt.plot(polydegree, variance, label='Variance')\n", - "plt.legend()\n", - "save_fig(\"baggingboot\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random forests\n", - "\n", - "Random forests provide an improvement over bagged trees by way of a\n", - "small tweak that decorrelates the trees. \n", - "\n", - "As in bagging, we build a\n", - "number of decision trees on bootstrapped training samples. But when\n", - "building these decision trees, each time a split in a tree is\n", - "considered, a random sample of $m$ predictors is chosen as split\n", - "candidates from the full set of $p$ predictors. The split is allowed to\n", - "use only one of those $m$ predictors. \n", - "\n", - "A fresh sample of $m$ predictors is\n", - "taken at each split, and typically we choose" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "m\\approx \\sqrt{p}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In building a random forest, at\n", - "each split in the tree, the algorithm is not even allowed to consider\n", - "a majority of the available predictors. \n", - "\n", - "The reason for this is rather clever. Suppose that there is one very\n", - "strong predictor in the data set, along with a number of other\n", - "moderately strong predictors. Then in the collection of bagged\n", - "variable importance random forest trees, most or all of the trees will\n", - "use this strong predictor in the top split. Consequently, all of the\n", - "bagged trees will look quite similar to each other. Hence the\n", - "predictions from the bagged trees will be highly correlated.\n", - "Unfortunately, averaging many highly correlated quantities does not\n", - "lead to as large of a reduction in variance as averaging many\n", - "uncorrelated quantities. In particular, this means that bagging will\n", - "not lead to a substantial reduction in variance over a single tree in\n", - "this setting.\n", - "\n", - "\n", - "The algorithm described here can be applied to both classification and regression problems.\n", - "\n", - "We will grow of forest of say $B$ trees.\n", - "1. For $b=1:B$\n", - "\n", - " * Draw a bootstrap sample from the training data organized in our $\\boldsymbol{X}$ matrix.\n", - "\n", - " * We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached\n", - "\n", - "1. we select $m \\le p$ variables at random from the $p$ predictors/features\n", - "\n", - "2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node\n", - "\n", - "3. split the node into daughter nodes\n", - "\n", - "\n", - "\n", - "4. Output then the ensemble of trees $\\{T_b\\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.svm import SVC\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.ensemble import BaggingClassifier\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "# Logistic Regression\n", - "logreg = LogisticRegression(solver='lbfgs')\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n", - "# Support vector machine\n", - "svm = SVC(gamma='auto', C=100)\n", - "svm.fit(X_train, y_train)\n", - "print(\"Test set accuracy with SVM: {:.2f}\".format(svm.score(X_test,y_test)))\n", - "# Decision Trees\n", - "deep_tree_clf = DecisionTreeClassifier(max_depth=None)\n", - "deep_tree_clf.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Decision Trees: {:.2f}\".format(deep_tree_clf.score(X_test,y_test)))\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Logistic Regression\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "# Support Vector Machine\n", - "svm.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy SVM with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "# Decision Trees\n", - "deep_tree_clf.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy with Decision Trees and scaled data: {:.2f}\".format(deep_tree_clf.score(X_test_scaled,y_test)))\n", - "\n", - "\n", - "from sklearn.ensemble import RandomForestClassifier\n", - "from sklearn.preprocessing import LabelEncoder\n", - "from sklearn.model_selection import cross_validate\n", - "# Data set not specificied\n", - "#Instantiate the model with 500 trees and entropy as splitting criteria\n", - "Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion=\"entropy\")\n", - "Random_Forest_model.fit(X_train_scaled, y_train)\n", - "#Cross validation\n", - "accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']\n", - "print(accuracy)\n", - "print(\"Test set accuracy with Random Forests and scaled data: {:.2f}\".format(Random_Forest_model.score(X_test_scaled,y_test)))\n", - "\n", - "\n", - "import scikitplot as skplt\n", - "y_pred = Random_Forest_model.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "plt.show()\n", - "y_probas = Random_Forest_model.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Recall that the cumulative gains curve shows the percentage of the\n", - "overall number of cases in a given category *gained* by targeting a\n", - "percentage of the total number of cases.\n", - "\n", - "Similarly, the receiver operating characteristic curve, or ROC curve,\n", - "displays the diagnostic ability of a binary classifier system as its\n", - "discrimination threshold is varied. It plots the true positive rate against the false positive rate.\n", - "\n", - "\n", - "### Compare Bagging on Trees with Random Forests" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "bag_clf = BaggingClassifier(\n", - " DecisionTreeClassifier(splitter=\"random\", max_leaf_nodes=16, random_state=42),\n", - " n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "bag_clf.fit(X_train, y_train)\n", - "y_pred = bag_clf.predict(X_test)\n", - "from sklearn.ensemble import RandomForestClassifier\n", - "rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)\n", - "rnd_clf.fit(X_train, y_train)\n", - "y_pred_rf = rnd_clf.predict(X_test)\n", - "np.sum(y_pred == y_pred_rf) / len(y_pred)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Boosting, a Bird's Eye View\n", - "\n", - "The basic idea is to combine weak classifiers in order to create a good\n", - "classifier. With a weak classifier we often intend a classifier which\n", - "produces results which are only slightly better than we would get by\n", - "random guesses.\n", - "\n", - "This is done by applying in an iterative way a weak (or a standard\n", - "classifier like decision trees) to modify the data. In each iteration\n", - "we emphasize those observations which are misclassified by weighting\n", - "them with a factor.\n", - "\n", - "\n", - "\n", - "Boosting is a way of fitting an additive expansion in a set of\n", - "elementary basis functions like for example some simple polynomials.\n", - "Assume for example that we have a function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\beta_m$ are the expansion parameters to be determined in a\n", - "minimization process and $b(x;\\gamma_m)$ are some simple functions of\n", - "the multivariable parameter $x$ which is characterized by the\n", - "parameters $\\gamma_m$.\n", - "\n", - "As an example, consider the Sigmoid function we used in logistic\n", - "regression. In that case, we can translate the function\n", - "$b(x;\\gamma_m)$ into the Sigmoid function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma(t) = \\frac{1}{1+\\exp{(-t)}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $t=\\gamma_0+\\gamma_1 x$ and the parameters $\\gamma_0$ and\n", - "$\\gamma_1$ were determined by the Logistic Regression fitting\n", - "algorithm.\n", - "\n", - "As another example, consider the cost function we defined for linear regression" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case the function $f(x)$ was replaced by the design matrix\n", - "$\\boldsymbol{X}$ and the unknown linear regression parameters $\\boldsymbol{\\beta}$,\n", - "that is $\\boldsymbol{f}=\\boldsymbol{X}\\boldsymbol{\\beta}$. In linear regression we can \n", - "simply invert a matrix and obtain the parameters $\\beta$ by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\\beta_m$ and $\\gamma_m$.\n", - "\n", - "\n", - "### Iterative Fitting, Regression and Squared-error Cost Function\n", - "\n", - "The way we proceed is as follows (here we specialize to the squared-error cost function)\n", - "\n", - "1. Establish a cost function, here $\\cal{C}(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ with $f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m)$.\n", - "\n", - "2. Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers.\n", - "\n", - "3. For $m=1:M$\n", - "\n", - "a. minimize $\\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta b(x;\\gamma))^2$ wrt $\\gamma$ and $\\beta$\n", - "\n", - "b. This gives the optimal values $\\beta_m$ and $\\gamma_m$\n", - "\n", - "c. Determine then the new values $f_m(x)=f_{m-1}(x) +\\beta_m b(x;\\gamma_m)$\n", - "\n", - "\n", - "We could use any of the algorithms we have discussed till now. If we\n", - "use trees, $\\gamma$ parameterizes the split variables and split points\n", - "at the internal nodes, and the predictions at the terminal nodes.\n", - "\n", - "\n", - "\n", - "To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.\n", - "\n", - "For simplicity we assume also that our functions $b(x;\\gamma)=1+\\gamma x$. \n", - "\n", - "This means that for every iteration $m$, we need to optimize" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\beta_m,\\gamma_m) = \\mathrm{argmin}_{\\beta,\\lambda}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta b(x;\\gamma))^2=\\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta(1+\\gamma x_i))^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We start our iteration by simply setting $f_0(x)=0$. \n", - "Taking the derivatives with respect to $\\beta$ and $\\gamma$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{C}}{\\partial \\beta} = -2\\sum_{i}(1+\\gamma x_i)(y_i-\\beta(1+\\gamma x_i))=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{C}}{\\partial \\gamma} =-2\\sum_{i}\\beta x_i(y_i-\\beta(1+\\gamma x_i))=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can then rewrite these equations as (defining $\\boldsymbol{w}=\\boldsymbol{e}+\\gamma \\boldsymbol{x})$ with $\\boldsymbol{e}$ being the unit vector)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma \\boldsymbol{w}^T(\\boldsymbol{y}-\\beta\\gamma \\boldsymbol{w})=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives us $\\beta = \\boldsymbol{w}^T\\boldsymbol{y}/(\\boldsymbol{w}^T\\boldsymbol{w})$. Similarly we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta\\gamma \\boldsymbol{x}^T(\\boldsymbol{y}-\\beta(1+\\gamma \\boldsymbol{x}))=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to $\\gamma =(\\boldsymbol{x}^T\\boldsymbol{y}-\\beta\\boldsymbol{x}^T\\boldsymbol{e})/(\\beta\\boldsymbol{x}^T\\boldsymbol{x})$. Inserting\n", - "for $\\beta$ gives us an equation for $\\gamma$. This is a non-linear equation in the unknown $\\gamma$ and has to be solved numerically. \n", - "\n", - "The solution to these two equations gives us in turn $\\beta_1$ and $\\gamma_1$ leading to the new expression for $f_1(x)$ as\n", - "$f_1(x) = \\beta_1(1+\\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$. \n", - "\n", - "\n", - "\n", - "### Iterative Fitting, Classification and AdaBoost\n", - "\n", - "Let us consider a binary classification problem with two outcomes $y_i \\in \\{-1,1\\}$ and $i=0,1,2,\\dots,n-1$ as our set of\n", - "observations. We define a classification function $G(x)$ which produces a prediction taking one or the other of the two values \n", - "$\\{-1,1\\}$.\n", - "\n", - "The error rate of the training sample is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{\\overline{err}}=\\frac{1}{n} \\sum_{i=0}^{n-1} I(y_i\\ne G(x_i)).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The iterative procedure starts with defining a weak classifier whose\n", - "error rate is barely better than random guessing. The iterative\n", - "procedure in boosting is to sequentially apply a weak\n", - "classification algorithm to repeatedly modified versions of the data\n", - "producing a sequence of weak classifiers $G_m(x)$.\n", - "\n", - "Here we will express our function $f(x)$ in terms of $G(x)$. That is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "will be a function of" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "G_M(x) = \\mathrm{sign} \\sum_{i=1}^M \\alpha_m G_m(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In our iterative procedure we define thus" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_m(x) = f_{m-1}(x)+\\beta_mG_m(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the\n", - "exponential cost/loss function defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{y},\\boldsymbol{f}) = \\sum_{i=0}^{n-1}\\exp{(-y_i(f_{m-1}(x_i)+\\beta G(x_i))}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We optimize $\\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case.\n", - "This is normally done in two steps. Let us however first rewrite the cost function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{y},\\boldsymbol{f}) = \\sum_{i=0}^{n-1}w_i^{m}\\exp{(-y_i\\beta G(x_i))},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined $w_i^m= \\exp{(-y_if_{m-1}(x_i))}$.\n", - "\n", - "\n", - "\n", - "First, for any $\\beta > 0$, we optimize $G$ by setting" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "G_m(x) = \\mathrm{sign} \\sum_{i=0}^{n-1} w_i^m I(y_i \\ne G_(x_i)),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is the classifier that minimizes the weighted error rate in predicting $y$.\n", - "\n", - "We can do this by rewriting" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\exp{-(\\beta)}\\sum_{y_i=G(x_i)}w_i^m+\\exp{(\\beta)}\\sum_{y_i\\ne G(x_i)}w_i^m,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which can be rewritten as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\exp{(\\beta)}-\\exp{-(\\beta)})\\sum_{i=0}^{n-1}w_i^mI(y_i\\ne G(x_i))+\\exp{(-\\beta)}\\sum_{i=0}^{n-1}w_i^m=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_m = \\frac{1}{2}\\log{\\frac{1-\\mathrm{\\overline{err}}}{\\mathrm{\\overline{err}}}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have redefined the error as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{\\overline{err}}_m=\\frac{1}{n}\\frac{\\sum_{i=0}^{n-1}w_i^mI(y_i\\ne G(x_i)}{\\sum_{i=0}^{n-1}w_i^m},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to an update of" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_m(x) = f_{m-1}(x) +\\beta_m G_m(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This leads to the new weights" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "w_i^{m+1} = w_i^m \\exp{(-y_i\\beta_m G_m(x_i))}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Adaptive boosting: AdaBoost, Basic Algorithm\n", - "\n", - "The algorithm here is rather straightforward. Assume that our weak\n", - "classifier is a decision tree and we consider a binary set of outputs\n", - "with $y_i \\in \\{-1,1\\}$ and $i=0,1,2,\\dots,n-1$ as our set of\n", - "observations. Our design matrix is given in terms of the\n", - "feature/predictor vectors\n", - "$\\boldsymbol{X}=[\\boldsymbol{x}_0\\boldsymbol{x}_1\\dots\\boldsymbol{x}_{p-1}]$. Finally, we define also a\n", - "classifier determined by our data via a function $G(x)$. This function tells us how well we are able to classify our outputs/targets $\\boldsymbol{y}$. \n", - "\n", - "We have already defined the misclassification error $\\mathrm{err}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{err}=\\frac{1}{n}\\sum_{i=0}^{n-1}I(y_i\\ne G(x_i)),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the function $I()$ is one if we misclassify and zero if we classify correctly. \n", - "\n", - "\n", - "With the above definitions we are now ready to set up the algorithm for AdaBoost.\n", - "The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.\n", - "1. We start by initializing all weights to $w_i = 1/n$, with $i=0,1,2,\\dots n-1$. It is easy to see that we must have $\\sum_{i=0}^{n-1}w_i = 1$.\n", - "\n", - "2. We rewrite the misclassification error as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{\\overline{err}}_m=\\frac{\\sum_{i=0}^{n-1}w_i^m I(y_i\\ne G(x_i))}{\\sum_{i=0}^{n-1}w_i},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "1. Then we start looping over all attempts at classifying, namely we start an iterative process for $m=1:M$, where $M$ is the final number of classifications. Our given classifier could for example be a plain decision tree.\n", - "\n", - "a. Fit then a given classifier to the training set using the weights $w_i$.\n", - "\n", - "b. Compute then $\\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly.\n", - "\n", - "c. Define a quantity $\\alpha_{m} = \\log{(1-\\mathrm{\\overline{err}}_m)/\\mathrm{\\overline{err}}_m}$\n", - "\n", - "d. Set the new weights to $w_i = w_i\\times \\exp{(\\alpha_m I(y_i\\ne G(x_i)}$.\n", - "\n", - "\n", - "5. Compute the new classifier $G(x)= \\sum_{i=0}^{n-1}\\alpha_m I(y_i\\ne G(x_i)$.\n", - "\n", - "For the iterations with $m \\le 2$ the weights are modified\n", - "individually at each steps. The observations which were misclassified\n", - "at iteration $m-1$ have a weight which is larger than those which were\n", - "classified properly. As this proceeds, the observations which were\n", - "difficult to classifiy correctly are given a larger influence. Each\n", - "new classification step $m$ is then forced to concentrate on those\n", - "observations that are missed in the previous iterations.\n", - "\n", - "\n", - "\n", - "\n", - "Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.ensemble import AdaBoostClassifier\n", - "\n", - "ada_clf = AdaBoostClassifier(\n", - " DecisionTreeClassifier(max_depth=1), n_estimators=200,\n", - " algorithm=\"SAMME.R\", learning_rate=0.5, random_state=42)\n", - "ada_clf.fit(X_train, y_train)\n", - "\n", - "from sklearn.ensemble import AdaBoostClassifier\n", - "\n", - "ada_clf = AdaBoostClassifier(\n", - " DecisionTreeClassifier(max_depth=1), n_estimators=200,\n", - " algorithm=\"SAMME.R\", learning_rate=0.5, random_state=42)\n", - "ada_clf.fit(X_train_scaled, y_train)\n", - "y_pred = ada_clf.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "plt.show()\n", - "y_probas = ada_clf.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent\n", - "\n", - "Gradient boosting is again a similar technique to Adaptive boosting,\n", - "it combines so-called weak classifiers or regressors into a strong\n", - "method via a series of iterations.\n", - "\n", - "In order to understand the method, let us illustrate its basics by\n", - "bringing back the essential steps in linear regression, where our cost\n", - "function was the least squares function.\n", - "\n", - "\n", - "We start again with our cost function $\\cal{C}(\\boldsymbol{y}m\\boldsymbol{f})=\\sum_{i=0}^{n-1}\\cal{L}(y_i, f(x_i))$ where we want to minimize\n", - "This means that for every iteration, we need to optimize" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\hat{\\boldsymbol{f}}) = \\mathrm{argmin}_{\\boldsymbol{f}}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_M(x) = \\sum_{m=0}^M h_m(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the steepest decent approach we approximate $h_m(x) = -\\rho_m g_m(x)$, where $\\rho_m$ is a scalar and $g_m(x)$ the gradient defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_m(x_i) = \\left[ \\frac{\\partial \\cal{L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{m-1}(x_i)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With the new gradient we can update $f_m(x) = f_{m-1}(x) -\\rho_m g_m(x)$. Using the above squared-error function we see that\n", - "the gradient is $g_m(x_i) = -2(y_i-f(x_i))$.\n", - "\n", - "Choosing $f_0(x)=0$ we obtain $g_m(x) = -2y_i$ and inserting this into the minimization problem for the cost function we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\rho_1) = \\mathrm{argmin}_{\\rho}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i+2\\rho y_i)^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Optimizing with respect to $\\rho$ we obtain (taking the derivative) that $\\rho_1 = -1/2$. We have then that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_1(x) = f_{0}(x) -\\rho_1 g_1(x)=-y_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can then proceed and compute" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_2(x_i) = \\left[ \\frac{\\partial \\cal{L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and find a new value for $\\rho_2=-1/2$ and continue till we have reached $m=M$. We can modify the steepest descent method, or steepest boosting, by introducing what is called **gradient boosting**. \n", - "\n", - "\n", - "Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points,\n", - "so we do not learn a function that can generalize. However, we can modify the algorithm by\n", - "fitting a weak learner to approximate the negative gradient signal. \n", - "\n", - "Suppose we have a cost function $C(f)=\\sum_{i=0}^{n-1}L(y_i, f(x_i))$ where $y_i$ is our target and $f(x_i)$ the function which is meant to model $y_i$. The above cost function could be our standard squared-error function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{y},\\boldsymbol{f})=\\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The way we proceed in an iterative fashion is to\n", - "1. Initialize our estimate $f_0(x)$.\n", - "\n", - "2. For $m=1:M$, we\n", - "\n", - "a. compute the negative gradient vector $\\boldsymbol{u}_m = -\\partial C(\\boldsymbol{y},\\boldsymbol{f})/\\partial \\boldsymbol{f}(x)$ at $f(x) = f_{m-1}(x)$;\n", - "\n", - "b. fit the so-called base-learner to the negative gradient $h_m(u_m,x)$;\n", - "\n", - "c. update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$;\n", - "\n", - "\n", - "4. The final estimate is then $f_M(x) = \\sum_{m=1}^M h_m(u_m,x)$.\n", - "\n", - "## Gradient Boosting, Examples of Regression" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.ensemble import GradientBoostingRegressor\n", - "from sklearn.preprocessing import StandardScaler\n", - "import scikitplot as skplt\n", - "from sklearn.metrics import mean_squared_error\n", - "\n", - "n = 100\n", - "maxdegree = 6\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "\n", - "error = np.zeros(maxdegree)\n", - "bias = np.zeros(maxdegree)\n", - "variance = np.zeros(maxdegree)\n", - "polydegree = np.zeros(maxdegree)\n", - "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "for degree in range(1,maxdegree):\n", - " model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0) \n", - " model.fit(X_train_scaled,y_train)\n", - " y_pred = model.predict(X_test_scaled)\n", - " polydegree[degree] = degree\n", - " error[degree] = np.mean( np.mean((y_test - y_pred)**2) )\n", - " bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )\n", - " variance[degree] = np.mean( np.var(y_pred) )\n", - " print('Max depth:', degree)\n", - " print('Error:', error[degree])\n", - " print('Bias^2:', bias[degree])\n", - " print('Var:', variance[degree])\n", - " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", - "\n", - "plt.xlim(1,maxdegree-1)\n", - "plt.plot(polydegree, error, label='Error')\n", - "plt.plot(polydegree, bias, label='bias')\n", - "plt.plot(polydegree, variance, label='Variance')\n", - "plt.legend()\n", - "save_fig(\"gdregression\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gradient Boosting, Classification Example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "import scikitplot as skplt\n", - "from sklearn.ensemble import GradientBoostingClassifier\n", - "from sklearn.model_selection import cross_validate\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0) \n", - "gd_clf.fit(X_train_scaled, y_train)\n", - "#Cross validation\n", - "accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']\n", - "print(accuracy)\n", - "print(\"Test set accuracy with Random Forests and scaled data: {:.2f}\".format(gd_clf.score(X_test_scaled,y_test)))\n", - "\n", - "import scikitplot as skplt\n", - "y_pred = gd_clf.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "save_fig(\"gdclassiffierconfusion\")\n", - "plt.show()\n", - "y_probas = gd_clf.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "save_fig(\"gdclassiffierroc\")\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "save_fig(\"gdclassiffiercgain\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## XGBoost: Extreme Gradient Boosting\n", - "\n", - "\n", - "[XGBoost](https://github.com/dmlc/xgboost) or Extreme Gradient\n", - "Boosting, is an optimized distributed gradient boosting library\n", - "designed to be highly efficient, flexible and portable. It implements\n", - "machine learning algorithms under the Gradient Boosting\n", - "framework. XGBoost provides a parallel tree boosting that solve many\n", - "data science problems in a fast and accurate way. See the [article by Chen and Guestrin](https://arxiv.org/abs/1603.02754).\n", - "\n", - "The authors design and build a highly scalable end-to-end tree\n", - "boosting system. It has a theoretically justified weighted quantile\n", - "sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.\n", - "\n", - "It is now the algorithm which wins essentially all ML competitions!!!\n", - "\n", - "## Regression Case" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split\n", - "import xgboost as xgb\n", - "from sklearn.preprocessing import StandardScaler\n", - "import scikitplot as skplt\n", - "from sklearn.metrics import mean_squared_error\n", - "\n", - "n = 100\n", - "maxdegree = 6\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "\n", - "error = np.zeros(maxdegree)\n", - "bias = np.zeros(maxdegree)\n", - "variance = np.zeros(maxdegree)\n", - "polydegree = np.zeros(maxdegree)\n", - "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "for degree in range(maxdegree):\n", - " model = xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200)\n", - "\n", - " model.fit(X_train_scaled,y_train)\n", - " y_pred = model.predict(X_test_scaled)\n", - " polydegree[degree] = degree\n", - " error[degree] = np.mean( np.mean((y_test - y_pred)**2) )\n", - " bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )\n", - " variance[degree] = np.mean( np.var(y_pred) )\n", - " print('Max depth:', degree)\n", - " print('Error:', error[degree])\n", - " print('Bias^2:', bias[degree])\n", - " print('Var:', variance[degree])\n", - " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", - "\n", - "plt.xlim(1,maxdegree-1)\n", - "plt.plot(polydegree, error, label='Error')\n", - "plt.plot(polydegree, bias, label='bias')\n", - "plt.plot(polydegree, variance, label='Variance')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.preprocessing import LabelEncoder\n", - "from sklearn.model_selection import cross_validate\n", - "import scikitplot as skplt\n", - "import xgboost as xgb\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "xg_clf = xgb.XGBClassifier()\n", - "xg_clf.fit(X_train_scaled,y_train)\n", - "\n", - "y_test = xg_clf.predict(X_test_scaled)\n", - "\n", - "print(\"Test set accuracy with Random Forests and scaled data: {:.2f}\".format(xg_clf.score(X_test_scaled,y_test)))\n", - "\n", - "import scikitplot as skplt\n", - "y_pred = xg_clf.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "save_fig(\"xdclassiffierconfusion\")\n", - "plt.show()\n", - "y_probas = xg_clf.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "save_fig(\"xdclassiffierroc\")\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "save_fig(\"gdclassiffiercgain\")\n", - "plt.show()\n", - "\n", - "\n", - "xgb.plot_tree(xg_clf,num_trees=0)\n", - "plt.rcParams['figure.figsize'] = [50, 10]\n", - "save_fig(\"xgtree\")\n", - "plt.show()\n", - "\n", - "xgb.plot_importance(xg_clf)\n", - "plt.rcParams['figure.figsize'] = [5, 5]\n", - "save_fig(\"xgparams\")\n", - "plt.show()" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter8.ipynb b/doc/LectureNotes/_build/html/_sources/chapter8.ipynb deleted file mode 100644 index 1ee4dd6cd..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter8.ipynb +++ /dev/null @@ -1,1359 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Basic ideas of the Principal Component Analysis (PCA)\n", - "\n", - "The principal component analysis deals with the problem of fitting a\n", - "low-dimensional affine subspace $S$ of dimension $d$ much smaller than\n", - "the total dimension $D$ of the problem at hand (our data\n", - "set). Mathematically it can be formulated as a statistical problem or\n", - "a geometric problem. In our discussion of the theorem for the\n", - "classical PCA, we will stay with a statistical approach. \n", - "Historically, the PCA was first formulated in a statistical setting in order to estimate the principal component of a multivariate random variable.\n", - "\n", - "We have a data set defined by a design/feature matrix $\\boldsymbol{X}$ (see below for its definition) \n", - "* Each data point is determined by $p$ extrinsic (measurement) variables\n", - "\n", - "* We may want to ask the following question: Are there fewer intrinsic variables (say $d << p$) that still approximately describe the data?\n", - "\n", - "* If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do. \n", - "\n", - "A good read is for example [Vidal, Ma and Sastry](https://www.springer.com/gp/book/9780387878102).\n", - "\n", - "\n", - "\n", - "## Introducing the Covariance and Correlation functions\n", - "\n", - "Before we discuss the PCA theorem, we need to remind ourselves about\n", - "the definition of the covariance and the correlation function. These are quantities \n", - "\n", - "Suppose we have defined two vectors\n", - "$\\hat{x}$ and $\\hat{y}$ with $n$ elements each. The covariance matrix $\\boldsymbol{C}$ is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{y}] \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where for example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With this definition and recalling that the variance is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{var}[\\boldsymbol{x}]=\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we can rewrite the covariance matrix as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] & \\mathrm{var}[\\boldsymbol{y}] \\\\\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The covariance takes values between zero and infinity and may thus\n", - "lead to problems with loss of numerical precision for particularly\n", - "large values. It is common to scale the covariance matrix by\n", - "introducing instead the correlation matrix defined via the so-called\n", - "correlation function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]=\\frac{\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}]}{\\sqrt{\\mathrm{var}[\\boldsymbol{x}] \\mathrm{var}[\\boldsymbol{y}]}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The correlation function is then given by values $\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]\n", - "\\in [-1,1]$. This avoids eventual problems with too large values. We\n", - "can then define the correlation matrix for the two vectors $\\boldsymbol{x}$\n", - "and $\\boldsymbol{y}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{K}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} 1 & \\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{corr}[\\boldsymbol{y},\\boldsymbol{x}] & 1 \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the above example this is the function we constructed using **pandas**.\n", - "\n", - "\n", - "In our derivation of the various regression algorithms like **Ordinary Least Squares** or **Ridge regression**\n", - "we defined the design/feature matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n", - "x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n", - "x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n", - "\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n", - "x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n", - "x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$ refering to the column numbers and the\n", - "entries $n$ being the row elements.\n", - "We can rewrite the design/feature matrix in terms of its column vectors as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix} \\boldsymbol{x}_0 & \\boldsymbol{x}_1 & \\boldsymbol{x}_2 & \\dots & \\dots & \\boldsymbol{x}_{p-1}\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with a given vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i^T = \\begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \\dots & \\dots x_{n-1,i}\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With these definitions, we can now rewrite our $2\\times 2$\n", - "correaltion/covariance matrix in terms of a moe general design/feature\n", - "matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$. This leads to a $p\\times p$\n", - "covariance matrix for the vectors $\\boldsymbol{x}_i$ with $i=0,1,\\dots,p-1$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\begin{bmatrix}\n", - "\\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & \\mathrm{var}[\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & \\mathrm{var}[\\boldsymbol{x}_{p-1}]\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the correlation matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{K}[\\boldsymbol{x}] = \\begin{bmatrix}\n", - "1 & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & 1 & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & 1 & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & 1\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The Numpy function **np.cov** calculates the covariance elements using\n", - "the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have\n", - "the exact mean values. The following simple function uses the\n", - "**np.vstack** function which takes each vector of dimension $1\\times n$\n", - "and produces a $2\\times n$ matrix $\\boldsymbol{W}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{W} = \\begin{bmatrix} x_0 & y_0 \\\\\n", - " x_1 & y_1 \\\\\n", - " x_2 & y_2\\\\\n", - " \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2}\\\\\n", - " x_{n-1} & y_{n-1} & \n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in turn is converted into into the $2\\times 2$ covariance matrix\n", - "$\\boldsymbol{C}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", - "the mean value of each set of samples $\\boldsymbol{x}$ etc using the Numpy\n", - "function **np.mean(x)**. We can also extract the eigenvalues of the\n", - "covariance matrix through the **np.linalg.eig()** function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "W = np.vstack((x, y))\n", - "C = np.cov(W)\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Correlation Matrix\n", - "\n", - "The previous example can be converted into the correlation matrix by\n", - "simply scaling the matrix elements with the variances. We should also\n", - "subtract the mean values for each column. This leads to the following\n", - "code which sets up the correlations matrix for the previous example in\n", - "a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\\times 2$ correlation matrix (since we have only two vectors)." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 100\n", - "# define two vectors \n", - "x = np.random.random(size=n)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "#scaling the x and y vectors \n", - "x = x - np.mean(x)\n", - "y = y - np.mean(y)\n", - "variance_x = np.sum(x@x)/n\n", - "variance_y = np.sum(y@y)/n\n", - "print(variance_x)\n", - "print(variance_y)\n", - "cov_xy = np.sum(x@y)/n\n", - "cov_xx = np.sum(x@x)/n\n", - "cov_yy = np.sum(y@y)/n\n", - "C = np.zeros((2,2))\n", - "C[0,0]= cov_xx/variance_x\n", - "C[1,1]= cov_yy/variance_y\n", - "C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)\n", - "C[1,0]= C[0,1]\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see that the matrix elements along the diagonal are one as they\n", - "should be and that the matrix is symmetric. Furthermore, diagonalizing\n", - "this matrix we easily see that it is a positive definite matrix.\n", - "\n", - "The above procedure with **numpy** can be made more compact if we use **pandas**.\n", - "\n", - "\n", - "We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import pandas as pd\n", - "n = 10\n", - "x = np.random.normal(size=n)\n", - "x = x - np.mean(x)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "y = y - np.mean(y)\n", - "X = (np.vstack((x, y))).T\n", - "print(X)\n", - "Xpd = pd.DataFrame(X)\n", - "print(Xpd)\n", - "correlation_matrix = Xpd.corr()\n", - "print(correlation_matrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We expand this model to the Franke function discussed above." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "\n", - "\n", - "def FrankeFunction(x,y):\n", - "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", - "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", - "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", - "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", - "\treturn term1 + term2 + term3 + term4\n", - "\n", - "\n", - "def create_X(x, y, n ):\n", - "\tif len(x.shape) > 1:\n", - "\t\tx = np.ravel(x)\n", - "\t\ty = np.ravel(y)\n", - "\n", - "\tN = len(x)\n", - "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", - "\tX = np.ones((N,l))\n", - "\n", - "\tfor i in range(1,n+1):\n", - "\t\tq = int((i)*(i+1)/2)\n", - "\t\tfor k in range(i+1):\n", - "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", - "\n", - "\treturn X\n", - "\n", - "\n", - "# Making meshgrid of datapoints and compute Franke's function\n", - "n = 4\n", - "N = 100\n", - "x = np.sort(np.random.uniform(0, 1, N))\n", - "y = np.sort(np.random.uniform(0, 1, N))\n", - "z = FrankeFunction(x, y)\n", - "X = create_X(x, y, n=n) \n", - "\n", - "Xpd = pd.DataFrame(X)\n", - "# subtract the mean values and set up the covariance matrix\n", - "Xpd = Xpd - Xpd.mean()\n", - "covariance_matrix = Xpd.cov()\n", - "print(covariance_matrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note here that the covariance is zero for the first rows and\n", - "columns since all matrix elements in the design matrix were set to one\n", - "(we are fitting the function in terms of a polynomial of degree $n$). We would however not include the intercept\n", - "and wee can simply\n", - "drop these elements and construct a correlation\n", - "matrix without them. \n", - "\n", - "\n", - "\n", - "We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}= \\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}].\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To see this let us simply look at a design matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{2\\times 2}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{00} & x_{01}\\\\\n", - "x_{10} & x_{11}\\\\\n", - "\\end{bmatrix}=\\begin{bmatrix}\n", - "\\boldsymbol{x}_{0} & \\boldsymbol{x}_{1}\\\\\n", - "\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we then compute the expectation value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\\\\n", - "x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is just" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]=\\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we wrote $$\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]$$ to indicate that this the covariance of the vectors $\\boldsymbol{x}$ of the design/feature matrix $\\boldsymbol{X}$.\n", - "\n", - "It is easy to generalize this to a matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$.\n", - "\n", - "\n", - "\n", - "## Towards the PCA theorem\n", - "\n", - "We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}= \\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}].\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices $\\boldsymbol{S}$.\n", - "These matrices are defined as $\\boldsymbol{S}\\in {\\mathbb{R}}^{p\\times p}$ and obey the orthogonality requirements $\\boldsymbol{S}\\boldsymbol{S}^T=\\boldsymbol{S}^T\\boldsymbol{S}=\\boldsymbol{I}$. The matrix can be written out in terms of the column vectors $\\boldsymbol{s}_i$ as $\\boldsymbol{S}=[\\boldsymbol{s}_0,\\boldsymbol{s}_1,\\dots,\\boldsymbol{s}_{p-1}]$ and $\\boldsymbol{s}_i \\in {\\mathbb{R}}^{p}$.\n", - "\n", - "Assume also that there is a transformation $\\boldsymbol{S}^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S}=\\boldsymbol{C}[\\boldsymbol{y}]$ such that the new matrix $\\boldsymbol{C}[\\boldsymbol{y}]$ is diagonal with elements $[\\lambda_0,\\lambda_1,\\lambda_2,\\dots,\\lambda_{p-1}]$. \n", - "\n", - "That is we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{y}] = \\mathbb{E}[\\boldsymbol{S}^T\\boldsymbol{X}^T\\boldsymbol{X}T\\boldsymbol{S}]=\\boldsymbol{S}^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "since the matrix $\\boldsymbol{S}$ is not a data dependent matrix. Multiplying with $\\boldsymbol{S}$ from the left we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{S}\\boldsymbol{C}[\\boldsymbol{y}] = \\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and since $\\boldsymbol{C}[\\boldsymbol{y}]$ is diagonal we have for a given eigenvalue $i$ of the covariance matrix that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{S}_i\\lambda_i = \\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is\n", - "$\\lambda_0 > \\lambda_1 > \\dots > \\lambda_{p-1}$. \n", - "\n", - "\n", - "The eigenvalues tell us then how much we need to stretch the\n", - "corresponding eigenvectors. Dimensions with large eigenvalues have\n", - "thus large variations (large variance) and define therefore useful\n", - "dimensions. The data points are more spread out in the direction of\n", - "these eigenvectors. Smaller eigenvalues mean on the other hand that\n", - "the corresponding eigenvectors are shrunk accordingly and the data\n", - "points are tightly bunched together and there is not much variation in\n", - "these specific directions. Hopefully then we could leave it out\n", - "dimensions where the eigenvalues are very small. If $p$ is very large,\n", - "we could then aim at reducing $p$ to $l << p$ and handle only $l$\n", - "features/predictors.\n", - "\n", - "### The Algorithm before theorem\n", - "\n", - "Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. \n", - "* Set up the datapoints for the design/feature matrix $\\boldsymbol{X}$ with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$ referring to the column numbers and the entries $n$ being the row elements." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n", - "x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n", - "x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n", - "\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n", - "x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n", - "x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "* Center the data by subtracting the mean value for each column. This leads to a new matrix $\\boldsymbol{X}\\rightarrow \\overline{\\boldsymbol{X}}$.\n", - "\n", - "* Compute then the covariance/correlation matrix $\\mathbb{E}[\\overline{\\boldsymbol{X}}^T\\overline{\\boldsymbol{X}}]$.\n", - "\n", - "* Find the eigenpairs of $\\boldsymbol{C}$ with eigenvalues $[\\lambda_0,\\lambda_1,\\dots,\\lambda_{p-1}]$ and eigenvectors $[\\boldsymbol{s}_0,\\boldsymbol{s}_1,\\dots,\\boldsymbol{s}_{p-1}]$.\n", - "\n", - "* Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.\n", - "\n", - "* Keep only those $l$ eigenvalues larger than a selected threshold value, discarding thus $p-l$ features since we expect small variations in the data here.\n", - "\n", - "### Writing our own PCA code\n", - "\n", - "We will use a simple example first with two-dimensional data\n", - "drawn from a multivariate normal distribution with the following mean and covariance matrix (we have fixed these quantities but will play around with them below):" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = (-1,2) \\qquad \\Sigma = \\begin{bmatrix} 4 & 2 \\\\\n", - "2 & 2\n", - "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that the mean refers to each column of data. \n", - "We will generate $n = 10000$ points $X = \\{ x_1, \\ldots, x_N \\}$ from\n", - "this distribution, and store them in the $1000 \\times 2$ matrix $\\boldsymbol{X}$. This is our design matrix where we have forced the covariance and mean values to take specific values.\n", - "\n", - "The following Python code aids in setting up the data and writing out the design matrix.\n", - "Note that the function **multivariate** returns also the covariance discussed above and that it is defined by dividing by $n-1$ instead of $n$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from IPython.display import display\n", - "n = 10000\n", - "mean = (-1, 2)\n", - "cov = [[4, 2], [2, 2]]\n", - "X = np.random.multivariate_normal(mean, cov, n)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now we are going to implement the PCA algorithm. We will break it down into various substeps.\n", - "\n", - "\n", - "The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu_n = \\frac{1}{n} \\sum_{i=1}^n x_i\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the mean-centered data $\\bar{X} = \\{ \\bar{x}_1, \\ldots, \\bar{x}_n \\}$ takes the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\bar{x}_i = x_i - \\mu_n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "When you are done with these steps, print out $\\mu_n$ to verify it is\n", - "close to $\\mu$ and plot your mean centered data to verify it is\n", - "centered at the origin! \n", - "The following code elements perform these operations using **pandas** or using our own functionality for doing so. The latter, using **numpy** is rather simple through the **mean()** function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "df = pd.DataFrame(X)\n", - "# Pandas does the centering for us\n", - "df = df -df.mean()\n", - "# we center it ourselves\n", - "X_centered = X - X.mean(axis=0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Alternatively, we could use the functions we discussed\n", - "earlier for scaling the data set. That is, we could have used the\n", - "**StandardScaler** function in **Scikit-Learn**, a function which ensures\n", - "that for each feature/predictor we study the mean value is zero and\n", - "the variance is one (every column in the design/feature matrix). You\n", - "would then not get the same results, since we divide by the\n", - "variance. The diagonal covariance matrix elements will then be one,\n", - "while the non-diagonal ones need to be divided by $2\\sqrt{2}$ for our\n", - "specific case.\n", - "\n", - "\n", - "Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\Sigma_n = \\frac{1}{n-1} \\sum_{i=1}^n \\bar{x}_i^T \\bar{x}_i = \\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\mu_n)^T (x_i - \\mu_n)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the data points $x_i \\in \\mathbb{R}^p$ (here in this example $p = 2$) are column vectors and $x^T$ is the transpose of $x$.\n", - "We can write our own code or simply use either the functionaly of **numpy** or that of **pandas**, as follows" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "print(df.cov())\n", - "print(np.cov(X_centered.T))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that the way we define the covariance matrix here has a factor $n-1$ instead of $n$. This is included in the **cov()** function by **numpy** and **pandas**. \n", - "Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific $2\\times 2$ covariance matrix." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# extract the relevant columns from the centered design matrix of dim n x 2\n", - "x = X_centered[:,0]\n", - "y = X_centered[:,1]\n", - "Cov = np.zeros((2,2))\n", - "Cov[0,1] = np.sum(x.T@y)/(n-1.0)\n", - "Cov[0,0] = np.sum(x.T@x)/(n-1.0)\n", - "Cov[1,1] = np.sum(y.T@y)/(n-1.0)\n", - "Cov[1,0]= Cov[0,1]\n", - "print(\"Centered covariance using own code\")\n", - "print(Cov)\n", - "plt.plot(x, y, 'x')\n", - "plt.axis('equal')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Depending on the number of points $n$, we will get results that are close to the covariance values defined above.\n", - "The plot shows how the data are clustered around a line with slope close to one. Is this expected? Try to change the covariance and the mean values. For example, try to make the variance of the first element much larger than that of the second diagonal element. Try also to shrink the covariance (the non-diagonal elements) and see how the data points are distributed. \n", - "\n", - "### Diagonalize the sample covariance matrix to obtain the principal components\n", - "\n", - "Now we are ready to solve for the principal components! To do so we\n", - "diagonalize the sample covariance matrix $\\Sigma$. We can use the\n", - "function **np.linalg.eig** to do so. It will return the eigenvalues and\n", - "eigenvectors of $\\Sigma$. Once we have these we can perform the \n", - "following tasks:\n", - "\n", - "* We compute the percentage of the total variance captured by the first principal component\n", - "\n", - "* We plot the mean centered data and lines along the first and second principal components\n", - "\n", - "* Then we project the mean centered data onto the first and second principal components, and plot the projected data. \n", - "\n", - "* Finally, we approximate the data as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "x_i \\approx \\tilde{x}_i = \\mu_n + \\langle x_i, v_0 \\rangle v_0\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $v_0$ is the first principal component. \n", - "\n", - "Collecting all these steps we can write our own PCA function and\n", - "compare this with the functionality included in **Scikit-Learn**. \n", - "\n", - "The code here outlines some of the elements we could include in the\n", - "analysis. Feel free to extend upon this in order to address the above\n", - "questions." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# diagonalize and obtain eigenvalues, not necessarily sorted\n", - "EigValues, EigVectors = np.linalg.eig(Cov)\n", - "# sort eigenvectors and eigenvalues\n", - "#permute = EigValues.argsort()\n", - "#EigValues = EigValues[permute]\n", - "#EigVectors = EigVectors[:,permute]\n", - "print(\"Eigenvalues of Covariance matrix\")\n", - "for i in range(2):\n", - " print(EigValues[i])\n", - "FirstEigvector = EigVectors[:,0]\n", - "SecondEigvector = EigVectors[:,1]\n", - "print(\"First eigenvector\")\n", - "print(FirstEigvector)\n", - "print(\"Second eigenvector\")\n", - "print(SecondEigvector)\n", - "#thereafter we do a PCA with Scikit-learn\n", - "from sklearn.decomposition import PCA\n", - "pca = PCA(n_components = 2)\n", - "X2Dsl = pca.fit_transform(X)\n", - "print(\"Eigenvector of largest eigenvalue\")\n", - "print(pca.components_.T[:, 0])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This code does not contain all the above elements, but it shows how we can use **Scikit-Learn** to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then? \n", - "\n", - "\n", - "## Classical PCA Theorem\n", - "\n", - "We assume now that we have a design matrix $\\boldsymbol{X}$ which has been\n", - "centered as discussed above. For the sake of simplicity we skip the\n", - "overline symbol. The matrix is defined in terms of the various column\n", - "vectors $[\\boldsymbol{x}_0,\\boldsymbol{x}_1,\\dots, \\boldsymbol{x}_{p-1}]$ each with dimension\n", - "$\\boldsymbol{x}\\in {\\mathbb{R}}^{n}$.\n", - "\n", - "\n", - "\n", - "The PCA theorem states that minimizing the above reconstruction error\n", - "corresponds to setting $\\boldsymbol{W}=\\boldsymbol{S}$, the orthogonal matrix which\n", - "diagonalizes the empirical covariance(correlation) matrix. The optimal\n", - "low-dimensional encoding of the data is then given by a set of vectors\n", - "$\\boldsymbol{z}_i$ with at most $l$ vectors, with $l << p$, defined by the\n", - "orthogonal projection of the data onto the columns spanned by the\n", - "eigenvectors of the covariance(correlations matrix).\n", - "\n", - "\n", - "\n", - "\n", - "To show the PCA theorem let us start with the assumption that there is one vector $\\boldsymbol{s}_0$ which corresponds to a solution which minimized the reconstruction error $J$. This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of $\\boldsymbol{w}_0$ and $\\boldsymbol{z}_0$ as\n", - "\n", - "\n", - "\n", - "We are almost there, we have obtained a relation between minimizing\n", - "the reconstruction error and the variance and the covariance\n", - "matrix. Minimizing the error is equivalent to maximizing the variance\n", - "of the projected data.\n", - "\n", - "\n", - "We could trivially maximize the variance of the projection (and\n", - "thereby minimize the error in the reconstruction function) by letting\n", - "the norm-2 of $\\boldsymbol{w}_0$ go to infinity. However, this norm since we\n", - "want the matrix $\\boldsymbol{W}$ to be an orthogonal matrix, is constrained by\n", - "$\\vert\\vert \\boldsymbol{w}_0 \\vert\\vert_2^2=1$. Imposing this condition via a\n", - "Lagrange multiplier we can then in turn maximize" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "J(\\boldsymbol{w}_0)= \\boldsymbol{w}_0^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0+\\lambda_0(1-\\boldsymbol{w}_0^T\\boldsymbol{w}_0).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Taking the derivative with respect to $\\boldsymbol{w}_0$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial J(\\boldsymbol{w}_0)}{\\partial \\boldsymbol{w}_0}= 2\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0-2\\lambda_0\\boldsymbol{w}_0=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "meaning that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0=\\lambda_0\\boldsymbol{w}_0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix**! If we left multiply with $\\boldsymbol{w}_0^T$ we have the variance of the projected data is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{w}_0^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0=\\lambda_0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we want to maximize the variance (minimize the construction error)\n", - "we simply pick the eigenvector of the covariance matrix with the\n", - "largest eigenvalue. This establishes the link between the minimization\n", - "of the reconstruction function $J$ in terms of an orthogonal matrix\n", - "and the maximization of the variance and thereby the covariance of our\n", - "observations encoded in the design/feature matrix $\\boldsymbol{X}$.\n", - "\n", - "The proof\n", - "for the other eigenvectors $\\boldsymbol{w}_1,\\boldsymbol{w}_2,\\dots$ can be\n", - "established by applying the above arguments and using the fact that\n", - "our basis of eigenvectors is orthogonal, see [Murphy chapter\n", - "12.2](https://mitpress.mit.edu/books/machine-learning-1). The\n", - "discussion in chapter 12.2 of Murphy's text has also a nice link with\n", - "the Singular Value Decomposition theorem. For categorical data, see\n", - "chapter 12.4 and discussion therein.\n", - "\n", - "For more details, see for example [Vidal, Ma and Sastry, chapter 2](https://www.springer.com/gp/book/9780387878102).\n", - "\n", - "\n", - "## Geometric Interpretation and link with Singular Value Decomposition\n", - "\n", - "For a detailed demonstration of the geometric interpretation, see [Vidal, Ma and Sastry, section 2.1.2](https://www.springer.com/gp/book/9780387878102).\n", - "\n", - "\n", - "Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.\n", - "First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.\n", - "\n", - "The following Python code uses NumPy’s **svd()** function to obtain all the principal components of the\n", - "training set, then extracts the first two principal components. First we center the data using either **pandas** or our own code" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import pandas as pd\n", - "from IPython.display import display\n", - "np.random.seed(100)\n", - "# setting up a 10 x 5 vanilla matrix \n", - "rows = 10\n", - "cols = 5\n", - "X = np.random.randn(rows,cols)\n", - "df = pd.DataFrame(X)\n", - "# Pandas does the centering for us\n", - "df = df -df.mean()\n", - "display(df)\n", - "\n", - "# we center it ourselves\n", - "X_centered = X - X.mean(axis=0)\n", - "# Then check the difference between pandas and our own set up\n", - "print(X_centered-df)\n", - "#Now we do an SVD\n", - "U, s, V = np.linalg.svd(X_centered)\n", - "c1 = V.T[:, 0]\n", - "c2 = V.T[:, 1]\n", - "W2 = V.T[:, :2]\n", - "X2D = X_centered.dot(W2)\n", - "print(X2D)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering\n", - "the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t\n", - "forget to center the data first.\n", - "\n", - "Once you have identified all the principal components, you can reduce the dimensionality of the dataset\n", - "down to $d$ dimensions by projecting it onto the hyperplane defined by the first $d$ principal components.\n", - "Selecting this hyperplane ensures that the projection will preserve as much variance as possible." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "W2 = V.T[:, :2]\n", - "X2D = X_centered.dot(W2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## PCA and scikit-learn\n", - "\n", - "Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The\n", - "following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note\n", - "that it automatically takes care of centering the data):" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "#thereafter we do a PCA with Scikit-learn\n", - "from sklearn.decomposition import PCA\n", - "pca = PCA(n_components = 2)\n", - "X2D = pca.fit_transform(X)\n", - "print(X2D)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "After fitting the PCA transformer to the dataset, you can access the principal components using the\n", - "components variable (note that it contains the PCs as horizontal vectors, so, for example, the first\n", - "principal component is equal to" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pca.components_.T[:, 0]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Another very useful piece of information is the explained variance ratio of each principal component,\n", - "available via the $explained\\_variance\\_ratio$ variable. It indicates the proportion of the dataset’s\n", - "variance that lies along the axis of each principal component. \n", - "\n", - "## Back to the Cancer Data\n", - "We can now repeat the above but applied to real data, in this case our breast cancer data.\n", - "Here we compute performance scores on the training data using logistic regression." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.linear_model import LogisticRegression\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "\n", - "logreg = LogisticRegression()\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Train set accuracy from Logistic Regression: {:.2f}\".format(logreg.score(X_train,y_train)))\n", - "# We scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Then perform again a log reg fit\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Train set accuracy scaled data: {:.2f}\".format(logreg.score(X_train_scaled,y_train)))\n", - "#thereafter we do a PCA with Scikit-learn\n", - "from sklearn.decomposition import PCA\n", - "pca = PCA(n_components = 2)\n", - "X2D_train = pca.fit_transform(X_train_scaled)\n", - "# and finally compute the log reg fit and the score on the training data\t\n", - "logreg.fit(X2D_train,y_train)\n", - "print(\"Train set accuracy scaled and PCA data: {:.2f}\".format(logreg.score(X2D_train,y_train)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see that our training data after the PCA decomposition has a performance similar to the non-scaled data. \n", - "\n", - "\n", - "Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to\n", - "choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).\n", - "Unless, of course, you are reducing dimensionality for data visualization — in that case you will\n", - "generally want to reduce the dimensionality down to 2 or 3.\n", - "The following code computes PCA without reducing dimensionality, then computes the minimum number\n", - "of dimensions required to preserve 95% of the training set’s variance:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pca = PCA()\n", - "pca.fit(X)\n", - "cumsum = np.cumsum(pca.explained_variance_ratio_)\n", - "d = np.argmax(cumsum >= 0.95) + 1" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "You could then set $n\\_components=d$ and run PCA again. However, there is a much better option: instead\n", - "of specifying the number of principal components you want to preserve, you can set $n\\_components$ to be\n", - "a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pca = PCA(n_components=0.95)\n", - "X_reduced = pca.fit_transform(X)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Incremental PCA\n", - "\n", - "One problem with the preceding implementation of PCA is that it requires the whole training set to fit in\n", - "memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have\n", - "been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch\n", - "at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new\n", - "instances arrive).\n", - "\n", - "\n", - "### Randomized PCA\n", - "\n", - "Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic\n", - "algorithm that quickly finds an approximation of the first d principal components. Its computational\n", - "complexity is $O(m \\times d^2)+O(d^3)$, instead of $O(m \\times n^2) + O(n^3)$, so it is dramatically faster than the\n", - "previous algorithms when $d$ is much smaller than $n$.\n", - "\n", - "\n", - "### Kernel PCA\n", - "\n", - "The kernel trick is a mathematical technique that implicitly maps instances into a\n", - "very high-dimensional space (called the feature space), enabling nonlinear classification and regression\n", - "with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature\n", - "space corresponds to a complex nonlinear decision boundary in the original space.\n", - "It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear\n", - "projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at\n", - "preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a\n", - "twisted manifold.\n", - "For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.decomposition import KernelPCA\n", - "rbf_pca = KernelPCA(n_components = 2, kernel=\"rbf\", gamma=0.04)\n", - "X_reduced = rbf_pca.fit_transform(X)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Other techniques\n", - "\n", - "\n", - "There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.\n", - "\n", - "Here are some of the most popular:\n", - "* **Multidimensional Scaling (MDS)** reduces dimensionality while trying to preserve the distances between the instances.\n", - "\n", - "* **Isomap** creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.\n", - "\n", - "* **t-Distributed Stochastic Neighbor Embedding** (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).\n", - "\n", - "* Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures." - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/chapter9.ipynb b/doc/LectureNotes/_build/html/_sources/chapter9.ipynb deleted file mode 100644 index 89b5cbf54..000000000 --- a/doc/LectureNotes/_build/html/_sources/chapter9.ipynb +++ /dev/null @@ -1,1283 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Neural networks\n", - "\n", - "Artificial neural networks are computational systems that can learn to\n", - "perform tasks by considering examples, generally without being\n", - "programmed with any task-specific rules. It is supposed to mimic a\n", - "biological system, wherein neurons interact by sending signals in the\n", - "form of mathematical functions between layers. All layers can contain\n", - "an arbitrary number of neurons, and each connection is represented by\n", - "a weight variable.\n", - "\n", - "\n", - "The field of artificial neural networks has a long history of\n", - "development, and is closely connected with the advancement of computer\n", - "science and computers in general. A model of artificial neurons was\n", - "first developed by McCulloch and Pitts in 1943 to study signal\n", - "processing in the brain and has later been refined by others. The\n", - "general idea is to mimic neural networks in the human brain, which is\n", - "composed of billions of neurons that communicate with each other by\n", - "sending electrical signals. Each neuron accumulates its incoming\n", - "signals, which must exceed an activation threshold to yield an\n", - "output. If the threshold is not overcome, the neuron remains inactive,\n", - "i.e. has zero output.\n", - "\n", - "This behaviour has inspired a simple mathematical model for an artificial neuron." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y = f\\left(\\sum_{i=1}^n w_ix_i\\right) = f(u)\n", - "\\label{artificialNeuron} \\tag{1}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here, the output $y$ of the neuron is the value of its activation function, which have as input\n", - "a weighted sum of signals $x_i, \\dots ,x_n$ received by $n$ other neurons.\n", - "\n", - "Conceptually, it is helpful to divide neural networks into four\n", - "categories:\n", - "1. general purpose neural networks for supervised learning,\n", - "\n", - "2. neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),\n", - "\n", - "3. neural networks for sequential data such as Recurrent Neural Networks (RNNs), and\n", - "\n", - "4. neural networks for unsupervised learning such as Deep Boltzmann Machines.\n", - "\n", - "In natural science, DNNs and CNNs have already found numerous\n", - "applications. In statistical physics, they have been applied to detect\n", - "phase transitions in 2D Ising and Potts models, lattice gauge\n", - "theories, and different phases of polymers, or solving the\n", - "Navier-Stokes equation in weather forecasting. Deep learning has also\n", - "found interesting applications in quantum physics. Various quantum\n", - "phase transitions can be detected and studied using DNNs and CNNs,\n", - "topological phases, and even non-equilibrium many-body\n", - "localization. Representing quantum states as DNNs quantum state\n", - "tomography are among some of the impressive achievements to reveal the\n", - "potential of DNNs to facilitate the study of quantum systems.\n", - "\n", - "In quantum information theory, it has been shown that one can perform\n", - "gate decompositions with the help of neural. \n", - "\n", - "The applications are not limited to the natural sciences. There is a\n", - "plethora of applications in essentially all disciplines, from the\n", - "humanities to life science and medicine.\n", - "\n", - "\n", - "An artificial neural network (ANN), is a computational model that\n", - "consists of layers of connected neurons, or nodes or units. We will\n", - "refer to these interchangeably as units or nodes, and sometimes as\n", - "neurons.\n", - "\n", - "It is supposed to mimic a biological nervous system by letting each\n", - "neuron interact with other neurons by sending signals in the form of\n", - "mathematical functions between layers. A wide variety of different\n", - "ANNs have been developed, but most of them consist of an input layer,\n", - "an output layer and eventual layers in-between, called *hidden\n", - "layers*. All layers can contain an arbitrary number of nodes, and each\n", - "connection between two nodes is associated with a weight variable.\n", - "\n", - "Neural networks (also called neural nets) are neural-inspired\n", - "nonlinear models for supervised learning. As we will see, neural nets\n", - "can be viewed as natural, more powerful extensions of supervised\n", - "learning methods such as linear and logistic regression and soft-max\n", - "methods we discussed earlier.\n", - "\n", - "\n", - "### Feed-forward neural networks\n", - "\n", - "The feed-forward neural network (FFNN) was the first and simplest type\n", - "of ANNs that were devised. In this network, the information moves in\n", - "only one direction: forward through the layers.\n", - "\n", - "Nodes are represented by circles, while the arrows display the\n", - "connections between the nodes, including the direction of information\n", - "flow. Additionally, each arrow corresponds to a weight variable\n", - "(figure to come). We observe that each node in a layer is connected\n", - "to *all* nodes in the subsequent layer, making this a so-called\n", - "*fully-connected* FFNN.\n", - "\n", - "\n", - "\n", - "### Convolutional Neural Network\n", - "\n", - "A different variant of FFNNs are *convolutional neural networks*\n", - "(CNNs), which have a connectivity pattern inspired by the animal\n", - "visual cortex. Individual neurons in the visual cortex only respond to\n", - "stimuli from small sub-regions of the visual field, called a receptive\n", - "field. This makes the neurons well-suited to exploit the strong\n", - "spatially local correlation present in natural images. The response of\n", - "each neuron can be approximated mathematically as a convolution\n", - "operation. (figure to come)\n", - "\n", - "Convolutional neural networks emulate the behaviour of neurons in the\n", - "visual cortex by enforcing a *local* connectivity pattern between\n", - "nodes of adjacent layers: Each node in a convolutional layer is\n", - "connected only to a subset of the nodes in the previous layer, in\n", - "contrast to the fully-connected FFNN. Often, CNNs consist of several\n", - "convolutional layers that learn local features of the input, with a\n", - "fully-connected layer at the end, which gathers all the local data and\n", - "produces the outputs. They have wide applications in image and video\n", - "recognition.\n", - "\n", - "### Recurrent neural networks\n", - "\n", - "So far we have only mentioned ANNs where information flows in one\n", - "direction: forward. *Recurrent neural networks* on the other hand,\n", - "have connections between nodes that form directed *cycles*. This\n", - "creates a form of internal memory which are able to capture\n", - "information on what has been calculated before; the output is\n", - "dependent on the previous computations. Recurrent NNs make use of\n", - "sequential information by performing the same task for every element\n", - "in a sequence, where each element depends on previous elements. An\n", - "example of such information is sentences, making recurrent NNs\n", - "especially well-suited for handwriting and speech recognition.\n", - "\n", - "### Other types of networks\n", - "\n", - "There are many other kinds of ANNs that have been developed. One type\n", - "that is specifically designed for interpolation in multidimensional\n", - "space is the radial basis function (RBF) network. RBFs are typically\n", - "made up of three layers: an input layer, a hidden layer with\n", - "non-linear radial symmetric activation functions and a linear output\n", - "layer (''linear'' here means that each node in the output layer has a\n", - "linear activation function). The layers are normally fully-connected\n", - "and there are no cycles, thus RBFs can be viewed as a type of\n", - "fully-connected FFNN. They are however usually treated as a separate\n", - "type of NN due the unusual activation functions.\n", - "\n", - "\n", - "## Multilayer perceptrons\n", - "\n", - "One uses often so-called fully-connected feed-forward neural networks\n", - "with three or more layers (an input layer, one or more hidden layers\n", - "and an output layer) consisting of neurons that have non-linear\n", - "activation functions.\n", - "\n", - "Such networks are often called *multilayer perceptrons* (MLPs).\n", - "\n", - "\n", - "According to the *Universal approximation theorem*, a feed-forward\n", - "neural network with just a single hidden layer containing a finite\n", - "number of neurons can approximate a continuous multidimensional\n", - "function to arbitrary accuracy, assuming the activation function for\n", - "the hidden layer is a **non-constant, bounded and\n", - "monotonically-increasing continuous function**.\n", - "\n", - "Note that the requirements on the activation function only applies to\n", - "the hidden layer, the output nodes are always assumed to be linear, so\n", - "as to not restrict the range of output values.\n", - "\n", - "\n", - "\n", - "The output $y$ is produced via the activation function $f$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y = f\\left(\\sum_{i=1}^n w_ix_i + b_i\\right) = f(z),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This function receives $x_i$ as inputs.\n", - "Here the activation $z=(\\sum_{i=1}^n w_ix_i+b_i)$. \n", - "In an FFNN of such neurons, the *inputs* $x_i$ are the *outputs* of\n", - "the neurons in the preceding layer. Furthermore, an MLP is\n", - "fully-connected, which means that each neuron receives a weighted sum\n", - "of the outputs of *all* neurons in the previous layer.\n", - "\n", - "\n", - "First, for each node $i$ in the first hidden layer, we calculate a weighted sum $z_i^1$ of the input coordinates $x_j$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} z_i^1 = \\sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1\n", - "\\label{_auto1} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here $b_i$ is the so-called bias which is normally needed in\n", - "case of zero activation weights or inputs. How to fix the biases and\n", - "the weights will be discussed below. The value of $z_i^1$ is the\n", - "argument to the activation function $f_i$ of each node $i$, The\n", - "variable $M$ stands for all possible inputs to a given node $i$ in the\n", - "first layer. We define the output $y_i^1$ of all neurons in layer 1 as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y_i^1 = f(z_i^1) = f\\left(\\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\\right)\n", - "\\label{outputLayer1} \\tag{3}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we assume that all nodes in the same layer have identical\n", - "activation functions, hence the notation $f$. In general, we could assume in the more general case that different layers have different activation functions.\n", - "In this case we would identify these functions with a superscript $l$ for the $l$-th layer," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y_i^l = f^l(u_i^l) = f^l\\left(\\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\\right)\n", - "\\label{generalLayer} \\tag{4}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $N_l$ is the number of nodes in layer $l$. When the output of\n", - "all the nodes in the first hidden layer are computed, the values of\n", - "the subsequent layer can be calculated and so forth until the output\n", - "is obtained.\n", - "\n", - "\n", - "\n", - "\n", - "The output of neuron $i$ in layer 2 is thus," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y_i^2 = f^2\\left(\\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\\right) \n", - "\\label{_auto2} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \n", - " = f^2\\left[\\sum_{j=1}^N w_{ij}^2f^1\\left(\\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\\right) + b_i^2\\right]\n", - "\\label{outputLayer2} \\tag{6}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have substituted $y_k^1$ with the inputs $x_k$. Finally, the ANN output reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y_i^3 = f^3\\left(\\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\\right) \n", - "\\label{_auto3} \\tag{7}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \n", - " = f_3\\left[\\sum_{j} w_{ij}^3 f^2\\left(\\sum_{k} w_{jk}^2 f^1\\left(\\sum_{m} w_{km}^1 x_m + b_k^1\\right) + b_j^2\\right)\n", - " + b_1^3\\right]\n", - "\\label{_auto4} \\tag{8}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can generalize this expression to an MLP with $l$ hidden\n", - "layers. The complete functional form is," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "y^{l+1}_i = f^{l+1}\\left[\\!\\sum_{j=1}^{N_l} w_{ij}^3 f^l\\left(\\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\\left(\\dots f^1\\left(\\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\\right)\\dots\\right)+b_k^2\\right)+b_1^3\\right] \n", - "\\label{completeNN} \\tag{9}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which illustrates a basic property of MLPs: The only independent\n", - "variables are the input values $x_n$.\n", - "\n", - "\n", - "This confirms that an MLP, despite its quite convoluted mathematical\n", - "form, is nothing more than an analytic function, specifically a\n", - "mapping of real-valued vectors $\\hat{x} \\in \\mathbb{R}^n \\rightarrow\n", - "\\hat{y} \\in \\mathbb{R}^m$.\n", - "\n", - "Furthermore, the flexibility and universality of an MLP can be\n", - "illustrated by realizing that the expression is essentially a nested\n", - "sum of scaled activation functions of the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " f(x) = c_1 f(c_2 x + c_3) + c_4\n", - "\\label{_auto5} \\tag{10}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the parameters $c_i$ are weights and biases. By adjusting these\n", - "parameters, the activation functions can be shifted up and down or\n", - "left and right, change slope or be rescaled which is the key to the\n", - "flexibility of a neural network.\n", - "\n", - "\n", - "We can introduce a more convenient notation for the activations in an A NN. \n", - "\n", - "Additionally, we can represent the biases and activations\n", - "as layer-wise column vectors $\\hat{b}_l$ and $\\hat{y}_l$, so that the $i$-th element of each vector \n", - "is the bias $b_i^l$ and activation $y_i^l$ of node $i$ in layer $l$ respectively. \n", - "\n", - "We have that $\\mathrm{W}_l$ is an $N_{l-1} \\times N_l$ matrix, while $\\hat{b}_l$ and $\\hat{y}_l$ are $N_l \\times 1$ column vectors. \n", - "With this notation, the sum becomes a matrix-vector multiplication, and we can write\n", - "the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\hat{y}_2 = f_2(\\mathrm{W}_2 \\hat{y}_{1} + \\hat{b}_{2}) = \n", - " f_2\\left(\\left[\\begin{array}{ccc}\n", - " w^2_{11} &w^2_{12} &w^2_{13} \\\\\n", - " w^2_{21} &w^2_{22} &w^2_{23} \\\\\n", - " w^2_{31} &w^2_{32} &w^2_{33} \\\\\n", - " \\end{array} \\right] \\cdot\n", - " \\left[\\begin{array}{c}\n", - " y^1_1 \\\\\n", - " y^1_2 \\\\\n", - " y^1_3 \\\\\n", - " \\end{array}\\right] + \n", - " \\left[\\begin{array}{c}\n", - " b^2_1 \\\\\n", - " b^2_2 \\\\\n", - " b^2_3 \\\\\n", - " \\end{array}\\right]\\right).\n", - "\\label{_auto6} \\tag{11}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Matrix-vector notation and activation\n", - "\n", - "The activation of node $i$ in layer 2 is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y^2_i = f_2\\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\\Bigr) = \n", - " f_2\\left(\\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\\right).\n", - "\\label{_auto7} \\tag{12}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This is not just a convenient and compact notation, but also a useful\n", - "and intuitive way to think about MLPs: The output is calculated by a\n", - "series of matrix-vector multiplications and vector additions that are\n", - "used as input to the activation functions. For each operation\n", - "$\\mathrm{W}_l \\hat{y}_{l-1}$ we move forward one layer.\n", - "\n", - "\n", - "\n", - "### Activation functions\n", - "\n", - "A property that characterizes a neural network, other than its\n", - "connectivity, is the choice of activation function(s). As described\n", - "in, the following restrictions are imposed on an activation function\n", - "for a FFNN to fulfill the universal approximation theorem\n", - "\n", - " * Non-constant\n", - "\n", - " * Bounded\n", - "\n", - " * Monotonically-increasing\n", - "\n", - " * Continuous\n", - "\n", - "The second requirement excludes all linear functions. Furthermore, in\n", - "a MLP with only linear activation functions, each layer simply\n", - "performs a linear transformation of its inputs.\n", - "\n", - "Regardless of the number of layers, the output of the NN will be\n", - "nothing but a linear function of the inputs. Thus we need to introduce\n", - "some kind of non-linearity to the NN to be able to fit non-linear\n", - "functions Typical examples are the logistic *Sigmoid*" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(x) = \\frac{1}{1 + e^{-x}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the *hyperbolic tangent* function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(x) = \\tanh(x)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The *sigmoid* function are more biologically plausible because the\n", - "output of inactive neurons are zero. Such activation function are\n", - "called *one-sided*. However, it has been shown that the hyperbolic\n", - "tangent performs better than the sigmoid for training MLPs. has\n", - "become the most popular for *deep neural networks*" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "\"\"\"The sigmoid function (or the logistic curve) is a \n", - "function that takes any real number, z, and outputs a number (0,1).\n", - "It is useful in neural networks for assigning weights on a relative scale.\n", - "The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n", - "\n", - "import numpy\n", - "import matplotlib.pyplot as plt\n", - "import math as mt\n", - "\n", - "z = numpy.arange(-5, 5, .1)\n", - "sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n", - "sigma = sigma_fn(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, sigma)\n", - "ax.set_ylim([-0.1, 1.1])\n", - "ax.set_xlim([-5,5])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('sigmoid function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"Step Function\"\"\"\n", - "z = numpy.arange(-5, 5, .02)\n", - "step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n", - "step = step_fn(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, step)\n", - "ax.set_ylim([-0.5, 1.5])\n", - "ax.set_xlim([-5,5])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('step function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"Sine Function\"\"\"\n", - "z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n", - "t = numpy.sin(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, t)\n", - "ax.set_ylim([-1.0, 1.0])\n", - "ax.set_xlim([-2*mt.pi,2*mt.pi])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('sine function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"Plots a graph of the squashing function used by a rectified linear\n", - "unit\"\"\"\n", - "z = numpy.arange(-2, 2, .1)\n", - "zero = numpy.zeros(len(z))\n", - "y = numpy.max([zero, z], axis=0)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, y)\n", - "ax.set_ylim([-2.0, 2.0])\n", - "ax.set_xlim([-2.0, 2.0])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('Rectified linear unit')\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The multilayer perceptron (MLP)\n", - "\n", - "The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of\n", - "1. A neural network with one or more layers of nodes between the input and the output nodes.\n", - "\n", - "2. The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.\n", - "\n", - "3. The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.\n", - "\n", - "As a convention it is normal to call a network with one layer of input units, one layer of hidden\n", - "units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.\n", - "\n", - "For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.\n", - "Hereafter we will call the various entities of a layer for nodes.\n", - "There are also no connections within a single layer.\n", - "\n", - "The number of input nodes does not need to equal the number of output\n", - "nodes. This applies also to the hidden layers. Each layer may have its\n", - "own number of nodes and activation functions.\n", - "\n", - "The hidden layers have their name from the fact that they are not\n", - "linked to observables and as we will see below when we define the\n", - "so-called activation $\\hat{z}$, we can think of this as a basis\n", - "expansion of the original inputs $\\hat{x}$. The difference however\n", - "between neural networks and say linear regression is that now these\n", - "basis functions (which will correspond to the weights in the network)\n", - "are learned from data. This results in an important difference between\n", - "neural networks and deep learning approaches on one side and methods\n", - "like logistic regression or linear regression and their modifications on the other side.\n", - "\n", - "\n", - "### From one to many layers, the universal approximation theorem\n", - "\n", - "A neural network with only one layer, what we called the simple\n", - "perceptron, is best suited if we have a standard binary model with\n", - "clear (linear) boundaries between the outcomes. As such it could\n", - "equally well be replaced by standard linear regression or logistic\n", - "regression. Networks with one or more hidden layers approximate\n", - "systems with more complex boundaries.\n", - "\n", - "As stated earlier, \n", - "an important theorem in studies of neural networks, restated without\n", - "proof here, is the [universal approximation\n", - "theorem](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf).\n", - "\n", - "It states that a feed-forward network with a single hidden layer\n", - "containing a finite number of neurons can approximate continuous\n", - "functions on compact subsets of real functions. The theorem thus\n", - "states that simple neural networks can represent a wide variety of\n", - "interesting functions when given appropriate parameters. It is the\n", - "multilayer feedforward architecture itself which gives neural networks\n", - "the potential of being universal approximators.\n", - "\n", - "\n", - "\n", - "## Deriving the back propagation code for a multilayer perceptron model\n", - "\n", - "\n", - "\n", - "As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.\n", - "The unknowwn quantities are our weights $w_{ij}$ and we need to find an algorithm for changing them so that our errors are as small as possible.\n", - "This leads us to the famous [back propagation algorithm](https://www.nature.com/articles/323533a0).\n", - "\n", - "The questions we want to ask are how do changes in the biases and the\n", - "weights in our network change the cost function and how can we use the\n", - "final output to modify the weights?\n", - "\n", - "To derive these equations let us start with a plain regression problem\n", - "and define our cost function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\cal C}(\\hat{W}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the $t_i$s are our $n$ targets (the values we want to\n", - "reproduce), while the outputs of the network after having propagated\n", - "all inputs $\\hat{x}$ are given by $y_i$. Below we will demonstrate\n", - "how the basic equations arising from the back propagation algorithm\n", - "can be modified in order to study classification problems with $K$\n", - "classes.\n", - "\n", - "\n", - "With our definition of the targets $\\hat{t}$, the outputs of the\n", - "network $\\hat{y}$ and the inputs $\\hat{x}$ we\n", - "define now the activation $z_j^l$ of node/neuron/unit $j$ of the\n", - "$l$-th layer as a function of the bias, the weights which add up from\n", - "the previous layer $l-1$ and the forward passes/outputs\n", - "$\\hat{a}^{l-1}$ from the previous layer as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "z_j^l = \\sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $b_k^l$ are the biases from layer $l$. Here $M_{l-1}$\n", - "represents the total number of nodes/neurons/units of layer $l-1$. The\n", - "figure here illustrates this equation. We can rewrite this in a more\n", - "compact form as the matrix-vector products we discussed earlier," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{z}^l = \\left(\\hat{W}^l\\right)^T\\hat{a}^{l-1}+\\hat{b}^l.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With the activation values $\\hat{z}^l$ we can in turn define the\n", - "output of layer $l$ as $\\hat{a}^l = f(\\hat{z}^l)$ where $f$ is our\n", - "activation function. In the examples here we will use the sigmoid\n", - "function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers\n", - "and their nodes. It means we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_j^l = f(z_j^l) = \\frac{1}{1+\\exp{-(z_j^l)}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Derivatives and the chain rule\n", - "\n", - "From the definition of the activation $z_j^l$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial z_j^l}{\\partial w_{ij}^l} = a_i^{l-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial z_j^l}{\\partial a_i^{l-1}} = w_{ji}^l.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With our definition of the activation function we have that (note that this function depends only on $z_j^l$)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial a_j^l}{\\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With these definitions we can now compute the derivative of the cost function in terms of the weights.\n", - "\n", - "Let us specialize to the output layer $l=L$. Our cost function is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\cal C}(\\hat{W^L}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2=\\frac{1}{2}\\sum_{i=1}^n\\left(a_i^L - t_i\\right)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The derivative of this function with respect to the weights is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\left(a_j^L - t_j\\right)\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The last partial derivative can easily be computed and reads (by applying the chain rule)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}} = \\frac{\\partial a_j^L}{\\partial z_{j}^{L}}\\frac{\\partial z_j^L}{\\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Bringing it together, first back propagation equation\n", - "\n", - "We have thus" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\left(a_j^L - t_j\\right)a_j^L(1-a_j^L)a_k^{L-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Defining" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^L = a_j^L(1-a_j^L)\\left(a_j^L - t_j\\right) = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and using the Hadamard product of two vectors we can write this as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{\\delta}^L = f'(\\hat{z}^L)\\circ\\frac{\\partial {\\cal C}}{\\partial (\\hat{a}^L)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This is an important expression. The second term on the right handside\n", - "measures how fast the cost function is changing as a function of the $j$th\n", - "output activation. If, for example, the cost function doesn't depend\n", - "much on a particular output node $j$, then $\\delta_j^L$ will be small,\n", - "which is what we would expect. The first term on the right, measures\n", - "how fast the activation function $f$ is changing at a given activation\n", - "value $z_j^L$.\n", - "\n", - "Notice that everything in the above equations is easily computed. In\n", - "particular, we compute $z_j^L$ while computing the behaviour of the\n", - "network, and it is only a small additional overhead to compute\n", - "$f'(z^L_j)$. The exact form of the derivative with respect to the\n", - "output depends on the form of the cost function.\n", - "However, provided the cost function is known there should be little\n", - "trouble in calculating" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With the definition of $\\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is also easy to see that our previous equation can be written as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^L =\\frac{\\partial {\\cal C}}{\\partial z_j^L}= \\frac{\\partial {\\cal C}}{\\partial a_j^L}\\frac{\\partial a_j^L}{\\partial z_j^L},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L}\\frac{\\partial b_j^L}{\\partial z_j^L}=\\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "That is, the error $\\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias. \n", - "\n", - "We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are\n", - "\n", - "**The starting equations.**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1},\n", - "\\label{_auto8} \\tag{13}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", - "\\label{_auto9} \\tag{14}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", - "\\label{_auto10} \\tag{15}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "An interesting consequence of the above equations is that when the\n", - "activation $a_k^{L-1}$ is small, the gradient term, that is the\n", - "derivative of the cost function with respect to the weights, will also\n", - "tend to be small. We say then that the weight learns slowly, meaning\n", - "that it changes slowly when we minimize the weights via say gradient\n", - "descent. In this case we say the system learns slowly.\n", - "\n", - "Another interesting feature is that is when the activation function,\n", - "represented by the sigmoid function here, is rather flat when we move towards\n", - "its end values $0$ and $1$ (see the above Python codes). In these\n", - "cases, the derivatives of the activation function will also be close\n", - "to zero, meaning again that the gradients will be small and the\n", - "network learns slowly again.\n", - "\n", - "\n", - "\n", - "We need a fourth equation and we are set. We are going to propagate\n", - "backwards in order to the determine the weights and biases. In order\n", - "to do so we need to represent the error in the layer before the final\n", - "one $L-1$ in terms of the errors in the final output layer.\n", - "\n", - "### Final back propagating equation\n", - "\n", - "We have that (replacing $L$ with a general layer $l$)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^l =\\frac{\\partial {\\cal C}}{\\partial z_j^l}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^l =\\sum_k \\frac{\\partial {\\cal C}}{\\partial z_k^{l+1}}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}}=\\sum_k \\delta_k^{l+1}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and recalling that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "z_j^{l+1} = \\sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $M_l$ being the number of nodes in layer $l$, we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^l =\\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This is our final equation.\n", - "\n", - "We are now ready to set up the algorithm for back propagation and learning the weights and biases.\n", - "\n", - "\n", - "### Setting up the Back propagation algorithm\n", - "\n", - "The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.\n", - "\n", - "First, we set up the input data $\\hat{x}$ and the activations\n", - "$\\hat{z}_1$ of the input layer and compute the activation function and\n", - "the pertinent outputs $\\hat{a}^1$.\n", - "\n", - "\n", - "\n", - "Secondly, we perform then the feed forward till we reach the output\n", - "layer and compute all $\\hat{z}_l$ of the input layer and compute the\n", - "activation function and the pertinent outputs $\\hat{a}^l$ for\n", - "$l=2,3,\\dots,L$.\n", - "\n", - "\n", - "\n", - "Thereafter we compute the ouput error $\\hat{\\delta}^L$ by computing all" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Then we compute the back propagate error for each $l=L-1,L-2,\\dots,2$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,2$ and update the weights and biases according to the rules" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "w_{jk}^l\\leftarrow = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The parameter $\\eta$ is the learning parameter discussed in connection with the gradient descent methods.\n", - "Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training." - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/content.md b/doc/LectureNotes/_build/html/_sources/content.md deleted file mode 100644 index 0f6aca77a..000000000 --- a/doc/LectureNotes/_build/html/_sources/content.md +++ /dev/null @@ -1,5 +0,0 @@ -Content in Jupyter Book -======================= - -There are many ways to write content in Jupyter Book. This short section -covers a few tips for how to do so. diff --git a/doc/LectureNotes/_build/html/_sources/intro.md b/doc/LectureNotes/_build/html/_sources/intro.md deleted file mode 100644 index 811e85085..000000000 --- a/doc/LectureNotes/_build/html/_sources/intro.md +++ /dev/null @@ -1,145 +0,0 @@ -# Applied Data Analysis and Machine Learning - - -## Introduction - -Probability theory and statistical methods play a central role in science. Nowadays we are -surrounded by huge amounts of data. For example, there are about one trillion web pages; more than one -hour of video is uploaded to YouTube every second, amounting to years of content every -day; the genomes of 1000s of people, each of which has a length of more than a billion base pairs, have -been sequenced by various labs and so on. This deluge of data calls for automated methods of data analysis, -which is exactly what machine learning aims at providing. - -## Learning outcomes - -This course aims at giving you insights and knowledge about many of the central algorithms used in Data Analysis and Machine Learning. The course is project based and through various numerical projects, normally three, you will be exposed to fundamental research problems in these fields, with the aim to reproduce state of the art scientific results. Both supervised and unsupervised methods will be covered. The emphasis is on a frequentist approach, although we will try to link it with a Bayesian approach as well. You will learn to develop and structure large codes for studying different cases where Machine Learning is applied to, get acquainted with computing facilities and learn to handle large scientific projects. A good scientific and ethical conduct is emphasized throughout the course. More specifically, after this course you will - -- Learn about basic data analysis, statistical analysis, Bayesian statistics, Monte Carlo sampling, data optimization and machine learning; -- Be capable of extending the acquired knowledge to other systems and cases; -- Have an understanding of central algorithms used in data analysis and machine learning; -- Understand linear methods for regression and classification, from ordinary least squares, via Lasso and Ridge to Logistic regression; -- Learn about neural networks and deep learning methods for supervised and unsupervised learning. Emphasis on feed forward neural networks, convolutional and recurrent neural networks; -- Learn about about decision trees, random forests, bagging and boosting methods; -- Learn about support vector machines and kernel transformations; -- Reduction of data sets, from PCA to clustering; -- Autoencoders and Reinforcement Learning; -- Work on numerical projects to illustrate the theory. The projects play a central role and you are expected to know modern programming languages like Python or C++ and/or Fortran (Fortran2003 or later). - -## Prerequisites - -Basic knowledge in programming and mathematics, with an emphasis on -linear algebra. Knowledge of Python or/and C++ as programming -languages is strongly recommended and experience with Jupiter notebook -is recommended. Required courses are the equivalents to the University -of Oslo mathematics courses MAT1100, MAT1110, MAT1120 and at least one -of the corresponding computing and programming courses INF1000/INF1110 -or MAT-INF1100/MAT-INF1100L/BIOS1100/KJM-INF1100. Most universities -offer nowadays a basic programming course (often compulsory) where -Python is the recurring programming language. - - -## The course has two central parts - -1. Statistical analysis and optimization of data -2. Machine learning - -These topics will be scattered thorughout the course and may not necessarily be taught separately. Rather, we will often take an approach (during the lectures and project/exercise sessions) where say elements from statistical data analysis are mixed with specific Machine Learning algorithms. - -### Statistical analysis and optimization of data - -The following topics will be covered -- Basic concepts, expectation values, variance, covariance, correlation functions and errors; -- Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions; -- Central elements of Bayesian statistics and modeling; -- Gradient methods for data optimization, -- Monte Carlo methods, Markov chains, Gibbs sampling and Metropolis-Hastings sampling; -- Estimation of errors and resampling techniques such as the cross-validation, blocking, bootstrapping and jackknife methods; -- Principal Component Analysis (PCA) and its mathematical foundation - -### Machine learning - -The following topics will be covered: -- Linear Regression and Logistic Regression; -- Neural networks and deep learning, including convolutional and recurrent neural networks -- Decisions trees, Random Forests, Bagging and Boosting -- Support vector machines -- Bayesian linear and logistic regression -- Boltzmann Machines -- Unsupervised learning Dimensionality reduction, from PCA to cluster models - -Hands-on demonstrations, exercises and projects aim at deepening your understanding of these topics. - -Computational aspects play a central role and you are -expected to work on numerical examples and projects which illustrate -the theory and varous algorithms discussed during the lectures. We recommend strongly to form small project groups of 2-3 participants, if possible. - -## Required Technologies - -Course participants are expected to have their own laptops/PCs. We use _Git_ as version control software and the usage of providers like _GitHub_, _GitLab_ or similar are strongly recommended. - -We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -Jupyter notebooks invaluable in your work. You can run _R_ -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be mainly -on Python. - - -If you have Python installed and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via _pip_ as - -* pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow - -For OSX users we recommend, after having installed Xcode, to -install _brew_. Brew allows for a seamless installation of additional -software via for example - -* brew install python3 - -For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use _pip_ as well and simply install Python as - -* sudo apt-get install python3 - -### Python installers - -If you don't want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely - -* Anaconda:https://docs.anaconda.com/, - -which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system _conda_. - -* Enthought canopy:https://www.enthought.com/product/canopy/ - -is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license. - -Furthermore, Google's Colab:https://colab.research.google.com/notebooks/welcome.ipynb is a free Jupyter notebook environment that requires -no setup and runs entirely in the cloud. Try it out! - -### Useful Python libraries -Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there) - -* _NumPy_:https://www.numpy.org/ is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays -* _The pandas_:https://pandas.pydata.org/ library provides high-performance, easy-to-use data structures and data analysis tools -* _Xarray_:http://xarray.pydata.org/en/stable/ is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun! -* _Scipy_:https://www.scipy.org/ (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. -* _Matplotlib_:https://matplotlib.org/ is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms. -* _Autograd_:https://github.com/HIPS/autograd can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives -* _SymPy_:https://www.sympy.org/en/index.html is a Python library for symbolic mathematics. -* _scikit-learn_:https://scikit-learn.org/stable/ has simple and efficient tools for machine learning, data mining and data analysis -* _TensorFlow_:https://www.tensorflow.org/ is a Python library for fast numerical computing created and released by Google -* _Keras_:https://keras.io/ is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano -* And many more such as _pytorch_:https://pytorch.org/, _Theano_:https://pypi.org/project/Theano/ etc - diff --git a/doc/LectureNotes/_build/html/_sources/linalg.ipynb b/doc/LectureNotes/_build/html/_sources/linalg.ipynb deleted file mode 100644 index 874a51b2e..000000000 --- a/doc/LectureNotes/_build/html/_sources/linalg.ipynb +++ /dev/null @@ -1,1442 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Linear Algebra, Handling of Arrays and more Python Features\n", - "\n", - "## Introduction\n", - "\n", - "The aim of this set of lectures is to review some central linear algebra algorithms that we will need in our \n", - "data analysis part and in the construction of Machine Learning algorithms (ML). \n", - "This will allow us to introduce some central programming features of high-level languages like Python and \n", - "compiled languages like C++ and/or Fortran. \n", - "\n", - "As discussed in the introductory notes, these series of lectures focuses both on using\n", - "central Python packages like **tensorflow** and **scikit-learn** as well\n", - "as writing your own codes for some central ML algorithms. The\n", - "latter can be written in a language of your choice, be it Python, Julia, R,\n", - "Rust, C++, Fortran etc. In order to avoid confusion however, in these lectures we will limit our\n", - "attention to Python, C++ and Fortran. \n", - "\n", - "\n", - "## Important Matrix and vector handling packages\n", - "\n", - "There are several central software packages for linear algebra and eigenvalue problems. Several of the more\n", - "popular ones have been wrapped into ofter software packages like those from the widely used text **Numerical Recipes**. The original source codes in many of the available packages are often taken from the widely used\n", - "software package LAPACK, which follows two other popular packages\n", - "developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.\n", - "\n", - " * LINPACK: package for linear equations and least square problems.\n", - "\n", - " * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.\n", - "\n", - " * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from .\n", - "\n", - "When dealing with matrices and vectors a central issue is memory\n", - "handling and allocation. If our code is written in Python the way we\n", - "declare these objects and the way they are handled, interpreted and\n", - "used by say a linear algebra library, requires codes that interface\n", - "our Python program with such libraries. For Python programmers,\n", - "**Numpy** is by now the standard Python package for numerical arrays in\n", - "Python as well as the source of functions which act on these\n", - "arrays. These functions span from eigenvalue solvers to functions that\n", - "compute the mean value, variance or the covariance matrix. If you are\n", - "not familiar with how arrays are handled in say Python or compiled\n", - "languages like C++ and Fortran, the sections in this chapter may be\n", - "useful. For C++ programmer, **Armadillo** is widely used library for\n", - "linear algebra and eigenvalue problems. In addition it offers a\n", - "convenient way to handle and organize arrays. We discuss this library\n", - "as well. Before we proceed we believe it may be convenient to repeat some basic features of \n", - " matrices and vectors.\n", - "\n", - "\n", - "## Basic Matrix Features\n", - "\n", - "Matrix properties reminder" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A} =\n", - " \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", - " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", - " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", - " a_{41} & a_{42} & a_{43} & a_{44}\n", - " \\end{bmatrix}\\qquad\n", - "\\mathbf{I} =\n", - " \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n", - " 0 & 1 & 0 & 0 \\\\\n", - " 0 & 0 & 1 & 0 \\\\\n", - " 0 & 0 & 0 & 1\n", - " \\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The inverse of a matrix is defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \\left (A^{T} \\right )^{-1}$ real orthogonal $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \\left (A^{\\dagger} \\right )^{-1}$ unitary $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$
\n", - "### Some famous Matrices\n", - "\n", - " * Diagonal if $a_{ij}=0$ for $i\\ne j$\n", - "\n", - " * Upper triangular if $a_{ij}=0$ for $i > j$\n", - "\n", - " * Lower triangular if $a_{ij}=0$ for $i < j$\n", - "\n", - " * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$\n", - "\n", - " * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$\n", - "\n", - " * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$\n", - "\n", - " * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$\n", - "\n", - " * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$\n", - "\n", - " * Banded, block upper triangular, block lower triangular....\n", - "\n", - "Some Equivalent Statements. For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", - "\n", - " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", - "\n", - " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", - "\n", - " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * $\\mathbf{A}$ is a product of elementary matrices.\n", - "\n", - " * $0$ is not eigenvalue of $\\mathbf{A}$.\n", - "\n", - "## Numpy and arrays\n", - "[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "x = np.random.normal(size=n)\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.\n", - "Another alternative is to declare a vector as follows" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.array([1, 2, 3])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++\n", - "start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \\dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4, 7, 8]))\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have used Numpy's unary function $np.log$. This function is\n", - "highly tuned to compute array elements since the code is vectorized\n", - "and does not require looping. We normaly recommend that you use the\n", - "Numpy intrinsic functions instead of the corresponding **log** function\n", - "from Python's **math** module. The looping is done explicitely by the\n", - "**np.log** function. The alternative, and slower way to compute the\n", - "logarithms of a vector would be to write" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "from math import log\n", - "x = np.array([4, 7, 8])\n", - "for i in range(0, len(x)):\n", - " x[i] = log(x[i])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note that our code is much longer already and we need to import the **log** function from the **math** module. \n", - "The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automacally our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4, 7, 8], dtype = np.float64))\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", - "print(x.itemsize)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Having defined vectors, we are now ready to try out matrices. We can define a $3 \\times 3 $ real matrix $\\hat{A}$\n", - "as (recall that we user lowercase letters for vectors and uppercase letters for matrices)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "# print the first column, row-major order and elements start with 0\n", - "print(A[:,0])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can continue this was by printing out other columns or rows. The example here prints out the second column" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "# print the first column, row-major order and elements start with 0\n", - "print(A[1,:])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to zero\n", - "A = np.zeros( (n, n) )\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or initializing all elements to" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to one\n", - "A = np.ones( (n, n) )\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or as unitarily distributed random numbers (see the material on random number generators in the statistics part)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \\in [0, 1]\n", - "A = np.random.rand(n, n)\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n", - "As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n", - "$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", - " \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n", - " \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where for example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma_{xy} =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. For a more in-depth discussion of the covariance and covariance matrix and its meaning, we refer you to the lectures on statistics. \n", - "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $ 3\\times n$ matrix $\\hat{W}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n", - " x_1 & y_1 & z_1 \\\\\n", - " x_2 & y_2 & z_2 \\\\\n", - " \\dots & \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2} & z_{n-2} \\\\\n", - " x_{n-1} & y_{n-1} & z_{n-1}\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in turn is converted into into the $3 times 3$ covariance matrix\n", - "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. In our review of\n", - "statistical functions and quantities we will discuss more about the\n", - "meaning of the covariance matrix. Here we note that we can calculate\n", - "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", - "function **np.mean(x)**. We can also extract the eigenvalues of the\n", - "covariance matrix through the **np.linalg.eig()** function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "z = x**3+np.random.normal(size=n)\n", - "print(np.mean(z))\n", - "W = np.vstack((x, y, z))\n", - "Sigma = np.cov(W)\n", - "print(Sigma)\n", - "Eigvals, Eigvecs = np.linalg.eig(Sigma)\n", - "print(Eigvals)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from scipy import sparse\n", - "eye = np.eye(4)\n", - "print(eye)\n", - "sparse_mtx = sparse.csr_matrix(eye)\n", - "print(sparse_mtx)\n", - "x = np.linspace(-10,10,100)\n", - "y = np.sin(x)\n", - "plt.plot(x,y,marker='x')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gaussian Elimination\n", - "\n", - "We start with the linear set of equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}\\mathbf{x} = \\mathbf{w}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We assume also that the matrix $\\mathbf{A}$ is non-singular and that the\n", - "matrix elements along the diagonal satisfy $a_{ii} \\ne 0$. Simple $4\\times 4 $ example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{bmatrix}\n", - " a_{11}& a_{12} &a_{13}& a_{14}\\\\\n", - " a_{21}& a_{22} &a_{23}& a_{24}\\\\\n", - " a_{31}& a_{32} &a_{33}& a_{34}\\\\\n", - " a_{41}& a_{42} &a_{43}& a_{44}\\\\\n", - " \\end{bmatrix} \\begin{bmatrix}\n", - " x_1\\\\\n", - " x_2\\\\\n", - " x_3 \\\\\n", - " x_4 \\\\\n", - " \\end{bmatrix}\n", - " =\\begin{bmatrix}\n", - " w_1\\\\\n", - " w_2\\\\\n", - " w_3 \\\\\n", - " w_4\\\\\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown $x_1$\n", - "from the remaining $n-1$ equations. Then we use the new second equation to eliminate the second unknown\n", - "$x_2$ from the remaining $n-2$ equations. With $n-1$ such eliminations\n", - "we obtain a so-called upper triangular set of equations of the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=y_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_{22}x_2 + b_{23}x_3 + b_{24}x_4=y_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_{33}x_3 + b_{34}x_4=y_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "b_{44}x_4=y_4. \\nonumber\n", - "\\label{eq:gaussbacksub} \\tag{1}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can solve this system of equations recursively starting from $x_n$ (in our case $x_4$) and proceed with\n", - "what is called a backward substitution. \n", - "\n", - "\n", - "This process can be expressed mathematically as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " x_m = \\frac{1}{b_{mm}}\\left(y_m-\\sum_{k=m+1}^nb_{mk}x_k\\right)\\quad m=n-1,n-2,\\dots,1.\n", - "\\label{_auto1} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To arrive at such an upper triangular system of equations, we start by eliminating\n", - "the unknown $x_1$ for $j=2,n$. We achieve this by multiplying the first equation by $a_{j1}/a_{11}$ and then subtract\n", - "the result from the $j$th equation. We assume obviously that $a_{11}\\ne 0$ and that\n", - "$\\mathbf{A}$ is not singular.\n", - "\n", - "\n", - "Our actual $4\\times 4$ example reads after the first operation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{bmatrix}\n", - " a_{11}& a_{12} &a_{13}& a_{14}\\\\\n", - " 0& (a_{22}-\\frac{a_{21}a_{12}}{a_{11}}) &(a_{23}-\\frac{a_{21}a_{13}}{a_{11}}) & (a_{24}-\\frac{a_{21}a_{14}}{a_{11}})\\\\\n", - "0& (a_{32}-\\frac{a_{31}a_{12}}{a_{11}})& (a_{33}-\\frac{a_{31}a_{13}}{a_{11}})& (a_{34}-\\frac{a_{31}a_{14}}{a_{11}})\\\\\n", - "0&(a_{42}-\\frac{a_{41}a_{12}}{a_{11}}) &(a_{43}-\\frac{a_{41}a_{13}}{a_{11}}) & (a_{44}-\\frac{a_{41}a_{14}}{a_{11}}) \\\\\n", - " \\end{bmatrix} \\begin{bmatrix}\n", - " x_1\\\\\n", - " x_2\\\\\n", - " x_3 \\\\\n", - " x_4 \\\\\n", - " \\end{bmatrix} \n", - " =\\begin{bmatrix}\n", - " y_1\\\\\n", - " w_2^{(2)}\\\\\n", - " w_3^{(2)} \\\\\n", - " w_4^{(2)}\\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=y_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a^{(2)}_{22}x_2 + a^{(2)}_{23}x_3 + a^{(2)}_{24}x_4=w^{(2)}_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a^{(2)}_{32}x_2 + a^{(2)}_{33}x_3 + a^{(2)}_{34}x_4=w^{(2)}_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a^{(2)}_{42}x_2 + a^{(2)}_{43}x_3 + a^{(2)}_{44}x_4=w^{(2)}_4, \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \n", - "\\label{_auto2} \\tag{3}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The new coefficients are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " b_{1k} = a_{1k}^{(1)} \\quad k=1,\\dots,n,\n", - "\\label{_auto3} \\tag{4}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where each $a_{1k}^{(1)}$ is equal to the original $a_{1k}$ element. The other coefficients are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "a_{jk}^{(2)} = a_{jk}^{(1)}-\\frac{a_{j1}^{(1)}a_{1k}^{(1)}}{a_{11}^{(1)}} \\quad j,k=2,\\dots,n,\n", - "\\label{_auto4} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with a new right-hand side given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "y_{1}=w_1^{(1)}, \\quad w_j^{(2)} =w_j^{(1)}-\\frac{a_{j1}^{(1)}w_1^{(1)}}{a_{11}^{(1)}} \\quad j=2,\\dots,n.\n", - "\\label{_auto5} \\tag{6}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We have also set $w_1^{(1)}=w_1$, the original vector element.\n", - "We see that the system of unknowns $x_1,\\dots,x_n$ is transformed into an $(n-1)\\times (n-1)$ problem.\n", - "\n", - "\n", - "\n", - "This step is called forward substitution.\n", - "Proceeding with these substitutions, we obtain the\n", - "general expressions for the new coefficients" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " a_{jk}^{(m+1)} = a_{jk}^{(m)}-\\frac{a_{jm}^{(m)}a_{mk}^{(m)}}{a_{mm}^{(m)}} \\quad j,k=m+1,\\dots,n,\n", - "\\label{_auto6} \\tag{7}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $m=1,\\dots,n-1$ and a\n", - "right-hand side given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " w_j^{(m+1)} =w_j^{(m)}-\\frac{a_{jm}^{(m)}w_m^{(m)}}{a_{mm}^{(m)}}\\quad j=m+1,\\dots,n.\n", - "\\label{_auto7} \\tag{8}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This set of $n-1$ elimations leads us to an equations which is solved by back substitution.\n", - "If the arithmetics is exact and the matrix $\\mathbf{A}$ is not singular, then the computed answer will be exact.\n", - "\n", - "Even though the matrix elements along the diagonal are not zero,\n", - "numerically small numbers may appear and subsequent divisions may lead to large numbers, which, if added\n", - "to a small number may yield losses of precision. Suppose for example that our first division in $(a_{22}-a_{21}a_{12}/a_{11})$\n", - "results in $-10^{-7}$ and that $a_{22}$ is one.\n", - "one. We are then\n", - "adding $10^7+1$. With single precision this results in $10^7$.\n", - "\n", - "\n", - "\n", - "\n", - " * Gaussian elimination, $O(2/3n^3)$ flops, general matrix\n", - "\n", - " * LU decomposition, upper triangular and lower tridiagonal matrices, $O(2/3n^3)$ flops, general matrix. Get easily the inverse, determinant and can solve linear equations with back-substitution only, $O(n^2)$ flops\n", - "\n", - " * Cholesky decomposition. Real symmetric or hermitian positive definite matrix, $O(1/3n^3)$ flops.\n", - "\n", - " * Tridiagonal linear systems, important for differential equations. Normally positive definite and non-singular. $O(8n)$ flops for symmetric. Special case of banded matrices.\n", - "\n", - " * Singular value decomposition\n", - "\n", - " * the QR method will be discussed in chapter 7 in connection with eigenvalue systems. $O(4/3n^3)$ flops.\n", - "\n", - "The LU decomposition method means that we can rewrite\n", - "this matrix as the product of two matrices $\\mathbf{L}$ and $\\mathbf{U}$\n", - "where" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{bmatrix}\n", - " a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", - " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", - " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", - " a_{41} & a_{42} & a_{43} & a_{44}\n", - " \\end{bmatrix}\n", - " = \\begin{bmatrix}\n", - " 1 & 0 & 0 & 0 \\\\\n", - " l_{21} & 1 & 0 & 0 \\\\\n", - " l_{31} & l_{32} & 1 & 0 \\\\\n", - " l_{41} & l_{42} & l_{43} & 1\n", - " \\end{bmatrix}\n", - " \\begin{bmatrix}\n", - " u_{11} & u_{12} & u_{13} & u_{14} \\\\\n", - " 0 & u_{22} & u_{23} & u_{24} \\\\\n", - " 0 & 0 & u_{33} & u_{34} \\\\\n", - " 0 & 0 & 0 & u_{44}\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "LU decomposition forms the backbone of other algorithms in linear algebra, such as the\n", - "solution of linear equations given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The above set of equations is conveniently solved by using LU decomposition as an intermediate step.\n", - "\n", - "The matrix $\\mathbf{A}\\in \\mathbb{R}^{n\\times n}$ has an LU factorization if the determinant\n", - "is different from zero. If the LU factorization exists and $\\mathbf{A}$ is non-singular, then the LU factorization\n", - "is unique and the determinant is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "det\\{\\mathbf{A}\\}=det\\{\\mathbf{LU}\\}= det\\{\\mathbf{L}\\}det\\{\\mathbf{U}\\}=u_{11}u_{22}\\dots u_{nn}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There are at least three main advantages with LU decomposition compared with standard Gaussian elimination:\n", - "\n", - " * It is straightforward to compute the determinant of a matrix\n", - "\n", - " * If we have to solve sets of linear equations with the same matrix but with different vectors $\\mathbf{y}$, the number of FLOPS is of the order $n^3$.\n", - "\n", - " * The inverse is such an operation \n", - "\n", - "With the LU decomposition it is rather\n", - "simple to solve a system of linear equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This can be written in matrix form as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{Ax}=\\mathbf{w}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\mathbf{A}$ and $\\mathbf{w}$ are known and we have to solve for\n", - "$\\mathbf{x}$. Using the LU dcomposition we write" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A} \\mathbf{x} \\equiv \\mathbf{L} \\mathbf{U} \\mathbf{x} =\\mathbf{w}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The previous equation can be calculated in two steps" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{L} \\mathbf{y} = \\mathbf{w};\\qquad \\mathbf{Ux}=\\mathbf{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To show that this is correct we use to the LU decomposition\n", - "to rewrite our system of linear equations as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{LUx}=\\mathbf{w},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and since the determinant of $\\mathbf{L}$ is equal to 1 (by construction\n", - "since the diagonals of $\\mathbf{L}$ equal 1) we can use the inverse of\n", - "$\\mathbf{L}$ to obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{Ux}=\\mathbf{L^{-1}w}=\\mathbf{y},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which yields the intermediate step" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{L^{-1}w}=\\mathbf{y}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and as soon as we have $\\mathbf{y}$ we can obtain $\\mathbf{x}$\n", - "through $\\mathbf{Ux}=\\mathbf{y}$.\n", - "\n", - "\n", - "For our four-dimentional example this takes the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_1=w_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "l_{21}y_1 + y_2=w_2\\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "l_{31}y_1 + l_{32}y_2 + y_3 =w_3\\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "l_{41}y_1 + l_{42}y_2 + l_{43}y_3 + y_4=w_4. \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u_{11}x_1 +u_{12}x_2 +u_{13}x_3 + u_{14}x_4=y_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u_{22}x_2 + u_{23}x_3 + u_{24}x_4=y_2\\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u_{33}x_3 + u_{34}x_4=y_3\\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u_{44}x_4=y_4 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This example shows the basis for the algorithm\n", - "needed to solve the set of $n$ linear equations.\n", - "\n", - "\n", - "\n", - "The algorithm goes as follows\n", - "\n", - " * Set up the matrix $\\bf A$ and the vector $\\bf w$ with their correct dimensions. This determines the dimensionality of the unknown vector $\\bf x$.\n", - "\n", - " * Then LU decompose the matrix $\\bf A$ through a call to the function `ludcmp(double a, int n, int indx, double &d)`. This functions returns the LU decomposed matrix $\\bf A$, its determinant and the vector indx which keeps track of the number of interchanges of rows. If the determinant is zero, the solution is malconditioned.\n", - "\n", - " * Thereafter you call the function `lubksb(double a, int n, int indx, double w)` which uses the LU decomposed matrix $\\bf A$ and the vector $\\bf w$ and returns $\\bf x$ in the same place as $\\bf w$. Upon exit the original content in $\\bf w$ is destroyed. If you wish to keep this information, you should make a backup of it in your calling function.\n", - "\n", - "### LU Decomposition, the inverse of a matrix\n", - "\n", - "If the inverse exists then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}^{-1}\\mathbf{A}=\\mathbf{I},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "the identity matrix. With an LU decomposed matrix we can rewrite the last equation as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{LU}\\mathbf{A}^{-1}=\\mathbf{I}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we assume that the first column (that is column 1) of the inverse matrix\n", - "can be written as a vector with unknown entries" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}_1^{-1}= \\begin{bmatrix}\n", - " a_{11}^{-1} \\\\\n", - " a_{21}^{-1} \\\\\n", - " \\dots \\\\\n", - " a_{n1}^{-1} \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "then we have a linear set of equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{LU}\\begin{bmatrix}\n", - " a_{11}^{-1} \\\\\n", - " a_{21}^{-1} \\\\\n", - " \\dots \\\\\n", - " a_{n1}^{-1} \\\\\n", - " \\end{bmatrix} =\\begin{bmatrix}\n", - " 1 \\\\\n", - " 0 \\\\\n", - " \\dots \\\\\n", - " 0 \\\\\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In a similar way we can compute the unknow entries of the second column," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{LU}\\begin{bmatrix}\n", - " a_{12}^{-1} \\\\\n", - " a_{22}^{-1} \\\\\n", - " \\dots \\\\\n", - " a_{n2}^{-1} \\\\\n", - " \\end{bmatrix}=\\begin{bmatrix}\n", - " 0 \\\\\n", - " 1 \\\\\n", - " \\dots \\\\\n", - " 0 \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and continue till we have solved all $n$ sets of linear equations." - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/schedule.md b/doc/LectureNotes/_build/html/_sources/schedule.md deleted file mode 100644 index 13527dc67..000000000 --- a/doc/LectureNotes/_build/html/_sources/schedule.md +++ /dev/null @@ -1,14 +0,0 @@ -# Teaching schedule with links to material - - -This course will be delivered in a hybrid mode, with online lectures and on site or online laboratory sessions. - -1. Four lectures per week, Fall semester, 10 ECTS. The lectures will be fully online. The lectures will be recorded and linked to this site and the official University of Oslo website for the course; -2. Two hours of laboratory sessions for work on computational projects and exercises for each group. Due to social distancing, at most 15 participants can attend. There will also be fully digital laboratory sessions for those who cannot attend; -3. Three projects which are graded and count 1/3 each of the final grade; -4. A selected number of weekly assignments; -5. The course is part of the CS Master of Science program, but is open to other bachelor and Master of Science students at the University of Oslo; -6. The course is offered as a FYS-MAT4155 (Master of Science level) and a FYS-MAT3155 (senior undergraduate) course; -7. Videos of teaching material are available via the links at https://compphysics.github.io/MachineLearning/doc/web/course.html; -8. Weekly emails with summary of activities will be mailed to all participants; - diff --git a/doc/LectureNotes/_build/html/_sources/statistics.ipynb b/doc/LectureNotes/_build/html/_sources/statistics.ipynb deleted file mode 100644 index 8690ffe7a..000000000 --- a/doc/LectureNotes/_build/html/_sources/statistics.ipynb +++ /dev/null @@ -1,2882 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Elements of Probability Theory and Statistical Data Analysis\n", - "\n", - "\n", - "## Domains and probabilities\n", - "Consider the following simple example, namely the tossing of two dice, resulting in the following possible values" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{2,3,4,5,6,7,8,9,10,11,12\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These values are called the *domain*. \n", - "To this domain we have the corresponding *probabilities*" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Tossing the dice\n", - "The numbers in the domain are the outcomes of the physical process of tossing say two dice.\n", - "We cannot tell beforehand whether the outcome is 3 or 5 or any other number in this domain.\n", - "This defines the randomness of the outcome, or unexpectedness or any other synonimous word which\n", - "encompasses the uncertitude of the final outcome. \n", - "\n", - "The only thing we can tell beforehand\n", - "is that say the outcome 2 has a certain probability. \n", - "If our favorite hobby is to spend an hour every evening throwing dice and \n", - "registering the sequence of outcomes, we will note that the numbers in the above domain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{2,3,4,5,6,7,8,9,10,11,12\\},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "appear in a random order. After 11 throws the results may look like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{10,8,6,3,6,9,11,8,12,4,5\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Stochastic variables\n", - "\n", - "**Random variables are characterized by a domain which contains all possible values that the random value may take. This domain has a corresponding probability distribution function(PDF)**.\n", - "\n", - "\n", - "\n", - "\n", - "## Stochastic variables and the main concepts, the discrete case\n", - "There are two main concepts associated with a stochastic variable. The\n", - "*domain* is the set $\\mathbb D = \\{x\\}$ of all accessible values\n", - "the variable can assume, so that $X \\in \\mathbb D$. An example of a\n", - "discrete domain is the set of six different numbers that we may get by\n", - "throwing of a dice, $x\\in\\{1,\\,2,\\,3,\\,4,\\,5,\\,6\\}$.\n", - "\n", - "The *probability distribution function (PDF)* is a function\n", - "$p(x)$ on the domain which, in the discrete case, gives us the\n", - "probability or relative frequency with which these values of $X$\n", - "occur" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\mathrm{Prob}(X=x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Stochastic variables and the main concepts, the continuous case\n", - "In the continuous case, the PDF does not directly depict the\n", - "actual probability. Instead we define the probability for the\n", - "stochastic variable to assume any value on an infinitesimal interval\n", - "around $x$ to be $p(x)dx$. The continuous function $p(x)$ then gives us\n", - "the *density* of the probability rather than the probability\n", - "itself. The probability for a stochastic variable to assume any value\n", - "on a non-infinitesimal interval $[a,\\,b]$ is then just the integral" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{Prob}(a\\leq X\\leq b) = \\int_a^b p(x)dx.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Qualitatively speaking, a stochastic variable represents the values of\n", - "numbers chosen as if by chance from some specified PDF so that the\n", - "selection of a large set of these numbers reproduces this PDF.\n", - "\n", - "\n", - "\n", - "\n", - "## The cumulative probability\n", - "Of interest to us is the *cumulative probability\n", - "distribution function* (**CDF**), $P(x)$, which is just the probability\n", - "for a stochastic variable $X$ to assume any value less than $x$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(x)=\\mathrm{Prob(}X\\leq x\\mathrm{)} =\n", - "\\int_{-\\infty}^x p(x^{\\prime})dx^{\\prime}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The relation between a CDF and its corresponding PDF is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\frac{d}{dx}P(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Properties of PDFs\n", - "\n", - "There are two properties that all PDFs must satisfy. The first one is\n", - "positivity (assuming that the PDF is normalized)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "0 \\leq p(x) \\leq 1.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Naturally, it would be nonsensical for any of the values of the domain\n", - "to occur with a probability greater than $1$ or less than $0$. Also,\n", - "the PDF must be normalized. That is, all the probabilities must add up\n", - "to unity. The probability of \"anything\" to happen is always unity. For\n", - "both discrete and continuous PDFs, this condition is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\sum_{x_i\\in\\mathbb D} p(x_i) & = 1,\\\\\n", - "\\int_{x\\in\\mathbb D} p(x)\\,dx & = 1.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Important distributions, the uniform distribution\n", - "The first one\n", - "is the most basic PDF; namely the uniform distribution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "p(x) = \\frac{1}{b-a}\\theta(x-a)\\theta(b-x).\n", - "\\label{eq:unifromPDF} \\tag{1}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For $a=0$ and $b=1$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{array}{ll}\n", - "p(x)dx = dx & \\in [0,1].\n", - "\\end{array}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The latter distribution is used to generate random numbers. For other PDFs, one needs normally a mapping from this distribution to say for example the exponential distribution.\n", - "\n", - "\n", - "\n", - "\n", - "## Gaussian distribution\n", - "The second one is the Gaussian Distribution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp{(-\\frac{(x-\\mu)^2}{2\\sigma^2})},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with mean value $\\mu$ and standard deviation $\\sigma$. If $\\mu=0$ and $\\sigma=1$, it is normally called the **standard normal distribution**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\frac{1}{\\sqrt{2\\pi}} \\exp{(-\\frac{x^2}{2})},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The following simple Python code plots the above distribution for different values of $\\mu$ and $\\sigma$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "from math import acos, exp, sqrt\n", - "from matplotlib import pyplot as plt\n", - "from matplotlib import rc, rcParams\n", - "import matplotlib.units as units\n", - "import matplotlib.ticker as ticker\n", - "rc('text',usetex=True)\n", - "rc('font',**{'family':'serif','serif':['Gaussian distribution']})\n", - "font = {'family' : 'serif',\n", - " 'color' : 'darkred',\n", - " 'weight' : 'normal',\n", - " 'size' : 16,\n", - " }\n", - "pi = acos(-1.0)\n", - "mu0 = 0.0\n", - "sigma0 = 1.0\n", - "mu1= 1.0\n", - "sigma1 = 2.0\n", - "mu2 = 2.0\n", - "sigma2 = 4.0\n", - "\n", - "x = np.linspace(-20.0, 20.0)\n", - "v0 = np.exp(-(x*x-2*x*mu0+mu0*mu0)/(2*sigma0*sigma0))/sqrt(2*pi*sigma0*sigma0)\n", - "v1 = np.exp(-(x*x-2*x*mu1+mu1*mu1)/(2*sigma1*sigma1))/sqrt(2*pi*sigma1*sigma1)\n", - "v2 = np.exp(-(x*x-2*x*mu2+mu2*mu2)/(2*sigma2*sigma2))/sqrt(2*pi*sigma2*sigma2)\n", - "plt.plot(x, v0, 'b-', x, v1, 'r-', x, v2, 'g-')\n", - "plt.title(r'{\\bf Gaussian distributions}', fontsize=20)\n", - "plt.text(-19, 0.3, r'Parameters: $\\mu = 0$, $\\sigma = 1$', fontdict=font)\n", - "plt.text(-19, 0.18, r'Parameters: $\\mu = 1$, $\\sigma = 2$', fontdict=font)\n", - "plt.text(-19, 0.08, r'Parameters: $\\mu = 2$, $\\sigma = 4$', fontdict=font)\n", - "plt.xlabel(r'$x$',fontsize=20)\n", - "plt.ylabel(r'$p(x)$ [MeV]',fontsize=20)\n", - "\n", - "# Tweak spacing to prevent clipping of ylabel \n", - "plt.subplots_adjust(left=0.15)\n", - "plt.savefig('gaussian.pdf', format='pdf')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Exponential distribution\n", - "Another important distribution in science is the exponential distribution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\alpha\\exp{-(\\alpha x)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Expectation values\n", - "Let $h(x)$ be an arbitrary continuous function on the domain of the stochastic\n", - "variable $X$ whose PDF is $p(x)$. We define the *expectation value*\n", - "of $h$ with respect to $p$ as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\langle h \\rangle_X \\equiv \\int\\! h(x)p(x)\\,dx\n", - "\\label{eq:expectation_value_of_h_wrt_p} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Whenever the PDF is known implicitly, like in this case, we will drop\n", - "the index $X$ for clarity. \n", - "A particularly useful class of special expectation values are the\n", - "*moments*. The $n$-th moment of the PDF $p$ is defined as\n", - "follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x^n \\rangle \\equiv \\int\\! x^n p(x)\\,dx\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Stochastic variables and the main concepts, mean values\n", - "The zero-th moment $\\langle 1\\rangle$ is just the normalization condition of\n", - "$p$. The first moment, $\\langle x\\rangle$, is called the *mean* of $p$\n", - "and often denoted by the letter $\\mu$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x\\rangle = \\mu \\equiv \\int x p(x)dx,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for a continuous distribution and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x\\rangle = \\mu \\equiv \\sum_{i=1}^N x_i p(x_i),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for a discrete distribution. \n", - "Qualitatively it represents the centroid or the average value of the\n", - "PDF and is therefore simply called the expectation value of $p(x)$.\n", - "\n", - "\n", - "\n", - "\n", - "## Stochastic variables and the main concepts, central moments, the variance\n", - "\n", - "A special version of the moments is the set of *central moments*, the n-th central moment defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle (x-\\langle x\\rangle )^n\\rangle \\equiv \\int\\! (x-\\langle x\\rangle)^n p(x)\\,dx\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The zero-th and first central moments are both trivial, equal $1$ and\n", - "$0$, respectively. But the second central moment, known as the\n", - "*variance* of $p$, is of particular interest. For the stochastic\n", - "variable $X$, the variance is denoted as $\\sigma^2_X$ or $\\mathrm{Var}(X)$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\sigma^2_X &=\\mathrm{Var}(X) = \\langle (x-\\langle x\\rangle)^2\\rangle =\n", - "\\int (x-\\langle x\\rangle)^2 p(x)dx\\\\\n", - "& = \\int\\left(x^2 - 2 x \\langle x\\rangle^{2} +\\langle x\\rangle^2\\right)p(x)dx\\\\\n", - "& = \\langle x^2\\rangle - 2 \\langle x\\rangle\\langle x\\rangle + \\langle x\\rangle^2\\\\\n", - "& = \\langle x^2 \\rangle - \\langle x\\rangle^2\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The square root of the variance, $\\sigma =\\sqrt{\\langle (x-\\langle x\\rangle)^2\\rangle}$ is called the \n", - "**standard deviation** of $p$. It is the RMS (root-mean-square)\n", - "value of the deviation of the PDF from its mean value, interpreted\n", - "qualitatively as the \"spread\" of $p$ around its mean.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions\n", - "\n", - "The following table collects properties of probability distribution functions.\n", - "In our notation we reserve the label $p(x)$ for the probability of a certain event,\n", - "while $P(x)$ is the cumulative probability. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
Discrete PDF Continuous PDF
Domain $\\left\\{x_1, x_2, x_3, \\dots, x_N\\right\\}$ $[a,b]$
Probability $p(x_i)$ $p(x)dx$
Cumulative $P_i=\\sum_{l=1}^ip(x_l)$ $P(x)=\\int_a^xp(t)dt$
Positivity $0 \\le p(x_i) \\le 1$ $p(x) \\ge 0$
Positivity $0 \\le P_i \\le 1$ $0 \\le P(x) \\le 1$
Monotonic $P_i \\ge P_j$ if $x_i \\ge x_j$ $P(x_i) \\ge P(x_j)$ if $x_i \\ge x_j$
Normalization $P_N=1$ $P(b)=1$
\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions\n", - "With a PDF we can compute expectation values of selected quantities such as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x^k\\rangle=\\sum_{i=1}^{N}x_i^kp(x_i),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "if we have a discrete PDF or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x^k\\rangle=\\int_a^b x^kp(x)dx,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "in the case of a continuous PDF. We have already defined the mean value $\\mu$\n", - "and the variance $\\sigma^2$.\n", - "\n", - "\n", - "\n", - "\n", - "## The three famous Probability Distribution Functions\n", - "\n", - "There are at least three PDFs which one may encounter. These are the\n", - "\n", - "**Uniform distribution**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x)=\\frac{1}{b-a}\\Theta(x-a)\\Theta(b-x),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "yielding probabilities different from zero in the interval $[a,b]$.\n", - "\n", - "**The exponential distribution**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x)=\\alpha \\exp{(-\\alpha x)},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "yielding probabilities different from zero in the interval $[0,\\infty)$ and with mean value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \\int_0^{\\infty}xp(x)dx=\\int_0^{\\infty}x\\alpha \\exp{(-\\alpha x)}dx=\\frac{1}{\\alpha},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with variance" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2=\\int_0^{\\infty}x^2p(x)dx-\\mu^2 = \\frac{1}{\\alpha^2}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Probability Distribution Functions, the normal distribution\n", - "Finally, we have the so-called univariate normal distribution, or just the **normal distribution**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x)=\\frac{1}{b\\sqrt{2\\pi}}\\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with probabilities different from zero in the interval $(-\\infty,\\infty)$.\n", - "The integral $\\int_{-\\infty}^{\\infty}\\exp{\\left(-(x^2\\right)}dx$ appears in many calculations, its value\n", - "is $\\sqrt{\\pi}$, a result we will need when we compute the mean value and the variance.\n", - "The mean value is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \\int_0^{\\infty}xp(x)dx=\\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}x \\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}dx,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which becomes with a suitable change of variables" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu =\\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}b\\sqrt{2}(a+b\\sqrt{2}y)\\exp{-y^2}dy=a.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Probability Distribution Functions, the normal distribution\n", - "Similarly, the variance becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2 = \\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}(x-\\mu)^2 \\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}dx,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and inserting the mean value and performing a variable change we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2 = \\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}b\\sqrt{2}(b\\sqrt{2}y)^2\\exp{\\left(-y^2\\right)}dy=\n", - "\\frac{2b^2}{\\sqrt{\\pi}}\\int_{-\\infty}^{\\infty}y^2\\exp{\\left(-y^2\\right)}dy,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and performing a final integration by parts we obtain the well-known result $\\sigma^2=b^2$.\n", - "It is useful to introduce the standard normal distribution as well, defined by $\\mu=a=0$, viz. a distribution\n", - "centered around zero and with a variance $\\sigma^2=1$, leading to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " p(x)=\\frac{1}{\\sqrt{2\\pi}}\\exp{\\left(-\\frac{x^2}{2}\\right)}.\n", - "\\label{_auto1} \\tag{3}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Probability Distribution Functions, the cumulative distribution\n", - "\n", - "The exponential and uniform distributions have simple cumulative functions,\n", - "whereas the normal distribution does not, being proportional to the so-called\n", - "error function $erf(x)$, given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(x) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^x\\exp{\\left(-\\frac{t^2}{2}\\right)}dt,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is difficult to evaluate in a quick way.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions, other important distribution\n", - "\n", - "Some other PDFs which one encounters often in the natural sciences are the binomial distribution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} \\hspace{0.5cm}x=0,1,\\dots,n,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $y$ is the probability for a specific event, such as the tossing of a coin or moving left or right\n", - "in case of a random walker. Note that $x$ is a discrete stochastic variable. \n", - "\n", - "The sequence of binomial trials is characterized by the following definitions\n", - "\n", - " * Every experiment is thought to consist of $N$ independent trials.\n", - "\n", - " * In every independent trial one registers if a specific situation happens or not, such as the jump to the left or right of a random walker.\n", - "\n", - " * The probability for every outcome in a single trial has the same value, for example the outcome of tossing (either heads or tails) a coin is always $1/2$.\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions, the binomial distribution\n", - "\n", - "In order to compute the mean and variance we need to recall Newton's binomial\n", - "formula" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(a+b)^m=\\sum_{n=0}^m \\left(\\begin{array}{c} m \\\\ n\\end{array}\\right)a^nb^{m-n},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which can be used to show that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sum_{x=0}^n\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "the PDF is normalized to one. \n", - "The mean value is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \\sum_{x=0}^n x\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} =\n", - "\\sum_{x=0}^n x\\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \n", - "\\sum_{x=0}^n x\\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which we rewrite as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu=ny\\sum_{\\nu=0}^n\\left(\\begin{array}{c} n-1 \\\\ \\nu\\end{array}\\right)y^{\\nu}(1-y)^{n-1-\\nu} =ny(y+1-y)^{n-1}=ny.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The variance is slightly trickier to get. It reads $\\sigma^2=ny(1-y)$. \n", - "\n", - "\n", - "## Probability Distribution Functions, Poisson's distribution\n", - "\n", - "Another important distribution with discrete stochastic variables $x$ is \n", - "the Poisson model, which resembles the exponential distribution and reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\frac{\\lambda^x}{x!} e^{-\\lambda} \\hspace{0.5cm}x=0,1,\\dots,;\\lambda > 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case both the mean value and the variance are easier to calculate," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \\sum_{x=0}^{\\infty} x \\frac{\\lambda^x}{x!} e^{-\\lambda} = \\lambda e^{-\\lambda}\\sum_{x=1}^{\\infty}\n", - "\\frac{\\lambda^{x-1}}{(x-1)!}=\\lambda,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the variance is $\\sigma^2=\\lambda$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions, Poisson's distribution\n", - "An example of applications of the Poisson distribution could be the counting\n", - "of the number of $\\alpha$-particles emitted from a radioactive source in a given time interval.\n", - "In the limit of $n\\rightarrow \\infty$ and for small probabilities $y$, the binomial distribution\n", - "approaches the Poisson distribution. Setting $\\lambda = ny$, with $y$ the probability for an event in\n", - "the binomial distribution we can show that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\lim_{n\\rightarrow \\infty}\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} e^{-\\lambda}=\\sum_{x=1}^{\\infty}\\frac{\\lambda^x}{x!} e^{-\\lambda}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Meet the covariance!\n", - "An important quantity in a statistical analysis is the so-called covariance. \n", - "\n", - "Consider the set $\\{X_i\\}$ of $n$\n", - "stochastic variables (not necessarily uncorrelated) with the\n", - "multivariate PDF $P(x_1,\\dots,x_n)$. The *covariance* of two\n", - "of the stochastic variables, $X_i$ and $X_j$, is defined as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\mathrm{Cov}(X_i,\\,X_j) = \\langle (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle \n", - "\\label{_auto2} \\tag{4}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \n", - "=\\int\\cdots\\int (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)P(x_1,\\dots,x_n)\\,dx_1\\dots dx_n,\n", - "\\label{eq:def_covariance} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x_i\\rangle =\n", - "\\int\\cdots\\int x_i P(x_1,\\dots,x_n)\\,dx_1\\dots dx_n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Meet the covariance in matrix disguise\n", - "If we consider the above covariance as a matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C_{ij} =\\mathrm{Cov}(X_i,\\,X_j),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "then the diagonal elements are just the familiar\n", - "variances, $C_{ii} = \\mathrm{Cov}(X_i,\\,X_i) = \\mathrm{Var}(X_i)$. It turns out that\n", - "all the off-diagonal elements are zero if the stochastic variables are\n", - "uncorrelated.\n", - "\n", - "\n", - "\n", - "\n", - "## Covariance" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "from math import exp, sqrt\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "\n", - "def covariance(x, y, n):\n", - " sum = 0.0\n", - " mean_x = np.mean(x)\n", - " mean_y = np.mean(y)\n", - " for i in range(0, n):\n", - " sum += (x[(i)]-mean_x)*(y[i]-mean_y)\n", - " return sum/n\n", - "\n", - "n = 10\n", - "\n", - "x=np.random.normal(size=n)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "covxy = covariance(x,y,n)\n", - "print(covxy)\n", - "z = np.vstack((x, y))\n", - "c = np.cov(z.T)\n", - "\n", - "print(c)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Meet the covariance, uncorrelated events\n", - "\n", - "Consider the stochastic variables $X_i$ and $X_j$, ($i\\neq j$). We have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "Cov(X_i,\\,X_j) &= \\langle (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle\\\\\n", - "&=\\langle x_i x_j - x_i\\langle x_j\\rangle - \\langle x_i\\rangle x_j + \\langle x_i\\rangle\\langle x_j\\rangle\\rangle\\\\\n", - "&=\\langle x_i x_j\\rangle - \\langle x_i\\langle x_j\\rangle\\rangle - \\langle \\langle x_i\\rangle x_j \\rangle +\n", - "\\langle \\langle x_i\\rangle\\langle x_j\\rangle\\rangle \\\\\n", - "&=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle +\n", - "\\langle x_i\\rangle\\langle x_j\\rangle \\\\\n", - "&=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If $X_i$ and $X_j$ are independent (assuming $i \\neq j$), we have that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x_i x_j\\rangle = \\langle x_i\\rangle\\langle x_j\\rangle,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "leading to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "Cov(X_i, X_j) = 0 \\hspace{0.1cm} (i\\neq j).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Numerical experiments and the covariance\n", - "\n", - "Now that we have constructed an idealized mathematical framework, let\n", - "us try to apply it to empirical observations. Examples of relevant\n", - "physical phenomena may be spontaneous decays of nuclei, or a purely\n", - "mathematical set of numbers produced by some deterministic\n", - "mechanism. It is the latter we will deal with, using so-called pseudo-random\n", - "number generators. In general our observations will contain only a limited set of\n", - "observables. We remind the reader that\n", - "a *stochastic process* is a process that produces sequentially a\n", - "chain of values" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{x_1, x_2,\\dots\\,x_k,\\dots\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Numerical experiments and the covariance\n", - "We will call these\n", - "values our *measurements* and the entire set as our measured\n", - "*sample*. The action of measuring all the elements of a sample\n", - "we will call a stochastic *experiment* (since, operationally,\n", - "they are often associated with results of empirical observation of\n", - "some physical or mathematical phenomena; precisely an experiment). We\n", - "assume that these values are distributed according to some \n", - "PDF $p_X^{\\phantom X}(x)$, where $X$ is just the formal symbol for the\n", - "stochastic variable whose PDF is $p_X^{\\phantom X}(x)$. Instead of\n", - "trying to determine the full distribution $p$ we are often only\n", - "interested in finding the few lowest moments, like the mean\n", - "$\\mu_X^{\\phantom X}$ and the variance $\\sigma_X^{\\phantom X}$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Numerical experiments and the covariance, actual situations\n", - "In practical situations however, a sample is always of finite size. Let that\n", - "size be $n$. The expectation value of a sample $\\alpha$, the **sample mean**, is then defined as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x_{\\alpha} \\rangle \\equiv \\frac{1}{n}\\sum_{k=1}^n x_{\\alpha,k}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The *sample variance* is:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{Var}(x) \\equiv \\frac{1}{n}\\sum_{k=1}^n (x_{\\alpha,k} - \\langle x_{\\alpha} \\rangle)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with its square root being the *standard deviation of the sample*.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Numerical experiments and the covariance, our observables\n", - "You can think of the above observables as a set of quantities which define\n", - "a given experiment. This experiment is then repeated several times, say $m$ times.\n", - "The total average is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\langle X_m \\rangle= \\frac{1}{m}\\sum_{\\alpha=1}^mx_{\\alpha}=\\frac{1}{mn}\\sum_{\\alpha, k} x_{\\alpha,k},\n", - "\\label{eq:exptmean} \\tag{6}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the last sums end at $m$ and $n$.\n", - "The total variance is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2_m= \\frac{1}{mn^2}\\sum_{\\alpha=1}^m(\\langle x_{\\alpha} \\rangle-\\langle X_m \\rangle)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which we rewrite as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\sigma^2_m=\\frac{1}{m}\\sum_{\\alpha=1}^m\\sum_{kl=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,l}-\\langle X_m \\rangle).\n", - "\\label{eq:exptvariance} \\tag{7}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Numerical experiments and the covariance, the sample variance\n", - "\n", - "We define also the sample variance $\\sigma^2$ of all $mn$ individual experiments as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\sigma^2=\\frac{1}{mn}\\sum_{\\alpha=1}^m\\sum_{k=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)^2.\n", - "\\label{eq:sampleexptvariance} \\tag{8}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These quantities, being known experimental values or the results from our calculations, \n", - "may differ, in some cases\n", - "significantly, from the similarly named\n", - "exact values for the mean value $\\mu_X$, the variance $\\mathrm{Var}(X)$\n", - "and the covariance $\\mathrm{Cov}(X,Y)$.\n", - "\n", - "\n", - "\n", - "\n", - "## Numerical experiments and the covariance, central limit theorem\n", - "\n", - "The central limit theorem states that the PDF $\\tilde{p}(z)$ of\n", - "the average of $m$ random values corresponding to a PDF $p(x)$ \n", - "is a normal distribution whose mean is the \n", - "mean value of the PDF $p(x)$ and whose variance is the variance\n", - "of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$.\n", - "\n", - "The central limit theorem leads then to the well-known expression for the\n", - "standard deviation, given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma_m=\n", - "\\frac{\\sigma}{\\sqrt{m}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic. We need therefore a more precise defintion of the error and the variance in our results.\n", - "\n", - "\n", - "\n", - "\n", - "## Definition of Correlation Functions and Standard Deviation\n", - "Our estimate of the true average $\\mu_{X}$ is the sample mean $\\langle X_m \\rangle$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu_{X}^{\\phantom X} \\approx X_m=\\frac{1}{mn}\\sum_{\\alpha=1}^m\\sum_{k=1}^n x_{\\alpha,k}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can then use Eq. ([7](#eq:exptvariance))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2_m=\\frac{1}{mn^2}\\sum_{\\alpha=1}^m\\sum_{kl=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,l}-\\langle X_m \\rangle),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and rewrite it as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2_m=\\frac{\\sigma^2}{n}+\\frac{2}{mn^2}\\sum_{\\alpha=1}^m\\sum_{k\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\kappa_d = \\frac{f_d}{\\sigma^2}\n", - "\\label{eq:autocorrelformal} \\tag{9}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives us a useful measure of the correlation pair correlation\n", - "starting always at $1$ for $d=0$.\n", - "\n", - "\n", - "\n", - "\n", - "## Definition of Correlation Functions and Standard Deviation, sample variance\n", - "\n", - "The sample variance of the $mn$ experiments can now be\n", - "written in terms of the autocorrelation function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\sigma_m^2=\\frac{\\sigma^2}{n}+\\frac{2}{n}\\cdot\\sigma^2\\sum_{d=1}^{n-1}\n", - "\\frac{f_d}{\\sigma^2}=\\left(1+2\\sum_{d=1}^{n-1}\\kappa_d\\right)\\frac{1}{n}\\sigma^2=\\frac{\\tau}{n}\\cdot\\sigma^2\n", - "\\label{eq:error_estimate_corr_time} \\tag{10}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we see that $\\sigma_m$ can be expressed in terms of the\n", - "uncorrelated sample variance times a correction factor $\\tau$ which\n", - "accounts for the correlation between measurements. We call this\n", - "correction factor the *autocorrelation time*" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\tau = 1+2\\sum_{d=1}^{n-1}\\kappa_d\n", - "\\label{eq:autocorrelation_time} \\tag{11}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "\n", - "For a correlation free experiment, $\\tau$\n", - "equals 1.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Definition of Correlation Functions and Standard Deviation\n", - "From the point of view of\n", - "Eq. ([10](#eq:error_estimate_corr_time)) we can interpret a sequential\n", - "correlation as an effective reduction of the number of measurements by\n", - "a factor $\\tau$. The effective number of measurements becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "n_\\mathrm{eff} = \\frac{n}{\\tau}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To neglect the autocorrelation time $\\tau$ will always cause our\n", - "simple uncorrelated estimate of $\\sigma_m^2\\approx \\sigma^2/n$ to\n", - "be less than the true sample error. The estimate of the error will be\n", - "too \"good\". On the other hand, the calculation of the full\n", - "autocorrelation time poses an efficiency problem if the set of\n", - "measurements is very large. The solution to this problem is given by \n", - "more practically oriented methods like the blocking technique.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Code to compute the Covariance matrix and the Covariance" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "from math import exp, sqrt\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "\n", - "# Sample covariance, note the factor 1/(n-1)\n", - "def covariance(x, y, n):\n", - " sum = 0.0\n", - " mean_x = np.mean(x)\n", - " mean_y = np.mean(y)\n", - " for i in range(0, n):\n", - " sum += (x[(i)]-mean_x)*(y[i]-mean_y)\n", - " return sum/(n-1.)\n", - "\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "z = x**3+np.random.normal(size=n)\n", - "print(np.mean(z))\n", - "covxx = covariance(x,x,n)\n", - "covyy = covariance(y,y,n)\n", - "covzz = covariance(z,z,n)\n", - "covxy = covariance(x,y,n)\n", - "covxz = covariance(x,z,n)\n", - "covyz = covariance(y,z,n)\n", - "print(covxx,covyy, covzz)\n", - "print(covxy,covxz, covyz)\n", - "w = np.vstack((x, y, z))\n", - "#print(w)\n", - "c = np.cov(w)\n", - "print(c)\n", - "#eigen = np.zeros(n)\n", - "Eigvals, Eigvecs = np.linalg.eig(c)\n", - "print(Eigvals)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random Numbers\n", - "\n", - "Uniform deviates are just random numbers that lie within a specified range\n", - "(typically 0 to 1), with any one number in the range just as likely as any other. They\n", - "are, in other words, what you probably think random numbers are. However,\n", - "we want to distinguish uniform deviates from other sorts of random numbers, for\n", - "example numbers drawn from a normal (Gaussian) distribution of specified mean\n", - "and standard deviation. These other sorts of deviates are almost always generated by\n", - "performing appropriate operations on one or more uniform deviates, as we will see\n", - "in subsequent sections. So, a reliable source of random uniform deviates, the subject\n", - "of this section, is an essential building block for any sort of stochastic modeling\n", - "or Monte Carlo computer work.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random Numbers, better name: pseudo random numbers\n", - "\n", - "A disclaimer is however appropriate. It should be fairly obvious that \n", - "something as deterministic as a computer cannot generate purely random numbers.\n", - "\n", - "Numbers generated by any of the standard algorithms are in reality pseudo random\n", - "numbers, hopefully abiding to the following criteria:\n", - "\n", - " * they produce a uniform distribution in the interval [0,1].\n", - "\n", - " * correlations between random numbers are negligible\n", - "\n", - " * the period before the same sequence of random numbers is repeated is as large as possible and finally\n", - "\n", - " * the algorithm should be fast.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG\n", - " The most common random number generators are based on so-called\n", - "Linear congruential relations of the type" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_i=(aN_{i-1}+c) \\mathrm{MOD} (M),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which yield a number in the interval [0,1] through" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "x_i=N_i/M\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The number \n", - "$M$ is called the period and it should be as large as possible \n", - " and \n", - "$N_0$ is the starting value, or seed. The function $\\mathrm{MOD}$ means the remainder,\n", - "that is if we were to evaluate $(13)\\mathrm{MOD}(9)$, the outcome is the remainder\n", - "of the division $13/9$, namely $4$.\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG and periodic outputs\n", - "\n", - "The problem with such generators is that their outputs are periodic;\n", - "they \n", - "will start to repeat themselves with a period that is at most $M$. If however\n", - "the parameters $a$ and $c$ are badly chosen, the period may be even shorter.\n", - "\n", - "Consider the following example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_i=(6N_{i-1}+7) \\mathrm{MOD} (5),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with a seed $N_0=2$. This generator produces the sequence\n", - "$4,1,3,0,2,4,1,3,0,2,...\\dots$, i.e., a sequence with period $5$.\n", - "However, increasing $M$ may not guarantee a larger period as the following\n", - "example shows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_i=(27N_{i-1}+11) \\mathrm{MOD} (54),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which still, with $N_0=2$, results in $11,38,11,38,11,38,\\dots$, a period of\n", - "just $2$.\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG and its period\n", - "Typical periods for the random generators provided in the program library \n", - "are of the order of $\\sim 10^9$ or larger. Other random number generators which have\n", - "become increasingly popular are so-called shift-register generators.\n", - "In these generators each successive number depends on many preceding\n", - "values (rather than the last values as in the linear congruential\n", - "generator).\n", - "For example, you could make a shift register generator whose $l$th \n", - "number is the sum of the $l-i$th and $l-j$th values with modulo $M$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_l=(aN_{l-i}+cN_{l-j})\\mathrm{MOD}(M).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random number generator RNG, other examples\n", - "Such a generator again produces a sequence of pseudorandom numbers\n", - "but this time with a period much larger than $M$.\n", - "It is also possible to construct more elaborate algorithms by including\n", - "more than two past terms in the sum of each iteration.\n", - "One example is the generator of [Marsaglia and Zaman](http://dl.acm.org/citation.cfm?id=187154)\n", - "which consists of two congruential relations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " N_l=(N_{l-3}-N_{l-1})\\mathrm{MOD}(2^{31}-69),\n", - "\\label{eq:mz1} \\tag{12}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "followed by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " N_l=(69069N_{l-1}+1013904243)\\mathrm{MOD}(2^{32}),\n", - "\\label{eq:mz2} \\tag{13}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which according to the authors has a period larger than $2^{94}$.\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, other examples\n", - "Instead of using modular addition, we could use the bitwise\n", - "exclusive-OR ($\\oplus$) operation so that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_l=(N_{l-i})\\oplus (N_{l-j})\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the bitwise action of $\\oplus$ means that if $N_{l-i}=N_{l-j}$ the result is\n", - "$0$ whereas if $N_{l-i}\\ne N_{l-j}$ the result is\n", - "$1$. As an example, consider the case where $N_{l-i}=6$ and $N_{l-j}=11$. The first\n", - "one has a bit representation (using 4 bits only) which reads $0110$ whereas the \n", - "second number is $1011$. Employing the $\\oplus$ operator yields \n", - "$1101$, or $2^3+2^2+2^0=13$.\n", - "\n", - "In Fortran90, the bitwise $\\oplus$ operation is coded through the intrinsic\n", - "function $\\mathrm{IEOR}(m,n)$ where $m$ and $n$ are the input numbers, while in $C$\n", - "it is given by $m\\wedge n$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, RAN0\n", - "\n", - "We show here how the linear congruential algorithm can be implemented, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_i=(aN_{i-1}) \\mathrm{MOD} (M).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "However, since $a$ and $N_{i-1}$ are integers and their multiplication \n", - "could become greater than the standard 32 bit integer, there is a trick via \n", - "Schrage's algorithm which approximates the multiplication\n", - "of large integers through the factorization" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "M=aq+r,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "q=[M/a],\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "r = M\\hspace{0.1cm}\\mathrm{MOD} \\hspace{0.1cm}a.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the brackets denote integer division. In the code below the numbers \n", - "$q$ and $r$ are chosen so that $r < q$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, RAN0\n", - "\n", - "To see how this works we note first that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "(aN_{i-1}) \\mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\\mathrm{MOD} (M),\n", - "\\label{eq:rntrick1} \\tag{14}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "since we can add or subtract any integer multiple of $M$ from $aN_{i-1}$.\n", - "The last term $[N_{i-1}/q]M\\mathrm{MOD}(M)$ is zero since the integer division \n", - "$[N_{i-1}/q]$ just yields a constant which is multiplied with $M$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, RAN0\n", - "We can now rewrite Eq. ([14](#eq:rntrick1)) as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "(aN_{i-1}) \\mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\\mathrm{MOD} (M),\n", - "\\label{eq:rntrick2} \\tag{15}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which results\n", - "in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "(aN_{i-1}) \\mathrm{MOD} (M)= \\left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\\right)\\mathrm{MOD} (M),\n", - "\\label{eq:rntrick3} \\tag{16}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "yielding" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "(aN_{i-1}) \\mathrm{MOD} (M)= \\left(a(N_{i-1}\\mathrm{MOD} (q)) -[N_{i-1}/q]r)\\right)\\mathrm{MOD} (M).\n", - "\\label{eq:rntrick4} \\tag{17}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random number generator RNG, RAN0\n", - "The term $[N_{i-1}/q]r$ is always smaller or equal $N_{i-1}(r/q)$ and with $r < q$ we obtain always a \n", - "number smaller than $N_{i-1}$, which is smaller than $M$. \n", - "And since the number $N_{i-1}\\mathrm{MOD} (q)$ is between zero and $q-1$ then\n", - "$a(N_{i-1}\\mathrm{MOD} (q))< aq$. Combined with our definition of $q=[M/a]$ ensures that \n", - "this term is also smaller than $M$ meaning that both terms fit into a\n", - "32-bit signed integer. None of these two terms can be negative, but their difference could.\n", - "The algorithm below adds $M$ if their difference is negative.\n", - "Note that the program uses the bitwise $\\oplus$ operator to generate\n", - "the starting point for each generation of a random number. The period\n", - "of $ran0$ is $\\sim 2.1\\times 10^{9}$. A special feature of this\n", - "algorithm is that is should never be called with the initial seed \n", - "set to $0$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, RAN0 code" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " /*\n", - " ** The function\n", - " ** ran0()\n", - " ** is an \"Minimal\" random number generator of Park and Miller\n", - " ** Set or reset the input value\n", - " ** idum to any integer value (except the unlikely value MASK)\n", - " ** to initialize the sequence; idum must not be altered between\n", - " ** calls for sucessive deviates in a sequence.\n", - " ** The function returns a uniform deviate between 0.0 and 1.0.\n", - " */\n", - " double ran0(long &idum)\n", - " {\n", - " const int a = 16807, m = 2147483647, q = 127773;\n", - " const int r = 2836, MASK = 123459876;\n", - " const double am = 1./m;\n", - " long k;\n", - " double ans;\n", - " idum ^= MASK;\n", - " k = (*idum)/q;\n", - " idum = a*(idum - k*q) - r*k;\n", - " // add m if negative difference\n", - " if(idum < 0) idum += m;\n", - " ans=am*(idum);\n", - " idum ^= MASK;\n", - " return ans;\n", - " } // End: function ran0() \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Properties of Selected Random Number Generators\n", - "\n", - "As mentioned previously, the underlying PDF for the generation of\n", - "random numbers is the uniform distribution, meaning that the \n", - "probability for finding a number $x$ in the interval [0,1] is $p(x)=1$.\n", - "\n", - "A random number generator should produce numbers which are uniformly distributed\n", - "in this interval. The table shows the distribution of $N=10000$ random\n", - "numbers generated by the functions in the program library.\n", - "We note in this table that the number of points in the various\n", - "intervals $0.0-0.1$, $0.1-0.2$ etc are fairly close to $1000$, with some minor\n", - "deviations. \n", - "\n", - "Two additional measures are the standard deviation $\\sigma$ and the mean\n", - "$\\mu=\\langle x\\rangle$.\n", - "\n", - "\n", - "\n", - "\n", - "## Properties of Selected Random Number Generators\n", - "For the uniform distribution, the mean value $\\mu$ is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu=\\langle x\\rangle=\\frac{1}{2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "while the standard deviation is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma=\\sqrt{\\langle x^2\\rangle-\\mu^2}=\\frac{1}{\\sqrt{12}}=0.2886.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Properties of Selected Random Number Generators\n", - "The various random number generators produce results which agree rather well with\n", - "these limiting values. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
$x$-bin ran0 ran1 ran2 ran3
0.0-0.1 1013 991 938 1047
0.1-0.2 1002 1009 1040 1030
0.2-0.3 989 999 1030 993
0.3-0.4 939 960 1023 937
0.4-0.5 1038 1001 1002 992
0.5-0.6 1037 1047 1009 1009
0.6-0.7 1005 989 1003 989
0.7-0.8 986 962 985 954
0.8-0.9 1000 1027 1009 1023
0.9-1.0 991 1015 961 1026
$\\mu$ 0.4997 0.5018 0.4992 0.4990
$\\sigma$ 0.2882 0.2892 0.2861 0.2915
\n", - "\n", - "\n", - "\n", - "\n", - "## Simple demonstration of RNGs using python\n", - "The following simple Python code plots the distribution of the produced random numbers using the linear congruential RNG employed by Python. The trend displayed in the previous table is seen rather clearly." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "#!/usr/bin/env python\n", - "import numpy as np\n", - "import matplotlib.mlab as mlab\n", - "import matplotlib.pyplot as plt\n", - "import random\n", - "\n", - "# initialize the rng with a seed\n", - "random.seed() \n", - "counts = 10000\n", - "values = np.zeros(counts) \n", - "for i in range (1, counts, 1):\n", - " values[i] = random.random()\n", - "\n", - "# the histogram of the data\n", - "n, bins, patches = plt.hist(values, 10, facecolor='green')\n", - "\n", - "plt.xlabel('$x$')\n", - "plt.ylabel('Number of counts')\n", - "plt.title(r'Test of uniform distribution')\n", - "plt.axis([0, 1, 0, 1100])\n", - "plt.grid(True)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Properties of Selected Random Number Generators\n", - "Since our random numbers, which are typically generated via a linear congruential algorithm,\n", - "are never fully independent, we can then define \n", - "an important test which measures the degree of correlation, namely the so-called \n", - "auto-correlation function defined previously, see again Eq. ([9](#eq:autocorrelformal)).\n", - "We rewrite it here as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C_k=\\frac{f_d}\n", - " {\\sigma^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $C_0=1$. Recall that \n", - "$\\sigma^2=\\langle x_i^2\\rangle-\\langle x_i\\rangle^2$ and that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_d = \\frac{1}{nm}\\sum_{\\alpha=1}^m\\sum_{k=1}^{n-d}(x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,k+d}-\\langle X_m \\rangle),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The non-vanishing of $C_k$ for $k\\ne 0$ means that the random\n", - "numbers are not independent. The independence of the random numbers is crucial \n", - "in the evaluation of other expectation values. If they are not independent, our\n", - "assumption for approximating $\\sigma_N$ is no longer valid.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Autocorrelation function\n", - "This program computes the autocorrelation function as discussed in the equation on the previous slide for random numbers generated with the normal distribution $N(0,1)$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "from math import exp, sqrt\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "\n", - "def autocovariance(x, n, k, mean_x):\n", - " sum = 0.0\n", - " for i in range(0, n-k):\n", - " sum += (x[(i+k)]-mean_x)*(x[i]-mean_x)\n", - " return sum/n\n", - "\n", - "n = 1000\n", - "x=np.random.normal(size=n)\n", - "autocor = np.zeros(n)\n", - "figaxis = np.zeros(n)\n", - "mean_x=np.mean(x)\n", - "var_x = np.var(x)\n", - "print(mean_x, var_x)\n", - "for i in range (0, n):\n", - " figaxis[i] = i\n", - " autocor[i]=(autocovariance(x, n, i, mean_x))/var_x \n", - "\n", - "plt.plot(figaxis, autocor, \"r-\")\n", - "plt.axis([0,n,-0.1, 1.0])\n", - "plt.xlabel(r'$i$')\n", - "plt.ylabel(r'$\\gamma_i$')\n", - "plt.title(r'Autocorrelation function')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As can be seen from the plot, the first point gives back the variance and a value of one. \n", - "For the remaining values we notice that there are still non-zero values for the auto-correlation function.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Correlation function and which random number generators should I use\n", - "The program here computes the correlation function for one of the standard functions included with the c++ compiler." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " // This function computes the autocorrelation function for \n", - " // the standard c++ random number generator\n", - " \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " using namespace std;\n", - " // output file as global variable\n", - " ofstream ofile; \n", - " \n", - " // Main function begins here \n", - " int main(int argc, char* argv[])\n", - " {\n", - " int n;\n", - " char *outfilename;\n", - " \n", - " cin >> n;\n", - " double MCint = 0.; double MCintsqr2=0.;\n", - " double invers_period = 1./RAND_MAX; // initialise the random number generator\n", - " srand(time(NULL)); // This produces the so-called seed in MC jargon\n", - " // Compute the variance and the mean value of the uniform distribution\n", - " // Compute also the specific values x for each cycle in order to be able to\n", - " // the covariance and the correlation function \n", - " // Read in output file, abort if there are too few command-line arguments\n", - " if( argc <= 2 ){\n", - " cout << \"Bad Usage: \" << argv[0] << \n", - " \t \" read also output file and number of cycles on same line\" << endl;\n", - " exit(1);\n", - " }\n", - " else{\n", - " outfilename=argv[1];\n", - " }\n", - " ofile.open(outfilename); \n", - " // Get the number of Monte-Carlo samples\n", - " n = atoi(argv[2]);\n", - " double *X; \n", - " X = new double[n];\n", - " for (int i = 0; i < n; i++){\n", - " double x = double(rand())*invers_period; \n", - " X[i] = x;\n", - " MCint += x;\n", - " MCintsqr2 += x*x;\n", - " }\n", - " double Mean = MCint/((double) n );\n", - " MCintsqr2 = MCintsqr2/((double) n );\n", - " double STDev = sqrt(MCintsqr2-Mean*Mean);\n", - " double Variance = MCintsqr2-Mean*Mean;\n", - " // Write mean value and standard deviation \n", - " cout << \" Standard deviation= \" << STDev << \" Integral = \" << Mean << endl;\n", - " \n", - " // Now we compute the autocorrelation function\n", - " double *autocor; autocor = new double[n];\n", - " for (int j = 0; j < n; j++){\n", - " double sum = 0.0;\n", - " for (int k = 0; k < (n-j); k++){\n", - " \t sum += (X[k]-Mean)*(X[k+j]-Mean); \n", - " }\n", - " autocor[j] = sum/Variance/((double) n );\n", - " ofile << setiosflags(ios::showpoint | ios::uppercase);\n", - " ofile << setw(15) << setprecision(8) << j;\n", - " ofile << setw(15) << setprecision(8) << autocor[j] << endl;\n", - " }\n", - " ofile.close(); // close output file\n", - " return 0;\n", - " } // end of main program \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Which RNG should I use?\n", - "* C++ has a class called **random**. The [random class](http://www.cplusplus.com/reference/random/) contains a large selection of RNGs and is highly recommended. Some of these RNGs have very large periods making it thereby very safe to use these RNGs in case one is performing large calculations. In particular, the [Mersenne twister random number engine](http://www.cplusplus.com/reference/random/mersenne_twister_engine/) has a period of $2^{19937}$. \n", - "\n", - "* Add RNGs in Python\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## How to use the Mersenne generator\n", - "The following part of a c++ code (from project 4) sets up the uniform distribution for $x\\in [0,1]$." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " /*\n", - " \n", - " // You need this \n", - " #include \n", - " \n", - " // Initialize the seed and call the Mersienne algo\n", - " std::random_device rd;\n", - " std::mt19937_64 gen(rd());\n", - " // Set up the uniform distribution for x \\in [[0, 1]\n", - " std::uniform_real_distribution RandomNumberGenerator(0.0,1.0);\n", - " \n", - " // Now use the RNG\n", - " int ix = (int) (RandomNumberGenerator(gen)*NSpins);\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Why blocking?\n", - "**Statistical analysis.**\n", - "\n", - " * Monte Carlo simulations can be treated as *computer experiments*\n", - "\n", - " * The results can be analysed with the same statistical tools as we would use analysing experimental data.\n", - "\n", - " * As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n", - "\n", - "A very good article which explains blocking is H. Flyvbjerg and H. G. Petersen, *Error estimates on averages of correlated data*, [Journal of Chemical Physics 91, 461-466 (1989)](http://scitation.aip.org/content/aip/journal/jcp/91/1/10.1063/1.457480).\n", - "\n", - " \n", - "\n", - "\n", - "\n", - "\n", - "## Why blocking?\n", - "**Statistical analysis.**\n", - "\n", - " * As in other experiments, Monte Carlo experiments have two classes of errors:\n", - "\n", - " * Statistical errors\n", - "\n", - " * Systematical errors\n", - "\n", - "\n", - " * Statistical errors can be estimated using standard tools from statistics\n", - "\n", - " * Systematical errors are method specific and must be treated differently from case to case. (In VMC a common source is the step length or time step in importance sampling)\n", - "\n", - " \n", - "\n", - "\n", - "\n", - "## Code to demonstrate the calculation of the autocorrelation function\n", - "The following code computes the autocorrelation function, the covariance and the standard deviation\n", - "for standard RNG. \n", - "The [following file](https://github.com/CompPhysics/ComputationalPhysics2/tree/gh-pages/doc/Programs/LecturePrograms/programs/Blocking/autocorrelation.cpp) gives the code." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " // This function computes the autocorrelation function for \n", - " // the Mersenne random number generator with a uniform distribution\n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " using namespace std;\n", - " using namespace arma;\n", - " // output file\n", - " ofstream ofile;\n", - " \n", - " // Main function begins here \n", - " int main(int argc, char* argv[])\n", - " {\n", - " int MonteCarloCycles;\n", - " string filename;\n", - " if (argc > 1) {\n", - " filename=argv[1];\n", - " MonteCarloCycles = atoi(argv[2]);\n", - " string fileout = filename;\n", - " string argument = to_string(MonteCarloCycles);\n", - " fileout.append(argument);\n", - " ofile.open(fileout);\n", - " }\n", - " \n", - " // Compute the variance and the mean value of the uniform distribution\n", - " // Compute also the specific values x for each cycle in order to be able to\n", - " // compute the covariance and the correlation function \n", - " \n", - " vec X = zeros(MonteCarloCycles);\n", - " double MCint = 0.; double MCintsqr2=0.;\n", - " std::random_device rd;\n", - " std::mt19937_64 gen(rd());\n", - " // Set up the uniform distribution for x \\in [[0, 1]\n", - " std::uniform_real_distribution RandomNumberGenerator(0.0,1.0);\n", - " for (int i = 0; i < MonteCarloCycles; i++){\n", - " double x = RandomNumberGenerator(gen); \n", - " X(i) = x;\n", - " MCint += x;\n", - " MCintsqr2 += x*x;\n", - " }\n", - " double Mean = MCint/((double) MonteCarloCycles );\n", - " MCintsqr2 = MCintsqr2/((double) MonteCarloCycles );\n", - " double STDev = sqrt(MCintsqr2-Mean*Mean);\n", - " double Variance = MCintsqr2-Mean*Mean;\n", - " // Write mean value and variance\n", - " cout << \" Sample variance= \" << Variance << \" Mean value = \" << Mean << endl;\n", - " // Now we compute the autocorrelation function\n", - " vec autocorrelation = zeros(MonteCarloCycles);\n", - " for (int j = 0; j < MonteCarloCycles; j++){\n", - " double sum = 0.0;\n", - " for (int k = 0; k < (MonteCarloCycles-j); k++){\n", - " sum += (X(k)-Mean)*(X(k+j)-Mean); \n", - " }\n", - " autocorrelation(j) = sum/Variance/((double) MonteCarloCycles );\n", - " ofile << setiosflags(ios::showpoint | ios::uppercase);\n", - " ofile << setw(15) << setprecision(8) << j;\n", - " ofile << setw(15) << setprecision(8) << autocorrelation(j) << endl;\n", - " }\n", - " // Now compute the exact covariance using the autocorrelation function\n", - " double Covariance = 0.0;\n", - " for (int j = 0; j < MonteCarloCycles; j++){\n", - " Covariance += autocorrelation(j);\n", - " }\n", - " Covariance *= 2.0/((double) MonteCarloCycles);\n", - " // Compute now the total variance, including the covariance, and obtain the standard deviation\n", - " double TotalVariance = (Variance/((double) MonteCarloCycles ))+Covariance;\n", - " cout << \"Covariance =\" << Covariance << \"Totalvariance= \" << TotalVariance << \"Sample Variance/n= \" << (Variance/((double) MonteCarloCycles )) << endl;\n", - " cout << \" STD from sample variance= \" << sqrt(Variance/((double) MonteCarloCycles )) << \" STD with covariance = \" << sqrt(TotalVariance) << endl;\n", - " \n", - " ofile.close(); // close output file\n", - " return 0;\n", - " } // end of main program \n", - " \n", - " \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## What is blocking?\n", - "**Blocking.**\n", - "\n", - " * Say that we have a set of samples from a Monte Carlo experiment\n", - "\n", - " * Assuming (wrongly) that our samples are uncorrelated our best estimate of the standard deviation of the mean $\\langle \\mathbf{M}\\rangle$ is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma=\\sqrt{\\frac{1}{n}\\left(\\langle \\mathbf{M}^2\\rangle-\\langle \\mathbf{M}\\rangle^2\\right)}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "* If the samples are correlated we can rewrite our results to show that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma=\\sqrt{\\frac{1+2\\tau/\\Delta t}{n}\\left(\\langle \\mathbf{M}^2\\rangle-\\langle \\mathbf{M}\\rangle^2\\right)}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\tau$ is the correlation time (the time between a sample and the next uncorrelated sample) and $\\Delta t$ is time between each sample\n", - "\n", - " \n", - "\n", - "\n", - "## What is blocking?\n", - "**Blocking.**\n", - "\n", - " * If $\\Delta t\\gg\\tau$ our first estimate of $\\sigma$ still holds\n", - "\n", - " * Much more common that $\\Delta t<\\tau$\n", - "\n", - " * In the method of data blocking we divide the sequence of samples into blocks\n", - "\n", - " * We then take the mean $\\langle \\mathbf{M}_i\\rangle$ of block $i=1\\ldots n_{blocks}$ to calculate the total mean and variance\n", - "\n", - " * The size of each block must be so large that sample $j$ of block $i$ is not correlated with sample $j$ of block $i+1$\n", - "\n", - " * The correlation time $\\tau$ would be a good choice\n", - "\n", - "\n", - "\n", - "\n", - "## What is blocking?\n", - "**Blocking.**\n", - "\n", - " * Problem: We don't know $\\tau$ or it is too expensive to compute\n", - "\n", - " * Solution: Make a plot of std. dev. as a function of blocksize\n", - "\n", - " * The estimate of std. dev. of correlated data is too low $\\to$ the error will increase with increasing block size until the blocks are uncorrelated, where we reach a plateau\n", - "\n", - " * When the std. dev. stops increasing the blocks are uncorrelated\n", - "\n", - "\n", - "\n", - "\n", - "## Implementation\n", - " * Do a Monte Carlo simulation, storing all samples to file\n", - "\n", - " * Do the statistical analysis on this file, independently of your Monte Carlo program\n", - "\n", - " * Read the file into an array\n", - "\n", - " * Loop over various block sizes\n", - "\n", - " * For each block size $n_b$, loop over the array in steps of $n_b$ taking the mean of elements $i n_b,\\ldots,(i+1) n_b$\n", - "\n", - " * Take the mean and variance of the resulting array\n", - "\n", - " * Write the results for each block size to file for later\n", - " analysis\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Actual implementation with code, main function\n", - "When the file gets large, it can be useful to write your data in binary mode instead of ascii characters.\n", - "The [following python file](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) reads data from file with the output from every Monte Carlo cycle." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Blocking\n", - " @timeFunction\n", - " def blocking(self, blockSizeMax = 500):\n", - " blockSizeMin = 1\n", - "\n", - " self.blockSizes = []\n", - " self.meanVec = []\n", - " self.varVec = []\n", - "\n", - " for i in range(blockSizeMin, blockSizeMax):\n", - " if(len(self.data) % i != 0):\n", - " pass#continue\n", - " blockSize = i\n", - " meanTempVec = []\n", - " varTempVec = []\n", - " startPoint = 0\n", - " endPoint = blockSize\n", - "\n", - " while endPoint <= len(self.data):\n", - " meanTempVec.append(np.average(self.data[startPoint:endPoint]))\n", - " startPoint = endPoint\n", - " endPoint += blockSize\n", - " mean, var = np.average(meanTempVec), np.var(meanTempVec)/len(meanTempVec)\n", - " self.meanVec.append(mean)\n", - " self.varVec.append(var)\n", - " self.blockSizes.append(blockSize)\n", - "\n", - " self.blockingAvg = np.average(self.meanVec[-200:])\n", - " self.blockingVar = (np.average(self.varVec[-200:]))\n", - " self.blockingStd = np.sqrt(self.blockingVar)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The Bootstrap method\n", - "\n", - "The Bootstrap resampling method is also very popular. It is very simple:\n", - "\n", - "1. Start with your sample of measurements and compute the sample variance and the mean values\n", - "\n", - "2. Then start again but pick in a random way the numbers in the sample and recalculate the mean and the sample variance.\n", - "\n", - "3. Repeat this $K$ times.\n", - "\n", - "It can be shown, see the article by [Efron](https://projecteuclid.org/download/pdf_1/euclid.aos/1176344552)\n", - "that it produces the correct standard deviation.\n", - "\n", - "This method is very useful for small ensembles of data points. \n", - "\n", - "\n", - "## Bootstrapping\n", - "Given a set of $N$ data, assume that we are interested in some \n", - "observable $\\theta$ which may be estimated from that set. This observable can also be for example the result of a fit based on all $N$ raw data. \n", - "Let us call the value of the observable obtained from the original \n", - "data set $\\hat{\\theta}$. One recreates from the sample repeatedly \n", - "other samples by choosing randomly $N$ data out of the original set. \n", - "This costs essentially nothing, since we just recycle the original data set for the building of new sets. \n", - "\n", - "\n", - "## Bootstrapping, recipe\n", - "Let us assume we have done this $K$ times and thus have $K$ sets of $N$ \n", - "data values each. \n", - "Of course some values will enter more than once in the new sets. For each of these sets one computes the observable $\\theta$ resulting in values $\\theta_k$ with $k = 1,...,K$. Then one determines" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\tilde{\\theta} = \\frac{1}{K} \\sum_{k=1}^K \\theta_k,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "sigma^2_{\\tilde{\\theta}} = \\frac{1}{K} \\sum_{k=1}^K \\left(\\theta_k-\\tilde{\\theta}\\right)^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These are estimators for $\\angle\\theta\\rangle$ and its variance. They are not unbiased and therefore \n", - "$\\tilde{\\theta}\\neq\\hat{\\theta}$ for finite K. \n", - "\n", - "The difference is called bias and gives an idea on how far away the result may be from \n", - "the true $\\angle\\theta\\rangle$. As final result for the observable one quotes $\\angle\\theta\\rangle = \\tilde{\\theta} \\pm \\sigma_{\\tilde{\\theta}}$ .\n", - "\n", - "\n", - "\n", - "## Bootstrapping, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " # Bootstrap\n", - " @timeFunction\n", - " def bootstrap(self, nBoots = 1000):\n", - " bootVec = np.zeros(nBoots)\n", - " for k in range(0,nBoots):\n", - " bootVec[k] = np.average(np.random.choice(self.data, len(self.data)))\n", - " self.bootAvg = np.average(bootVec)\n", - " self.bootVar = np.var(bootVec)\n", - " self.bootStd = np.std(bootVec)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Jackknife, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " # Jackknife\n", - " @timeFunction\n", - " def jackknife(self):\n", - " jackknVec = np.zeros(len(self.data))\n", - " for k in range(0,len(self.data)):\n", - " jackknVec[k] = np.average(np.delete(self.data, k))\n", - " self.jackknAvg = self.avg - (len(self.data) - 1) * (np.average(jackknVec) - self.avg)\n", - " self.jackknVar = float(len(self.data) - 1) * np.var(jackknVec)\n", - " self.jackknStd = np.sqrt(self.jackknVar)\n" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/_build/html/_sources/teachers.md b/doc/LectureNotes/_build/html/_sources/teachers.md deleted file mode 100644 index 4b934a68b..000000000 --- a/doc/LectureNotes/_build/html/_sources/teachers.md +++ /dev/null @@ -1,23 +0,0 @@ -# Teachers and Grading - - -## Instructor information -* _Name_: Morten Hjorth-Jensen -* _Email_: morten.hjorth-jensen@fys.uio.no -* _Phone_: +47-48257387 -* _Office_: Department of Physics, University of Oslo, Eastern wing, room FØ470 -* _Office hours_: *Anytime*! In Fall Semester 2020 (FS20), as a rule of thumb office hours are planned via computer or telephone. Individual or group office hours will be performed via zoom. Feel free to send an email for planning. In person meetings may also be possible if allowed by the University of Oslo's COVID-19 instructions (see below for links). - - -## Grading -Grading scale: Grades are awarded on a scale from A to F, where A is the best grade and F is a fail. There are three projects which are graded and each project counts 1/3 of the final grade. The total score is thus the average from all three projects. - -The final number of points is based on the average of all projects (including eventual additional points) and the grade follows the following table: - - * 92-100 points: A - * 77-91 points: B - * 58-76 points: C - * 46-57 points: D - * 40-45 points: E - * 0-39 points: F-failed - diff --git a/doc/LectureNotes/_build/html/_sources/textbooks.md b/doc/LectureNotes/_build/html/_sources/textbooks.md deleted file mode 100644 index 92d980b58..000000000 --- a/doc/LectureNotes/_build/html/_sources/textbooks.md +++ /dev/null @@ -1,38 +0,0 @@ -# Textbooks - - -_Recommended textbooks_: -- Christopher M. Bishop, Pattern Recognition and Machine Learning, Springer, https://www.springer.com/gp/book/9780387310732. This is the main textbook and this course covers chapters 1-7, 11 and 12. -- Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer, https://www.springer.com/gp/book/9780387848570. This is a well-known text and serves as additional text. -- Aurelien Geron, Hands‑On Machine Learning with Scikit‑Learn and TensorFlow, O'Reilly, https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/. This text is very useful since it contains many code examples. - -The books by Bishop and Hastie et al. can be downloaded for free if you access the university library via an IP number of your home university. - - - -_General learning book on statistical analysis_: -- Christian Robert and George Casella, Monte Carlo Statistical Methods, Springer -- Peter Hoff, A first course in Bayesian statistical models, Springer - -_General Machine Learning Books_: -- Kevin Murphy, Machine Learning: A Probabilistic Perspective, MIT Press -- Christopher M. Bishop, Pattern Recognition and Machine Learning, Springer -- David J.C. MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge University Press -- David Barber, Bayesian Reasoning and Machine Learning, Cambridge University Press - -## Links to relevant courses at the University of Oslo -The link here https://www.mn.uio.no/english/research/about/centre-focus/innovation/data-science/studies/ gives an excellent overview of courses on Machine learning at UiO. - -- _STK2100 Machine learning and statistical methods for prediction and classification_ http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html. -- _IN3050 Introduction to Artificial Intelligence and Machine Learning_ https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html. Introductory course in machine learning and AI with an algorithmic approach. -- _STK-INF3000/4000 Selected Topics in Data Science_ http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html. The course provides insight into selected contemporary relevant topics within Data Science. -- _IN4080 Natural Language Processing_ https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html. Probabilistic and machine learning techniques applied to natural language processing. -- _STK-IN4300 Statistical learning methods in Data Science_ https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background. -- _INF4490 Biologically Inspired Computing_ http://www.uio.no/studier/emner/matnat/ifi/INF4490/. An introduction to self-adapting methods also called artificial intelligence or machine learning. -- _IN-STK5000 Adaptive Methods for Data-Based Decision Making_ https://www.uio.no/studier/emner/matnat/ifi/IN-STK5000/index-eng.html. Methods for adaptive collection and processing of data based on machine learning techniques. -- _IN5400/INF5860 Machine Learning for Image Analysis_ https://www.uio.no/studier/emner/matnat/ifi/IN5400/. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too. -- _TEK5040 Deep learning for autonomous systems_ https://www.uio.no/studier/emner/matnat/its/TEK5040/. The course addresses advanced algorithms and architectures for deep learning with neural networks. The course provides an introduction to how deep-learning techniques can be used in the construction of key parts of advanced autonomous systems that exist in physical environments and cyber environments. -- _STK4051 Computational Statistics_ https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html -- _STK4021 Applied Bayesian Analysis and Numerical Methods_ https://www.uio.no/studier/emner/matnat/math/STK4021/ - - diff --git a/doc/LectureNotes/_build/html/_static/__init__.py b/doc/LectureNotes/_build/html/_static/__init__.py deleted file mode 100644 index e69de29bb..000000000 diff --git a/doc/LectureNotes/_build/html/_static/__pycache__/__init__.cpython-38.pyc b/doc/LectureNotes/_build/html/_static/__pycache__/__init__.cpython-38.pyc deleted file mode 100644 index dce0d97cb2a0cc96a29090e51370f0c4fbbac3fb..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 185 zcmYj~F%H5o5Ck2G0wM7b3UWmn3Ix0$4N8k~k`p#b&Q|V_$dmXIEl;4M!mePYoz-r$ z)pEH|QF~jSQ@#@ZmBn(1=2=9mj%t;a4>hLwhtCNr#*tyLS0qLP9|R1U##3tw=v@tA z66>kRH^5GC9Zb`i3o>x9j_$hlzSClHKwvTA8qnI26RqGvJ2zT6Qo AJOBUy diff --git a/doc/LectureNotes/_build/html/_static/basic.css b/doc/LectureNotes/_build/html/_static/basic.css deleted file mode 100644 index fb51eb711..000000000 --- a/doc/LectureNotes/_build/html/_static/basic.css +++ /dev/null @@ -1,856 +0,0 @@ -/* - * basic.css - * ~~~~~~~~~ - * - * Sphinx stylesheet -- basic theme. - * - * :copyright: Copyright 2007-2020 by the Sphinx team, see AUTHORS. - * :license: BSD, see LICENSE for details. - * - */ - -/* -- main layout ----------------------------------------------------------- */ - -div.clearer { - clear: both; 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- clear: right; - overflow-x: auto; -} - -p.sidebar-title { - font-weight: bold; -} - -div.admonition, div.topic, blockquote { - clear: left; -} - -/* -- topics ---------------------------------------------------------------- */ - -div.topic { - border: 1px solid #ccc; - padding: 7px; - margin: 10px 0 10px 0; -} - -p.topic-title { - font-size: 1.1em; - font-weight: bold; - margin-top: 10px; -} - -/* -- admonitions ----------------------------------------------------------- */ - -div.admonition { - margin-top: 10px; - margin-bottom: 10px; - padding: 7px; -} - -div.admonition dt { - font-weight: bold; -} - -p.admonition-title { - margin: 0px 10px 5px 0px; - font-weight: bold; -} - -div.body p.centered { - text-align: center; - margin-top: 25px; -} - -/* -- content of sidebars/topics/admonitions -------------------------------- */ - -div.sidebar > :last-child, -div.topic > :last-child, -div.admonition > :last-child { - margin-bottom: 0; -} - -div.sidebar::after, -div.topic::after, -div.admonition::after, -blockquote::after { - display: block; - content: ''; - clear: both; -} - -/* -- tables ---------------------------------------------------------------- */ - -table.docutils { - margin-top: 10px; - margin-bottom: 10px; - border: 0; - border-collapse: collapse; -} - -table.align-center { - margin-left: auto; - margin-right: auto; -} - -table.align-default { - margin-left: auto; - margin-right: auto; -} - -table caption span.caption-number { - font-style: italic; -} - -table caption span.caption-text { -} - -table.docutils td, table.docutils th { - padding: 1px 8px 1px 5px; - border-top: 0; - border-left: 0; - border-right: 0; - border-bottom: 1px solid #aaa; -} - -table.footnote td, table.footnote th { - border: 0 !important; -} - -th { - text-align: left; - padding-right: 5px; -} - -table.citation { - border-left: solid 1px gray; - margin-left: 1px; -} - -table.citation td { - border-bottom: none; -} - -th > :first-child, -td > :first-child { - margin-top: 0px; -} - -th > :last-child, -td > :last-child { - margin-bottom: 0px; 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- top: .2em; - right: .2em; - width: 1em; - height: 1em; - opacity: .3; - transition: opacity 0.5s; - border: none; - user-select: none; -} - -div.highlight { - position: relative; -} - -a.copybtn > img { - vertical-align: top; - margin: 0; - top: 0; - left: 0; - position: absolute; -} - -.highlight:hover .copybtn { - opacity: 1; -} - -/** - * A minimal CSS-only tooltip copied from: - * https://codepen.io/mildrenben/pen/rVBrpK - * - * To use, write HTML like the following: - * - *

Short

- */ - .o-tooltip--left { - position: relative; - } - - .o-tooltip--left:after { - opacity: 0; - visibility: hidden; - position: absolute; - content: attr(data-tooltip); - padding: 2px; - top: 0; - left: -.2em; - background: grey; - font-size: 1rem; - color: white; - white-space: nowrap; - z-index: 2; - border-radius: 2px; - transform: translateX(-102%) translateY(0); - transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1); -} - -.o-tooltip--left:hover:after { - display: block; - opacity: 1; - visibility: visible; - transform: translateX(-100%) translateY(0); - transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1); - transition-delay: .5s; -} diff --git a/doc/LectureNotes/_build/html/_static/copybutton.js b/doc/LectureNotes/_build/html/_static/copybutton.js deleted file mode 100644 index 65a59167a..000000000 --- a/doc/LectureNotes/_build/html/_static/copybutton.js +++ /dev/null @@ -1,153 +0,0 @@ -// Localization support -const messages = { - 'en': { - 'copy': 'Copy', - 'copy_to_clipboard': 'Copy to clipboard', - 'copy_success': 'Copied!', - 'copy_failure': 'Failed to copy', - }, - 'es' : { - 'copy': 'Copiar', - 'copy_to_clipboard': 'Copiar al portapapeles', - 'copy_success': '¡Copiado!', - 'copy_failure': 'Error al copiar', - }, - 'de' : { - 'copy': 'Kopieren', - 'copy_to_clipboard': 'In die Zwischenablage kopieren', - 'copy_success': 'Kopiert!', - 'copy_failure': 'Fehler beim Kopieren', - } -} - -let locale = 'en' -if( document.documentElement.lang !== undefined - && messages[document.documentElement.lang] !== undefined ) { - locale = document.documentElement.lang -} - -/** - * Set up copy/paste for code blocks - */ - -const runWhenDOMLoaded = cb => { - if (document.readyState != 'loading') { - cb() - } else if (document.addEventListener) { - document.addEventListener('DOMContentLoaded', cb) - } else { - document.attachEvent('onreadystatechange', function() { - if (document.readyState == 'complete') cb() - }) - } -} - -const codeCellId = index => `codecell${index}` - -// Clears selected text since ClipboardJS will select the text when copying -const clearSelection = () => { - if (window.getSelection) { - window.getSelection().removeAllRanges() - } else if (document.selection) { - document.selection.empty() - } -} - -// Changes tooltip text for two seconds, then changes it back -const temporarilyChangeTooltip = (el, newText) => { - const oldText = el.getAttribute('data-tooltip') - el.setAttribute('data-tooltip', newText) - setTimeout(() => el.setAttribute('data-tooltip', oldText), 2000) -} - -const addCopyButtonToCodeCells = () => { - // If ClipboardJS hasn't loaded, wait a bit and try again. This - // happens because we load ClipboardJS asynchronously. - if (window.ClipboardJS === undefined) { - setTimeout(addCopyButtonToCodeCells, 250) - return - } - - // Add copybuttons to all of our code cells - const codeCells = document.querySelectorAll('div.highlight pre') - codeCells.forEach((codeCell, index) => { - const id = codeCellId(index) - codeCell.setAttribute('id', id) - const pre_bg = getComputedStyle(codeCell).backgroundColor; - - const clipboardButton = id => - ` - ${messages[locale]['copy_to_clipboard']} - ` - codeCell.insertAdjacentHTML('afterend', clipboardButton(id)) - }) - -function escapeRegExp(string) { - return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string -} - -// Callback when a copy button is clicked. Will be passed the node that was clicked -// should then grab the text and replace pieces of text that shouldn't be used in output -function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true) { - - var regexp; - var match; - - // create regexp to capture prompt and remaining line - if (isRegexp) { - regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)') - } else { - regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)') - } - - const outputLines = []; - var promptFound = false; - for (const line of textContent.split('\n')) { - match = line.match(regexp) - if (match) { - promptFound = true - if (removePrompts) { - outputLines.push(match[2]) - } else { - outputLines.push(line) - } - } else { - if (!onlyCopyPromptLines) { - outputLines.push(line) - } - } - } - - // If no lines with the prompt were found then just use original lines - if (promptFound) { - textContent = outputLines.join('\n'); - } - - // Remove a trailing newline to avoid auto-running when pasting - if (textContent.endsWith("\n")) { - textContent = textContent.slice(0, -1) - } - return textContent -} - - -var copyTargetText = (trigger) => { - var target = document.querySelector(trigger.attributes['data-clipboard-target'].value); - return formatCopyText(target.innerText, '', false, true, true) -} - - // Initialize with a callback so we can modify the text before copy - const clipboard = new ClipboardJS('.copybtn', {text: copyTargetText}) - - // Update UI with error/success messages - clipboard.on('success', event => { - clearSelection() - temporarilyChangeTooltip(event.trigger, messages[locale]['copy_success']) - }) - - clipboard.on('error', event => { - temporarilyChangeTooltip(event.trigger, messages[locale]['copy_failure']) - }) -} - -runWhenDOMLoaded(addCopyButtonToCodeCells) \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/_static/copybutton_funcs.js b/doc/LectureNotes/_build/html/_static/copybutton_funcs.js deleted file mode 100644 index 57caa5585..000000000 --- a/doc/LectureNotes/_build/html/_static/copybutton_funcs.js +++ /dev/null @@ -1,47 +0,0 @@ -function escapeRegExp(string) { - return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string -} - -// Callback when a copy button is clicked. Will be passed the node that was clicked -// should then grab the text and replace pieces of text that shouldn't be used in output -export function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true) { - - var regexp; - var match; - - // create regexp to capture prompt and remaining line - if (isRegexp) { - regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)') - } else { - regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)') - } - - const outputLines = []; - var promptFound = false; - for (const line of textContent.split('\n')) { - match = line.match(regexp) - if (match) { - promptFound = true - if (removePrompts) { - outputLines.push(match[2]) - } else { - outputLines.push(line) - } - } else { - if (!onlyCopyPromptLines) { - outputLines.push(line) - } - } - } - - // If no lines with the prompt were found then just use original lines - if (promptFound) { - textContent = outputLines.join('\n'); - } - - // Remove a trailing newline to avoid auto-running when pasting - if (textContent.endsWith("\n")) { - textContent = textContent.slice(0, -1) - } - return textContent -} diff --git a/doc/LectureNotes/_build/html/_static/css/index.f658d18f9b420779cfdf24aa0a7e2d77.css b/doc/LectureNotes/_build/html/_static/css/index.f658d18f9b420779cfdf24aa0a7e2d77.css deleted file mode 100644 index 7fd19a770..000000000 --- a/doc/LectureNotes/_build/html/_static/css/index.f658d18f9b420779cfdf24aa0a7e2d77.css +++ /dev/null @@ -1,6 +0,0 @@ -/*! - * Bootstrap v4.5.0 (https://getbootstrap.com/) - * Copyright 2011-2020 The Bootstrap Authors - * Copyright 2011-2020 Twitter, Inc. - * Licensed under MIT (https://github.com/twbs/bootstrap/blob/master/LICENSE) - 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CommonJS-like environments where a proper `window` - // is present, execute the factory and get jQuery. - // For environments that do not have a `window` with a `document` - // (such as Node.js), expose a factory as module.exports. - // This accentuates the need for the creation of a real `window`. - // e.g. var jQuery = require("jquery")(window); - // See ticket #14549 for more info. - module.exports = global.document ? - factory( global, true ) : - function( w ) { - if ( !w.document ) { - throw new Error( "jQuery requires a window with a document" ); - } - return factory( w ); - }; - } else { - factory( global ); - } - -// Pass this if window is not defined yet -} )( typeof window !== "undefined" ? window : this, function( window, noGlobal ) { - -// Edge <= 12 - 13+, Firefox <=18 - 45+, IE 10 - 11, Safari 5.1 - 9+, iOS 6 - 9.1 -// throw exceptions when non-strict code (e.g., ASP.NET 4.5) accesses strict mode -// arguments.callee.caller (trac-13335). But as of jQuery 3.0 (2016), strict mode should be common -// enough that all such attempts are guarded in a try block. -"use strict"; - -var arr = []; - -var getProto = Object.getPrototypeOf; - -var slice = arr.slice; - -var flat = arr.flat ? function( array ) { - return arr.flat.call( array ); -} : function( array ) { - return arr.concat.apply( [], array ); -}; - - -var push = arr.push; - -var indexOf = arr.indexOf; - -var class2type = {}; - -var toString = class2type.toString; - -var hasOwn = class2type.hasOwnProperty; - -var fnToString = hasOwn.toString; - -var ObjectFunctionString = fnToString.call( Object ); - -var support = {}; - -var isFunction = function isFunction( obj ) { - - // Support: Chrome <=57, Firefox <=52 - // In some browsers, typeof returns "function" for HTML elements - // (i.e., `typeof document.createElement( "object" ) === "function"`). - // We don't want to classify *any* DOM node as a function. - return typeof obj === "function" && typeof obj.nodeType !== "number"; - }; - - -var isWindow = function isWindow( obj ) { - return obj != null && obj === obj.window; - }; - - -var document = window.document; - - - - var preservedScriptAttributes = { - type: true, - src: true, - nonce: true, - noModule: true - }; - - function DOMEval( code, node, doc ) { - doc = doc || document; - - var i, val, - script = doc.createElement( "script" ); - - script.text = code; - if ( node ) { - for ( i in preservedScriptAttributes ) { - - // Support: Firefox 64+, Edge 18+ - // Some browsers don't support the "nonce" property on scripts. - // On the other hand, just using `getAttribute` is not enough as - // the `nonce` attribute is reset to an empty string whenever it - // becomes browsing-context connected. - // See https://github.com/whatwg/html/issues/2369 - // See https://html.spec.whatwg.org/#nonce-attributes - // The `node.getAttribute` check was added for the sake of - // `jQuery.globalEval` so that it can fake a nonce-containing node - // via an object. - val = node[ i ] || node.getAttribute && node.getAttribute( i ); - if ( val ) { - script.setAttribute( i, val ); - } - } - } - doc.head.appendChild( script ).parentNode.removeChild( script ); - } - - -function toType( obj ) { - if ( obj == null ) { - return obj + ""; - } - - // Support: Android <=2.3 only (functionish RegExp) - return typeof obj === "object" || typeof obj === "function" ? - class2type[ toString.call( obj ) ] || "object" : - typeof obj; -} -/* global Symbol */ -// Defining this global in .eslintrc.json would create a danger of using the global -// unguarded in another place, it seems safer to define global only for this module - - - -var - version = "3.5.1", - - // Define a local copy of jQuery - jQuery = function( selector, context ) { - - // The jQuery object is actually just the init constructor 'enhanced' - // Need init if jQuery is called (just allow error to be thrown if not included) - return new jQuery.fn.init( selector, context ); - }; - -jQuery.fn = jQuery.prototype = { - - // The current version of jQuery being used - jquery: version, - - constructor: jQuery, - - // The default length of a jQuery object is 0 - length: 0, - - toArray: function() { - return slice.call( this ); - }, - - // Get the Nth element in the matched element set OR - // Get the whole matched element set as a clean array - get: function( num ) { - - // Return all the elements in a clean array - if ( num == null ) { - return slice.call( this ); - } - - // Return just the one element from the set - return num < 0 ? this[ num + this.length ] : this[ num ]; - }, - - // Take an array of elements and push it onto the stack - // (returning the new matched element set) - pushStack: function( elems ) { - - // Build a new jQuery matched element set - var ret = jQuery.merge( this.constructor(), elems ); - - // Add the old object onto the stack (as a reference) - ret.prevObject = this; - - // Return the newly-formed element set - return ret; - }, - - // Execute a callback for every element in the matched set. - each: function( callback ) { - return jQuery.each( this, callback ); - }, - - map: function( callback ) { - return this.pushStack( jQuery.map( this, function( elem, i ) { - return callback.call( elem, i, elem ); - } ) ); - }, - - slice: function() { - return this.pushStack( slice.apply( this, arguments ) ); - }, - - first: function() { - return this.eq( 0 ); - }, - - last: function() { - return this.eq( -1 ); - }, - - even: function() { - return this.pushStack( jQuery.grep( this, function( _elem, i ) { - return ( i + 1 ) % 2; - } ) ); - }, - - odd: function() { - return this.pushStack( jQuery.grep( this, function( _elem, i ) { - return i % 2; - } ) ); - }, - - eq: function( i ) { - var len = this.length, - j = +i + ( i < 0 ? len : 0 ); - return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] ); - }, - - end: function() { - return this.prevObject || this.constructor(); - }, - - // For internal use only. - // Behaves like an Array's method, not like a jQuery method. - push: push, - sort: arr.sort, - splice: arr.splice -}; - -jQuery.extend = jQuery.fn.extend = function() { - var options, name, src, copy, copyIsArray, clone, - target = arguments[ 0 ] || {}, - i = 1, - length = arguments.length, - deep = false; - - // Handle a deep copy situation - if ( typeof target === "boolean" ) { - deep = target; - - // Skip the boolean and the target - target = arguments[ i ] || {}; - i++; - } - - // Handle case when target is a string or something (possible in deep copy) - if ( typeof target !== "object" && !isFunction( target ) ) { - target = {}; - } - - // Extend jQuery itself if only one argument is passed - if ( i === length ) { - target = this; - i--; - } - - for ( ; i < length; i++ ) { - - // Only deal with non-null/undefined values - if ( ( options = arguments[ i ] ) != null ) { - - // Extend the base object - for ( name in options ) { - copy = options[ name ]; - - // Prevent Object.prototype pollution - // Prevent never-ending loop - if ( name === "__proto__" || target === copy ) { - continue; - } - - // Recurse if we're merging plain objects or arrays - if ( deep && copy && ( jQuery.isPlainObject( copy ) || - ( copyIsArray = Array.isArray( copy ) ) ) ) { - src = target[ name ]; - - // Ensure proper type for the source value - if ( copyIsArray && !Array.isArray( src ) ) { - clone = []; - } else if ( !copyIsArray && !jQuery.isPlainObject( src ) ) { - clone = {}; - } else { - clone = src; - } - copyIsArray = false; - - // Never move original objects, clone them - target[ name ] = jQuery.extend( deep, clone, copy ); - - // Don't bring in undefined values - } else if ( copy !== undefined ) { - target[ name ] = copy; - } - } - } - } - - // Return the modified object - return target; -}; - -jQuery.extend( { - - // Unique for each copy of jQuery on the page - expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ), - - // Assume jQuery is ready without the ready module - isReady: true, - - error: function( msg ) { - throw new Error( msg ); - }, - - noop: function() {}, - - isPlainObject: function( obj ) { - var proto, Ctor; - - // Detect obvious negatives - // Use toString instead of jQuery.type to catch host objects - if ( !obj || toString.call( obj ) !== "[object Object]" ) { - return false; - } - - proto = getProto( obj ); - - // Objects with no prototype (e.g., `Object.create( null )`) are plain - if ( !proto ) { - return true; - } - - // Objects with prototype are plain iff they were constructed by a global Object function - Ctor = hasOwn.call( proto, "constructor" ) && proto.constructor; - return typeof Ctor === "function" && fnToString.call( Ctor ) === ObjectFunctionString; - }, - - isEmptyObject: function( obj ) { - var name; - - for ( name in obj ) { - return false; - } - return true; - }, - - // Evaluates a script in a provided context; falls back to the global one - // if not specified. - globalEval: function( code, options, doc ) { - DOMEval( code, { nonce: options && options.nonce }, doc ); - }, - - each: function( obj, callback ) { - var length, i = 0; - - if ( isArrayLike( obj ) ) { - length = obj.length; - for ( ; i < length; i++ ) { - if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { - break; - } - } - } else { - for ( i in obj ) { - if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { - break; - } - } - } - - return obj; - }, - - // results is for internal usage only - makeArray: function( arr, results ) { - var ret = results || []; - - if ( arr != null ) { - if ( isArrayLike( Object( arr ) ) ) { - jQuery.merge( ret, - typeof arr === "string" ? - [ arr ] : arr - ); - } else { - push.call( ret, arr ); - } - } - - return ret; - }, - - inArray: function( elem, arr, i ) { - return arr == null ? -1 : indexOf.call( arr, elem, i ); - }, - - // Support: Android <=4.0 only, PhantomJS 1 only - // push.apply(_, arraylike) throws on ancient WebKit - merge: function( first, second ) { - var len = +second.length, - j = 0, - i = first.length; - - for ( ; j < len; j++ ) { - first[ i++ ] = second[ j ]; - } - - first.length = i; - - return first; - }, - - grep: function( elems, callback, invert ) { - var callbackInverse, - matches = [], - i = 0, - length = elems.length, - callbackExpect = !invert; - - // Go through the array, only saving the items - // that pass the validator function - for ( ; i < length; i++ ) { - callbackInverse = !callback( elems[ i ], i ); - if ( callbackInverse !== callbackExpect ) { - matches.push( elems[ i ] ); - } - } - - return matches; - }, - - // arg is for internal usage only - map: function( elems, callback, arg ) { - var length, value, - i = 0, - ret = []; - - // Go through the array, translating each of the items to their new values - if ( isArrayLike( elems ) ) { - length = elems.length; - for ( ; i < length; i++ ) { - value = callback( elems[ i ], i, arg ); - - if ( value != null ) { - ret.push( value ); - } - } - - // Go through every key on the object, - } else { - for ( i in elems ) { - value = callback( elems[ i ], i, arg ); - - if ( value != null ) { - ret.push( value ); - } - } - } - - // Flatten any nested arrays - return flat( ret ); - }, - - // A global GUID counter for objects - guid: 1, - - // jQuery.support is not used in Core but other projects attach their - // properties to it so it needs to exist. - support: support -} ); - -if ( typeof Symbol === "function" ) { - jQuery.fn[ Symbol.iterator ] = arr[ Symbol.iterator ]; -} - -// Populate the class2type map -jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ), -function( _i, name ) { - class2type[ "[object " + name + "]" ] = name.toLowerCase(); -} ); - -function isArrayLike( obj ) { - - // Support: real iOS 8.2 only (not reproducible in simulator) - // `in` check used to prevent JIT error (gh-2145) - // hasOwn isn't used here due to false negatives - // regarding Nodelist length in IE - var length = !!obj && "length" in obj && obj.length, - type = toType( obj ); - - if ( isFunction( obj ) || isWindow( obj ) ) { - return false; - } - - return type === "array" || length === 0 || - typeof length === "number" && length > 0 && ( length - 1 ) in obj; -} -var Sizzle = -/*! - * Sizzle CSS Selector Engine v2.3.5 - * https://sizzlejs.com/ - * - * Copyright JS Foundation and other contributors - * Released under the MIT license - * https://js.foundation/ - * - * Date: 2020-03-14 - */ -( function( window ) { -var i, - support, - Expr, - getText, - isXML, - tokenize, - compile, - select, - outermostContext, - sortInput, - hasDuplicate, - - // Local document vars - setDocument, - document, - docElem, - documentIsHTML, - rbuggyQSA, - rbuggyMatches, - matches, - contains, - - // Instance-specific data - expando = "sizzle" + 1 * new Date(), - preferredDoc = window.document, - dirruns = 0, - done = 0, - classCache = createCache(), - tokenCache = createCache(), - compilerCache = createCache(), - nonnativeSelectorCache = createCache(), - sortOrder = function( a, b ) { - if ( a === b ) { - hasDuplicate = true; - } - return 0; - }, - - // Instance methods - hasOwn = ( {} ).hasOwnProperty, - arr = [], - pop = arr.pop, - pushNative = arr.push, - push = arr.push, - slice = arr.slice, - - // Use a stripped-down indexOf as it's faster than native - // https://jsperf.com/thor-indexof-vs-for/5 - indexOf = function( list, elem ) { - var i = 0, - len = list.length; - for ( ; i < len; i++ ) { - if ( list[ i ] === elem ) { - return i; - } - } - return -1; - }, - - booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|" + - "ismap|loop|multiple|open|readonly|required|scoped", - - // Regular expressions - - // http://www.w3.org/TR/css3-selectors/#whitespace - whitespace = "[\\x20\\t\\r\\n\\f]", - - // https://www.w3.org/TR/css-syntax-3/#ident-token-diagram - identifier = "(?:\\\\[\\da-fA-F]{1,6}" + whitespace + - "?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+", - - // Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors - attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace + - - // Operator (capture 2) - "*([*^$|!~]?=)" + whitespace + - - // "Attribute values must be CSS identifiers [capture 5] - // or strings [capture 3 or capture 4]" - "*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" + - whitespace + "*\\]", - - pseudos = ":(" + identifier + ")(?:\\((" + - - // To reduce the number of selectors needing tokenize in the preFilter, prefer arguments: - // 1. quoted (capture 3; capture 4 or capture 5) - "('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" + - - // 2. simple (capture 6) - "((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" + - - // 3. anything else (capture 2) - ".*" + - ")\\)|)", - - // Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter - rwhitespace = new RegExp( whitespace + "+", "g" ), - rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" + - whitespace + "+$", "g" ), - - rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ), - rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace + - "*" ), - rdescend = new RegExp( whitespace + "|>" ), - - rpseudo = new RegExp( pseudos ), - ridentifier = new RegExp( "^" + identifier + "$" ), - - matchExpr = { - "ID": new RegExp( "^#(" + identifier + ")" ), - "CLASS": new RegExp( "^\\.(" + identifier + ")" ), - "TAG": new RegExp( "^(" + identifier + "|[*])" ), - "ATTR": new RegExp( "^" + attributes ), - "PSEUDO": new RegExp( "^" + pseudos ), - "CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\\(" + - whitespace + "*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" + - whitespace + "*(\\d+)|))" + whitespace + "*\\)|)", "i" ), - "bool": new RegExp( "^(?:" + booleans + ")$", "i" ), - - // For use in libraries implementing .is() - // We use this for POS matching in `select` - "needsContext": new RegExp( "^" + whitespace + - "*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\\(" + whitespace + - "*((?:-\\d)?\\d*)" + whitespace + "*\\)|)(?=[^-]|$)", "i" ) - }, - - rhtml = /HTML$/i, - rinputs = /^(?:input|select|textarea|button)$/i, - rheader = /^h\d$/i, - - rnative = /^[^{]+\{\s*\[native \w/, - - // Easily-parseable/retrievable ID or TAG or CLASS selectors - rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/, - - rsibling = /[+~]/, - - // CSS escapes - // http://www.w3.org/TR/CSS21/syndata.html#escaped-characters - runescape = new RegExp( "\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\([^\\r\\n\\f])", "g" ), - funescape = function( escape, nonHex ) { - var high = "0x" + escape.slice( 1 ) - 0x10000; - - return nonHex ? - - // Strip the backslash prefix from a non-hex escape sequence - nonHex : - - // Replace a hexadecimal escape sequence with the encoded Unicode code point - // Support: IE <=11+ - // For values outside the Basic Multilingual Plane (BMP), manually construct a - // surrogate pair - high < 0 ? - String.fromCharCode( high + 0x10000 ) : - String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 ); - }, - - // CSS string/identifier serialization - // https://drafts.csswg.org/cssom/#common-serializing-idioms - rcssescape = /([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g, - fcssescape = function( ch, asCodePoint ) { - if ( asCodePoint ) { - - // U+0000 NULL becomes U+FFFD REPLACEMENT CHARACTER - if ( ch === "\0" ) { - return "\uFFFD"; - } - - // Control characters and (dependent upon position) numbers get escaped as code points - return ch.slice( 0, -1 ) + "\\" + - ch.charCodeAt( ch.length - 1 ).toString( 16 ) + " "; - } - - // Other potentially-special ASCII characters get backslash-escaped - return "\\" + ch; - }, - - // Used for iframes - // See setDocument() - // Removing the function wrapper causes a "Permission Denied" - // error in IE - unloadHandler = function() { - setDocument(); - }, - - inDisabledFieldset = addCombinator( - function( elem ) { - return elem.disabled === true && elem.nodeName.toLowerCase() === "fieldset"; - }, - { dir: "parentNode", next: "legend" } - ); - -// Optimize for push.apply( _, NodeList ) -try { - push.apply( - ( arr = slice.call( preferredDoc.childNodes ) ), - preferredDoc.childNodes - ); - - // Support: Android<4.0 - // Detect silently failing push.apply - // eslint-disable-next-line no-unused-expressions - arr[ preferredDoc.childNodes.length ].nodeType; -} catch ( e ) { - push = { apply: arr.length ? - - // Leverage slice if possible - function( target, els ) { - pushNative.apply( target, slice.call( els ) ); - } : - - // Support: IE<9 - // Otherwise append directly - function( target, els ) { - var j = target.length, - i = 0; - - // Can't trust NodeList.length - while ( ( target[ j++ ] = els[ i++ ] ) ) {} - target.length = j - 1; - } - }; -} - -function Sizzle( selector, context, results, seed ) { - var m, i, elem, nid, match, groups, newSelector, - newContext = context && context.ownerDocument, - - // nodeType defaults to 9, since context defaults to document - nodeType = context ? context.nodeType : 9; - - results = results || []; - - // Return early from calls with invalid selector or context - if ( typeof selector !== "string" || !selector || - nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) { - - return results; - } - - // Try to shortcut find operations (as opposed to filters) in HTML documents - if ( !seed ) { - setDocument( context ); - context = context || document; - - if ( documentIsHTML ) { - - // If the selector is sufficiently simple, try using a "get*By*" DOM method - // (excepting DocumentFragment context, where the methods don't exist) - if ( nodeType !== 11 && ( match = rquickExpr.exec( selector ) ) ) { - - // ID selector - if ( ( m = match[ 1 ] ) ) { - - // Document context - if ( nodeType === 9 ) { - if ( ( elem = context.getElementById( m ) ) ) { - - // Support: IE, Opera, Webkit - // TODO: identify versions - // getElementById can match elements by name instead of ID - if ( elem.id === m ) { - results.push( elem ); - return results; - } - } else { - return results; - } - - // Element context - } else { - - // Support: IE, Opera, Webkit - // TODO: identify versions - // getElementById can match elements by name instead of ID - if ( newContext && ( elem = newContext.getElementById( m ) ) && - contains( context, elem ) && - elem.id === m ) { - - results.push( elem ); - return results; - } - } - - // Type selector - } else if ( match[ 2 ] ) { - push.apply( results, context.getElementsByTagName( selector ) ); - return results; - - // Class selector - } else if ( ( m = match[ 3 ] ) && support.getElementsByClassName && - context.getElementsByClassName ) { - - push.apply( results, context.getElementsByClassName( m ) ); - return results; - } - } - - // Take advantage of querySelectorAll - if ( support.qsa && - !nonnativeSelectorCache[ selector + " " ] && - ( !rbuggyQSA || !rbuggyQSA.test( selector ) ) && - - // Support: IE 8 only - // Exclude object elements - ( nodeType !== 1 || context.nodeName.toLowerCase() !== "object" ) ) { - - newSelector = selector; - newContext = context; - - // qSA considers elements outside a scoping root when evaluating child or - // descendant combinators, which is not what we want. - // In such cases, we work around the behavior by prefixing every selector in the - // list with an ID selector referencing the scope context. - // The technique has to be used as well when a leading combinator is used - // as such selectors are not recognized by querySelectorAll. - // Thanks to Andrew Dupont for this technique. - if ( nodeType === 1 && - ( rdescend.test( selector ) || rcombinators.test( selector ) ) ) { - - // Expand context for sibling selectors - newContext = rsibling.test( selector ) && testContext( context.parentNode ) || - context; - - // We can use :scope instead of the ID hack if the browser - // supports it & if we're not changing the context. - if ( newContext !== context || !support.scope ) { - - // Capture the context ID, setting it first if necessary - if ( ( nid = context.getAttribute( "id" ) ) ) { - nid = nid.replace( rcssescape, fcssescape ); - } else { - context.setAttribute( "id", ( nid = expando ) ); - } - } - - // Prefix every selector in the list - groups = tokenize( selector ); - i = groups.length; - while ( i-- ) { - groups[ i ] = ( nid ? "#" + nid : ":scope" ) + " " + - toSelector( groups[ i ] ); - } - newSelector = groups.join( "," ); - } - - try { - push.apply( results, - newContext.querySelectorAll( newSelector ) - ); - return results; - } catch ( qsaError ) { - nonnativeSelectorCache( selector, true ); - } finally { - if ( nid === expando ) { - context.removeAttribute( "id" ); - } - } - } - } - } - - // All others - return select( selector.replace( rtrim, "$1" ), context, results, seed ); -} - -/** - * Create key-value caches of limited size - * @returns {function(string, object)} Returns the Object data after storing it on itself with - * property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength) - * deleting the oldest entry - */ -function createCache() { - var keys = []; - - function cache( key, value ) { - - // Use (key + " ") to avoid collision with native prototype properties (see Issue #157) - if ( keys.push( key + " " ) > Expr.cacheLength ) { - - // Only keep the most recent entries - delete cache[ keys.shift() ]; - } - return ( cache[ key + " " ] = value ); - } - return cache; -} - -/** - * Mark a function for special use by Sizzle - * @param {Function} fn The function to mark - */ -function markFunction( fn ) { - fn[ expando ] = true; - return fn; -} - -/** - * Support testing using an element - * @param {Function} fn Passed the created element and returns a boolean result - */ -function assert( fn ) { - var el = document.createElement( "fieldset" ); - - try { - return !!fn( el ); - } catch ( e ) { - return false; - } finally { - - // Remove from its parent by default - if ( el.parentNode ) { - el.parentNode.removeChild( el ); - } - - // release memory in IE - el = null; - } -} - -/** - * Adds the same handler for all of the specified attrs - * @param {String} attrs Pipe-separated list of attributes - * @param {Function} handler The method that will be applied - */ -function addHandle( attrs, handler ) { - var arr = attrs.split( "|" ), - i = arr.length; - - while ( i-- ) { - Expr.attrHandle[ arr[ i ] ] = handler; - } -} - -/** - * Checks document order of two siblings - * @param {Element} a - * @param {Element} b - * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b - */ -function siblingCheck( a, b ) { - var cur = b && a, - diff = cur && a.nodeType === 1 && b.nodeType === 1 && - a.sourceIndex - b.sourceIndex; - - // Use IE sourceIndex if available on both nodes - if ( diff ) { - return diff; - } - - // Check if b follows a - if ( cur ) { - while ( ( cur = cur.nextSibling ) ) { - if ( cur === b ) { - return -1; - } - } - } - - return a ? 1 : -1; -} - -/** - * Returns a function to use in pseudos for input types - * @param {String} type - */ -function createInputPseudo( type ) { - return function( elem ) { - var name = elem.nodeName.toLowerCase(); - return name === "input" && elem.type === type; - }; -} - -/** - * Returns a function to use in pseudos for buttons - * @param {String} type - */ -function createButtonPseudo( type ) { - return function( elem ) { - var name = elem.nodeName.toLowerCase(); - return ( name === "input" || name === "button" ) && elem.type === type; - }; -} - -/** - * Returns a function to use in pseudos for :enabled/:disabled - * @param {Boolean} disabled true for :disabled; false for :enabled - */ -function createDisabledPseudo( disabled ) { - - // Known :disabled false positives: fieldset[disabled] > legend:nth-of-type(n+2) :can-disable - return function( elem ) { - - // Only certain elements can match :enabled or :disabled - // https://html.spec.whatwg.org/multipage/scripting.html#selector-enabled - // https://html.spec.whatwg.org/multipage/scripting.html#selector-disabled - if ( "form" in elem ) { - - // Check for inherited disabledness on relevant non-disabled elements: - // * listed form-associated elements in a disabled fieldset - // https://html.spec.whatwg.org/multipage/forms.html#category-listed - // https://html.spec.whatwg.org/multipage/forms.html#concept-fe-disabled - // * option elements in a disabled optgroup - // https://html.spec.whatwg.org/multipage/forms.html#concept-option-disabled - // All such elements have a "form" property. - if ( elem.parentNode && elem.disabled === false ) { - - // Option elements defer to a parent optgroup if present - if ( "label" in elem ) { - if ( "label" in elem.parentNode ) { - return elem.parentNode.disabled === disabled; - } else { - return elem.disabled === disabled; - } - } - - // Support: IE 6 - 11 - // Use the isDisabled shortcut property to check for disabled fieldset ancestors - return elem.isDisabled === disabled || - - // Where there is no isDisabled, check manually - /* jshint -W018 */ - elem.isDisabled !== !disabled && - inDisabledFieldset( elem ) === disabled; - } - - return elem.disabled === disabled; - - // Try to winnow out elements that can't be disabled before trusting the disabled property. - // Some victims get caught in our net (label, legend, menu, track), but it shouldn't - // even exist on them, let alone have a boolean value. - } else if ( "label" in elem ) { - return elem.disabled === disabled; - } - - // Remaining elements are neither :enabled nor :disabled - return false; - }; -} - -/** - * Returns a function to use in pseudos for positionals - * @param {Function} fn - */ -function createPositionalPseudo( fn ) { - return markFunction( function( argument ) { - argument = +argument; - return markFunction( function( seed, matches ) { - var j, - matchIndexes = fn( [], seed.length, argument ), - i = matchIndexes.length; - - // Match elements found at the specified indexes - while ( i-- ) { - if ( seed[ ( j = matchIndexes[ i ] ) ] ) { - seed[ j ] = !( matches[ j ] = seed[ j ] ); - } - } - } ); - } ); -} - -/** - * Checks a node for validity as a Sizzle context - * @param {Element|Object=} context - * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value - */ -function testContext( context ) { - return context && typeof context.getElementsByTagName !== "undefined" && context; -} - -// Expose support vars for convenience -support = Sizzle.support = {}; - -/** - * Detects XML nodes - * @param {Element|Object} elem An element or a document - * @returns {Boolean} True iff elem is a non-HTML XML node - */ -isXML = Sizzle.isXML = function( elem ) { - var namespace = elem.namespaceURI, - docElem = ( elem.ownerDocument || elem ).documentElement; - - // Support: IE <=8 - // Assume HTML when documentElement doesn't yet exist, such as inside loading iframes - // https://bugs.jquery.com/ticket/4833 - return !rhtml.test( namespace || docElem && docElem.nodeName || "HTML" ); -}; - -/** - * Sets document-related variables once based on the current document - * @param {Element|Object} [doc] An element or document object to use to set the document - * @returns {Object} Returns the current document - */ -setDocument = Sizzle.setDocument = function( node ) { - var hasCompare, subWindow, - doc = node ? node.ownerDocument || node : preferredDoc; - - // Return early if doc is invalid or already selected - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( doc == document || doc.nodeType !== 9 || !doc.documentElement ) { - return document; - } - - // Update global variables - document = doc; - docElem = document.documentElement; - documentIsHTML = !isXML( document ); - - // Support: IE 9 - 11+, Edge 12 - 18+ - // Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936) - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( preferredDoc != document && - ( subWindow = document.defaultView ) && subWindow.top !== subWindow ) { - - // Support: IE 11, Edge - if ( subWindow.addEventListener ) { - subWindow.addEventListener( "unload", unloadHandler, false ); - - // Support: IE 9 - 10 only - } else if ( subWindow.attachEvent ) { - subWindow.attachEvent( "onunload", unloadHandler ); - } - } - - // Support: IE 8 - 11+, Edge 12 - 18+, Chrome <=16 - 25 only, Firefox <=3.6 - 31 only, - // Safari 4 - 5 only, Opera <=11.6 - 12.x only - // IE/Edge & older browsers don't support the :scope pseudo-class. - // Support: Safari 6.0 only - // Safari 6.0 supports :scope but it's an alias of :root there. - support.scope = assert( function( el ) { - docElem.appendChild( el ).appendChild( document.createElement( "div" ) ); - return typeof el.querySelectorAll !== "undefined" && - !el.querySelectorAll( ":scope fieldset div" ).length; - } ); - - /* Attributes - ---------------------------------------------------------------------- */ - - // Support: IE<8 - // Verify that getAttribute really returns attributes and not properties - // (excepting IE8 booleans) - support.attributes = assert( function( el ) { - el.className = "i"; - return !el.getAttribute( "className" ); - } ); - - /* getElement(s)By* - ---------------------------------------------------------------------- */ - - // Check if getElementsByTagName("*") returns only elements - support.getElementsByTagName = assert( function( el ) { - el.appendChild( document.createComment( "" ) ); - return !el.getElementsByTagName( "*" ).length; - } ); - - // Support: IE<9 - support.getElementsByClassName = rnative.test( document.getElementsByClassName ); - - // Support: IE<10 - // Check if getElementById returns elements by name - // The broken getElementById methods don't pick up programmatically-set names, - // so use a roundabout getElementsByName test - support.getById = assert( function( el ) { - docElem.appendChild( el ).id = expando; - return !document.getElementsByName || !document.getElementsByName( expando ).length; - } ); - - // ID filter and find - if ( support.getById ) { - Expr.filter[ "ID" ] = function( id ) { - var attrId = id.replace( runescape, funescape ); - return function( elem ) { - return elem.getAttribute( "id" ) === attrId; - }; - }; - Expr.find[ "ID" ] = function( id, context ) { - if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { - var elem = context.getElementById( id ); - return elem ? [ elem ] : []; - } - }; - } else { - Expr.filter[ "ID" ] = function( id ) { - var attrId = id.replace( runescape, funescape ); - return function( elem ) { - var node = typeof elem.getAttributeNode !== "undefined" && - elem.getAttributeNode( "id" ); - return node && node.value === attrId; - }; - }; - - // Support: IE 6 - 7 only - // getElementById is not reliable as a find shortcut - Expr.find[ "ID" ] = function( id, context ) { - if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { - var node, i, elems, - elem = context.getElementById( id ); - - if ( elem ) { - - // Verify the id attribute - node = elem.getAttributeNode( "id" ); - if ( node && node.value === id ) { - return [ elem ]; - } - - // Fall back on getElementsByName - elems = context.getElementsByName( id ); - i = 0; - while ( ( elem = elems[ i++ ] ) ) { - node = elem.getAttributeNode( "id" ); - if ( node && node.value === id ) { - return [ elem ]; - } - } - } - - return []; - } - }; - } - - // Tag - Expr.find[ "TAG" ] = support.getElementsByTagName ? - function( tag, context ) { - if ( typeof context.getElementsByTagName !== "undefined" ) { - return context.getElementsByTagName( tag ); - - // DocumentFragment nodes don't have gEBTN - } else if ( support.qsa ) { - return context.querySelectorAll( tag ); - } - } : - - function( tag, context ) { - var elem, - tmp = [], - i = 0, - - // By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too - results = context.getElementsByTagName( tag ); - - // Filter out possible comments - if ( tag === "*" ) { - while ( ( elem = results[ i++ ] ) ) { - if ( elem.nodeType === 1 ) { - tmp.push( elem ); - } - } - - return tmp; - } - return results; - }; - - // Class - Expr.find[ "CLASS" ] = support.getElementsByClassName && function( className, context ) { - if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) { - return context.getElementsByClassName( className ); - } - }; - - /* QSA/matchesSelector - ---------------------------------------------------------------------- */ - - // QSA and matchesSelector support - - // matchesSelector(:active) reports false when true (IE9/Opera 11.5) - rbuggyMatches = []; - - // qSa(:focus) reports false when true (Chrome 21) - // We allow this because of a bug in IE8/9 that throws an error - // whenever `document.activeElement` is accessed on an iframe - // So, we allow :focus to pass through QSA all the time to avoid the IE error - // See https://bugs.jquery.com/ticket/13378 - rbuggyQSA = []; - - if ( ( support.qsa = rnative.test( document.querySelectorAll ) ) ) { - - // Build QSA regex - // Regex strategy adopted from Diego Perini - assert( function( el ) { - - var input; - - // Select is set to empty string on purpose - // This is to test IE's treatment of not explicitly - // setting a boolean content attribute, - // since its presence should be enough - // https://bugs.jquery.com/ticket/12359 - docElem.appendChild( el ).innerHTML = "" + - ""; - - // Support: IE8, Opera 11-12.16 - // Nothing should be selected when empty strings follow ^= or $= or *= - // The test attribute must be unknown in Opera but "safe" for WinRT - // https://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section - if ( el.querySelectorAll( "[msallowcapture^='']" ).length ) { - rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" ); - } - - // Support: IE8 - // Boolean attributes and "value" are not treated correctly - if ( !el.querySelectorAll( "[selected]" ).length ) { - rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" ); - } - - // Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+ - if ( !el.querySelectorAll( "[id~=" + expando + "-]" ).length ) { - rbuggyQSA.push( "~=" ); - } - - // Support: IE 11+, Edge 15 - 18+ - // IE 11/Edge don't find elements on a `[name='']` query in some cases. - // Adding a temporary attribute to the document before the selection works - // around the issue. - // Interestingly, IE 10 & older don't seem to have the issue. - input = document.createElement( "input" ); - input.setAttribute( "name", "" ); - el.appendChild( input ); - if ( !el.querySelectorAll( "[name='']" ).length ) { - rbuggyQSA.push( "\\[" + whitespace + "*name" + whitespace + "*=" + - whitespace + "*(?:''|\"\")" ); - } - - // Webkit/Opera - :checked should return selected option elements - // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked - // IE8 throws error here and will not see later tests - if ( !el.querySelectorAll( ":checked" ).length ) { - rbuggyQSA.push( ":checked" ); - } - - // Support: Safari 8+, iOS 8+ - // https://bugs.webkit.org/show_bug.cgi?id=136851 - // In-page `selector#id sibling-combinator selector` fails - if ( !el.querySelectorAll( "a#" + expando + "+*" ).length ) { - rbuggyQSA.push( ".#.+[+~]" ); - } - - // Support: Firefox <=3.6 - 5 only - // Old Firefox doesn't throw on a badly-escaped identifier. - el.querySelectorAll( "\\\f" ); - rbuggyQSA.push( "[\\r\\n\\f]" ); - } ); - - assert( function( el ) { - el.innerHTML = "" + - ""; - - // Support: Windows 8 Native Apps - // The type and name attributes are restricted during .innerHTML assignment - var input = document.createElement( "input" ); - input.setAttribute( "type", "hidden" ); - el.appendChild( input ).setAttribute( "name", "D" ); - - // Support: IE8 - // Enforce case-sensitivity of name attribute - if ( el.querySelectorAll( "[name=d]" ).length ) { - rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" ); - } - - // FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled) - // IE8 throws error here and will not see later tests - if ( el.querySelectorAll( ":enabled" ).length !== 2 ) { - rbuggyQSA.push( ":enabled", ":disabled" ); - } - - // Support: IE9-11+ - // IE's :disabled selector does not pick up the children of disabled fieldsets - docElem.appendChild( el ).disabled = true; - if ( el.querySelectorAll( ":disabled" ).length !== 2 ) { - rbuggyQSA.push( ":enabled", ":disabled" ); - } - - // Support: Opera 10 - 11 only - // Opera 10-11 does not throw on post-comma invalid pseudos - el.querySelectorAll( "*,:x" ); - rbuggyQSA.push( ",.*:" ); - } ); - } - - if ( ( support.matchesSelector = rnative.test( ( matches = docElem.matches || - docElem.webkitMatchesSelector || - docElem.mozMatchesSelector || - docElem.oMatchesSelector || - docElem.msMatchesSelector ) ) ) ) { - - assert( function( el ) { - - // Check to see if it's possible to do matchesSelector - // on a disconnected node (IE 9) - support.disconnectedMatch = matches.call( el, "*" ); - - // This should fail with an exception - // Gecko does not error, returns false instead - matches.call( el, "[s!='']:x" ); - rbuggyMatches.push( "!=", pseudos ); - } ); - } - - rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join( "|" ) ); - rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join( "|" ) ); - - /* Contains - ---------------------------------------------------------------------- */ - hasCompare = rnative.test( docElem.compareDocumentPosition ); - - // Element contains another - // Purposefully self-exclusive - // As in, an element does not contain itself - contains = hasCompare || rnative.test( docElem.contains ) ? - function( a, b ) { - var adown = a.nodeType === 9 ? a.documentElement : a, - bup = b && b.parentNode; - return a === bup || !!( bup && bup.nodeType === 1 && ( - adown.contains ? - adown.contains( bup ) : - a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16 - ) ); - } : - function( a, b ) { - if ( b ) { - while ( ( b = b.parentNode ) ) { - if ( b === a ) { - return true; - } - } - } - return false; - }; - - /* Sorting - ---------------------------------------------------------------------- */ - - // Document order sorting - sortOrder = hasCompare ? - function( a, b ) { - - // Flag for duplicate removal - if ( a === b ) { - hasDuplicate = true; - return 0; - } - - // Sort on method existence if only one input has compareDocumentPosition - var compare = !a.compareDocumentPosition - !b.compareDocumentPosition; - if ( compare ) { - return compare; - } - - // Calculate position if both inputs belong to the same document - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - compare = ( a.ownerDocument || a ) == ( b.ownerDocument || b ) ? - a.compareDocumentPosition( b ) : - - // Otherwise we know they are disconnected - 1; - - // Disconnected nodes - if ( compare & 1 || - ( !support.sortDetached && b.compareDocumentPosition( a ) === compare ) ) { - - // Choose the first element that is related to our preferred document - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( a == document || a.ownerDocument == preferredDoc && - contains( preferredDoc, a ) ) { - return -1; - } - - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( b == document || b.ownerDocument == preferredDoc && - contains( preferredDoc, b ) ) { - return 1; - } - - // Maintain original order - return sortInput ? - ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : - 0; - } - - return compare & 4 ? -1 : 1; - } : - function( a, b ) { - - // Exit early if the nodes are identical - if ( a === b ) { - hasDuplicate = true; - return 0; - } - - var cur, - i = 0, - aup = a.parentNode, - bup = b.parentNode, - ap = [ a ], - bp = [ b ]; - - // Parentless nodes are either documents or disconnected - if ( !aup || !bup ) { - - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - /* eslint-disable eqeqeq */ - return a == document ? -1 : - b == document ? 1 : - /* eslint-enable eqeqeq */ - aup ? -1 : - bup ? 1 : - sortInput ? - ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : - 0; - - // If the nodes are siblings, we can do a quick check - } else if ( aup === bup ) { - return siblingCheck( a, b ); - } - - // Otherwise we need full lists of their ancestors for comparison - cur = a; - while ( ( cur = cur.parentNode ) ) { - ap.unshift( cur ); - } - cur = b; - while ( ( cur = cur.parentNode ) ) { - bp.unshift( cur ); - } - - // Walk down the tree looking for a discrepancy - while ( ap[ i ] === bp[ i ] ) { - i++; - } - - return i ? - - // Do a sibling check if the nodes have a common ancestor - siblingCheck( ap[ i ], bp[ i ] ) : - - // Otherwise nodes in our document sort first - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - /* eslint-disable eqeqeq */ - ap[ i ] == preferredDoc ? -1 : - bp[ i ] == preferredDoc ? 1 : - /* eslint-enable eqeqeq */ - 0; - }; - - return document; -}; - -Sizzle.matches = function( expr, elements ) { - return Sizzle( expr, null, null, elements ); -}; - -Sizzle.matchesSelector = function( elem, expr ) { - setDocument( elem ); - - if ( support.matchesSelector && documentIsHTML && - !nonnativeSelectorCache[ expr + " " ] && - ( !rbuggyMatches || !rbuggyMatches.test( expr ) ) && - ( !rbuggyQSA || !rbuggyQSA.test( expr ) ) ) { - - try { - var ret = matches.call( elem, expr ); - - // IE 9's matchesSelector returns false on disconnected nodes - if ( ret || support.disconnectedMatch || - - // As well, disconnected nodes are said to be in a document - // fragment in IE 9 - elem.document && elem.document.nodeType !== 11 ) { - return ret; - } - } catch ( e ) { - nonnativeSelectorCache( expr, true ); - } - } - - return Sizzle( expr, document, null, [ elem ] ).length > 0; -}; - -Sizzle.contains = function( context, elem ) { - - // Set document vars if needed - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( ( context.ownerDocument || context ) != document ) { - setDocument( context ); - } - return contains( context, elem ); -}; - -Sizzle.attr = function( elem, name ) { - - // Set document vars if needed - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( ( elem.ownerDocument || elem ) != document ) { - setDocument( elem ); - } - - var fn = Expr.attrHandle[ name.toLowerCase() ], - - // Don't get fooled by Object.prototype properties (jQuery #13807) - val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ? - fn( elem, name, !documentIsHTML ) : - undefined; - - return val !== undefined ? - val : - support.attributes || !documentIsHTML ? - elem.getAttribute( name ) : - ( val = elem.getAttributeNode( name ) ) && val.specified ? - val.value : - null; -}; - -Sizzle.escape = function( sel ) { - return ( sel + "" ).replace( rcssescape, fcssescape ); -}; - -Sizzle.error = function( msg ) { - throw new Error( "Syntax error, unrecognized expression: " + msg ); -}; - -/** - * Document sorting and removing duplicates - * @param {ArrayLike} results - */ -Sizzle.uniqueSort = function( results ) { - var elem, - duplicates = [], - j = 0, - i = 0; - - // Unless we *know* we can detect duplicates, assume their presence - hasDuplicate = !support.detectDuplicates; - sortInput = !support.sortStable && results.slice( 0 ); - results.sort( sortOrder ); - - if ( hasDuplicate ) { - while ( ( elem = results[ i++ ] ) ) { - if ( elem === results[ i ] ) { - j = duplicates.push( i ); - } - } - while ( j-- ) { - results.splice( duplicates[ j ], 1 ); - } - } - - // Clear input after sorting to release objects - // See https://github.com/jquery/sizzle/pull/225 - sortInput = null; - - return results; -}; - -/** - * Utility function for retrieving the text value of an array of DOM nodes - * @param {Array|Element} elem - */ -getText = Sizzle.getText = function( elem ) { - var node, - ret = "", - i = 0, - nodeType = elem.nodeType; - - if ( !nodeType ) { - - // If no nodeType, this is expected to be an array - while ( ( node = elem[ i++ ] ) ) { - - // Do not traverse comment nodes - ret += getText( node ); - } - } else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) { - - // Use textContent for elements - // innerText usage removed for consistency of new lines (jQuery #11153) - if ( typeof elem.textContent === "string" ) { - return elem.textContent; - } else { - - // Traverse its children - for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { - ret += getText( elem ); - } - } - } else if ( nodeType === 3 || nodeType === 4 ) { - return elem.nodeValue; - } - - // Do not include comment or processing instruction nodes - - return ret; -}; - -Expr = Sizzle.selectors = { - - // Can be adjusted by the user - cacheLength: 50, - - createPseudo: markFunction, - - match: matchExpr, - - attrHandle: {}, - - find: {}, - - relative: { - ">": { dir: "parentNode", first: true }, - " ": { dir: "parentNode" }, - "+": { dir: "previousSibling", first: true }, - "~": { dir: "previousSibling" } - }, - - preFilter: { - "ATTR": function( match ) { - match[ 1 ] = match[ 1 ].replace( runescape, funescape ); - - // Move the given value to match[3] whether quoted or unquoted - match[ 3 ] = ( match[ 3 ] || match[ 4 ] || - match[ 5 ] || "" ).replace( runescape, funescape ); - - if ( match[ 2 ] === "~=" ) { - match[ 3 ] = " " + match[ 3 ] + " "; - } - - return match.slice( 0, 4 ); - }, - - "CHILD": function( match ) { - - /* matches from matchExpr["CHILD"] - 1 type (only|nth|...) - 2 what (child|of-type) - 3 argument (even|odd|\d*|\d*n([+-]\d+)?|...) - 4 xn-component of xn+y argument ([+-]?\d*n|) - 5 sign of xn-component - 6 x of xn-component - 7 sign of y-component - 8 y of y-component - */ - match[ 1 ] = match[ 1 ].toLowerCase(); - - if ( match[ 1 ].slice( 0, 3 ) === "nth" ) { - - // nth-* requires argument - if ( !match[ 3 ] ) { - Sizzle.error( match[ 0 ] ); - } - - // numeric x and y parameters for Expr.filter.CHILD - // remember that false/true cast respectively to 0/1 - match[ 4 ] = +( match[ 4 ] ? - match[ 5 ] + ( match[ 6 ] || 1 ) : - 2 * ( match[ 3 ] === "even" || match[ 3 ] === "odd" ) ); - match[ 5 ] = +( ( match[ 7 ] + match[ 8 ] ) || match[ 3 ] === "odd" ); - - // other types prohibit arguments - } else if ( match[ 3 ] ) { - Sizzle.error( match[ 0 ] ); - } - - return match; - }, - - "PSEUDO": function( match ) { - var excess, - unquoted = !match[ 6 ] && match[ 2 ]; - - if ( matchExpr[ "CHILD" ].test( match[ 0 ] ) ) { - return null; - } - - // Accept quoted arguments as-is - if ( match[ 3 ] ) { - match[ 2 ] = match[ 4 ] || match[ 5 ] || ""; - - // Strip excess characters from unquoted arguments - } else if ( unquoted && rpseudo.test( unquoted ) && - - // Get excess from tokenize (recursively) - ( excess = tokenize( unquoted, true ) ) && - - // advance to the next closing parenthesis - ( excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length ) ) { - - // excess is a negative index - match[ 0 ] = match[ 0 ].slice( 0, excess ); - match[ 2 ] = unquoted.slice( 0, excess ); - } - - // Return only captures needed by the pseudo filter method (type and argument) - return match.slice( 0, 3 ); - } - }, - - filter: { - - "TAG": function( nodeNameSelector ) { - var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase(); - return nodeNameSelector === "*" ? - function() { - return true; - } : - function( elem ) { - return elem.nodeName && elem.nodeName.toLowerCase() === nodeName; - }; - }, - - "CLASS": function( className ) { - var pattern = classCache[ className + " " ]; - - return pattern || - ( pattern = new RegExp( "(^|" + whitespace + - ")" + className + "(" + whitespace + "|$)" ) ) && classCache( - className, function( elem ) { - return pattern.test( - typeof elem.className === "string" && elem.className || - typeof elem.getAttribute !== "undefined" && - elem.getAttribute( "class" ) || - "" - ); - } ); - }, - - "ATTR": function( name, operator, check ) { - return function( elem ) { - var result = Sizzle.attr( elem, name ); - - if ( result == null ) { - return operator === "!="; - } - if ( !operator ) { - return true; - } - - result += ""; - - /* eslint-disable max-len */ - - return operator === "=" ? result === check : - operator === "!=" ? result !== check : - operator === "^=" ? check && result.indexOf( check ) === 0 : - operator === "*=" ? check && result.indexOf( check ) > -1 : - operator === "$=" ? check && result.slice( -check.length ) === check : - operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 : - operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" : - false; - /* eslint-enable max-len */ - - }; - }, - - "CHILD": function( type, what, _argument, first, last ) { - var simple = type.slice( 0, 3 ) !== "nth", - forward = type.slice( -4 ) !== "last", - ofType = what === "of-type"; - - return first === 1 && last === 0 ? - - // Shortcut for :nth-*(n) - function( elem ) { - return !!elem.parentNode; - } : - - function( elem, _context, xml ) { - var cache, uniqueCache, outerCache, node, nodeIndex, start, - dir = simple !== forward ? "nextSibling" : "previousSibling", - parent = elem.parentNode, - name = ofType && elem.nodeName.toLowerCase(), - useCache = !xml && !ofType, - diff = false; - - if ( parent ) { - - // :(first|last|only)-(child|of-type) - if ( simple ) { - while ( dir ) { - node = elem; - while ( ( node = node[ dir ] ) ) { - if ( ofType ? - node.nodeName.toLowerCase() === name : - node.nodeType === 1 ) { - - return false; - } - } - - // Reverse direction for :only-* (if we haven't yet done so) - start = dir = type === "only" && !start && "nextSibling"; - } - return true; - } - - start = [ forward ? parent.firstChild : parent.lastChild ]; - - // non-xml :nth-child(...) stores cache data on `parent` - if ( forward && useCache ) { - - // Seek `elem` from a previously-cached index - - // ...in a gzip-friendly way - node = parent; - outerCache = node[ expando ] || ( node[ expando ] = {} ); - - // Support: IE <9 only - // Defend against cloned attroperties (jQuery gh-1709) - uniqueCache = outerCache[ node.uniqueID ] || - ( outerCache[ node.uniqueID ] = {} ); - - cache = uniqueCache[ type ] || []; - nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; - diff = nodeIndex && cache[ 2 ]; - node = nodeIndex && parent.childNodes[ nodeIndex ]; - - while ( ( node = ++nodeIndex && node && node[ dir ] || - - // Fallback to seeking `elem` from the start - ( diff = nodeIndex = 0 ) || start.pop() ) ) { - - // When found, cache indexes on `parent` and break - if ( node.nodeType === 1 && ++diff && node === elem ) { - uniqueCache[ type ] = [ dirruns, nodeIndex, diff ]; - break; - } - } - - } else { - - // Use previously-cached element index if available - if ( useCache ) { - - // ...in a gzip-friendly way - node = elem; - outerCache = node[ expando ] || ( node[ expando ] = {} ); - - // Support: IE <9 only - // Defend against cloned attroperties (jQuery gh-1709) - uniqueCache = outerCache[ node.uniqueID ] || - ( outerCache[ node.uniqueID ] = {} ); - - cache = uniqueCache[ type ] || []; - nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; - diff = nodeIndex; - } - - // xml :nth-child(...) - // or :nth-last-child(...) or :nth(-last)?-of-type(...) - if ( diff === false ) { - - // Use the same loop as above to seek `elem` from the start - while ( ( node = ++nodeIndex && node && node[ dir ] || - ( diff = nodeIndex = 0 ) || start.pop() ) ) { - - if ( ( ofType ? - node.nodeName.toLowerCase() === name : - node.nodeType === 1 ) && - ++diff ) { - - // Cache the index of each encountered element - if ( useCache ) { - outerCache = node[ expando ] || - ( node[ expando ] = {} ); - - // Support: IE <9 only - // Defend against cloned attroperties (jQuery gh-1709) - uniqueCache = outerCache[ node.uniqueID ] || - ( outerCache[ node.uniqueID ] = {} ); - - uniqueCache[ type ] = [ dirruns, diff ]; - } - - if ( node === elem ) { - break; - } - } - } - } - } - - // Incorporate the offset, then check against cycle size - diff -= last; - return diff === first || ( diff % first === 0 && diff / first >= 0 ); - } - }; - }, - - "PSEUDO": function( pseudo, argument ) { - - // pseudo-class names are case-insensitive - // http://www.w3.org/TR/selectors/#pseudo-classes - // Prioritize by case sensitivity in case custom pseudos are added with uppercase letters - // Remember that setFilters inherits from pseudos - var args, - fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] || - Sizzle.error( "unsupported pseudo: " + pseudo ); - - // The user may use createPseudo to indicate that - // arguments are needed to create the filter function - // just as Sizzle does - if ( fn[ expando ] ) { - return fn( argument ); - } - - // But maintain support for old signatures - if ( fn.length > 1 ) { - args = [ pseudo, pseudo, "", argument ]; - return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ? - markFunction( function( seed, matches ) { - var idx, - matched = fn( seed, argument ), - i = matched.length; - while ( i-- ) { - idx = indexOf( seed, matched[ i ] ); - seed[ idx ] = !( matches[ idx ] = matched[ i ] ); - } - } ) : - function( elem ) { - return fn( elem, 0, args ); - }; - } - - return fn; - } - }, - - pseudos: { - - // Potentially complex pseudos - "not": markFunction( function( selector ) { - - // Trim the selector passed to compile - // to avoid treating leading and trailing - // spaces as combinators - var input = [], - results = [], - matcher = compile( selector.replace( rtrim, "$1" ) ); - - return matcher[ expando ] ? - markFunction( function( seed, matches, _context, xml ) { - var elem, - unmatched = matcher( seed, null, xml, [] ), - i = seed.length; - - // Match elements unmatched by `matcher` - while ( i-- ) { - if ( ( elem = unmatched[ i ] ) ) { - seed[ i ] = !( matches[ i ] = elem ); - } - } - } ) : - function( elem, _context, xml ) { - input[ 0 ] = elem; - matcher( input, null, xml, results ); - - // Don't keep the element (issue #299) - input[ 0 ] = null; - return !results.pop(); - }; - } ), - - "has": markFunction( function( selector ) { - return function( elem ) { - return Sizzle( selector, elem ).length > 0; - }; - } ), - - "contains": markFunction( function( text ) { - text = text.replace( runescape, funescape ); - return function( elem ) { - return ( elem.textContent || getText( elem ) ).indexOf( text ) > -1; - }; - } ), - - // "Whether an element is represented by a :lang() selector - // is based solely on the element's language value - // being equal to the identifier C, - // or beginning with the identifier C immediately followed by "-". - // The matching of C against the element's language value is performed case-insensitively. - // The identifier C does not have to be a valid language name." - // http://www.w3.org/TR/selectors/#lang-pseudo - "lang": markFunction( function( lang ) { - - // lang value must be a valid identifier - if ( !ridentifier.test( lang || "" ) ) { - Sizzle.error( "unsupported lang: " + lang ); - } - lang = lang.replace( runescape, funescape ).toLowerCase(); - return function( elem ) { - var elemLang; - do { - if ( ( elemLang = documentIsHTML ? - elem.lang : - elem.getAttribute( "xml:lang" ) || elem.getAttribute( "lang" ) ) ) { - - elemLang = elemLang.toLowerCase(); - return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0; - } - } while ( ( elem = elem.parentNode ) && elem.nodeType === 1 ); - return false; - }; - } ), - - // Miscellaneous - "target": function( elem ) { - var hash = window.location && window.location.hash; - return hash && hash.slice( 1 ) === elem.id; - }, - - "root": function( elem ) { - return elem === docElem; - }, - - "focus": function( elem ) { - return elem === document.activeElement && - ( !document.hasFocus || document.hasFocus() ) && - !!( elem.type || elem.href || ~elem.tabIndex ); - }, - - // Boolean properties - "enabled": createDisabledPseudo( false ), - "disabled": createDisabledPseudo( true ), - - "checked": function( elem ) { - - // In CSS3, :checked should return both checked and selected elements - // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked - var nodeName = elem.nodeName.toLowerCase(); - return ( nodeName === "input" && !!elem.checked ) || - ( nodeName === "option" && !!elem.selected ); - }, - - "selected": function( elem ) { - - // Accessing this property makes selected-by-default - // options in Safari work properly - if ( elem.parentNode ) { - // eslint-disable-next-line no-unused-expressions - elem.parentNode.selectedIndex; - } - - return elem.selected === true; - }, - - // Contents - "empty": function( elem ) { - - // http://www.w3.org/TR/selectors/#empty-pseudo - // :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5), - // but not by others (comment: 8; processing instruction: 7; etc.) - // nodeType < 6 works because attributes (2) do not appear as children - for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { - if ( elem.nodeType < 6 ) { - return false; - } - } - return true; - }, - - "parent": function( elem ) { - return !Expr.pseudos[ "empty" ]( elem ); - }, - - // Element/input types - "header": function( elem ) { - return rheader.test( elem.nodeName ); - }, - - "input": function( elem ) { - return rinputs.test( elem.nodeName ); - }, - - "button": function( elem ) { - var name = elem.nodeName.toLowerCase(); - return name === "input" && elem.type === "button" || name === "button"; - }, - - "text": function( elem ) { - var attr; - return elem.nodeName.toLowerCase() === "input" && - elem.type === "text" && - - // Support: IE<8 - // New HTML5 attribute values (e.g., "search") appear with elem.type === "text" - ( ( attr = elem.getAttribute( "type" ) ) == null || - attr.toLowerCase() === "text" ); - }, - - // Position-in-collection - "first": createPositionalPseudo( function() { - return [ 0 ]; - } ), - - "last": createPositionalPseudo( function( _matchIndexes, length ) { - return [ length - 1 ]; - } ), - - "eq": createPositionalPseudo( function( _matchIndexes, length, argument ) { - return [ argument < 0 ? argument + length : argument ]; - } ), - - "even": createPositionalPseudo( function( matchIndexes, length ) { - var i = 0; - for ( ; i < length; i += 2 ) { - matchIndexes.push( i ); - } - return matchIndexes; - } ), - - "odd": createPositionalPseudo( function( matchIndexes, length ) { - var i = 1; - for ( ; i < length; i += 2 ) { - matchIndexes.push( i ); - } - return matchIndexes; - } ), - - "lt": createPositionalPseudo( function( matchIndexes, length, argument ) { - var i = argument < 0 ? - argument + length : - argument > length ? - length : - argument; - for ( ; --i >= 0; ) { - matchIndexes.push( i ); - } - return matchIndexes; - } ), - - "gt": createPositionalPseudo( function( matchIndexes, length, argument ) { - var i = argument < 0 ? argument + length : argument; - for ( ; ++i < length; ) { - matchIndexes.push( i ); - } - return matchIndexes; - } ) - } -}; - -Expr.pseudos[ "nth" ] = Expr.pseudos[ "eq" ]; - -// Add button/input type pseudos -for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) { - Expr.pseudos[ i ] = createInputPseudo( i ); -} -for ( i in { submit: true, reset: true } ) { - Expr.pseudos[ i ] = createButtonPseudo( i ); -} - -// Easy API for creating new setFilters -function setFilters() {} -setFilters.prototype = Expr.filters = Expr.pseudos; -Expr.setFilters = new setFilters(); - -tokenize = Sizzle.tokenize = function( selector, parseOnly ) { - var matched, match, tokens, type, - soFar, groups, preFilters, - cached = tokenCache[ selector + " " ]; - - if ( cached ) { - return parseOnly ? 0 : cached.slice( 0 ); - } - - soFar = selector; - groups = []; - preFilters = Expr.preFilter; - - while ( soFar ) { - - // Comma and first run - if ( !matched || ( match = rcomma.exec( soFar ) ) ) { - if ( match ) { - - // Don't consume trailing commas as valid - soFar = soFar.slice( match[ 0 ].length ) || soFar; - } - groups.push( ( tokens = [] ) ); - } - - matched = false; - - // Combinators - if ( ( match = rcombinators.exec( soFar ) ) ) { - matched = match.shift(); - tokens.push( { - value: matched, - - // Cast descendant combinators to space - type: match[ 0 ].replace( rtrim, " " ) - } ); - soFar = soFar.slice( matched.length ); - } - - // Filters - for ( type in Expr.filter ) { - if ( ( match = matchExpr[ type ].exec( soFar ) ) && ( !preFilters[ type ] || - ( match = preFilters[ type ]( match ) ) ) ) { - matched = match.shift(); - tokens.push( { - value: matched, - type: type, - matches: match - } ); - soFar = soFar.slice( matched.length ); - } - } - - if ( !matched ) { - break; - } - } - - // Return the length of the invalid excess - // if we're just parsing - // Otherwise, throw an error or return tokens - return parseOnly ? - soFar.length : - soFar ? - Sizzle.error( selector ) : - - // Cache the tokens - tokenCache( selector, groups ).slice( 0 ); -}; - -function toSelector( tokens ) { - var i = 0, - len = tokens.length, - selector = ""; - for ( ; i < len; i++ ) { - selector += tokens[ i ].value; - } - return selector; -} - -function addCombinator( matcher, combinator, base ) { - var dir = combinator.dir, - skip = combinator.next, - key = skip || dir, - checkNonElements = base && key === "parentNode", - doneName = done++; - - return combinator.first ? - - // Check against closest ancestor/preceding element - function( elem, context, xml ) { - while ( ( elem = elem[ dir ] ) ) { - if ( elem.nodeType === 1 || checkNonElements ) { - return matcher( elem, context, xml ); - } - } - return false; - } : - - // Check against all ancestor/preceding elements - function( elem, context, xml ) { - var oldCache, uniqueCache, outerCache, - newCache = [ dirruns, doneName ]; - - // We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching - if ( xml ) { - while ( ( elem = elem[ dir ] ) ) { - if ( elem.nodeType === 1 || checkNonElements ) { - if ( matcher( elem, context, xml ) ) { - return true; - } - } - } - } else { - while ( ( elem = elem[ dir ] ) ) { - if ( elem.nodeType === 1 || checkNonElements ) { - outerCache = elem[ expando ] || ( elem[ expando ] = {} ); - - // Support: IE <9 only - // Defend against cloned attroperties (jQuery gh-1709) - uniqueCache = outerCache[ elem.uniqueID ] || - ( outerCache[ elem.uniqueID ] = {} ); - - if ( skip && skip === elem.nodeName.toLowerCase() ) { - elem = elem[ dir ] || elem; - } else if ( ( oldCache = uniqueCache[ key ] ) && - oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) { - - // Assign to newCache so results back-propagate to previous elements - return ( newCache[ 2 ] = oldCache[ 2 ] ); - } else { - - // Reuse newcache so results back-propagate to previous elements - uniqueCache[ key ] = newCache; - - // A match means we're done; a fail means we have to keep checking - if ( ( newCache[ 2 ] = matcher( elem, context, xml ) ) ) { - return true; - } - } - } - } - } - return false; - }; -} - -function elementMatcher( matchers ) { - return matchers.length > 1 ? - function( elem, context, xml ) { - var i = matchers.length; - while ( i-- ) { - if ( !matchers[ i ]( elem, context, xml ) ) { - return false; - } - } - return true; - } : - matchers[ 0 ]; -} - -function multipleContexts( selector, contexts, results ) { - var i = 0, - len = contexts.length; - for ( ; i < len; i++ ) { - Sizzle( selector, contexts[ i ], results ); - } - return results; -} - -function condense( unmatched, map, filter, context, xml ) { - var elem, - newUnmatched = [], - i = 0, - len = unmatched.length, - mapped = map != null; - - for ( ; i < len; i++ ) { - if ( ( elem = unmatched[ i ] ) ) { - if ( !filter || filter( elem, context, xml ) ) { - newUnmatched.push( elem ); - if ( mapped ) { - map.push( i ); - } - } - } - } - - return newUnmatched; -} - -function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) { - if ( postFilter && !postFilter[ expando ] ) { - postFilter = setMatcher( postFilter ); - } - if ( postFinder && !postFinder[ expando ] ) { - postFinder = setMatcher( postFinder, postSelector ); - } - return markFunction( function( seed, results, context, xml ) { - var temp, i, elem, - preMap = [], - postMap = [], - preexisting = results.length, - - // Get initial elements from seed or context - elems = seed || multipleContexts( - selector || "*", - context.nodeType ? [ context ] : context, - [] - ), - - // Prefilter to get matcher input, preserving a map for seed-results synchronization - matcherIn = preFilter && ( seed || !selector ) ? - condense( elems, preMap, preFilter, context, xml ) : - elems, - - matcherOut = matcher ? - - // If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results, - postFinder || ( seed ? preFilter : preexisting || postFilter ) ? - - // ...intermediate processing is necessary - [] : - - // ...otherwise use results directly - results : - matcherIn; - - // Find primary matches - if ( matcher ) { - matcher( matcherIn, matcherOut, context, xml ); - } - - // Apply postFilter - if ( postFilter ) { - temp = condense( matcherOut, postMap ); - postFilter( temp, [], context, xml ); - - // Un-match failing elements by moving them back to matcherIn - i = temp.length; - while ( i-- ) { - if ( ( elem = temp[ i ] ) ) { - matcherOut[ postMap[ i ] ] = !( matcherIn[ postMap[ i ] ] = elem ); - } - } - } - - if ( seed ) { - if ( postFinder || preFilter ) { - if ( postFinder ) { - - // Get the final matcherOut by condensing this intermediate into postFinder contexts - temp = []; - i = matcherOut.length; - while ( i-- ) { - if ( ( elem = matcherOut[ i ] ) ) { - - // Restore matcherIn since elem is not yet a final match - temp.push( ( matcherIn[ i ] = elem ) ); - } - } - postFinder( null, ( matcherOut = [] ), temp, xml ); - } - - // Move matched elements from seed to results to keep them synchronized - i = matcherOut.length; - while ( i-- ) { - if ( ( elem = matcherOut[ i ] ) && - ( temp = postFinder ? indexOf( seed, elem ) : preMap[ i ] ) > -1 ) { - - seed[ temp ] = !( results[ temp ] = elem ); - } - } - } - - // Add elements to results, through postFinder if defined - } else { - matcherOut = condense( - matcherOut === results ? - matcherOut.splice( preexisting, matcherOut.length ) : - matcherOut - ); - if ( postFinder ) { - postFinder( null, results, matcherOut, xml ); - } else { - push.apply( results, matcherOut ); - } - } - } ); -} - -function matcherFromTokens( tokens ) { - var checkContext, matcher, j, - len = tokens.length, - leadingRelative = Expr.relative[ tokens[ 0 ].type ], - implicitRelative = leadingRelative || Expr.relative[ " " ], - i = leadingRelative ? 1 : 0, - - // The foundational matcher ensures that elements are reachable from top-level context(s) - matchContext = addCombinator( function( elem ) { - return elem === checkContext; - }, implicitRelative, true ), - matchAnyContext = addCombinator( function( elem ) { - return indexOf( checkContext, elem ) > -1; - }, implicitRelative, true ), - matchers = [ function( elem, context, xml ) { - var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || ( - ( checkContext = context ).nodeType ? - matchContext( elem, context, xml ) : - matchAnyContext( elem, context, xml ) ); - - // Avoid hanging onto element (issue #299) - checkContext = null; - return ret; - } ]; - - for ( ; i < len; i++ ) { - if ( ( matcher = Expr.relative[ tokens[ i ].type ] ) ) { - matchers = [ addCombinator( elementMatcher( matchers ), matcher ) ]; - } else { - matcher = Expr.filter[ tokens[ i ].type ].apply( null, tokens[ i ].matches ); - - // Return special upon seeing a positional matcher - if ( matcher[ expando ] ) { - - // Find the next relative operator (if any) for proper handling - j = ++i; - for ( ; j < len; j++ ) { - if ( Expr.relative[ tokens[ j ].type ] ) { - break; - } - } - return setMatcher( - i > 1 && elementMatcher( matchers ), - i > 1 && toSelector( - - // If the preceding token was a descendant combinator, insert an implicit any-element `*` - tokens - .slice( 0, i - 1 ) - .concat( { value: tokens[ i - 2 ].type === " " ? "*" : "" } ) - ).replace( rtrim, "$1" ), - matcher, - i < j && matcherFromTokens( tokens.slice( i, j ) ), - j < len && matcherFromTokens( ( tokens = tokens.slice( j ) ) ), - j < len && toSelector( tokens ) - ); - } - matchers.push( matcher ); - } - } - - return elementMatcher( matchers ); -} - -function matcherFromGroupMatchers( elementMatchers, setMatchers ) { - var bySet = setMatchers.length > 0, - byElement = elementMatchers.length > 0, - superMatcher = function( seed, context, xml, results, outermost ) { - var elem, j, matcher, - matchedCount = 0, - i = "0", - unmatched = seed && [], - setMatched = [], - contextBackup = outermostContext, - - // We must always have either seed elements or outermost context - elems = seed || byElement && Expr.find[ "TAG" ]( "*", outermost ), - - // Use integer dirruns iff this is the outermost matcher - dirrunsUnique = ( dirruns += contextBackup == null ? 1 : Math.random() || 0.1 ), - len = elems.length; - - if ( outermost ) { - - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - outermostContext = context == document || context || outermost; - } - - // Add elements passing elementMatchers directly to results - // Support: IE<9, Safari - // Tolerate NodeList properties (IE: "length"; Safari: ) matching elements by id - for ( ; i !== len && ( elem = elems[ i ] ) != null; i++ ) { - if ( byElement && elem ) { - j = 0; - - // Support: IE 11+, Edge 17 - 18+ - // IE/Edge sometimes throw a "Permission denied" error when strict-comparing - // two documents; shallow comparisons work. - // eslint-disable-next-line eqeqeq - if ( !context && elem.ownerDocument != document ) { - setDocument( elem ); - xml = !documentIsHTML; - } - while ( ( matcher = elementMatchers[ j++ ] ) ) { - if ( matcher( elem, context || document, xml ) ) { - results.push( elem ); - break; - } - } - if ( outermost ) { - dirruns = dirrunsUnique; - } - } - - // Track unmatched elements for set filters - if ( bySet ) { - - // They will have gone through all possible matchers - if ( ( elem = !matcher && elem ) ) { - matchedCount--; - } - - // Lengthen the array for every element, matched or not - if ( seed ) { - unmatched.push( elem ); - } - } - } - - // `i` is now the count of elements visited above, and adding it to `matchedCount` - // makes the latter nonnegative. - matchedCount += i; - - // Apply set filters to unmatched elements - // NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount` - // equals `i`), unless we didn't visit _any_ elements in the above loop because we have - // no element matchers and no seed. - // Incrementing an initially-string "0" `i` allows `i` to remain a string only in that - // case, which will result in a "00" `matchedCount` that differs from `i` but is also - // numerically zero. - if ( bySet && i !== matchedCount ) { - j = 0; - while ( ( matcher = setMatchers[ j++ ] ) ) { - matcher( unmatched, setMatched, context, xml ); - } - - if ( seed ) { - - // Reintegrate element matches to eliminate the need for sorting - if ( matchedCount > 0 ) { - while ( i-- ) { - if ( !( unmatched[ i ] || setMatched[ i ] ) ) { - setMatched[ i ] = pop.call( results ); - } - } - } - - // Discard index placeholder values to get only actual matches - setMatched = condense( setMatched ); - } - - // Add matches to results - push.apply( results, setMatched ); - - // Seedless set matches succeeding multiple successful matchers stipulate sorting - if ( outermost && !seed && setMatched.length > 0 && - ( matchedCount + setMatchers.length ) > 1 ) { - - Sizzle.uniqueSort( results ); - } - } - - // Override manipulation of globals by nested matchers - if ( outermost ) { - dirruns = dirrunsUnique; - outermostContext = contextBackup; - } - - return unmatched; - }; - - return bySet ? - markFunction( superMatcher ) : - superMatcher; -} - -compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) { - var i, - setMatchers = [], - elementMatchers = [], - cached = compilerCache[ selector + " " ]; - - if ( !cached ) { - - // Generate a function of recursive functions that can be used to check each element - if ( !match ) { - match = tokenize( selector ); - } - i = match.length; - while ( i-- ) { - cached = matcherFromTokens( match[ i ] ); - if ( cached[ expando ] ) { - setMatchers.push( cached ); - } else { - elementMatchers.push( cached ); - } - } - - // Cache the compiled function - cached = compilerCache( - selector, - matcherFromGroupMatchers( elementMatchers, setMatchers ) - ); - - // Save selector and tokenization - cached.selector = selector; - } - return cached; -}; - -/** - * A low-level selection function that works with Sizzle's compiled - * selector functions - * @param {String|Function} selector A selector or a pre-compiled - * selector function built with Sizzle.compile - * @param {Element} context - * @param {Array} [results] - * @param {Array} [seed] A set of elements to match against - */ -select = Sizzle.select = function( selector, context, results, seed ) { - var i, tokens, token, type, find, - compiled = typeof selector === "function" && selector, - match = !seed && tokenize( ( selector = compiled.selector || selector ) ); - - results = results || []; - - // Try to minimize operations if there is only one selector in the list and no seed - // (the latter of which guarantees us context) - if ( match.length === 1 ) { - - // Reduce context if the leading compound selector is an ID - tokens = match[ 0 ] = match[ 0 ].slice( 0 ); - if ( tokens.length > 2 && ( token = tokens[ 0 ] ).type === "ID" && - context.nodeType === 9 && documentIsHTML && Expr.relative[ tokens[ 1 ].type ] ) { - - context = ( Expr.find[ "ID" ]( token.matches[ 0 ] - .replace( runescape, funescape ), context ) || [] )[ 0 ]; - if ( !context ) { - return results; - - // Precompiled matchers will still verify ancestry, so step up a level - } else if ( compiled ) { - context = context.parentNode; - } - - selector = selector.slice( tokens.shift().value.length ); - } - - // Fetch a seed set for right-to-left matching - i = matchExpr[ "needsContext" ].test( selector ) ? 0 : tokens.length; - while ( i-- ) { - token = tokens[ i ]; - - // Abort if we hit a combinator - if ( Expr.relative[ ( type = token.type ) ] ) { - break; - } - if ( ( find = Expr.find[ type ] ) ) { - - // Search, expanding context for leading sibling combinators - if ( ( seed = find( - token.matches[ 0 ].replace( runescape, funescape ), - rsibling.test( tokens[ 0 ].type ) && testContext( context.parentNode ) || - context - ) ) ) { - - // If seed is empty or no tokens remain, we can return early - tokens.splice( i, 1 ); - selector = seed.length && toSelector( tokens ); - if ( !selector ) { - push.apply( results, seed ); - return results; - } - - break; - } - } - } - } - - // Compile and execute a filtering function if one is not provided - // Provide `match` to avoid retokenization if we modified the selector above - ( compiled || compile( selector, match ) )( - seed, - context, - !documentIsHTML, - results, - !context || rsibling.test( selector ) && testContext( context.parentNode ) || context - ); - return results; -}; - -// One-time assignments - -// Sort stability -support.sortStable = expando.split( "" ).sort( sortOrder ).join( "" ) === expando; - -// Support: Chrome 14-35+ -// Always assume duplicates if they aren't passed to the comparison function -support.detectDuplicates = !!hasDuplicate; - -// Initialize against the default document -setDocument(); - -// Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27) -// Detached nodes confoundingly follow *each other* -support.sortDetached = assert( function( el ) { - - // Should return 1, but returns 4 (following) - return el.compareDocumentPosition( document.createElement( "fieldset" ) ) & 1; -} ); - -// Support: IE<8 -// Prevent attribute/property "interpolation" -// https://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx -if ( !assert( function( el ) { - el.innerHTML = ""; - return el.firstChild.getAttribute( "href" ) === "#"; -} ) ) { - addHandle( "type|href|height|width", function( elem, name, isXML ) { - if ( !isXML ) { - return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 ); - } - } ); -} - -// Support: IE<9 -// Use defaultValue in place of getAttribute("value") -if ( !support.attributes || !assert( function( el ) { - el.innerHTML = ""; - el.firstChild.setAttribute( "value", "" ); - return el.firstChild.getAttribute( "value" ) === ""; -} ) ) { - addHandle( "value", function( elem, _name, isXML ) { - if ( !isXML && elem.nodeName.toLowerCase() === "input" ) { - return elem.defaultValue; - } - } ); -} - -// Support: IE<9 -// Use getAttributeNode to fetch booleans when getAttribute lies -if ( !assert( function( el ) { - return el.getAttribute( "disabled" ) == null; -} ) ) { - addHandle( booleans, function( elem, name, isXML ) { - var val; - if ( !isXML ) { - return elem[ name ] === true ? name.toLowerCase() : - ( val = elem.getAttributeNode( name ) ) && val.specified ? - val.value : - null; - } - } ); -} - -return Sizzle; - -} )( window ); - - - -jQuery.find = Sizzle; -jQuery.expr = Sizzle.selectors; - -// Deprecated -jQuery.expr[ ":" ] = jQuery.expr.pseudos; -jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort; -jQuery.text = Sizzle.getText; -jQuery.isXMLDoc = Sizzle.isXML; -jQuery.contains = Sizzle.contains; -jQuery.escapeSelector = Sizzle.escape; - - - - -var dir = function( elem, dir, until ) { - var matched = [], - truncate = until !== undefined; - - while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) { - if ( elem.nodeType === 1 ) { - if ( truncate && jQuery( elem ).is( until ) ) { - break; - } - matched.push( elem ); - } - } - return matched; -}; - - -var siblings = function( n, elem ) { - var matched = []; - - for ( ; n; n = n.nextSibling ) { - if ( n.nodeType === 1 && n !== elem ) { - matched.push( n ); - } - } - - return matched; -}; - - -var rneedsContext = jQuery.expr.match.needsContext; - - - -function nodeName( elem, name ) { - - return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); - -}; -var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i ); - - - -// Implement the identical functionality for filter and not -function winnow( elements, qualifier, not ) { - if ( isFunction( qualifier ) ) { - return jQuery.grep( elements, function( elem, i ) { - return !!qualifier.call( elem, i, elem ) !== not; - } ); - } - - // Single element - if ( qualifier.nodeType ) { - return jQuery.grep( elements, function( elem ) { - return ( elem === qualifier ) !== not; - } ); - } - - // Arraylike of elements (jQuery, arguments, Array) - if ( typeof qualifier !== "string" ) { - return jQuery.grep( elements, function( elem ) { - return ( indexOf.call( qualifier, elem ) > -1 ) !== not; - } ); - } - - // Filtered directly for both simple and complex selectors - return jQuery.filter( qualifier, elements, not ); -} - -jQuery.filter = function( expr, elems, not ) { - var elem = elems[ 0 ]; - - if ( not ) { - expr = ":not(" + expr + ")"; - } - - if ( elems.length === 1 && elem.nodeType === 1 ) { - return jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : []; - } - - return jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) { - return elem.nodeType === 1; - } ) ); -}; - -jQuery.fn.extend( { - find: function( selector ) { - var i, ret, - len = this.length, - self = this; - - if ( typeof selector !== "string" ) { - return this.pushStack( jQuery( selector ).filter( function() { - for ( i = 0; i < len; i++ ) { - if ( jQuery.contains( self[ i ], this ) ) { - return true; - } - } - } ) ); - } - - ret = this.pushStack( [] ); - - for ( i = 0; i < len; i++ ) { - jQuery.find( selector, self[ i ], ret ); - } - - return len > 1 ? jQuery.uniqueSort( ret ) : ret; - }, - filter: function( selector ) { - return this.pushStack( winnow( this, selector || [], false ) ); - }, - not: function( selector ) { - return this.pushStack( winnow( this, selector || [], true ) ); - }, - is: function( selector ) { - return !!winnow( - this, - - // If this is a positional/relative selector, check membership in the returned set - // so $("p:first").is("p:last") won't return true for a doc with two "p". - typeof selector === "string" && rneedsContext.test( selector ) ? - jQuery( selector ) : - selector || [], - false - ).length; - } -} ); - - -// Initialize a jQuery object - - -// A central reference to the root jQuery(document) -var rootjQuery, - - // A simple way to check for HTML strings - // Prioritize #id over to avoid XSS via location.hash (#9521) - // Strict HTML recognition (#11290: must start with <) - // Shortcut simple #id case for speed - rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/, - - init = jQuery.fn.init = function( selector, context, root ) { - var match, elem; - - // HANDLE: $(""), $(null), $(undefined), $(false) - if ( !selector ) { - return this; - } - - // Method init() accepts an alternate rootjQuery - // so migrate can support jQuery.sub (gh-2101) - root = root || rootjQuery; - - // Handle HTML strings - if ( typeof selector === "string" ) { - if ( selector[ 0 ] === "<" && - selector[ selector.length - 1 ] === ">" && - selector.length >= 3 ) { - - // Assume that strings that start and end with <> are HTML and skip the regex check - match = [ null, selector, null ]; - - } else { - match = rquickExpr.exec( selector ); - } - - // Match html or make sure no context is specified for #id - if ( match && ( match[ 1 ] || !context ) ) { - - // HANDLE: $(html) -> $(array) - if ( match[ 1 ] ) { - context = context instanceof jQuery ? context[ 0 ] : context; - - // Option to run scripts is true for back-compat - // Intentionally let the error be thrown if parseHTML is not present - jQuery.merge( this, jQuery.parseHTML( - match[ 1 ], - context && context.nodeType ? context.ownerDocument || context : document, - true - ) ); - - // HANDLE: $(html, props) - if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) { - for ( match in context ) { - - // Properties of context are called as methods if possible - if ( isFunction( this[ match ] ) ) { - this[ match ]( context[ match ] ); - - // ...and otherwise set as attributes - } else { - this.attr( match, context[ match ] ); - } - } - } - - return this; - - // HANDLE: $(#id) - } else { - elem = document.getElementById( match[ 2 ] ); - - if ( elem ) { - - // Inject the element directly into the jQuery object - this[ 0 ] = elem; - this.length = 1; - } - return this; - } - - // HANDLE: $(expr, $(...)) - } else if ( !context || context.jquery ) { - return ( context || root ).find( selector ); - - // HANDLE: $(expr, context) - // (which is just equivalent to: $(context).find(expr) - } else { - return this.constructor( context ).find( selector ); - } - - // HANDLE: $(DOMElement) - } else if ( selector.nodeType ) { - this[ 0 ] = selector; - this.length = 1; - return this; - - // HANDLE: $(function) - // Shortcut for document ready - } else if ( isFunction( selector ) ) { - return root.ready !== undefined ? - root.ready( selector ) : - - // Execute immediately if ready is not present - selector( jQuery ); - } - - return jQuery.makeArray( selector, this ); - }; - -// Give the init function the jQuery prototype for later instantiation -init.prototype = jQuery.fn; - -// Initialize central reference -rootjQuery = jQuery( document ); - - -var rparentsprev = /^(?:parents|prev(?:Until|All))/, - - // Methods guaranteed to produce a unique set when starting from a unique set - guaranteedUnique = { - children: true, - contents: true, - next: true, - prev: true - }; - -jQuery.fn.extend( { - has: function( target ) { - var targets = jQuery( target, this ), - l = targets.length; - - return this.filter( function() { - var i = 0; - for ( ; i < l; i++ ) { - if ( jQuery.contains( this, targets[ i ] ) ) { - return true; - } - } - } ); - }, - - closest: function( selectors, context ) { - var cur, - i = 0, - l = this.length, - matched = [], - targets = typeof selectors !== "string" && jQuery( selectors ); - - // Positional selectors never match, since there's no _selection_ context - if ( !rneedsContext.test( selectors ) ) { - for ( ; i < l; i++ ) { - for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) { - - // Always skip document fragments - if ( cur.nodeType < 11 && ( targets ? - targets.index( cur ) > -1 : - - // Don't pass non-elements to Sizzle - cur.nodeType === 1 && - jQuery.find.matchesSelector( cur, selectors ) ) ) { - - matched.push( cur ); - break; - } - } - } - } - - return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched ); - }, - - // Determine the position of an element within the set - index: function( elem ) { - - // No argument, return index in parent - if ( !elem ) { - return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1; - } - - // Index in selector - if ( typeof elem === "string" ) { - return indexOf.call( jQuery( elem ), this[ 0 ] ); - } - - // Locate the position of the desired element - return indexOf.call( this, - - // If it receives a jQuery object, the first element is used - elem.jquery ? elem[ 0 ] : elem - ); - }, - - add: function( selector, context ) { - return this.pushStack( - jQuery.uniqueSort( - jQuery.merge( this.get(), jQuery( selector, context ) ) - ) - ); - }, - - addBack: function( selector ) { - return this.add( selector == null ? - this.prevObject : this.prevObject.filter( selector ) - ); - } -} ); - -function sibling( cur, dir ) { - while ( ( cur = cur[ dir ] ) && cur.nodeType !== 1 ) {} - return cur; -} - -jQuery.each( { - parent: function( elem ) { - var parent = elem.parentNode; - return parent && parent.nodeType !== 11 ? parent : null; - }, - parents: function( elem ) { - return dir( elem, "parentNode" ); - }, - parentsUntil: function( elem, _i, until ) { - return dir( elem, "parentNode", until ); - }, - next: function( elem ) { - return sibling( elem, "nextSibling" ); - }, - prev: function( elem ) { - return sibling( elem, "previousSibling" ); - }, - nextAll: function( elem ) { - return dir( elem, "nextSibling" ); - }, - prevAll: function( elem ) { - return dir( elem, "previousSibling" ); - }, - nextUntil: function( elem, _i, until ) { - return dir( elem, "nextSibling", until ); - }, - prevUntil: function( elem, _i, until ) { - return dir( elem, "previousSibling", until ); - }, - siblings: function( elem ) { - return siblings( ( elem.parentNode || {} ).firstChild, elem ); - }, - children: function( elem ) { - return siblings( elem.firstChild ); - }, - contents: function( elem ) { - if ( elem.contentDocument != null && - - // Support: IE 11+ - // elements with no `data` attribute has an object - // `contentDocument` with a `null` prototype. - getProto( elem.contentDocument ) ) { - - return elem.contentDocument; - } - - // Support: IE 9 - 11 only, iOS 7 only, Android Browser <=4.3 only - // Treat the template element as a regular one in browsers that - // don't support it. - if ( nodeName( elem, "template" ) ) { - elem = elem.content || elem; - } - - return jQuery.merge( [], elem.childNodes ); - } -}, function( name, fn ) { - jQuery.fn[ name ] = function( until, selector ) { - var matched = jQuery.map( this, fn, until ); - - if ( name.slice( -5 ) !== "Until" ) { - selector = until; - } - - if ( selector && typeof selector === "string" ) { - matched = jQuery.filter( selector, matched ); - } - - if ( this.length > 1 ) { - - // Remove duplicates - if ( !guaranteedUnique[ name ] ) { - jQuery.uniqueSort( matched ); - } - - // Reverse order for parents* and prev-derivatives - if ( rparentsprev.test( name ) ) { - matched.reverse(); - } - } - - return this.pushStack( matched ); - }; -} ); -var rnothtmlwhite = ( /[^\x20\t\r\n\f]+/g ); - - - -// Convert String-formatted options into Object-formatted ones -function createOptions( options ) { - var object = {}; - jQuery.each( options.match( rnothtmlwhite ) || [], function( _, flag ) { - object[ flag ] = true; - } ); - return object; -} - -/* - * Create a callback list using the following parameters: - * - * options: an optional list of space-separated options that will change how - * the callback list behaves or a more traditional option object - * - * By default a callback list will act like an event callback list and can be - * "fired" multiple times. - * - * Possible options: - * - * once: will ensure the callback list can only be fired once (like a Deferred) - * - * memory: will keep track of previous values and will call any callback added - * after the list has been fired right away with the latest "memorized" - * values (like a Deferred) - * - * unique: will ensure a callback can only be added once (no duplicate in the list) - * - * stopOnFalse: interrupt callings when a callback returns false - * - */ -jQuery.Callbacks = function( options ) { - - // Convert options from String-formatted to Object-formatted if needed - // (we check in cache first) - options = typeof options === "string" ? - createOptions( options ) : - jQuery.extend( {}, options ); - - var // Flag to know if list is currently firing - firing, - - // Last fire value for non-forgettable lists - memory, - - // Flag to know if list was already fired - fired, - - // Flag to prevent firing - locked, - - // Actual callback list - list = [], - - // Queue of execution data for repeatable lists - queue = [], - - // Index of currently firing callback (modified by add/remove as needed) - firingIndex = -1, - - // Fire callbacks - fire = function() { - - // Enforce single-firing - locked = locked || options.once; - - // Execute callbacks for all pending executions, - // respecting firingIndex overrides and runtime changes - fired = firing = true; - for ( ; queue.length; firingIndex = -1 ) { - memory = queue.shift(); - while ( ++firingIndex < list.length ) { - - // Run callback and check for early termination - if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false && - options.stopOnFalse ) { - - // Jump to end and forget the data so .add doesn't re-fire - firingIndex = list.length; - memory = false; - } - } - } - - // Forget the data if we're done with it - if ( !options.memory ) { - memory = false; - } - - firing = false; - - // Clean up if we're done firing for good - if ( locked ) { - - // Keep an empty list if we have data for future add calls - if ( memory ) { - list = []; - - // Otherwise, this object is spent - } else { - list = ""; - } - } - }, - - // Actual Callbacks object - self = { - - // Add a callback or a collection of callbacks to the list - add: function() { - if ( list ) { - - // If we have memory from a past run, we should fire after adding - if ( memory && !firing ) { - firingIndex = list.length - 1; - queue.push( memory ); - } - - ( function add( args ) { - jQuery.each( args, function( _, arg ) { - if ( isFunction( arg ) ) { - if ( !options.unique || !self.has( arg ) ) { - list.push( arg ); - } - } else if ( arg && arg.length && toType( arg ) !== "string" ) { - - // Inspect recursively - add( arg ); - } - } ); - } )( arguments ); - - if ( memory && !firing ) { - fire(); - } - } - return this; - }, - - // Remove a callback from the list - remove: function() { - jQuery.each( arguments, function( _, arg ) { - var index; - while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) { - list.splice( index, 1 ); - - // Handle firing indexes - if ( index <= firingIndex ) { - firingIndex--; - } - } - } ); - return this; - }, - - // Check if a given callback is in the list. - // If no argument is given, return whether or not list has callbacks attached. - has: function( fn ) { - return fn ? - jQuery.inArray( fn, list ) > -1 : - list.length > 0; - }, - - // Remove all callbacks from the list - empty: function() { - if ( list ) { - list = []; - } - return this; - }, - - // Disable .fire and .add - // Abort any current/pending executions - // Clear all callbacks and values - disable: function() { - locked = queue = []; - list = memory = ""; - return this; - }, - disabled: function() { - return !list; - }, - - // Disable .fire - // Also disable .add unless we have memory (since it would have no effect) - // Abort any pending executions - lock: function() { - locked = queue = []; - if ( !memory && !firing ) { - list = memory = ""; - } - return this; - }, - locked: function() { - return !!locked; - }, - - // Call all callbacks with the given context and arguments - fireWith: function( context, args ) { - if ( !locked ) { - args = args || []; - args = [ context, args.slice ? args.slice() : args ]; - queue.push( args ); - if ( !firing ) { - fire(); - } - } - return this; - }, - - // Call all the callbacks with the given arguments - fire: function() { - self.fireWith( this, arguments ); - return this; - }, - - // To know if the callbacks have already been called at least once - fired: function() { - return !!fired; - } - }; - - return self; -}; - - -function Identity( v ) { - return v; -} -function Thrower( ex ) { - throw ex; -} - -function adoptValue( value, resolve, reject, noValue ) { - var method; - - try { - - // Check for promise aspect first to privilege synchronous behavior - if ( value && isFunction( ( method = value.promise ) ) ) { - method.call( value ).done( resolve ).fail( reject ); - - // Other thenables - } else if ( value && isFunction( ( method = value.then ) ) ) { - method.call( value, resolve, reject ); - - // Other non-thenables - } else { - - // Control `resolve` arguments by letting Array#slice cast boolean `noValue` to integer: - // * false: [ value ].slice( 0 ) => resolve( value ) - // * true: [ value ].slice( 1 ) => resolve() - resolve.apply( undefined, [ value ].slice( noValue ) ); - } - - // For Promises/A+, convert exceptions into rejections - // Since jQuery.when doesn't unwrap thenables, we can skip the extra checks appearing in - // Deferred#then to conditionally suppress rejection. - } catch ( value ) { - - // Support: Android 4.0 only - // Strict mode functions invoked without .call/.apply get global-object context - reject.apply( undefined, [ value ] ); - } -} - -jQuery.extend( { - - Deferred: function( func ) { - var tuples = [ - - // action, add listener, callbacks, - // ... .then handlers, argument index, [final state] - [ "notify", "progress", jQuery.Callbacks( "memory" ), - jQuery.Callbacks( "memory" ), 2 ], - [ "resolve", "done", jQuery.Callbacks( "once memory" ), - jQuery.Callbacks( "once memory" ), 0, "resolved" ], - [ "reject", "fail", jQuery.Callbacks( "once memory" ), - jQuery.Callbacks( "once memory" ), 1, "rejected" ] - ], - state = "pending", - promise = { - state: function() { - return state; - }, - always: function() { - deferred.done( arguments ).fail( arguments ); - return this; - }, - "catch": function( fn ) { - return promise.then( null, fn ); - }, - - // Keep pipe for back-compat - pipe: function( /* fnDone, fnFail, fnProgress */ ) { - var fns = arguments; - - return jQuery.Deferred( function( newDefer ) { - jQuery.each( tuples, function( _i, tuple ) { - - // Map tuples (progress, done, fail) to arguments (done, fail, progress) - var fn = isFunction( fns[ tuple[ 4 ] ] ) && fns[ tuple[ 4 ] ]; - - // deferred.progress(function() { bind to newDefer or newDefer.notify }) - // deferred.done(function() { bind to newDefer or newDefer.resolve }) - // deferred.fail(function() { bind to newDefer or newDefer.reject }) - deferred[ tuple[ 1 ] ]( function() { - var returned = fn && fn.apply( this, arguments ); - if ( returned && isFunction( returned.promise ) ) { - returned.promise() - .progress( newDefer.notify ) - .done( newDefer.resolve ) - .fail( newDefer.reject ); - } else { - newDefer[ tuple[ 0 ] + "With" ]( - this, - fn ? [ returned ] : arguments - ); - } - } ); - } ); - fns = null; - } ).promise(); - }, - then: function( onFulfilled, onRejected, onProgress ) { - var maxDepth = 0; - function resolve( depth, deferred, handler, special ) { - return function() { - var that = this, - args = arguments, - mightThrow = function() { - var returned, then; - - // Support: Promises/A+ section 2.3.3.3.3 - // https://promisesaplus.com/#point-59 - // Ignore double-resolution attempts - if ( depth < maxDepth ) { - return; - } - - returned = handler.apply( that, args ); - - // Support: Promises/A+ section 2.3.1 - // https://promisesaplus.com/#point-48 - if ( returned === deferred.promise() ) { - throw new TypeError( "Thenable self-resolution" ); - } - - // Support: Promises/A+ sections 2.3.3.1, 3.5 - // https://promisesaplus.com/#point-54 - // https://promisesaplus.com/#point-75 - // Retrieve `then` only once - then = returned && - - // Support: Promises/A+ section 2.3.4 - // https://promisesaplus.com/#point-64 - // Only check objects and functions for thenability - ( typeof returned === "object" || - typeof returned === "function" ) && - returned.then; - - // Handle a returned thenable - if ( isFunction( then ) ) { - - // Special processors (notify) just wait for resolution - if ( special ) { - then.call( - returned, - resolve( maxDepth, deferred, Identity, special ), - resolve( maxDepth, deferred, Thrower, special ) - ); - - // Normal processors (resolve) also hook into progress - } else { - - // ...and disregard older resolution values - maxDepth++; - - then.call( - returned, - resolve( maxDepth, deferred, Identity, special ), - resolve( maxDepth, deferred, Thrower, special ), - resolve( maxDepth, deferred, Identity, - deferred.notifyWith ) - ); - } - - // Handle all other returned values - } else { - - // Only substitute handlers pass on context - // and multiple values (non-spec behavior) - if ( handler !== Identity ) { - that = undefined; - args = [ returned ]; - } - - // Process the value(s) - // Default process is resolve - ( special || deferred.resolveWith )( that, args ); - } - }, - - // Only normal processors (resolve) catch and reject exceptions - process = special ? - mightThrow : - function() { - try { - mightThrow(); - } catch ( e ) { - - if ( jQuery.Deferred.exceptionHook ) { - jQuery.Deferred.exceptionHook( e, - process.stackTrace ); - } - - // Support: Promises/A+ section 2.3.3.3.4.1 - // https://promisesaplus.com/#point-61 - // Ignore post-resolution exceptions - if ( depth + 1 >= maxDepth ) { - - // Only substitute handlers pass on context - // and multiple values (non-spec behavior) - if ( handler !== Thrower ) { - that = undefined; - args = [ e ]; - } - - deferred.rejectWith( that, args ); - } - } - }; - - // Support: Promises/A+ section 2.3.3.3.1 - // https://promisesaplus.com/#point-57 - // Re-resolve promises immediately to dodge false rejection from - // subsequent errors - if ( depth ) { - process(); - } else { - - // Call an optional hook to record the stack, in case of exception - // since it's otherwise lost when execution goes async - if ( jQuery.Deferred.getStackHook ) { - process.stackTrace = jQuery.Deferred.getStackHook(); - } - window.setTimeout( process ); - } - }; - } - - return jQuery.Deferred( function( newDefer ) { - - // progress_handlers.add( ... ) - tuples[ 0 ][ 3 ].add( - resolve( - 0, - newDefer, - isFunction( onProgress ) ? - onProgress : - Identity, - newDefer.notifyWith - ) - ); - - // fulfilled_handlers.add( ... ) - tuples[ 1 ][ 3 ].add( - resolve( - 0, - newDefer, - isFunction( onFulfilled ) ? - onFulfilled : - Identity - ) - ); - - // rejected_handlers.add( ... ) - tuples[ 2 ][ 3 ].add( - resolve( - 0, - newDefer, - isFunction( onRejected ) ? - onRejected : - Thrower - ) - ); - } ).promise(); - }, - - // Get a promise for this deferred - // If obj is provided, the promise aspect is added to the object - promise: function( obj ) { - return obj != null ? jQuery.extend( obj, promise ) : promise; - } - }, - deferred = {}; - - // Add list-specific methods - jQuery.each( tuples, function( i, tuple ) { - var list = tuple[ 2 ], - stateString = tuple[ 5 ]; - - // promise.progress = list.add - // promise.done = list.add - // promise.fail = list.add - promise[ tuple[ 1 ] ] = list.add; - - // Handle state - if ( stateString ) { - list.add( - function() { - - // state = "resolved" (i.e., fulfilled) - // state = "rejected" - state = stateString; - }, - - // rejected_callbacks.disable - // fulfilled_callbacks.disable - tuples[ 3 - i ][ 2 ].disable, - - // rejected_handlers.disable - // fulfilled_handlers.disable - tuples[ 3 - i ][ 3 ].disable, - - // progress_callbacks.lock - tuples[ 0 ][ 2 ].lock, - - // progress_handlers.lock - tuples[ 0 ][ 3 ].lock - ); - } - - // progress_handlers.fire - // fulfilled_handlers.fire - // rejected_handlers.fire - list.add( tuple[ 3 ].fire ); - - // deferred.notify = function() { deferred.notifyWith(...) } - // deferred.resolve = function() { deferred.resolveWith(...) } - // deferred.reject = function() { deferred.rejectWith(...) } - deferred[ tuple[ 0 ] ] = function() { - deferred[ tuple[ 0 ] + "With" ]( this === deferred ? undefined : this, arguments ); - return this; - }; - - // deferred.notifyWith = list.fireWith - // deferred.resolveWith = list.fireWith - // deferred.rejectWith = list.fireWith - deferred[ tuple[ 0 ] + "With" ] = list.fireWith; - } ); - - // Make the deferred a promise - promise.promise( deferred ); - - // Call given func if any - if ( func ) { - func.call( deferred, deferred ); - } - - // All done! - return deferred; - }, - - // Deferred helper - when: function( singleValue ) { - var - - // count of uncompleted subordinates - remaining = arguments.length, - - // count of unprocessed arguments - i = remaining, - - // subordinate fulfillment data - resolveContexts = Array( i ), - resolveValues = slice.call( arguments ), - - // the master Deferred - master = jQuery.Deferred(), - - // subordinate callback factory - updateFunc = function( i ) { - return function( value ) { - resolveContexts[ i ] = this; - resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value; - if ( !( --remaining ) ) { - master.resolveWith( resolveContexts, resolveValues ); - } - }; - }; - - // Single- and empty arguments are adopted like Promise.resolve - if ( remaining <= 1 ) { - adoptValue( singleValue, master.done( updateFunc( i ) ).resolve, master.reject, - !remaining ); - - // Use .then() to unwrap secondary thenables (cf. gh-3000) - if ( master.state() === "pending" || - isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { - - return master.then(); - } - } - - // Multiple arguments are aggregated like Promise.all array elements - while ( i-- ) { - adoptValue( resolveValues[ i ], updateFunc( i ), master.reject ); - } - - return master.promise(); - } -} ); - - -// These usually indicate a programmer mistake during development, -// warn about them ASAP rather than swallowing them by default. -var rerrorNames = /^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/; - -jQuery.Deferred.exceptionHook = function( error, stack ) { - - // Support: IE 8 - 9 only - // Console exists when dev tools are open, which can happen at any time - if ( window.console && window.console.warn && error && rerrorNames.test( error.name ) ) { - window.console.warn( "jQuery.Deferred exception: " + error.message, error.stack, stack ); - } -}; - - - - -jQuery.readyException = function( error ) { - window.setTimeout( function() { - throw error; - } ); -}; - - - - -// The deferred used on DOM ready -var readyList = jQuery.Deferred(); - -jQuery.fn.ready = function( fn ) { - - readyList - .then( fn ) - - // Wrap jQuery.readyException in a function so that the lookup - // happens at the time of error handling instead of callback - // registration. - .catch( function( error ) { - jQuery.readyException( error ); - } ); - - return this; -}; - -jQuery.extend( { - - // Is the DOM ready to be used? Set to true once it occurs. - isReady: false, - - // A counter to track how many items to wait for before - // the ready event fires. See #6781 - readyWait: 1, - - // Handle when the DOM is ready - ready: function( wait ) { - - // Abort if there are pending holds or we're already ready - if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) { - return; - } - - // Remember that the DOM is ready - jQuery.isReady = true; - - // If a normal DOM Ready event fired, decrement, and wait if need be - if ( wait !== true && --jQuery.readyWait > 0 ) { - return; - } - - // If there are functions bound, to execute - readyList.resolveWith( document, [ jQuery ] ); - } -} ); - -jQuery.ready.then = readyList.then; - -// The ready event handler and self cleanup method -function completed() { - document.removeEventListener( "DOMContentLoaded", completed ); - window.removeEventListener( "load", completed ); - jQuery.ready(); -} - -// Catch cases where $(document).ready() is called -// after the browser event has already occurred. -// Support: IE <=9 - 10 only -// Older IE sometimes signals "interactive" too soon -if ( document.readyState === "complete" || - ( document.readyState !== "loading" && !document.documentElement.doScroll ) ) { - - // Handle it asynchronously to allow scripts the opportunity to delay ready - window.setTimeout( jQuery.ready ); - -} else { - - // Use the handy event callback - document.addEventListener( "DOMContentLoaded", completed ); - - // A fallback to window.onload, that will always work - window.addEventListener( "load", completed ); -} - - - - -// Multifunctional method to get and set values of a collection -// The value/s can optionally be executed if it's a function -var access = function( elems, fn, key, value, chainable, emptyGet, raw ) { - var i = 0, - len = elems.length, - bulk = key == null; - - // Sets many values - if ( toType( key ) === "object" ) { - chainable = true; - for ( i in key ) { - access( elems, fn, i, key[ i ], true, emptyGet, raw ); - } - - // Sets one value - } else if ( value !== undefined ) { - chainable = true; - - if ( !isFunction( value ) ) { - raw = true; - } - - if ( bulk ) { - - // Bulk operations run against the entire set - if ( raw ) { - fn.call( elems, value ); - fn = null; - - // ...except when executing function values - } else { - bulk = fn; - fn = function( elem, _key, value ) { - return bulk.call( jQuery( elem ), value ); - }; - } - } - - if ( fn ) { - for ( ; i < len; i++ ) { - fn( - elems[ i ], key, raw ? - value : - value.call( elems[ i ], i, fn( elems[ i ], key ) ) - ); - } - } - } - - if ( chainable ) { - return elems; - } - - // Gets - if ( bulk ) { - return fn.call( elems ); - } - - return len ? fn( elems[ 0 ], key ) : emptyGet; -}; - - -// Matches dashed string for camelizing -var rmsPrefix = /^-ms-/, - rdashAlpha = /-([a-z])/g; - -// Used by camelCase as callback to replace() -function fcamelCase( _all, letter ) { - return letter.toUpperCase(); -} - -// Convert dashed to camelCase; used by the css and data modules -// Support: IE <=9 - 11, Edge 12 - 15 -// Microsoft forgot to hump their vendor prefix (#9572) -function camelCase( string ) { - return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase ); -} -var acceptData = function( owner ) { - - // Accepts only: - // - Node - // - Node.ELEMENT_NODE - // - Node.DOCUMENT_NODE - // - Object - // - Any - return owner.nodeType === 1 || owner.nodeType === 9 || !( +owner.nodeType ); -}; - - - - -function Data() { - this.expando = jQuery.expando + Data.uid++; -} - -Data.uid = 1; - -Data.prototype = { - - cache: function( owner ) { - - // Check if the owner object already has a cache - var value = owner[ this.expando ]; - - // If not, create one - if ( !value ) { - value = {}; - - // We can accept data for non-element nodes in modern browsers, - // but we should not, see #8335. - // Always return an empty object. - if ( acceptData( owner ) ) { - - // If it is a node unlikely to be stringify-ed or looped over - // use plain assignment - if ( owner.nodeType ) { - owner[ this.expando ] = value; - - // Otherwise secure it in a non-enumerable property - // configurable must be true to allow the property to be - // deleted when data is removed - } else { - Object.defineProperty( owner, this.expando, { - value: value, - configurable: true - } ); - } - } - } - - return value; - }, - set: function( owner, data, value ) { - var prop, - cache = this.cache( owner ); - - // Handle: [ owner, key, value ] args - // Always use camelCase key (gh-2257) - if ( typeof data === "string" ) { - cache[ camelCase( data ) ] = value; - - // Handle: [ owner, { properties } ] args - } else { - - // Copy the properties one-by-one to the cache object - for ( prop in data ) { - cache[ camelCase( prop ) ] = data[ prop ]; - } - } - return cache; - }, - get: function( owner, key ) { - return key === undefined ? - this.cache( owner ) : - - // Always use camelCase key (gh-2257) - owner[ this.expando ] && owner[ this.expando ][ camelCase( key ) ]; - }, - access: function( owner, key, value ) { - - // In cases where either: - // - // 1. No key was specified - // 2. A string key was specified, but no value provided - // - // Take the "read" path and allow the get method to determine - // which value to return, respectively either: - // - // 1. The entire cache object - // 2. The data stored at the key - // - if ( key === undefined || - ( ( key && typeof key === "string" ) && value === undefined ) ) { - - return this.get( owner, key ); - } - - // When the key is not a string, or both a key and value - // are specified, set or extend (existing objects) with either: - // - // 1. An object of properties - // 2. A key and value - // - this.set( owner, key, value ); - - // Since the "set" path can have two possible entry points - // return the expected data based on which path was taken[*] - return value !== undefined ? value : key; - }, - remove: function( owner, key ) { - var i, - cache = owner[ this.expando ]; - - if ( cache === undefined ) { - return; - } - - if ( key !== undefined ) { - - // Support array or space separated string of keys - if ( Array.isArray( key ) ) { - - // If key is an array of keys... - // We always set camelCase keys, so remove that. - key = key.map( camelCase ); - } else { - key = camelCase( key ); - - // If a key with the spaces exists, use it. - // Otherwise, create an array by matching non-whitespace - key = key in cache ? - [ key ] : - ( key.match( rnothtmlwhite ) || [] ); - } - - i = key.length; - - while ( i-- ) { - delete cache[ key[ i ] ]; - } - } - - // Remove the expando if there's no more data - if ( key === undefined || jQuery.isEmptyObject( cache ) ) { - - // Support: Chrome <=35 - 45 - // Webkit & Blink performance suffers when deleting properties - // from DOM nodes, so set to undefined instead - // https://bugs.chromium.org/p/chromium/issues/detail?id=378607 (bug restricted) - if ( owner.nodeType ) { - owner[ this.expando ] = undefined; - } else { - delete owner[ this.expando ]; - } - } - }, - hasData: function( owner ) { - var cache = owner[ this.expando ]; - return cache !== undefined && !jQuery.isEmptyObject( cache ); - } -}; -var dataPriv = new Data(); - -var dataUser = new Data(); - - - -// Implementation Summary -// -// 1. Enforce API surface and semantic compatibility with 1.9.x branch -// 2. Improve the module's maintainability by reducing the storage -// paths to a single mechanism. -// 3. Use the same single mechanism to support "private" and "user" data. -// 4. _Never_ expose "private" data to user code (TODO: Drop _data, _removeData) -// 5. Avoid exposing implementation details on user objects (eg. expando properties) -// 6. Provide a clear path for implementation upgrade to WeakMap in 2014 - -var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/, - rmultiDash = /[A-Z]/g; - -function getData( data ) { - if ( data === "true" ) { - return true; - } - - if ( data === "false" ) { - return false; - } - - if ( data === "null" ) { - return null; - } - - // Only convert to a number if it doesn't change the string - if ( data === +data + "" ) { - return +data; - } - - if ( rbrace.test( data ) ) { - return JSON.parse( data ); - } - - return data; -} - -function dataAttr( elem, key, data ) { - var name; - - // If nothing was found internally, try to fetch any - // data from the HTML5 data-* attribute - if ( data === undefined && elem.nodeType === 1 ) { - name = "data-" + key.replace( rmultiDash, "-$&" ).toLowerCase(); - data = elem.getAttribute( name ); - - if ( typeof data === "string" ) { - try { - data = getData( data ); - } catch ( e ) {} - - // Make sure we set the data so it isn't changed later - dataUser.set( elem, key, data ); - } else { - data = undefined; - } - } - return data; -} - -jQuery.extend( { - hasData: function( elem ) { - return dataUser.hasData( elem ) || dataPriv.hasData( elem ); - }, - - data: function( elem, name, data ) { - return dataUser.access( elem, name, data ); - }, - - removeData: function( elem, name ) { - dataUser.remove( elem, name ); - }, - - // TODO: Now that all calls to _data and _removeData have been replaced - // with direct calls to dataPriv methods, these can be deprecated. - _data: function( elem, name, data ) { - return dataPriv.access( elem, name, data ); - }, - - _removeData: function( elem, name ) { - dataPriv.remove( elem, name ); - } -} ); - -jQuery.fn.extend( { - data: function( key, value ) { - var i, name, data, - elem = this[ 0 ], - attrs = elem && elem.attributes; - - // Gets all values - if ( key === undefined ) { - if ( this.length ) { - data = dataUser.get( elem ); - - if ( elem.nodeType === 1 && !dataPriv.get( elem, "hasDataAttrs" ) ) { - i = attrs.length; - while ( i-- ) { - - // Support: IE 11 only - // The attrs elements can be null (#14894) - if ( attrs[ i ] ) { - name = attrs[ i ].name; - if ( name.indexOf( "data-" ) === 0 ) { - name = camelCase( name.slice( 5 ) ); - dataAttr( elem, name, data[ name ] ); - } - } - } - dataPriv.set( elem, "hasDataAttrs", true ); - } - } - - return data; - } - - // Sets multiple values - if ( typeof key === "object" ) { - return this.each( function() { - dataUser.set( this, key ); - } ); - } - - return access( this, function( value ) { - var data; - - // The calling jQuery object (element matches) is not empty - // (and therefore has an element appears at this[ 0 ]) and the - // `value` parameter was not undefined. An empty jQuery object - // will result in `undefined` for elem = this[ 0 ] which will - // throw an exception if an attempt to read a data cache is made. - if ( elem && value === undefined ) { - - // Attempt to get data from the cache - // The key will always be camelCased in Data - data = dataUser.get( elem, key ); - if ( data !== undefined ) { - return data; - } - - // Attempt to "discover" the data in - // HTML5 custom data-* attrs - data = dataAttr( elem, key ); - if ( data !== undefined ) { - return data; - } - - // We tried really hard, but the data doesn't exist. - return; - } - - // Set the data... - this.each( function() { - - // We always store the camelCased key - dataUser.set( this, key, value ); - } ); - }, null, value, arguments.length > 1, null, true ); - }, - - removeData: function( key ) { - return this.each( function() { - dataUser.remove( this, key ); - } ); - } -} ); - - -jQuery.extend( { - queue: function( elem, type, data ) { - var queue; - - if ( elem ) { - type = ( type || "fx" ) + "queue"; - queue = dataPriv.get( elem, type ); - - // Speed up dequeue by getting out quickly if this is just a lookup - if ( data ) { - if ( !queue || Array.isArray( data ) ) { - queue = dataPriv.access( elem, type, jQuery.makeArray( data ) ); - } else { - queue.push( data ); - } - } - return queue || []; - } - }, - - dequeue: function( elem, type ) { - type = type || "fx"; - - var queue = jQuery.queue( elem, type ), - startLength = queue.length, - fn = queue.shift(), - hooks = jQuery._queueHooks( elem, type ), - next = function() { - jQuery.dequeue( elem, type ); - }; - - // If the fx queue is dequeued, always remove the progress sentinel - if ( fn === "inprogress" ) { - fn = queue.shift(); - startLength--; - } - - if ( fn ) { - - // Add a progress sentinel to prevent the fx queue from being - // automatically dequeued - if ( type === "fx" ) { - queue.unshift( "inprogress" ); - } - - // Clear up the last queue stop function - delete hooks.stop; - fn.call( elem, next, hooks ); - } - - if ( !startLength && hooks ) { - hooks.empty.fire(); - } - }, - - // Not public - generate a queueHooks object, or return the current one - _queueHooks: function( elem, type ) { - var key = type + "queueHooks"; - return dataPriv.get( elem, key ) || dataPriv.access( elem, key, { - empty: jQuery.Callbacks( "once memory" ).add( function() { - dataPriv.remove( elem, [ type + "queue", key ] ); - } ) - } ); - } -} ); - -jQuery.fn.extend( { - queue: function( type, data ) { - var setter = 2; - - if ( typeof type !== "string" ) { - data = type; - type = "fx"; - setter--; - } - - if ( arguments.length < setter ) { - return jQuery.queue( this[ 0 ], type ); - } - - return data === undefined ? - this : - this.each( function() { - var queue = jQuery.queue( this, type, data ); - - // Ensure a hooks for this queue - jQuery._queueHooks( this, type ); - - if ( type === "fx" && queue[ 0 ] !== "inprogress" ) { - jQuery.dequeue( this, type ); - } - } ); - }, - dequeue: function( type ) { - return this.each( function() { - jQuery.dequeue( this, type ); - } ); - }, - clearQueue: function( type ) { - return this.queue( type || "fx", [] ); - }, - - // Get a promise resolved when queues of a certain type - // are emptied (fx is the type by default) - promise: function( type, obj ) { - var tmp, - count = 1, - defer = jQuery.Deferred(), - elements = this, - i = this.length, - resolve = function() { - if ( !( --count ) ) { - defer.resolveWith( elements, [ elements ] ); - } - }; - - if ( typeof type !== "string" ) { - obj = type; - type = undefined; - } - type = type || "fx"; - - while ( i-- ) { - tmp = dataPriv.get( elements[ i ], type + "queueHooks" ); - if ( tmp && tmp.empty ) { - count++; - tmp.empty.add( resolve ); - } - } - resolve(); - return defer.promise( obj ); - } -} ); -var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source; - -var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" ); - - -var cssExpand = [ "Top", "Right", "Bottom", "Left" ]; - -var documentElement = document.documentElement; - - - - var isAttached = function( elem ) { - return jQuery.contains( elem.ownerDocument, elem ); - }, - composed = { composed: true }; - - // Support: IE 9 - 11+, Edge 12 - 18+, iOS 10.0 - 10.2 only - // Check attachment across shadow DOM boundaries when possible (gh-3504) - // Support: iOS 10.0-10.2 only - // Early iOS 10 versions support `attachShadow` but not `getRootNode`, - // leading to errors. We need to check for `getRootNode`. - if ( documentElement.getRootNode ) { - isAttached = function( elem ) { - return jQuery.contains( elem.ownerDocument, elem ) || - elem.getRootNode( composed ) === elem.ownerDocument; - }; - } -var isHiddenWithinTree = function( elem, el ) { - - // isHiddenWithinTree might be called from jQuery#filter function; - // in that case, element will be second argument - elem = el || elem; - - // Inline style trumps all - return elem.style.display === "none" || - elem.style.display === "" && - - // Otherwise, check computed style - // Support: Firefox <=43 - 45 - // Disconnected elements can have computed display: none, so first confirm that elem is - // in the document. - isAttached( elem ) && - - jQuery.css( elem, "display" ) === "none"; - }; - - - -function adjustCSS( elem, prop, valueParts, tween ) { - var adjusted, scale, - maxIterations = 20, - currentValue = tween ? - function() { - return tween.cur(); - } : - function() { - return jQuery.css( elem, prop, "" ); - }, - initial = currentValue(), - unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ), - - // Starting value computation is required for potential unit mismatches - initialInUnit = elem.nodeType && - ( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) && - rcssNum.exec( jQuery.css( elem, prop ) ); - - if ( initialInUnit && initialInUnit[ 3 ] !== unit ) { - - // Support: Firefox <=54 - // Halve the iteration target value to prevent interference from CSS upper bounds (gh-2144) - initial = initial / 2; - - // Trust units reported by jQuery.css - unit = unit || initialInUnit[ 3 ]; - - // Iteratively approximate from a nonzero starting point - initialInUnit = +initial || 1; - - while ( maxIterations-- ) { - - // Evaluate and update our best guess (doubling guesses that zero out). - // Finish if the scale equals or crosses 1 (making the old*new product non-positive). - jQuery.style( elem, prop, initialInUnit + unit ); - if ( ( 1 - scale ) * ( 1 - ( scale = currentValue() / initial || 0.5 ) ) <= 0 ) { - maxIterations = 0; - } - initialInUnit = initialInUnit / scale; - - } - - initialInUnit = initialInUnit * 2; - jQuery.style( elem, prop, initialInUnit + unit ); - - // Make sure we update the tween properties later on - valueParts = valueParts || []; - } - - if ( valueParts ) { - initialInUnit = +initialInUnit || +initial || 0; - - // Apply relative offset (+=/-=) if specified - adjusted = valueParts[ 1 ] ? - initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] : - +valueParts[ 2 ]; - if ( tween ) { - tween.unit = unit; - tween.start = initialInUnit; - tween.end = adjusted; - } - } - return adjusted; -} - - -var defaultDisplayMap = {}; - -function getDefaultDisplay( elem ) { - var temp, - doc = elem.ownerDocument, - nodeName = elem.nodeName, - display = defaultDisplayMap[ nodeName ]; - - if ( display ) { - return display; - } - - temp = doc.body.appendChild( doc.createElement( nodeName ) ); - display = jQuery.css( temp, "display" ); - - temp.parentNode.removeChild( temp ); - - if ( display === "none" ) { - display = "block"; - } - defaultDisplayMap[ nodeName ] = display; - - return display; -} - -function showHide( elements, show ) { - var display, elem, - values = [], - index = 0, - length = elements.length; - - // Determine new display value for elements that need to change - for ( ; index < length; index++ ) { - elem = elements[ index ]; - if ( !elem.style ) { - continue; - } - - display = elem.style.display; - if ( show ) { - - // Since we force visibility upon cascade-hidden elements, an immediate (and slow) - // check is required in this first loop unless we have a nonempty display value (either - // inline or about-to-be-restored) - if ( display === "none" ) { - values[ index ] = dataPriv.get( elem, "display" ) || null; - if ( !values[ index ] ) { - elem.style.display = ""; - } - } - if ( elem.style.display === "" && isHiddenWithinTree( elem ) ) { - values[ index ] = getDefaultDisplay( elem ); - } - } else { - if ( display !== "none" ) { - values[ index ] = "none"; - - // Remember what we're overwriting - dataPriv.set( elem, "display", display ); - } - } - } - - // Set the display of the elements in a second loop to avoid constant reflow - for ( index = 0; index < length; index++ ) { - if ( values[ index ] != null ) { - elements[ index ].style.display = values[ index ]; - } - } - - return elements; -} - -jQuery.fn.extend( { - show: function() { - return showHide( this, true ); - }, - hide: function() { - return showHide( this ); - }, - toggle: function( state ) { - if ( typeof state === "boolean" ) { - return state ? this.show() : this.hide(); - } - - return this.each( function() { - if ( isHiddenWithinTree( this ) ) { - jQuery( this ).show(); - } else { - jQuery( this ).hide(); - } - } ); - } -} ); -var rcheckableType = ( /^(?:checkbox|radio)$/i ); - -var rtagName = ( /<([a-z][^\/\0>\x20\t\r\n\f]*)/i ); - -var rscriptType = ( /^$|^module$|\/(?:java|ecma)script/i ); - - - -( function() { - var fragment = document.createDocumentFragment(), - div = fragment.appendChild( document.createElement( "div" ) ), - input = document.createElement( "input" ); - - // Support: Android 4.0 - 4.3 only - // Check state lost if the name is set (#11217) - // Support: Windows Web Apps (WWA) - // `name` and `type` must use .setAttribute for WWA (#14901) - input.setAttribute( "type", "radio" ); - input.setAttribute( "checked", "checked" ); - input.setAttribute( "name", "t" ); - - div.appendChild( input ); - - // Support: Android <=4.1 only - // Older WebKit doesn't clone checked state correctly in fragments - support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked; - - // Support: IE <=11 only - // Make sure textarea (and checkbox) defaultValue is properly cloned - div.innerHTML = ""; - support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue; - - // Support: IE <=9 only - // IE <=9 replaces "; - support.option = !!div.lastChild; -} )(); - - -// We have to close these tags to support XHTML (#13200) -var wrapMap = { - - // XHTML parsers do not magically insert elements in the - // same way that tag soup parsers do. So we cannot shorten - // this by omitting or other required elements. - thead: [ 1, "", "
" ], - col: [ 2, "", "
" ], - tr: [ 2, "", "
" ], - td: [ 3, "", "
" ], - - _default: [ 0, "", "" ] -}; - -wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead; -wrapMap.th = wrapMap.td; - -// Support: IE <=9 only -if ( !support.option ) { - wrapMap.optgroup = wrapMap.option = [ 1, "" ]; -} - - -function getAll( context, tag ) { - - // Support: IE <=9 - 11 only - // Use typeof to avoid zero-argument method invocation on host objects (#15151) - var ret; - - if ( typeof context.getElementsByTagName !== "undefined" ) { - ret = context.getElementsByTagName( tag || "*" ); - - } else if ( typeof context.querySelectorAll !== "undefined" ) { - ret = context.querySelectorAll( tag || "*" ); - - } else { - ret = []; - } - - if ( tag === undefined || tag && nodeName( context, tag ) ) { - return jQuery.merge( [ context ], ret ); - } - - return ret; -} - - -// Mark scripts as having already been evaluated -function setGlobalEval( elems, refElements ) { - var i = 0, - l = elems.length; - - for ( ; i < l; i++ ) { - dataPriv.set( - elems[ i ], - "globalEval", - !refElements || dataPriv.get( refElements[ i ], "globalEval" ) - ); - } -} - - -var rhtml = /<|&#?\w+;/; - -function buildFragment( elems, context, scripts, selection, ignored ) { - var elem, tmp, tag, wrap, attached, j, - fragment = context.createDocumentFragment(), - nodes = [], - i = 0, - l = elems.length; - - for ( ; i < l; i++ ) { - elem = elems[ i ]; - - if ( elem || elem === 0 ) { - - // Add nodes directly - if ( toType( elem ) === "object" ) { - - // Support: Android <=4.0 only, PhantomJS 1 only - // push.apply(_, arraylike) throws on ancient WebKit - jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem ); - - // Convert non-html into a text node - } else if ( !rhtml.test( elem ) ) { - nodes.push( context.createTextNode( elem ) ); - - // Convert html into DOM nodes - } else { - tmp = tmp || fragment.appendChild( context.createElement( "div" ) ); - - // Deserialize a standard representation - tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase(); - wrap = wrapMap[ tag ] || wrapMap._default; - tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ]; - - // Descend through wrappers to the right content - j = wrap[ 0 ]; - while ( j-- ) { - tmp = tmp.lastChild; - } - - // Support: Android <=4.0 only, PhantomJS 1 only - // push.apply(_, arraylike) throws on ancient WebKit - jQuery.merge( nodes, tmp.childNodes ); - - // Remember the top-level container - tmp = fragment.firstChild; - - // Ensure the created nodes are orphaned (#12392) - tmp.textContent = ""; - } - } - } - - // Remove wrapper from fragment - fragment.textContent = ""; - - i = 0; - while ( ( elem = nodes[ i++ ] ) ) { - - // Skip elements already in the context collection (trac-4087) - if ( selection && jQuery.inArray( elem, selection ) > -1 ) { - if ( ignored ) { - ignored.push( elem ); - } - continue; - } - - attached = isAttached( elem ); - - // Append to fragment - tmp = getAll( fragment.appendChild( elem ), "script" ); - - // Preserve script evaluation history - if ( attached ) { - setGlobalEval( tmp ); - } - - // Capture executables - if ( scripts ) { - j = 0; - while ( ( elem = tmp[ j++ ] ) ) { - if ( rscriptType.test( elem.type || "" ) ) { - scripts.push( elem ); - } - } - } - } - - return fragment; -} - - -var - rkeyEvent = /^key/, - rmouseEvent = /^(?:mouse|pointer|contextmenu|drag|drop)|click/, - rtypenamespace = /^([^.]*)(?:\.(.+)|)/; - -function returnTrue() { - return true; -} - -function returnFalse() { - return false; -} - -// Support: IE <=9 - 11+ -// focus() and blur() are asynchronous, except when they are no-op. -// So expect focus to be synchronous when the element is already active, -// and blur to be synchronous when the element is not already active. -// (focus and blur are always synchronous in other supported browsers, -// this just defines when we can count on it). -function expectSync( elem, type ) { - return ( elem === safeActiveElement() ) === ( type === "focus" ); -} - -// Support: IE <=9 only -// Accessing document.activeElement can throw unexpectedly -// https://bugs.jquery.com/ticket/13393 -function safeActiveElement() { - try { - return document.activeElement; - } catch ( err ) { } -} - -function on( elem, types, selector, data, fn, one ) { - var origFn, type; - - // Types can be a map of types/handlers - if ( typeof types === "object" ) { - - // ( types-Object, selector, data ) - if ( typeof selector !== "string" ) { - - // ( types-Object, data ) - data = data || selector; - selector = undefined; - } - for ( type in types ) { - on( elem, type, selector, data, types[ type ], one ); - } - return elem; - } - - if ( data == null && fn == null ) { - - // ( types, fn ) - fn = selector; - data = selector = undefined; - } else if ( fn == null ) { - if ( typeof selector === "string" ) { - - // ( types, selector, fn ) - fn = data; - data = undefined; - } else { - - // ( types, data, fn ) - fn = data; - data = selector; - selector = undefined; - } - } - if ( fn === false ) { - fn = returnFalse; - } else if ( !fn ) { - return elem; - } - - if ( one === 1 ) { - origFn = fn; - fn = function( event ) { - - // Can use an empty set, since event contains the info - jQuery().off( event ); - return origFn.apply( this, arguments ); - }; - - // Use same guid so caller can remove using origFn - fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ ); - } - return elem.each( function() { - jQuery.event.add( this, types, fn, data, selector ); - } ); -} - -/* - * Helper functions for managing events -- not part of the public interface. - * Props to Dean Edwards' addEvent library for many of the ideas. - */ -jQuery.event = { - - global: {}, - - add: function( elem, types, handler, data, selector ) { - - var handleObjIn, eventHandle, tmp, - events, t, handleObj, - special, handlers, type, namespaces, origType, - elemData = dataPriv.get( elem ); - - // Only attach events to objects that accept data - if ( !acceptData( elem ) ) { - return; - } - - // Caller can pass in an object of custom data in lieu of the handler - if ( handler.handler ) { - handleObjIn = handler; - handler = handleObjIn.handler; - selector = handleObjIn.selector; - } - - // Ensure that invalid selectors throw exceptions at attach time - // Evaluate against documentElement in case elem is a non-element node (e.g., document) - if ( selector ) { - jQuery.find.matchesSelector( documentElement, selector ); - } - - // Make sure that the handler has a unique ID, used to find/remove it later - if ( !handler.guid ) { - handler.guid = jQuery.guid++; - } - - // Init the element's event structure and main handler, if this is the first - if ( !( events = elemData.events ) ) { - events = elemData.events = Object.create( null ); - } - if ( !( eventHandle = elemData.handle ) ) { - eventHandle = elemData.handle = function( e ) { - - // Discard the second event of a jQuery.event.trigger() and - // when an event is called after a page has unloaded - return typeof jQuery !== "undefined" && jQuery.event.triggered !== e.type ? - jQuery.event.dispatch.apply( elem, arguments ) : undefined; - }; - } - - // Handle multiple events separated by a space - types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; - t = types.length; - while ( t-- ) { - tmp = rtypenamespace.exec( types[ t ] ) || []; - type = origType = tmp[ 1 ]; - namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); - - // There *must* be a type, no attaching namespace-only handlers - if ( !type ) { - continue; - } - - // If event changes its type, use the special event handlers for the changed type - special = jQuery.event.special[ type ] || {}; - - // If selector defined, determine special event api type, otherwise given type - type = ( selector ? special.delegateType : special.bindType ) || type; - - // Update special based on newly reset type - special = jQuery.event.special[ type ] || {}; - - // handleObj is passed to all event handlers - handleObj = jQuery.extend( { - type: type, - origType: origType, - data: data, - handler: handler, - guid: handler.guid, - selector: selector, - needsContext: selector && jQuery.expr.match.needsContext.test( selector ), - namespace: namespaces.join( "." ) - }, handleObjIn ); - - // Init the event handler queue if we're the first - if ( !( handlers = events[ type ] ) ) { - handlers = events[ type ] = []; - handlers.delegateCount = 0; - - // Only use addEventListener if the special events handler returns false - if ( !special.setup || - special.setup.call( elem, data, namespaces, eventHandle ) === false ) { - - if ( elem.addEventListener ) { - elem.addEventListener( type, eventHandle ); - } - } - } - - if ( special.add ) { - special.add.call( elem, handleObj ); - - if ( !handleObj.handler.guid ) { - handleObj.handler.guid = handler.guid; - } - } - - // Add to the element's handler list, delegates in front - if ( selector ) { - handlers.splice( handlers.delegateCount++, 0, handleObj ); - } else { - handlers.push( handleObj ); - } - - // Keep track of which events have ever been used, for event optimization - jQuery.event.global[ type ] = true; - } - - }, - - // Detach an event or set of events from an element - remove: function( elem, types, handler, selector, mappedTypes ) { - - var j, origCount, tmp, - events, t, handleObj, - special, handlers, type, namespaces, origType, - elemData = dataPriv.hasData( elem ) && dataPriv.get( elem ); - - if ( !elemData || !( events = elemData.events ) ) { - return; - } - - // Once for each type.namespace in types; type may be omitted - types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; - t = types.length; - while ( t-- ) { - tmp = rtypenamespace.exec( types[ t ] ) || []; - type = origType = tmp[ 1 ]; - namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); - - // Unbind all events (on this namespace, if provided) for the element - if ( !type ) { - for ( type in events ) { - jQuery.event.remove( elem, type + types[ t ], handler, selector, true ); - } - continue; - } - - special = jQuery.event.special[ type ] || {}; - type = ( selector ? special.delegateType : special.bindType ) || type; - handlers = events[ type ] || []; - tmp = tmp[ 2 ] && - new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ); - - // Remove matching events - origCount = j = handlers.length; - while ( j-- ) { - handleObj = handlers[ j ]; - - if ( ( mappedTypes || origType === handleObj.origType ) && - ( !handler || handler.guid === handleObj.guid ) && - ( !tmp || tmp.test( handleObj.namespace ) ) && - ( !selector || selector === handleObj.selector || - selector === "**" && handleObj.selector ) ) { - handlers.splice( j, 1 ); - - if ( handleObj.selector ) { - handlers.delegateCount--; - } - if ( special.remove ) { - special.remove.call( elem, handleObj ); - } - } - } - - // Remove generic event handler if we removed something and no more handlers exist - // (avoids potential for endless recursion during removal of special event handlers) - if ( origCount && !handlers.length ) { - if ( !special.teardown || - special.teardown.call( elem, namespaces, elemData.handle ) === false ) { - - jQuery.removeEvent( elem, type, elemData.handle ); - } - - delete events[ type ]; - } - } - - // Remove data and the expando if it's no longer used - if ( jQuery.isEmptyObject( events ) ) { - dataPriv.remove( elem, "handle events" ); - } - }, - - dispatch: function( nativeEvent ) { - - var i, j, ret, matched, handleObj, handlerQueue, - args = new Array( arguments.length ), - - // Make a writable jQuery.Event from the native event object - event = jQuery.event.fix( nativeEvent ), - - handlers = ( - dataPriv.get( this, "events" ) || Object.create( null ) - )[ event.type ] || [], - special = jQuery.event.special[ event.type ] || {}; - - // Use the fix-ed jQuery.Event rather than the (read-only) native event - args[ 0 ] = event; - - for ( i = 1; i < arguments.length; i++ ) { - args[ i ] = arguments[ i ]; - } - - event.delegateTarget = this; - - // Call the preDispatch hook for the mapped type, and let it bail if desired - if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) { - return; - } - - // Determine handlers - handlerQueue = jQuery.event.handlers.call( this, event, handlers ); - - // Run delegates first; they may want to stop propagation beneath us - i = 0; - while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) { - event.currentTarget = matched.elem; - - j = 0; - while ( ( handleObj = matched.handlers[ j++ ] ) && - !event.isImmediatePropagationStopped() ) { - - // If the event is namespaced, then each handler is only invoked if it is - // specially universal or its namespaces are a superset of the event's. - if ( !event.rnamespace || handleObj.namespace === false || - event.rnamespace.test( handleObj.namespace ) ) { - - event.handleObj = handleObj; - event.data = handleObj.data; - - ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle || - handleObj.handler ).apply( matched.elem, args ); - - if ( ret !== undefined ) { - if ( ( event.result = ret ) === false ) { - event.preventDefault(); - event.stopPropagation(); - } - } - } - } - } - - // Call the postDispatch hook for the mapped type - if ( special.postDispatch ) { - special.postDispatch.call( this, event ); - } - - return event.result; - }, - - handlers: function( event, handlers ) { - var i, handleObj, sel, matchedHandlers, matchedSelectors, - handlerQueue = [], - delegateCount = handlers.delegateCount, - cur = event.target; - - // Find delegate handlers - if ( delegateCount && - - // Support: IE <=9 - // Black-hole SVG instance trees (trac-13180) - cur.nodeType && - - // Support: Firefox <=42 - // Suppress spec-violating clicks indicating a non-primary pointer button (trac-3861) - // https://www.w3.org/TR/DOM-Level-3-Events/#event-type-click - // Support: IE 11 only - // ...but not arrow key "clicks" of radio inputs, which can have `button` -1 (gh-2343) - !( event.type === "click" && event.button >= 1 ) ) { - - for ( ; cur !== this; cur = cur.parentNode || this ) { - - // Don't check non-elements (#13208) - // Don't process clicks on disabled elements (#6911, #8165, #11382, #11764) - if ( cur.nodeType === 1 && !( event.type === "click" && cur.disabled === true ) ) { - matchedHandlers = []; - matchedSelectors = {}; - for ( i = 0; i < delegateCount; i++ ) { - handleObj = handlers[ i ]; - - // Don't conflict with Object.prototype properties (#13203) - sel = handleObj.selector + " "; - - if ( matchedSelectors[ sel ] === undefined ) { - matchedSelectors[ sel ] = handleObj.needsContext ? - jQuery( sel, this ).index( cur ) > -1 : - jQuery.find( sel, this, null, [ cur ] ).length; - } - if ( matchedSelectors[ sel ] ) { - matchedHandlers.push( handleObj ); - } - } - if ( matchedHandlers.length ) { - handlerQueue.push( { elem: cur, handlers: matchedHandlers } ); - } - } - } - } - - // Add the remaining (directly-bound) handlers - cur = this; - if ( delegateCount < handlers.length ) { - handlerQueue.push( { elem: cur, handlers: handlers.slice( delegateCount ) } ); - } - - return handlerQueue; - }, - - addProp: function( name, hook ) { - Object.defineProperty( jQuery.Event.prototype, name, { - enumerable: true, - configurable: true, - - get: isFunction( hook ) ? - function() { - if ( this.originalEvent ) { - return hook( this.originalEvent ); - } - } : - function() { - if ( this.originalEvent ) { - return this.originalEvent[ name ]; - } - }, - - set: function( value ) { - Object.defineProperty( this, name, { - enumerable: true, - configurable: true, - writable: true, - value: value - } ); - } - } ); - }, - - fix: function( originalEvent ) { - return originalEvent[ jQuery.expando ] ? - originalEvent : - new jQuery.Event( originalEvent ); - }, - - special: { - load: { - - // Prevent triggered image.load events from bubbling to window.load - noBubble: true - }, - click: { - - // Utilize native event to ensure correct state for checkable inputs - setup: function( data ) { - - // For mutual compressibility with _default, replace `this` access with a local var. - // `|| data` is dead code meant only to preserve the variable through minification. - var el = this || data; - - // Claim the first handler - if ( rcheckableType.test( el.type ) && - el.click && nodeName( el, "input" ) ) { - - // dataPriv.set( el, "click", ... ) - leverageNative( el, "click", returnTrue ); - } - - // Return false to allow normal processing in the caller - return false; - }, - trigger: function( data ) { - - // For mutual compressibility with _default, replace `this` access with a local var. - // `|| data` is dead code meant only to preserve the variable through minification. - var el = this || data; - - // Force setup before triggering a click - if ( rcheckableType.test( el.type ) && - el.click && nodeName( el, "input" ) ) { - - leverageNative( el, "click" ); - } - - // Return non-false to allow normal event-path propagation - return true; - }, - - // For cross-browser consistency, suppress native .click() on links - // Also prevent it if we're currently inside a leveraged native-event stack - _default: function( event ) { - var target = event.target; - return rcheckableType.test( target.type ) && - target.click && nodeName( target, "input" ) && - dataPriv.get( target, "click" ) || - nodeName( target, "a" ); - } - }, - - beforeunload: { - postDispatch: function( event ) { - - // Support: Firefox 20+ - // Firefox doesn't alert if the returnValue field is not set. - if ( event.result !== undefined && event.originalEvent ) { - event.originalEvent.returnValue = event.result; - } - } - } - } -}; - -// Ensure the presence of an event listener that handles manually-triggered -// synthetic events by interrupting progress until reinvoked in response to -// *native* events that it fires directly, ensuring that state changes have -// already occurred before other listeners are invoked. -function leverageNative( el, type, expectSync ) { - - // Missing expectSync indicates a trigger call, which must force setup through jQuery.event.add - if ( !expectSync ) { - if ( dataPriv.get( el, type ) === undefined ) { - jQuery.event.add( el, type, returnTrue ); - } - return; - } - - // Register the controller as a special universal handler for all event namespaces - dataPriv.set( el, type, false ); - jQuery.event.add( el, type, { - namespace: false, - handler: function( event ) { - var notAsync, result, - saved = dataPriv.get( this, type ); - - if ( ( event.isTrigger & 1 ) && this[ type ] ) { - - // Interrupt processing of the outer synthetic .trigger()ed event - // Saved data should be false in such cases, but might be a leftover capture object - // from an async native handler (gh-4350) - if ( !saved.length ) { - - // Store arguments for use when handling the inner native event - // There will always be at least one argument (an event object), so this array - // will not be confused with a leftover capture object. - saved = slice.call( arguments ); - dataPriv.set( this, type, saved ); - - // Trigger the native event and capture its result - // Support: IE <=9 - 11+ - // focus() and blur() are asynchronous - notAsync = expectSync( this, type ); - this[ type ](); - result = dataPriv.get( this, type ); - if ( saved !== result || notAsync ) { - dataPriv.set( this, type, false ); - } else { - result = {}; - } - if ( saved !== result ) { - - // Cancel the outer synthetic event - event.stopImmediatePropagation(); - event.preventDefault(); - return result.value; - } - - // If this is an inner synthetic event for an event with a bubbling surrogate - // (focus or blur), assume that the surrogate already propagated from triggering the - // native event and prevent that from happening again here. - // This technically gets the ordering wrong w.r.t. to `.trigger()` (in which the - // bubbling surrogate propagates *after* the non-bubbling base), but that seems - // less bad than duplication. - } else if ( ( jQuery.event.special[ type ] || {} ).delegateType ) { - event.stopPropagation(); - } - - // If this is a native event triggered above, everything is now in order - // Fire an inner synthetic event with the original arguments - } else if ( saved.length ) { - - // ...and capture the result - dataPriv.set( this, type, { - value: jQuery.event.trigger( - - // Support: IE <=9 - 11+ - // Extend with the prototype to reset the above stopImmediatePropagation() - jQuery.extend( saved[ 0 ], jQuery.Event.prototype ), - saved.slice( 1 ), - this - ) - } ); - - // Abort handling of the native event - event.stopImmediatePropagation(); - } - } - } ); -} - -jQuery.removeEvent = function( elem, type, handle ) { - - // This "if" is needed for plain objects - if ( elem.removeEventListener ) { - elem.removeEventListener( type, handle ); - } -}; - -jQuery.Event = function( src, props ) { - - // Allow instantiation without the 'new' keyword - if ( !( this instanceof jQuery.Event ) ) { - return new jQuery.Event( src, props ); - } - - // Event object - if ( src && src.type ) { - this.originalEvent = src; - this.type = src.type; - - // Events bubbling up the document may have been marked as prevented - // by a handler lower down the tree; reflect the correct value. - this.isDefaultPrevented = src.defaultPrevented || - src.defaultPrevented === undefined && - - // Support: Android <=2.3 only - src.returnValue === false ? - returnTrue : - returnFalse; - - // Create target properties - // Support: Safari <=6 - 7 only - // Target should not be a text node (#504, #13143) - this.target = ( src.target && src.target.nodeType === 3 ) ? - src.target.parentNode : - src.target; - - this.currentTarget = src.currentTarget; - this.relatedTarget = src.relatedTarget; - - // Event type - } else { - this.type = src; - } - - // Put explicitly provided properties onto the event object - if ( props ) { - jQuery.extend( this, props ); - } - - // Create a timestamp if incoming event doesn't have one - this.timeStamp = src && src.timeStamp || Date.now(); - - // Mark it as fixed - this[ jQuery.expando ] = true; -}; - -// jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding -// https://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html -jQuery.Event.prototype = { - constructor: jQuery.Event, - isDefaultPrevented: returnFalse, - isPropagationStopped: returnFalse, - isImmediatePropagationStopped: returnFalse, - isSimulated: false, - - preventDefault: function() { - var e = this.originalEvent; - - this.isDefaultPrevented = returnTrue; - - if ( e && !this.isSimulated ) { - e.preventDefault(); - } - }, - stopPropagation: function() { - var e = this.originalEvent; - - this.isPropagationStopped = returnTrue; - - if ( e && !this.isSimulated ) { - e.stopPropagation(); - } - }, - stopImmediatePropagation: function() { - var e = this.originalEvent; - - this.isImmediatePropagationStopped = returnTrue; - - if ( e && !this.isSimulated ) { - e.stopImmediatePropagation(); - } - - this.stopPropagation(); - } -}; - -// Includes all common event props including KeyEvent and MouseEvent specific props -jQuery.each( { - altKey: true, - bubbles: true, - cancelable: true, - changedTouches: true, - ctrlKey: true, - detail: true, - eventPhase: true, - metaKey: true, - pageX: true, - pageY: true, - shiftKey: true, - view: true, - "char": true, - code: true, - charCode: true, - key: true, - keyCode: true, - button: true, - buttons: true, - clientX: true, - clientY: true, - offsetX: true, - offsetY: true, - pointerId: true, - pointerType: true, - screenX: true, - screenY: true, - targetTouches: true, - toElement: true, - touches: true, - - which: function( event ) { - var button = event.button; - - // Add which for key events - if ( event.which == null && rkeyEvent.test( event.type ) ) { - return event.charCode != null ? event.charCode : event.keyCode; - } - - // Add which for click: 1 === left; 2 === middle; 3 === right - if ( !event.which && button !== undefined && rmouseEvent.test( event.type ) ) { - if ( button & 1 ) { - return 1; - } - - if ( button & 2 ) { - return 3; - } - - if ( button & 4 ) { - return 2; - } - - return 0; - } - - return event.which; - } -}, jQuery.event.addProp ); - -jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) { - jQuery.event.special[ type ] = { - - // Utilize native event if possible so blur/focus sequence is correct - setup: function() { - - // Claim the first handler - // dataPriv.set( this, "focus", ... ) - // dataPriv.set( this, "blur", ... ) - leverageNative( this, type, expectSync ); - - // Return false to allow normal processing in the caller - return false; - }, - trigger: function() { - - // Force setup before trigger - leverageNative( this, type ); - - // Return non-false to allow normal event-path propagation - return true; - }, - - delegateType: delegateType - }; -} ); - -// Create mouseenter/leave events using mouseover/out and event-time checks -// so that event delegation works in jQuery. -// Do the same for pointerenter/pointerleave and pointerover/pointerout -// -// Support: Safari 7 only -// Safari sends mouseenter too often; see: -// https://bugs.chromium.org/p/chromium/issues/detail?id=470258 -// for the description of the bug (it existed in older Chrome versions as well). -jQuery.each( { - mouseenter: "mouseover", - mouseleave: "mouseout", - pointerenter: "pointerover", - pointerleave: "pointerout" -}, function( orig, fix ) { - jQuery.event.special[ orig ] = { - delegateType: fix, - bindType: fix, - - handle: function( event ) { - var ret, - target = this, - related = event.relatedTarget, - handleObj = event.handleObj; - - // For mouseenter/leave call the handler if related is outside the target. - // NB: No relatedTarget if the mouse left/entered the browser window - if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) { - event.type = handleObj.origType; - ret = handleObj.handler.apply( this, arguments ); - event.type = fix; - } - return ret; - } - }; -} ); - -jQuery.fn.extend( { - - on: function( types, selector, data, fn ) { - return on( this, types, selector, data, fn ); - }, - one: function( types, selector, data, fn ) { - return on( this, types, selector, data, fn, 1 ); - }, - off: function( types, selector, fn ) { - var handleObj, type; - if ( types && types.preventDefault && types.handleObj ) { - - // ( event ) dispatched jQuery.Event - handleObj = types.handleObj; - jQuery( types.delegateTarget ).off( - handleObj.namespace ? - handleObj.origType + "." + handleObj.namespace : - handleObj.origType, - handleObj.selector, - handleObj.handler - ); - return this; - } - if ( typeof types === "object" ) { - - // ( types-object [, selector] ) - for ( type in types ) { - this.off( type, selector, types[ type ] ); - } - return this; - } - if ( selector === false || typeof selector === "function" ) { - - // ( types [, fn] ) - fn = selector; - selector = undefined; - } - if ( fn === false ) { - fn = returnFalse; - } - return this.each( function() { - jQuery.event.remove( this, types, fn, selector ); - } ); - } -} ); - - -var - - // Support: IE <=10 - 11, Edge 12 - 13 only - // In IE/Edge using regex groups here causes severe slowdowns. - // See https://connect.microsoft.com/IE/feedback/details/1736512/ - rnoInnerhtml = /\s*$/g; - -// Prefer a tbody over its parent table for containing new rows -function manipulationTarget( elem, content ) { - if ( nodeName( elem, "table" ) && - nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ) { - - return jQuery( elem ).children( "tbody" )[ 0 ] || elem; - } - - return elem; -} - -// Replace/restore the type attribute of script elements for safe DOM manipulation -function disableScript( elem ) { - elem.type = ( elem.getAttribute( "type" ) !== null ) + "/" + elem.type; - return elem; -} -function restoreScript( elem ) { - if ( ( elem.type || "" ).slice( 0, 5 ) === "true/" ) { - elem.type = elem.type.slice( 5 ); - } else { - elem.removeAttribute( "type" ); - } - - return elem; -} - -function cloneCopyEvent( src, dest ) { - var i, l, type, pdataOld, udataOld, udataCur, events; - - if ( dest.nodeType !== 1 ) { - return; - } - - // 1. Copy private data: events, handlers, etc. - if ( dataPriv.hasData( src ) ) { - pdataOld = dataPriv.get( src ); - events = pdataOld.events; - - if ( events ) { - dataPriv.remove( dest, "handle events" ); - - for ( type in events ) { - for ( i = 0, l = events[ type ].length; i < l; i++ ) { - jQuery.event.add( dest, type, events[ type ][ i ] ); - } - } - } - } - - // 2. Copy user data - if ( dataUser.hasData( src ) ) { - udataOld = dataUser.access( src ); - udataCur = jQuery.extend( {}, udataOld ); - - dataUser.set( dest, udataCur ); - } -} - -// Fix IE bugs, see support tests -function fixInput( src, dest ) { - var nodeName = dest.nodeName.toLowerCase(); - - // Fails to persist the checked state of a cloned checkbox or radio button. - if ( nodeName === "input" && rcheckableType.test( src.type ) ) { - dest.checked = src.checked; - - // Fails to return the selected option to the default selected state when cloning options - } else if ( nodeName === "input" || nodeName === "textarea" ) { - dest.defaultValue = src.defaultValue; - } -} - -function domManip( collection, args, callback, ignored ) { - - // Flatten any nested arrays - args = flat( args ); - - var fragment, first, scripts, hasScripts, node, doc, - i = 0, - l = collection.length, - iNoClone = l - 1, - value = args[ 0 ], - valueIsFunction = isFunction( value ); - - // We can't cloneNode fragments that contain checked, in WebKit - if ( valueIsFunction || - ( l > 1 && typeof value === "string" && - !support.checkClone && rchecked.test( value ) ) ) { - return collection.each( function( index ) { - var self = collection.eq( index ); - if ( valueIsFunction ) { - args[ 0 ] = value.call( this, index, self.html() ); - } - domManip( self, args, callback, ignored ); - } ); - } - - if ( l ) { - fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored ); - first = fragment.firstChild; - - if ( fragment.childNodes.length === 1 ) { - fragment = first; - } - - // Require either new content or an interest in ignored elements to invoke the callback - if ( first || ignored ) { - scripts = jQuery.map( getAll( fragment, "script" ), disableScript ); - hasScripts = scripts.length; - - // Use the original fragment for the last item - // instead of the first because it can end up - // being emptied incorrectly in certain situations (#8070). - for ( ; i < l; i++ ) { - node = fragment; - - if ( i !== iNoClone ) { - node = jQuery.clone( node, true, true ); - - // Keep references to cloned scripts for later restoration - if ( hasScripts ) { - - // Support: Android <=4.0 only, PhantomJS 1 only - // push.apply(_, arraylike) throws on ancient WebKit - jQuery.merge( scripts, getAll( node, "script" ) ); - } - } - - callback.call( collection[ i ], node, i ); - } - - if ( hasScripts ) { - doc = scripts[ scripts.length - 1 ].ownerDocument; - - // Reenable scripts - jQuery.map( scripts, restoreScript ); - - // Evaluate executable scripts on first document insertion - for ( i = 0; i < hasScripts; i++ ) { - node = scripts[ i ]; - if ( rscriptType.test( node.type || "" ) && - !dataPriv.access( node, "globalEval" ) && - jQuery.contains( doc, node ) ) { - - if ( node.src && ( node.type || "" ).toLowerCase() !== "module" ) { - - // Optional AJAX dependency, but won't run scripts if not present - if ( jQuery._evalUrl && !node.noModule ) { - jQuery._evalUrl( node.src, { - nonce: node.nonce || node.getAttribute( "nonce" ) - }, doc ); - } - } else { - DOMEval( node.textContent.replace( rcleanScript, "" ), node, doc ); - } - } - } - } - } - } - - return collection; -} - -function remove( elem, selector, keepData ) { - var node, - nodes = selector ? jQuery.filter( selector, elem ) : elem, - i = 0; - - for ( ; ( node = nodes[ i ] ) != null; i++ ) { - if ( !keepData && node.nodeType === 1 ) { - jQuery.cleanData( getAll( node ) ); - } - - if ( node.parentNode ) { - if ( keepData && isAttached( node ) ) { - setGlobalEval( getAll( node, "script" ) ); - } - node.parentNode.removeChild( node ); - } - } - - return elem; -} - -jQuery.extend( { - htmlPrefilter: function( html ) { - return html; - }, - - clone: function( elem, dataAndEvents, deepDataAndEvents ) { - var i, l, srcElements, destElements, - clone = elem.cloneNode( true ), - inPage = isAttached( elem ); - - // Fix IE cloning issues - if ( !support.noCloneChecked && ( elem.nodeType === 1 || elem.nodeType === 11 ) && - !jQuery.isXMLDoc( elem ) ) { - - // We eschew Sizzle here for performance reasons: https://jsperf.com/getall-vs-sizzle/2 - destElements = getAll( clone ); - srcElements = getAll( elem ); - - for ( i = 0, l = srcElements.length; i < l; i++ ) { - fixInput( srcElements[ i ], destElements[ i ] ); - } - } - - // Copy the events from the original to the clone - if ( dataAndEvents ) { - if ( deepDataAndEvents ) { - srcElements = srcElements || getAll( elem ); - destElements = destElements || getAll( clone ); - - for ( i = 0, l = srcElements.length; i < l; i++ ) { - cloneCopyEvent( srcElements[ i ], destElements[ i ] ); - } - } else { - cloneCopyEvent( elem, clone ); - } - } - - // Preserve script evaluation history - destElements = getAll( clone, "script" ); - if ( destElements.length > 0 ) { - setGlobalEval( destElements, !inPage && getAll( elem, "script" ) ); - } - - // Return the cloned set - return clone; - }, - - cleanData: function( elems ) { - var data, elem, type, - special = jQuery.event.special, - i = 0; - - for ( ; ( elem = elems[ i ] ) !== undefined; i++ ) { - if ( acceptData( elem ) ) { - if ( ( data = elem[ dataPriv.expando ] ) ) { - if ( data.events ) { - for ( type in data.events ) { - if ( special[ type ] ) { - jQuery.event.remove( elem, type ); - - // This is a shortcut to avoid jQuery.event.remove's overhead - } else { - jQuery.removeEvent( elem, type, data.handle ); - } - } - } - - // Support: Chrome <=35 - 45+ - // Assign undefined instead of using delete, see Data#remove - elem[ dataPriv.expando ] = undefined; - } - if ( elem[ dataUser.expando ] ) { - - // Support: Chrome <=35 - 45+ - // Assign undefined instead of using delete, see Data#remove - elem[ dataUser.expando ] = undefined; - } - } - } - } -} ); - -jQuery.fn.extend( { - detach: function( selector ) { - return remove( this, selector, true ); - }, - - remove: function( selector ) { - return remove( this, selector ); - }, - - text: function( value ) { - return access( this, function( value ) { - return value === undefined ? - jQuery.text( this ) : - this.empty().each( function() { - if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { - this.textContent = value; - } - } ); - }, null, value, arguments.length ); - }, - - append: function() { - return domManip( this, arguments, function( elem ) { - if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { - var target = manipulationTarget( this, elem ); - target.appendChild( elem ); - } - } ); - }, - - prepend: function() { - return domManip( this, arguments, function( elem ) { - if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { - var target = manipulationTarget( this, elem ); - target.insertBefore( elem, target.firstChild ); - } - } ); - }, - - before: function() { - return domManip( this, arguments, function( elem ) { - if ( this.parentNode ) { - this.parentNode.insertBefore( elem, this ); - } - } ); - }, - - after: function() { - return domManip( this, arguments, function( elem ) { - if ( this.parentNode ) { - this.parentNode.insertBefore( elem, this.nextSibling ); - } - } ); - }, - - empty: function() { - var elem, - i = 0; - - for ( ; ( elem = this[ i ] ) != null; i++ ) { - if ( elem.nodeType === 1 ) { - - // Prevent memory leaks - jQuery.cleanData( getAll( elem, false ) ); - - // Remove any remaining nodes - elem.textContent = ""; - } - } - - return this; - }, - - clone: function( dataAndEvents, deepDataAndEvents ) { - dataAndEvents = dataAndEvents == null ? false : dataAndEvents; - deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents; - - return this.map( function() { - return jQuery.clone( this, dataAndEvents, deepDataAndEvents ); - } ); - }, - - html: function( value ) { - return access( this, function( value ) { - var elem = this[ 0 ] || {}, - i = 0, - l = this.length; - - if ( value === undefined && elem.nodeType === 1 ) { - return elem.innerHTML; - } - - // See if we can take a shortcut and just use innerHTML - if ( typeof value === "string" && !rnoInnerhtml.test( value ) && - !wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) { - - value = jQuery.htmlPrefilter( value ); - - try { - for ( ; i < l; i++ ) { - elem = this[ i ] || {}; - - // Remove element nodes and prevent memory leaks - if ( elem.nodeType === 1 ) { - jQuery.cleanData( getAll( elem, false ) ); - elem.innerHTML = value; - } - } - - elem = 0; - - // If using innerHTML throws an exception, use the fallback method - } catch ( e ) {} - } - - if ( elem ) { - this.empty().append( value ); - } - }, null, value, arguments.length ); - }, - - replaceWith: function() { - var ignored = []; - - // Make the changes, replacing each non-ignored context element with the new content - return domManip( this, arguments, function( elem ) { - var parent = this.parentNode; - - if ( jQuery.inArray( this, ignored ) < 0 ) { - jQuery.cleanData( getAll( this ) ); - if ( parent ) { - parent.replaceChild( elem, this ); - } - } - - // Force callback invocation - }, ignored ); - } -} ); - -jQuery.each( { - appendTo: "append", - prependTo: "prepend", - insertBefore: "before", - insertAfter: "after", - replaceAll: "replaceWith" -}, function( name, original ) { - jQuery.fn[ name ] = function( selector ) { - var elems, - ret = [], - insert = jQuery( selector ), - last = insert.length - 1, - i = 0; - - for ( ; i <= last; i++ ) { - elems = i === last ? this : this.clone( true ); - jQuery( insert[ i ] )[ original ]( elems ); - - // Support: Android <=4.0 only, PhantomJS 1 only - // .get() because push.apply(_, arraylike) throws on ancient WebKit - push.apply( ret, elems.get() ); - } - - return this.pushStack( ret ); - }; -} ); -var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" ); - -var getStyles = function( elem ) { - - // Support: IE <=11 only, Firefox <=30 (#15098, #14150) - // IE throws on elements created in popups - // FF meanwhile throws on frame elements through "defaultView.getComputedStyle" - var view = elem.ownerDocument.defaultView; - - if ( !view || !view.opener ) { - view = window; - } - - return view.getComputedStyle( elem ); - }; - -var swap = function( elem, options, callback ) { - var ret, name, - old = {}; - - // Remember the old values, and insert the new ones - for ( name in options ) { - old[ name ] = elem.style[ name ]; - elem.style[ name ] = options[ name ]; - } - - ret = callback.call( elem ); - - // Revert the old values - for ( name in options ) { - elem.style[ name ] = old[ name ]; - } - - return ret; -}; - - -var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" ); - - - -( function() { - - // Executing both pixelPosition & boxSizingReliable tests require only one layout - // so they're executed at the same time to save the second computation. - function computeStyleTests() { - - // This is a singleton, we need to execute it only once - if ( !div ) { - return; - } - - container.style.cssText = "position:absolute;left:-11111px;width:60px;" + - "margin-top:1px;padding:0;border:0"; - div.style.cssText = - "position:relative;display:block;box-sizing:border-box;overflow:scroll;" + - "margin:auto;border:1px;padding:1px;" + - "width:60%;top:1%"; - documentElement.appendChild( container ).appendChild( div ); - - var divStyle = window.getComputedStyle( div ); - pixelPositionVal = divStyle.top !== "1%"; - - // Support: Android 4.0 - 4.3 only, Firefox <=3 - 44 - reliableMarginLeftVal = roundPixelMeasures( divStyle.marginLeft ) === 12; - - // Support: Android 4.0 - 4.3 only, Safari <=9.1 - 10.1, iOS <=7.0 - 9.3 - // Some styles come back with percentage values, even though they shouldn't - div.style.right = "60%"; - pixelBoxStylesVal = roundPixelMeasures( divStyle.right ) === 36; - - // Support: IE 9 - 11 only - // Detect misreporting of content dimensions for box-sizing:border-box elements - boxSizingReliableVal = roundPixelMeasures( divStyle.width ) === 36; - - // Support: IE 9 only - // Detect overflow:scroll screwiness (gh-3699) - // Support: Chrome <=64 - // Don't get tricked when zoom affects offsetWidth (gh-4029) - div.style.position = "absolute"; - scrollboxSizeVal = roundPixelMeasures( div.offsetWidth / 3 ) === 12; - - documentElement.removeChild( container ); - - // Nullify the div so it wouldn't be stored in the memory and - // it will also be a sign that checks already performed - div = null; - } - - function roundPixelMeasures( measure ) { - return Math.round( parseFloat( measure ) ); - } - - var pixelPositionVal, boxSizingReliableVal, scrollboxSizeVal, pixelBoxStylesVal, - reliableTrDimensionsVal, reliableMarginLeftVal, - container = document.createElement( "div" ), - div = document.createElement( "div" ); - - // Finish early in limited (non-browser) environments - if ( !div.style ) { - return; - } - - // Support: IE <=9 - 11 only - // Style of cloned element affects source element cloned (#8908) - div.style.backgroundClip = "content-box"; - div.cloneNode( true ).style.backgroundClip = ""; - support.clearCloneStyle = div.style.backgroundClip === "content-box"; - - jQuery.extend( support, { - boxSizingReliable: function() { - computeStyleTests(); - return boxSizingReliableVal; - }, - pixelBoxStyles: function() { - computeStyleTests(); - return pixelBoxStylesVal; - }, - pixelPosition: function() { - computeStyleTests(); - return pixelPositionVal; - }, - reliableMarginLeft: function() { - computeStyleTests(); - return reliableMarginLeftVal; - }, - scrollboxSize: function() { - computeStyleTests(); - return scrollboxSizeVal; - }, - - // Support: IE 9 - 11+, Edge 15 - 18+ - // IE/Edge misreport `getComputedStyle` of table rows with width/height - // set in CSS while `offset*` properties report correct values. - // Behavior in IE 9 is more subtle than in newer versions & it passes - // some versions of this test; make sure not to make it pass there! - reliableTrDimensions: function() { - var table, tr, trChild, trStyle; - if ( reliableTrDimensionsVal == null ) { - table = document.createElement( "table" ); - tr = document.createElement( "tr" ); - trChild = document.createElement( "div" ); - - table.style.cssText = "position:absolute;left:-11111px"; - tr.style.height = "1px"; - trChild.style.height = "9px"; - - documentElement - .appendChild( table ) - .appendChild( tr ) - .appendChild( trChild ); - - trStyle = window.getComputedStyle( tr ); - reliableTrDimensionsVal = parseInt( trStyle.height ) > 3; - - documentElement.removeChild( table ); - } - return reliableTrDimensionsVal; - } - } ); -} )(); - - -function curCSS( elem, name, computed ) { - var width, minWidth, maxWidth, ret, - - // Support: Firefox 51+ - // Retrieving style before computed somehow - // fixes an issue with getting wrong values - // on detached elements - style = elem.style; - - computed = computed || getStyles( elem ); - - // getPropertyValue is needed for: - // .css('filter') (IE 9 only, #12537) - // .css('--customProperty) (#3144) - if ( computed ) { - ret = computed.getPropertyValue( name ) || computed[ name ]; - - if ( ret === "" && !isAttached( elem ) ) { - ret = jQuery.style( elem, name ); - } - - // A tribute to the "awesome hack by Dean Edwards" - // Android Browser returns percentage for some values, - // but width seems to be reliably pixels. - // This is against the CSSOM draft spec: - // https://drafts.csswg.org/cssom/#resolved-values - if ( !support.pixelBoxStyles() && rnumnonpx.test( ret ) && rboxStyle.test( name ) ) { - - // Remember the original values - width = style.width; - minWidth = style.minWidth; - maxWidth = style.maxWidth; - - // Put in the new values to get a computed value out - style.minWidth = style.maxWidth = style.width = ret; - ret = computed.width; - - // Revert the changed values - style.width = width; - style.minWidth = minWidth; - style.maxWidth = maxWidth; - } - } - - return ret !== undefined ? - - // Support: IE <=9 - 11 only - // IE returns zIndex value as an integer. - ret + "" : - ret; -} - - -function addGetHookIf( conditionFn, hookFn ) { - - // Define the hook, we'll check on the first run if it's really needed. - return { - get: function() { - if ( conditionFn() ) { - - // Hook not needed (or it's not possible to use it due - // to missing dependency), remove it. - delete this.get; - return; - } - - // Hook needed; redefine it so that the support test is not executed again. - return ( this.get = hookFn ).apply( this, arguments ); - } - }; -} - - -var cssPrefixes = [ "Webkit", "Moz", "ms" ], - emptyStyle = document.createElement( "div" ).style, - vendorProps = {}; - -// Return a vendor-prefixed property or undefined -function vendorPropName( name ) { - - // Check for vendor prefixed names - var capName = name[ 0 ].toUpperCase() + name.slice( 1 ), - i = cssPrefixes.length; - - while ( i-- ) { - name = cssPrefixes[ i ] + capName; - if ( name in emptyStyle ) { - return name; - } - } -} - -// Return a potentially-mapped jQuery.cssProps or vendor prefixed property -function finalPropName( name ) { - var final = jQuery.cssProps[ name ] || vendorProps[ name ]; - - if ( final ) { - return final; - } - if ( name in emptyStyle ) { - return name; - } - return vendorProps[ name ] = vendorPropName( name ) || name; -} - - -var - - // Swappable if display is none or starts with table - // except "table", "table-cell", or "table-caption" - // See here for display values: https://developer.mozilla.org/en-US/docs/CSS/display - rdisplayswap = /^(none|table(?!-c[ea]).+)/, - rcustomProp = /^--/, - cssShow = { position: "absolute", visibility: "hidden", display: "block" }, - cssNormalTransform = { - letterSpacing: "0", - fontWeight: "400" - }; - -function setPositiveNumber( _elem, value, subtract ) { - - // Any relative (+/-) values have already been - // normalized at this point - var matches = rcssNum.exec( value ); - return matches ? - - // Guard against undefined "subtract", e.g., when used as in cssHooks - Math.max( 0, matches[ 2 ] - ( subtract || 0 ) ) + ( matches[ 3 ] || "px" ) : - value; -} - -function boxModelAdjustment( elem, dimension, box, isBorderBox, styles, computedVal ) { - var i = dimension === "width" ? 1 : 0, - extra = 0, - delta = 0; - - // Adjustment may not be necessary - if ( box === ( isBorderBox ? "border" : "content" ) ) { - return 0; - } - - for ( ; i < 4; i += 2 ) { - - // Both box models exclude margin - if ( box === "margin" ) { - delta += jQuery.css( elem, box + cssExpand[ i ], true, styles ); - } - - // If we get here with a content-box, we're seeking "padding" or "border" or "margin" - if ( !isBorderBox ) { - - // Add padding - delta += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); - - // For "border" or "margin", add border - if ( box !== "padding" ) { - delta += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); - - // But still keep track of it otherwise - } else { - extra += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); - } - - // If we get here with a border-box (content + padding + border), we're seeking "content" or - // "padding" or "margin" - } else { - - // For "content", subtract padding - if ( box === "content" ) { - delta -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); - } - - // For "content" or "padding", subtract border - if ( box !== "margin" ) { - delta -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); - } - } - } - - // Account for positive content-box scroll gutter when requested by providing computedVal - if ( !isBorderBox && computedVal >= 0 ) { - - // offsetWidth/offsetHeight is a rounded sum of content, padding, scroll gutter, and border - // Assuming integer scroll gutter, subtract the rest and round down - delta += Math.max( 0, Math.ceil( - elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - - computedVal - - delta - - extra - - 0.5 - - // If offsetWidth/offsetHeight is unknown, then we can't determine content-box scroll gutter - // Use an explicit zero to avoid NaN (gh-3964) - ) ) || 0; - } - - return delta; -} - -function getWidthOrHeight( elem, dimension, extra ) { - - // Start with computed style - var styles = getStyles( elem ), - - // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-4322). - // Fake content-box until we know it's needed to know the true value. - boxSizingNeeded = !support.boxSizingReliable() || extra, - isBorderBox = boxSizingNeeded && - jQuery.css( elem, "boxSizing", false, styles ) === "border-box", - valueIsBorderBox = isBorderBox, - - val = curCSS( elem, dimension, styles ), - offsetProp = "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ); - - // Support: Firefox <=54 - // Return a confounding non-pixel value or feign ignorance, as appropriate. - if ( rnumnonpx.test( val ) ) { - if ( !extra ) { - return val; - } - val = "auto"; - } - - - // Support: IE 9 - 11 only - // Use offsetWidth/offsetHeight for when box sizing is unreliable. - // In those cases, the computed value can be trusted to be border-box. - if ( ( !support.boxSizingReliable() && isBorderBox || - - // Support: IE 10 - 11+, Edge 15 - 18+ - // IE/Edge misreport `getComputedStyle` of table rows with width/height - // set in CSS while `offset*` properties report correct values. - // Interestingly, in some cases IE 9 doesn't suffer from this issue. - !support.reliableTrDimensions() && nodeName( elem, "tr" ) || - - // Fall back to offsetWidth/offsetHeight when value is "auto" - // This happens for inline elements with no explicit setting (gh-3571) - val === "auto" || - - // Support: Android <=4.1 - 4.3 only - // Also use offsetWidth/offsetHeight for misreported inline dimensions (gh-3602) - !parseFloat( val ) && jQuery.css( elem, "display", false, styles ) === "inline" ) && - - // Make sure the element is visible & connected - elem.getClientRects().length ) { - - isBorderBox = jQuery.css( elem, "boxSizing", false, styles ) === "border-box"; - - // Where available, offsetWidth/offsetHeight approximate border box dimensions. - // Where not available (e.g., SVG), assume unreliable box-sizing and interpret the - // retrieved value as a content box dimension. - valueIsBorderBox = offsetProp in elem; - if ( valueIsBorderBox ) { - val = elem[ offsetProp ]; - } - } - - // Normalize "" and auto - val = parseFloat( val ) || 0; - - // Adjust for the element's box model - return ( val + - boxModelAdjustment( - elem, - dimension, - extra || ( isBorderBox ? "border" : "content" ), - valueIsBorderBox, - styles, - - // Provide the current computed size to request scroll gutter calculation (gh-3589) - val - ) - ) + "px"; -} - -jQuery.extend( { - - // Add in style property hooks for overriding the default - // behavior of getting and setting a style property - cssHooks: { - opacity: { - get: function( elem, computed ) { - if ( computed ) { - - // We should always get a number back from opacity - var ret = curCSS( elem, "opacity" ); - return ret === "" ? "1" : ret; - } - } - } - }, - - // Don't automatically add "px" to these possibly-unitless properties - cssNumber: { - "animationIterationCount": true, - "columnCount": true, - "fillOpacity": true, - "flexGrow": true, - "flexShrink": true, - "fontWeight": true, - "gridArea": true, - "gridColumn": true, - "gridColumnEnd": true, - "gridColumnStart": true, - "gridRow": true, - "gridRowEnd": true, - "gridRowStart": true, - "lineHeight": true, - "opacity": true, - "order": true, - "orphans": true, - "widows": true, - "zIndex": true, - "zoom": true - }, - - // Add in properties whose names you wish to fix before - // setting or getting the value - cssProps: {}, - - // Get and set the style property on a DOM Node - style: function( elem, name, value, extra ) { - - // Don't set styles on text and comment nodes - if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) { - return; - } - - // Make sure that we're working with the right name - var ret, type, hooks, - origName = camelCase( name ), - isCustomProp = rcustomProp.test( name ), - style = elem.style; - - // Make sure that we're working with the right name. We don't - // want to query the value if it is a CSS custom property - // since they are user-defined. - if ( !isCustomProp ) { - name = finalPropName( origName ); - } - - // Gets hook for the prefixed version, then unprefixed version - hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; - - // Check if we're setting a value - if ( value !== undefined ) { - type = typeof value; - - // Convert "+=" or "-=" to relative numbers (#7345) - if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) { - value = adjustCSS( elem, name, ret ); - - // Fixes bug #9237 - type = "number"; - } - - // Make sure that null and NaN values aren't set (#7116) - if ( value == null || value !== value ) { - return; - } - - // If a number was passed in, add the unit (except for certain CSS properties) - // The isCustomProp check can be removed in jQuery 4.0 when we only auto-append - // "px" to a few hardcoded values. - if ( type === "number" && !isCustomProp ) { - value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" ); - } - - // background-* props affect original clone's values - if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) { - style[ name ] = "inherit"; - } - - // If a hook was provided, use that value, otherwise just set the specified value - if ( !hooks || !( "set" in hooks ) || - ( value = hooks.set( elem, value, extra ) ) !== undefined ) { - - if ( isCustomProp ) { - style.setProperty( name, value ); - } else { - style[ name ] = value; - } - } - - } else { - - // If a hook was provided get the non-computed value from there - if ( hooks && "get" in hooks && - ( ret = hooks.get( elem, false, extra ) ) !== undefined ) { - - return ret; - } - - // Otherwise just get the value from the style object - return style[ name ]; - } - }, - - css: function( elem, name, extra, styles ) { - var val, num, hooks, - origName = camelCase( name ), - isCustomProp = rcustomProp.test( name ); - - // Make sure that we're working with the right name. We don't - // want to modify the value if it is a CSS custom property - // since they are user-defined. - if ( !isCustomProp ) { - name = finalPropName( origName ); - } - - // Try prefixed name followed by the unprefixed name - hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; - - // If a hook was provided get the computed value from there - if ( hooks && "get" in hooks ) { - val = hooks.get( elem, true, extra ); - } - - // Otherwise, if a way to get the computed value exists, use that - if ( val === undefined ) { - val = curCSS( elem, name, styles ); - } - - // Convert "normal" to computed value - if ( val === "normal" && name in cssNormalTransform ) { - val = cssNormalTransform[ name ]; - } - - // Make numeric if forced or a qualifier was provided and val looks numeric - if ( extra === "" || extra ) { - num = parseFloat( val ); - return extra === true || isFinite( num ) ? num || 0 : val; - } - - return val; - } -} ); - -jQuery.each( [ "height", "width" ], function( _i, dimension ) { - jQuery.cssHooks[ dimension ] = { - get: function( elem, computed, extra ) { - if ( computed ) { - - // Certain elements can have dimension info if we invisibly show them - // but it must have a current display style that would benefit - return rdisplayswap.test( jQuery.css( elem, "display" ) ) && - - // Support: Safari 8+ - // Table columns in Safari have non-zero offsetWidth & zero - // getBoundingClientRect().width unless display is changed. - // Support: IE <=11 only - // Running getBoundingClientRect on a disconnected node - // in IE throws an error. - ( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ? - swap( elem, cssShow, function() { - return getWidthOrHeight( elem, dimension, extra ); - } ) : - getWidthOrHeight( elem, dimension, extra ); - } - }, - - set: function( elem, value, extra ) { - var matches, - styles = getStyles( elem ), - - // Only read styles.position if the test has a chance to fail - // to avoid forcing a reflow. - scrollboxSizeBuggy = !support.scrollboxSize() && - styles.position === "absolute", - - // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-3991) - boxSizingNeeded = scrollboxSizeBuggy || extra, - isBorderBox = boxSizingNeeded && - jQuery.css( elem, "boxSizing", false, styles ) === "border-box", - subtract = extra ? - boxModelAdjustment( - elem, - dimension, - extra, - isBorderBox, - styles - ) : - 0; - - // Account for unreliable border-box dimensions by comparing offset* to computed and - // faking a content-box to get border and padding (gh-3699) - if ( isBorderBox && scrollboxSizeBuggy ) { - subtract -= Math.ceil( - elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - - parseFloat( styles[ dimension ] ) - - boxModelAdjustment( elem, dimension, "border", false, styles ) - - 0.5 - ); - } - - // Convert to pixels if value adjustment is needed - if ( subtract && ( matches = rcssNum.exec( value ) ) && - ( matches[ 3 ] || "px" ) !== "px" ) { - - elem.style[ dimension ] = value; - value = jQuery.css( elem, dimension ); - } - - return setPositiveNumber( elem, value, subtract ); - } - }; -} ); - -jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft, - function( elem, computed ) { - if ( computed ) { - return ( parseFloat( curCSS( elem, "marginLeft" ) ) || - elem.getBoundingClientRect().left - - swap( elem, { marginLeft: 0 }, function() { - return elem.getBoundingClientRect().left; - } ) - ) + "px"; - } - } -); - -// These hooks are used by animate to expand properties -jQuery.each( { - margin: "", - padding: "", - border: "Width" -}, function( prefix, suffix ) { - jQuery.cssHooks[ prefix + suffix ] = { - expand: function( value ) { - var i = 0, - expanded = {}, - - // Assumes a single number if not a string - parts = typeof value === "string" ? value.split( " " ) : [ value ]; - - for ( ; i < 4; i++ ) { - expanded[ prefix + cssExpand[ i ] + suffix ] = - parts[ i ] || parts[ i - 2 ] || parts[ 0 ]; - } - - return expanded; - } - }; - - if ( prefix !== "margin" ) { - jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber; - } -} ); - -jQuery.fn.extend( { - css: function( name, value ) { - return access( this, function( elem, name, value ) { - var styles, len, - map = {}, - i = 0; - - if ( Array.isArray( name ) ) { - styles = getStyles( elem ); - len = name.length; - - for ( ; i < len; i++ ) { - map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles ); - } - - return map; - } - - return value !== undefined ? - jQuery.style( elem, name, value ) : - jQuery.css( elem, name ); - }, name, value, arguments.length > 1 ); - } -} ); - - -function Tween( elem, options, prop, end, easing ) { - return new Tween.prototype.init( elem, options, prop, end, easing ); -} -jQuery.Tween = Tween; - -Tween.prototype = { - constructor: Tween, - init: function( elem, options, prop, end, easing, unit ) { - this.elem = elem; - this.prop = prop; - this.easing = easing || jQuery.easing._default; - this.options = options; - this.start = this.now = this.cur(); - this.end = end; - this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" ); - }, - cur: function() { - var hooks = Tween.propHooks[ this.prop ]; - - return hooks && hooks.get ? - hooks.get( this ) : - Tween.propHooks._default.get( this ); - }, - run: function( percent ) { - var eased, - hooks = Tween.propHooks[ this.prop ]; - - if ( this.options.duration ) { - this.pos = eased = jQuery.easing[ this.easing ]( - percent, this.options.duration * percent, 0, 1, this.options.duration - ); - } else { - this.pos = eased = percent; - } - this.now = ( this.end - this.start ) * eased + this.start; - - if ( this.options.step ) { - this.options.step.call( this.elem, this.now, this ); - } - - if ( hooks && hooks.set ) { - hooks.set( this ); - } else { - Tween.propHooks._default.set( this ); - } - return this; - } -}; - -Tween.prototype.init.prototype = Tween.prototype; - -Tween.propHooks = { - _default: { - get: function( tween ) { - var result; - - // Use a property on the element directly when it is not a DOM element, - // or when there is no matching style property that exists. - if ( tween.elem.nodeType !== 1 || - tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) { - return tween.elem[ tween.prop ]; - } - - // Passing an empty string as a 3rd parameter to .css will automatically - // attempt a parseFloat and fallback to a string if the parse fails. - // Simple values such as "10px" are parsed to Float; - // complex values such as "rotate(1rad)" are returned as-is. - result = jQuery.css( tween.elem, tween.prop, "" ); - - // Empty strings, null, undefined and "auto" are converted to 0. - return !result || result === "auto" ? 0 : result; - }, - set: function( tween ) { - - // Use step hook for back compat. - // Use cssHook if its there. - // Use .style if available and use plain properties where available. - if ( jQuery.fx.step[ tween.prop ] ) { - jQuery.fx.step[ tween.prop ]( tween ); - } else if ( tween.elem.nodeType === 1 && ( - jQuery.cssHooks[ tween.prop ] || - tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) { - jQuery.style( tween.elem, tween.prop, tween.now + tween.unit ); - } else { - tween.elem[ tween.prop ] = tween.now; - } - } - } -}; - -// Support: IE <=9 only -// Panic based approach to setting things on disconnected nodes -Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = { - set: function( tween ) { - if ( tween.elem.nodeType && tween.elem.parentNode ) { - tween.elem[ tween.prop ] = tween.now; - } - } -}; - -jQuery.easing = { - linear: function( p ) { - return p; - }, - swing: function( p ) { - return 0.5 - Math.cos( p * Math.PI ) / 2; - }, - _default: "swing" -}; - -jQuery.fx = Tween.prototype.init; - -// Back compat <1.8 extension point -jQuery.fx.step = {}; - - - - -var - fxNow, inProgress, - rfxtypes = /^(?:toggle|show|hide)$/, - rrun = /queueHooks$/; - -function schedule() { - if ( inProgress ) { - if ( document.hidden === false && window.requestAnimationFrame ) { - window.requestAnimationFrame( schedule ); - } else { - window.setTimeout( schedule, jQuery.fx.interval ); - } - - jQuery.fx.tick(); - } -} - -// Animations created synchronously will run synchronously -function createFxNow() { - window.setTimeout( function() { - fxNow = undefined; - } ); - return ( fxNow = Date.now() ); -} - -// Generate parameters to create a standard animation -function genFx( type, includeWidth ) { - var which, - i = 0, - attrs = { height: type }; - - // If we include width, step value is 1 to do all cssExpand values, - // otherwise step value is 2 to skip over Left and Right - includeWidth = includeWidth ? 1 : 0; - for ( ; i < 4; i += 2 - includeWidth ) { - which = cssExpand[ i ]; - attrs[ "margin" + which ] = attrs[ "padding" + which ] = type; - } - - if ( includeWidth ) { - attrs.opacity = attrs.width = type; - } - - return attrs; -} - -function createTween( value, prop, animation ) { - var tween, - collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ), - index = 0, - length = collection.length; - for ( ; index < length; index++ ) { - if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) { - - // We're done with this property - return tween; - } - } -} - -function defaultPrefilter( elem, props, opts ) { - var prop, value, toggle, hooks, oldfire, propTween, restoreDisplay, display, - isBox = "width" in props || "height" in props, - anim = this, - orig = {}, - style = elem.style, - hidden = elem.nodeType && isHiddenWithinTree( elem ), - dataShow = dataPriv.get( elem, "fxshow" ); - - // Queue-skipping animations hijack the fx hooks - if ( !opts.queue ) { - hooks = jQuery._queueHooks( elem, "fx" ); - if ( hooks.unqueued == null ) { - hooks.unqueued = 0; - oldfire = hooks.empty.fire; - hooks.empty.fire = function() { - if ( !hooks.unqueued ) { - oldfire(); - } - }; - } - hooks.unqueued++; - - anim.always( function() { - - // Ensure the complete handler is called before this completes - anim.always( function() { - hooks.unqueued--; - if ( !jQuery.queue( elem, "fx" ).length ) { - hooks.empty.fire(); - } - } ); - } ); - } - - // Detect show/hide animations - for ( prop in props ) { - value = props[ prop ]; - if ( rfxtypes.test( value ) ) { - delete props[ prop ]; - toggle = toggle || value === "toggle"; - if ( value === ( hidden ? "hide" : "show" ) ) { - - // Pretend to be hidden if this is a "show" and - // there is still data from a stopped show/hide - if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) { - hidden = true; - - // Ignore all other no-op show/hide data - } else { - continue; - } - } - orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop ); - } - } - - // Bail out if this is a no-op like .hide().hide() - propTween = !jQuery.isEmptyObject( props ); - if ( !propTween && jQuery.isEmptyObject( orig ) ) { - return; - } - - // Restrict "overflow" and "display" styles during box animations - if ( isBox && elem.nodeType === 1 ) { - - // Support: IE <=9 - 11, Edge 12 - 15 - // Record all 3 overflow attributes because IE does not infer the shorthand - // from identically-valued overflowX and overflowY and Edge just mirrors - // the overflowX value there. - opts.overflow = [ style.overflow, style.overflowX, style.overflowY ]; - - // Identify a display type, preferring old show/hide data over the CSS cascade - restoreDisplay = dataShow && dataShow.display; - if ( restoreDisplay == null ) { - restoreDisplay = dataPriv.get( elem, "display" ); - } - display = jQuery.css( elem, "display" ); - if ( display === "none" ) { - if ( restoreDisplay ) { - display = restoreDisplay; - } else { - - // Get nonempty value(s) by temporarily forcing visibility - showHide( [ elem ], true ); - restoreDisplay = elem.style.display || restoreDisplay; - display = jQuery.css( elem, "display" ); - showHide( [ elem ] ); - } - } - - // Animate inline elements as inline-block - if ( display === "inline" || display === "inline-block" && restoreDisplay != null ) { - if ( jQuery.css( elem, "float" ) === "none" ) { - - // Restore the original display value at the end of pure show/hide animations - if ( !propTween ) { - anim.done( function() { - style.display = restoreDisplay; - } ); - if ( restoreDisplay == null ) { - display = style.display; - restoreDisplay = display === "none" ? "" : display; - } - } - style.display = "inline-block"; - } - } - } - - if ( opts.overflow ) { - style.overflow = "hidden"; - anim.always( function() { - style.overflow = opts.overflow[ 0 ]; - style.overflowX = opts.overflow[ 1 ]; - style.overflowY = opts.overflow[ 2 ]; - } ); - } - - // Implement show/hide animations - propTween = false; - for ( prop in orig ) { - - // General show/hide setup for this element animation - if ( !propTween ) { - if ( dataShow ) { - if ( "hidden" in dataShow ) { - hidden = dataShow.hidden; - } - } else { - dataShow = dataPriv.access( elem, "fxshow", { display: restoreDisplay } ); - } - - // Store hidden/visible for toggle so `.stop().toggle()` "reverses" - if ( toggle ) { - dataShow.hidden = !hidden; - } - - // Show elements before animating them - if ( hidden ) { - showHide( [ elem ], true ); - } - - /* eslint-disable no-loop-func */ - - anim.done( function() { - - /* eslint-enable no-loop-func */ - - // The final step of a "hide" animation is actually hiding the element - if ( !hidden ) { - showHide( [ elem ] ); - } - dataPriv.remove( elem, "fxshow" ); - for ( prop in orig ) { - jQuery.style( elem, prop, orig[ prop ] ); - } - } ); - } - - // Per-property setup - propTween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim ); - if ( !( prop in dataShow ) ) { - dataShow[ prop ] = propTween.start; - if ( hidden ) { - propTween.end = propTween.start; - propTween.start = 0; - } - } - } -} - -function propFilter( props, specialEasing ) { - var index, name, easing, value, hooks; - - // camelCase, specialEasing and expand cssHook pass - for ( index in props ) { - name = camelCase( index ); - easing = specialEasing[ name ]; - value = props[ index ]; - if ( Array.isArray( value ) ) { - easing = value[ 1 ]; - value = props[ index ] = value[ 0 ]; - } - - if ( index !== name ) { - props[ name ] = value; - delete props[ index ]; - } - - hooks = jQuery.cssHooks[ name ]; - if ( hooks && "expand" in hooks ) { - value = hooks.expand( value ); - delete props[ name ]; - - // Not quite $.extend, this won't overwrite existing keys. - // Reusing 'index' because we have the correct "name" - for ( index in value ) { - if ( !( index in props ) ) { - props[ index ] = value[ index ]; - specialEasing[ index ] = easing; - } - } - } else { - specialEasing[ name ] = easing; - } - } -} - -function Animation( elem, properties, options ) { - var result, - stopped, - index = 0, - length = Animation.prefilters.length, - deferred = jQuery.Deferred().always( function() { - - // Don't match elem in the :animated selector - delete tick.elem; - } ), - tick = function() { - if ( stopped ) { - return false; - } - var currentTime = fxNow || createFxNow(), - remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ), - - // Support: Android 2.3 only - // Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497) - temp = remaining / animation.duration || 0, - percent = 1 - temp, - index = 0, - length = animation.tweens.length; - - for ( ; index < length; index++ ) { - animation.tweens[ index ].run( percent ); - } - - deferred.notifyWith( elem, [ animation, percent, remaining ] ); - - // If there's more to do, yield - if ( percent < 1 && length ) { - return remaining; - } - - // If this was an empty animation, synthesize a final progress notification - if ( !length ) { - deferred.notifyWith( elem, [ animation, 1, 0 ] ); - } - - // Resolve the animation and report its conclusion - deferred.resolveWith( elem, [ animation ] ); - return false; - }, - animation = deferred.promise( { - elem: elem, - props: jQuery.extend( {}, properties ), - opts: jQuery.extend( true, { - specialEasing: {}, - easing: jQuery.easing._default - }, options ), - originalProperties: properties, - originalOptions: options, - startTime: fxNow || createFxNow(), - duration: options.duration, - tweens: [], - createTween: function( prop, end ) { - var tween = jQuery.Tween( elem, animation.opts, prop, end, - animation.opts.specialEasing[ prop ] || animation.opts.easing ); - animation.tweens.push( tween ); - return tween; - }, - stop: function( gotoEnd ) { - var index = 0, - - // If we are going to the end, we want to run all the tweens - // otherwise we skip this part - length = gotoEnd ? animation.tweens.length : 0; - if ( stopped ) { - return this; - } - stopped = true; - for ( ; index < length; index++ ) { - animation.tweens[ index ].run( 1 ); - } - - // Resolve when we played the last frame; otherwise, reject - if ( gotoEnd ) { - deferred.notifyWith( elem, [ animation, 1, 0 ] ); - deferred.resolveWith( elem, [ animation, gotoEnd ] ); - } else { - deferred.rejectWith( elem, [ animation, gotoEnd ] ); - } - return this; - } - } ), - props = animation.props; - - propFilter( props, animation.opts.specialEasing ); - - for ( ; index < length; index++ ) { - result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts ); - if ( result ) { - if ( isFunction( result.stop ) ) { - jQuery._queueHooks( animation.elem, animation.opts.queue ).stop = - result.stop.bind( result ); - } - return result; - } - } - - jQuery.map( props, createTween, animation ); - - if ( isFunction( animation.opts.start ) ) { - animation.opts.start.call( elem, animation ); - } - - // Attach callbacks from options - animation - .progress( animation.opts.progress ) - .done( animation.opts.done, animation.opts.complete ) - .fail( animation.opts.fail ) - .always( animation.opts.always ); - - jQuery.fx.timer( - jQuery.extend( tick, { - elem: elem, - anim: animation, - queue: animation.opts.queue - } ) - ); - - return animation; -} - -jQuery.Animation = jQuery.extend( Animation, { - - tweeners: { - "*": [ function( prop, value ) { - var tween = this.createTween( prop, value ); - adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween ); - return tween; - } ] - }, - - tweener: function( props, callback ) { - if ( isFunction( props ) ) { - callback = props; - props = [ "*" ]; - } else { - props = props.match( rnothtmlwhite ); - } - - var prop, - index = 0, - length = props.length; - - for ( ; index < length; index++ ) { - prop = props[ index ]; - Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || []; - Animation.tweeners[ prop ].unshift( callback ); - } - }, - - prefilters: [ defaultPrefilter ], - - prefilter: function( callback, prepend ) { - if ( prepend ) { - Animation.prefilters.unshift( callback ); - } else { - Animation.prefilters.push( callback ); - } - } -} ); - -jQuery.speed = function( speed, easing, fn ) { - var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : { - complete: fn || !fn && easing || - isFunction( speed ) && speed, - duration: speed, - easing: fn && easing || easing && !isFunction( easing ) && easing - }; - - // Go to the end state if fx are off - if ( jQuery.fx.off ) { - opt.duration = 0; - - } else { - if ( typeof opt.duration !== "number" ) { - if ( opt.duration in jQuery.fx.speeds ) { - opt.duration = jQuery.fx.speeds[ opt.duration ]; - - } else { - opt.duration = jQuery.fx.speeds._default; - } - } - } - - // Normalize opt.queue - true/undefined/null -> "fx" - if ( opt.queue == null || opt.queue === true ) { - opt.queue = "fx"; - } - - // Queueing - opt.old = opt.complete; - - opt.complete = function() { - if ( isFunction( opt.old ) ) { - opt.old.call( this ); - } - - if ( opt.queue ) { - jQuery.dequeue( this, opt.queue ); - } - }; - - return opt; -}; - -jQuery.fn.extend( { - fadeTo: function( speed, to, easing, callback ) { - - // Show any hidden elements after setting opacity to 0 - return this.filter( isHiddenWithinTree ).css( "opacity", 0 ).show() - - // Animate to the value specified - .end().animate( { opacity: to }, speed, easing, callback ); - }, - animate: function( prop, speed, easing, callback ) { - var empty = jQuery.isEmptyObject( prop ), - optall = jQuery.speed( speed, easing, callback ), - doAnimation = function() { - - // Operate on a copy of prop so per-property easing won't be lost - var anim = Animation( this, jQuery.extend( {}, prop ), optall ); - - // Empty animations, or finishing resolves immediately - if ( empty || dataPriv.get( this, "finish" ) ) { - anim.stop( true ); - } - }; - doAnimation.finish = doAnimation; - - return empty || optall.queue === false ? - this.each( doAnimation ) : - this.queue( optall.queue, doAnimation ); - }, - stop: function( type, clearQueue, gotoEnd ) { - var stopQueue = function( hooks ) { - var stop = hooks.stop; - delete hooks.stop; - stop( gotoEnd ); - }; - - if ( typeof type !== "string" ) { - gotoEnd = clearQueue; - clearQueue = type; - type = undefined; - } - if ( clearQueue ) { - this.queue( type || "fx", [] ); - } - - return this.each( function() { - var dequeue = true, - index = type != null && type + "queueHooks", - timers = jQuery.timers, - data = dataPriv.get( this ); - - if ( index ) { - if ( data[ index ] && data[ index ].stop ) { - stopQueue( data[ index ] ); - } - } else { - for ( index in data ) { - if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) { - stopQueue( data[ index ] ); - } - } - } - - for ( index = timers.length; index--; ) { - if ( timers[ index ].elem === this && - ( type == null || timers[ index ].queue === type ) ) { - - timers[ index ].anim.stop( gotoEnd ); - dequeue = false; - timers.splice( index, 1 ); - } - } - - // Start the next in the queue if the last step wasn't forced. - // Timers currently will call their complete callbacks, which - // will dequeue but only if they were gotoEnd. - if ( dequeue || !gotoEnd ) { - jQuery.dequeue( this, type ); - } - } ); - }, - finish: function( type ) { - if ( type !== false ) { - type = type || "fx"; - } - return this.each( function() { - var index, - data = dataPriv.get( this ), - queue = data[ type + "queue" ], - hooks = data[ type + "queueHooks" ], - timers = jQuery.timers, - length = queue ? queue.length : 0; - - // Enable finishing flag on private data - data.finish = true; - - // Empty the queue first - jQuery.queue( this, type, [] ); - - if ( hooks && hooks.stop ) { - hooks.stop.call( this, true ); - } - - // Look for any active animations, and finish them - for ( index = timers.length; index--; ) { - if ( timers[ index ].elem === this && timers[ index ].queue === type ) { - timers[ index ].anim.stop( true ); - timers.splice( index, 1 ); - } - } - - // Look for any animations in the old queue and finish them - for ( index = 0; index < length; index++ ) { - if ( queue[ index ] && queue[ index ].finish ) { - queue[ index ].finish.call( this ); - } - } - - // Turn off finishing flag - delete data.finish; - } ); - } -} ); - -jQuery.each( [ "toggle", "show", "hide" ], function( _i, name ) { - var cssFn = jQuery.fn[ name ]; - jQuery.fn[ name ] = function( speed, easing, callback ) { - return speed == null || typeof speed === "boolean" ? - cssFn.apply( this, arguments ) : - this.animate( genFx( name, true ), speed, easing, callback ); - }; -} ); - -// Generate shortcuts for custom animations -jQuery.each( { - slideDown: genFx( "show" ), - slideUp: genFx( "hide" ), - slideToggle: genFx( "toggle" ), - fadeIn: { opacity: "show" }, - fadeOut: { opacity: "hide" }, - fadeToggle: { opacity: "toggle" } -}, function( name, props ) { - jQuery.fn[ name ] = function( speed, easing, callback ) { - return this.animate( props, speed, easing, callback ); - }; -} ); - -jQuery.timers = []; -jQuery.fx.tick = function() { - var timer, - i = 0, - timers = jQuery.timers; - - fxNow = Date.now(); - - for ( ; i < timers.length; i++ ) { - timer = timers[ i ]; - - // Run the timer and safely remove it when done (allowing for external removal) - if ( !timer() && timers[ i ] === timer ) { - timers.splice( i--, 1 ); - } - } - - if ( !timers.length ) { - jQuery.fx.stop(); - } - fxNow = undefined; -}; - -jQuery.fx.timer = function( timer ) { - jQuery.timers.push( timer ); - jQuery.fx.start(); -}; - -jQuery.fx.interval = 13; -jQuery.fx.start = function() { - if ( inProgress ) { - return; - } - - inProgress = true; - schedule(); -}; - -jQuery.fx.stop = function() { - inProgress = null; -}; - -jQuery.fx.speeds = { - slow: 600, - fast: 200, - - // Default speed - _default: 400 -}; - - -// Based off of the plugin by Clint Helfers, with permission. -// https://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/ -jQuery.fn.delay = function( time, type ) { - time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time; - type = type || "fx"; - - return this.queue( type, function( next, hooks ) { - var timeout = window.setTimeout( next, time ); - hooks.stop = function() { - window.clearTimeout( timeout ); - }; - } ); -}; - - -( function() { - var input = document.createElement( "input" ), - select = document.createElement( "select" ), - opt = select.appendChild( document.createElement( "option" ) ); - - input.type = "checkbox"; - - // Support: Android <=4.3 only - // Default value for a checkbox should be "on" - support.checkOn = input.value !== ""; - - // Support: IE <=11 only - // Must access selectedIndex to make default options select - support.optSelected = opt.selected; - - // Support: IE <=11 only - // An input loses its value after becoming a radio - input = document.createElement( "input" ); - input.value = "t"; - input.type = "radio"; - support.radioValue = input.value === "t"; -} )(); - - -var boolHook, - attrHandle = jQuery.expr.attrHandle; - -jQuery.fn.extend( { - attr: function( name, value ) { - return access( this, jQuery.attr, name, value, arguments.length > 1 ); - }, - - removeAttr: function( name ) { - return this.each( function() { - jQuery.removeAttr( this, name ); - } ); - } -} ); - -jQuery.extend( { - attr: function( elem, name, value ) { - var ret, hooks, - nType = elem.nodeType; - - // Don't get/set attributes on text, comment and attribute nodes - if ( nType === 3 || nType === 8 || nType === 2 ) { - return; - } - - // Fallback to prop when attributes are not supported - if ( typeof elem.getAttribute === "undefined" ) { - return jQuery.prop( elem, name, value ); - } - - // Attribute hooks are determined by the lowercase version - // Grab necessary hook if one is defined - if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { - hooks = jQuery.attrHooks[ name.toLowerCase() ] || - ( jQuery.expr.match.bool.test( name ) ? boolHook : undefined ); - } - - if ( value !== undefined ) { - if ( value === null ) { - jQuery.removeAttr( elem, name ); - return; - } - - if ( hooks && "set" in hooks && - ( ret = hooks.set( elem, value, name ) ) !== undefined ) { - return ret; - } - - elem.setAttribute( name, value + "" ); - return value; - } - - if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { - return ret; - } - - ret = jQuery.find.attr( elem, name ); - - // Non-existent attributes return null, we normalize to undefined - return ret == null ? undefined : ret; - }, - - attrHooks: { - type: { - set: function( elem, value ) { - if ( !support.radioValue && value === "radio" && - nodeName( elem, "input" ) ) { - var val = elem.value; - elem.setAttribute( "type", value ); - if ( val ) { - elem.value = val; - } - return value; - } - } - } - }, - - removeAttr: function( elem, value ) { - var name, - i = 0, - - // Attribute names can contain non-HTML whitespace characters - // https://html.spec.whatwg.org/multipage/syntax.html#attributes-2 - attrNames = value && value.match( rnothtmlwhite ); - - if ( attrNames && elem.nodeType === 1 ) { - while ( ( name = attrNames[ i++ ] ) ) { - elem.removeAttribute( name ); - } - } - } -} ); - -// Hooks for boolean attributes -boolHook = { - set: function( elem, value, name ) { - if ( value === false ) { - - // Remove boolean attributes when set to false - jQuery.removeAttr( elem, name ); - } else { - elem.setAttribute( name, name ); - } - return name; - } -}; - -jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( _i, name ) { - var getter = attrHandle[ name ] || jQuery.find.attr; - - attrHandle[ name ] = function( elem, name, isXML ) { - var ret, handle, - lowercaseName = name.toLowerCase(); - - if ( !isXML ) { - - // Avoid an infinite loop by temporarily removing this function from the getter - handle = attrHandle[ lowercaseName ]; - attrHandle[ lowercaseName ] = ret; - ret = getter( elem, name, isXML ) != null ? - lowercaseName : - null; - attrHandle[ lowercaseName ] = handle; - } - return ret; - }; -} ); - - - - -var rfocusable = /^(?:input|select|textarea|button)$/i, - rclickable = /^(?:a|area)$/i; - -jQuery.fn.extend( { - prop: function( name, value ) { - return access( this, jQuery.prop, name, value, arguments.length > 1 ); - }, - - removeProp: function( name ) { - return this.each( function() { - delete this[ jQuery.propFix[ name ] || name ]; - } ); - } -} ); - -jQuery.extend( { - prop: function( elem, name, value ) { - var ret, hooks, - nType = elem.nodeType; - - // Don't get/set properties on text, comment and attribute nodes - if ( nType === 3 || nType === 8 || nType === 2 ) { - return; - } - - if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { - - // Fix name and attach hooks - name = jQuery.propFix[ name ] || name; - hooks = jQuery.propHooks[ name ]; - } - - if ( value !== undefined ) { - if ( hooks && "set" in hooks && - ( ret = hooks.set( elem, value, name ) ) !== undefined ) { - return ret; - } - - return ( elem[ name ] = value ); - } - - if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { - return ret; - } - - return elem[ name ]; - }, - - propHooks: { - tabIndex: { - get: function( elem ) { - - // Support: IE <=9 - 11 only - // elem.tabIndex doesn't always return the - // correct value when it hasn't been explicitly set - // https://web.archive.org/web/20141116233347/http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/ - // Use proper attribute retrieval(#12072) - var tabindex = jQuery.find.attr( elem, "tabindex" ); - - if ( tabindex ) { - return parseInt( tabindex, 10 ); - } - - if ( - rfocusable.test( elem.nodeName ) || - rclickable.test( elem.nodeName ) && - elem.href - ) { - return 0; - } - - return -1; - } - } - }, - - propFix: { - "for": "htmlFor", - "class": "className" - } -} ); - -// Support: IE <=11 only -// Accessing the selectedIndex property -// forces the browser to respect setting selected -// on the option -// The getter ensures a default option is selected -// when in an optgroup -// eslint rule "no-unused-expressions" is disabled for this code -// since it considers such accessions noop -if ( !support.optSelected ) { - jQuery.propHooks.selected = { - get: function( elem ) { - - /* eslint no-unused-expressions: "off" */ - - var parent = elem.parentNode; - if ( parent && parent.parentNode ) { - parent.parentNode.selectedIndex; - } - return null; - }, - set: function( elem ) { - - /* eslint no-unused-expressions: "off" */ - - var parent = elem.parentNode; - if ( parent ) { - parent.selectedIndex; - - if ( parent.parentNode ) { - parent.parentNode.selectedIndex; - } - } - } - }; -} - -jQuery.each( [ - "tabIndex", - "readOnly", - "maxLength", - "cellSpacing", - "cellPadding", - "rowSpan", - "colSpan", - "useMap", - "frameBorder", - "contentEditable" -], function() { - jQuery.propFix[ this.toLowerCase() ] = this; -} ); - - - - - // Strip and collapse whitespace according to HTML spec - // https://infra.spec.whatwg.org/#strip-and-collapse-ascii-whitespace - function stripAndCollapse( value ) { - var tokens = value.match( rnothtmlwhite ) || []; - return tokens.join( " " ); - } - - -function getClass( elem ) { - return elem.getAttribute && elem.getAttribute( "class" ) || ""; -} - -function classesToArray( value ) { - if ( Array.isArray( value ) ) { - return value; - } - if ( typeof value === "string" ) { - return value.match( rnothtmlwhite ) || []; - } - return []; -} - -jQuery.fn.extend( { - addClass: function( value ) { - var classes, elem, cur, curValue, clazz, j, finalValue, - i = 0; - - if ( isFunction( value ) ) { - return this.each( function( j ) { - jQuery( this ).addClass( value.call( this, j, getClass( this ) ) ); - } ); - } - - classes = classesToArray( value ); - - if ( classes.length ) { - while ( ( elem = this[ i++ ] ) ) { - curValue = getClass( elem ); - cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); - - if ( cur ) { - j = 0; - while ( ( clazz = classes[ j++ ] ) ) { - if ( cur.indexOf( " " + clazz + " " ) < 0 ) { - cur += clazz + " "; - } - } - - // Only assign if different to avoid unneeded rendering. - finalValue = stripAndCollapse( cur ); - if ( curValue !== finalValue ) { - elem.setAttribute( "class", finalValue ); - } - } - } - } - - return this; - }, - - removeClass: function( value ) { - var classes, elem, cur, curValue, clazz, j, finalValue, - i = 0; - - if ( isFunction( value ) ) { - return this.each( function( j ) { - jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) ); - } ); - } - - if ( !arguments.length ) { - return this.attr( "class", "" ); - } - - classes = classesToArray( value ); - - if ( classes.length ) { - while ( ( elem = this[ i++ ] ) ) { - curValue = getClass( elem ); - - // This expression is here for better compressibility (see addClass) - cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); - - if ( cur ) { - j = 0; - while ( ( clazz = classes[ j++ ] ) ) { - - // Remove *all* instances - while ( cur.indexOf( " " + clazz + " " ) > -1 ) { - cur = cur.replace( " " + clazz + " ", " " ); - } - } - - // Only assign if different to avoid unneeded rendering. - finalValue = stripAndCollapse( cur ); - if ( curValue !== finalValue ) { - elem.setAttribute( "class", finalValue ); - } - } - } - } - - return this; - }, - - toggleClass: function( value, stateVal ) { - var type = typeof value, - isValidValue = type === "string" || Array.isArray( value ); - - if ( typeof stateVal === "boolean" && isValidValue ) { - return stateVal ? this.addClass( value ) : this.removeClass( value ); - } - - if ( isFunction( value ) ) { - return this.each( function( i ) { - jQuery( this ).toggleClass( - value.call( this, i, getClass( this ), stateVal ), - stateVal - ); - } ); - } - - return this.each( function() { - var className, i, self, classNames; - - if ( isValidValue ) { - - // Toggle individual class names - i = 0; - self = jQuery( this ); - classNames = classesToArray( value ); - - while ( ( className = classNames[ i++ ] ) ) { - - // Check each className given, space separated list - if ( self.hasClass( className ) ) { - self.removeClass( className ); - } else { - self.addClass( className ); - } - } - - // Toggle whole class name - } else if ( value === undefined || type === "boolean" ) { - className = getClass( this ); - if ( className ) { - - // Store className if set - dataPriv.set( this, "__className__", className ); - } - - // If the element has a class name or if we're passed `false`, - // then remove the whole classname (if there was one, the above saved it). - // Otherwise bring back whatever was previously saved (if anything), - // falling back to the empty string if nothing was stored. - if ( this.setAttribute ) { - this.setAttribute( "class", - className || value === false ? - "" : - dataPriv.get( this, "__className__" ) || "" - ); - } - } - } ); - }, - - hasClass: function( selector ) { - var className, elem, - i = 0; - - className = " " + selector + " "; - while ( ( elem = this[ i++ ] ) ) { - if ( elem.nodeType === 1 && - ( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) { - return true; - } - } - - return false; - } -} ); - - - - -var rreturn = /\r/g; - -jQuery.fn.extend( { - val: function( value ) { - var hooks, ret, valueIsFunction, - elem = this[ 0 ]; - - if ( !arguments.length ) { - if ( elem ) { - hooks = jQuery.valHooks[ elem.type ] || - jQuery.valHooks[ elem.nodeName.toLowerCase() ]; - - if ( hooks && - "get" in hooks && - ( ret = hooks.get( elem, "value" ) ) !== undefined - ) { - return ret; - } - - ret = elem.value; - - // Handle most common string cases - if ( typeof ret === "string" ) { - return ret.replace( rreturn, "" ); - } - - // Handle cases where value is null/undef or number - return ret == null ? "" : ret; - } - - return; - } - - valueIsFunction = isFunction( value ); - - return this.each( function( i ) { - var val; - - if ( this.nodeType !== 1 ) { - return; - } - - if ( valueIsFunction ) { - val = value.call( this, i, jQuery( this ).val() ); - } else { - val = value; - } - - // Treat null/undefined as ""; convert numbers to string - if ( val == null ) { - val = ""; - - } else if ( typeof val === "number" ) { - val += ""; - - } else if ( Array.isArray( val ) ) { - val = jQuery.map( val, function( value ) { - return value == null ? "" : value + ""; - } ); - } - - hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ]; - - // If set returns undefined, fall back to normal setting - if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) { - this.value = val; - } - } ); - } -} ); - -jQuery.extend( { - valHooks: { - option: { - get: function( elem ) { - - var val = jQuery.find.attr( elem, "value" ); - return val != null ? - val : - - // Support: IE <=10 - 11 only - // option.text throws exceptions (#14686, #14858) - // Strip and collapse whitespace - // https://html.spec.whatwg.org/#strip-and-collapse-whitespace - stripAndCollapse( jQuery.text( elem ) ); - } - }, - select: { - get: function( elem ) { - var value, option, i, - options = elem.options, - index = elem.selectedIndex, - one = elem.type === "select-one", - values = one ? null : [], - max = one ? index + 1 : options.length; - - if ( index < 0 ) { - i = max; - - } else { - i = one ? index : 0; - } - - // Loop through all the selected options - for ( ; i < max; i++ ) { - option = options[ i ]; - - // Support: IE <=9 only - // IE8-9 doesn't update selected after form reset (#2551) - if ( ( option.selected || i === index ) && - - // Don't return options that are disabled or in a disabled optgroup - !option.disabled && - ( !option.parentNode.disabled || - !nodeName( option.parentNode, "optgroup" ) ) ) { - - // Get the specific value for the option - value = jQuery( option ).val(); - - // We don't need an array for one selects - if ( one ) { - return value; - } - - // Multi-Selects return an array - values.push( value ); - } - } - - return values; - }, - - set: function( elem, value ) { - var optionSet, option, - options = elem.options, - values = jQuery.makeArray( value ), - i = options.length; - - while ( i-- ) { - option = options[ i ]; - - /* eslint-disable no-cond-assign */ - - if ( option.selected = - jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1 - ) { - optionSet = true; - } - - /* eslint-enable no-cond-assign */ - } - - // Force browsers to behave consistently when non-matching value is set - if ( !optionSet ) { - elem.selectedIndex = -1; - } - return values; - } - } - } -} ); - -// Radios and checkboxes getter/setter -jQuery.each( [ "radio", "checkbox" ], function() { - jQuery.valHooks[ this ] = { - set: function( elem, value ) { - if ( Array.isArray( value ) ) { - return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 ); - } - } - }; - if ( !support.checkOn ) { - jQuery.valHooks[ this ].get = function( elem ) { - return elem.getAttribute( "value" ) === null ? "on" : elem.value; - }; - } -} ); - - - - -// Return jQuery for attributes-only inclusion - - -support.focusin = "onfocusin" in window; - - -var rfocusMorph = /^(?:focusinfocus|focusoutblur)$/, - stopPropagationCallback = function( e ) { - e.stopPropagation(); - }; - -jQuery.extend( jQuery.event, { - - trigger: function( event, data, elem, onlyHandlers ) { - - var i, cur, tmp, bubbleType, ontype, handle, special, lastElement, - eventPath = [ elem || document ], - type = hasOwn.call( event, "type" ) ? event.type : event, - namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : []; - - cur = lastElement = tmp = elem = elem || document; - - // Don't do events on text and comment nodes - if ( elem.nodeType === 3 || elem.nodeType === 8 ) { - return; - } - - // focus/blur morphs to focusin/out; ensure we're not firing them right now - if ( rfocusMorph.test( type + jQuery.event.triggered ) ) { - return; - } - - if ( type.indexOf( "." ) > -1 ) { - - // Namespaced trigger; create a regexp to match event type in handle() - namespaces = type.split( "." ); - type = namespaces.shift(); - namespaces.sort(); - } - ontype = type.indexOf( ":" ) < 0 && "on" + type; - - // Caller can pass in a jQuery.Event object, Object, or just an event type string - event = event[ jQuery.expando ] ? - event : - new jQuery.Event( type, typeof event === "object" && event ); - - // Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true) - event.isTrigger = onlyHandlers ? 2 : 3; - event.namespace = namespaces.join( "." ); - event.rnamespace = event.namespace ? - new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) : - null; - - // Clean up the event in case it is being reused - event.result = undefined; - if ( !event.target ) { - event.target = elem; - } - - // Clone any incoming data and prepend the event, creating the handler arg list - data = data == null ? - [ event ] : - jQuery.makeArray( data, [ event ] ); - - // Allow special events to draw outside the lines - special = jQuery.event.special[ type ] || {}; - if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) { - return; - } - - // Determine event propagation path in advance, per W3C events spec (#9951) - // Bubble up to document, then to window; watch for a global ownerDocument var (#9724) - if ( !onlyHandlers && !special.noBubble && !isWindow( elem ) ) { - - bubbleType = special.delegateType || type; - if ( !rfocusMorph.test( bubbleType + type ) ) { - cur = cur.parentNode; - } - for ( ; cur; cur = cur.parentNode ) { - eventPath.push( cur ); - tmp = cur; - } - - // Only add window if we got to document (e.g., not plain obj or detached DOM) - if ( tmp === ( elem.ownerDocument || document ) ) { - eventPath.push( tmp.defaultView || tmp.parentWindow || window ); - } - } - - // Fire handlers on the event path - i = 0; - while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) { - lastElement = cur; - event.type = i > 1 ? - bubbleType : - special.bindType || type; - - // jQuery handler - handle = ( - dataPriv.get( cur, "events" ) || Object.create( null ) - )[ event.type ] && - dataPriv.get( cur, "handle" ); - if ( handle ) { - handle.apply( cur, data ); - } - - // Native handler - handle = ontype && cur[ ontype ]; - if ( handle && handle.apply && acceptData( cur ) ) { - event.result = handle.apply( cur, data ); - if ( event.result === false ) { - event.preventDefault(); - } - } - } - event.type = type; - - // If nobody prevented the default action, do it now - if ( !onlyHandlers && !event.isDefaultPrevented() ) { - - if ( ( !special._default || - special._default.apply( eventPath.pop(), data ) === false ) && - acceptData( elem ) ) { - - // Call a native DOM method on the target with the same name as the event. - // Don't do default actions on window, that's where global variables be (#6170) - if ( ontype && isFunction( elem[ type ] ) && !isWindow( elem ) ) { - - // Don't re-trigger an onFOO event when we call its FOO() method - tmp = elem[ ontype ]; - - if ( tmp ) { - elem[ ontype ] = null; - } - - // Prevent re-triggering of the same event, since we already bubbled it above - jQuery.event.triggered = type; - - if ( event.isPropagationStopped() ) { - lastElement.addEventListener( type, stopPropagationCallback ); - } - - elem[ type ](); - - if ( event.isPropagationStopped() ) { - lastElement.removeEventListener( type, stopPropagationCallback ); - } - - jQuery.event.triggered = undefined; - - if ( tmp ) { - elem[ ontype ] = tmp; - } - } - } - } - - return event.result; - }, - - // Piggyback on a donor event to simulate a different one - // Used only for `focus(in | out)` events - simulate: function( type, elem, event ) { - var e = jQuery.extend( - new jQuery.Event(), - event, - { - type: type, - isSimulated: true - } - ); - - jQuery.event.trigger( e, null, elem ); - } - -} ); - -jQuery.fn.extend( { - - trigger: function( type, data ) { - return this.each( function() { - jQuery.event.trigger( type, data, this ); - } ); - }, - triggerHandler: function( type, data ) { - var elem = this[ 0 ]; - if ( elem ) { - return jQuery.event.trigger( type, data, elem, true ); - } - } -} ); - - -// Support: Firefox <=44 -// Firefox doesn't have focus(in | out) events -// Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787 -// -// Support: Chrome <=48 - 49, Safari <=9.0 - 9.1 -// focus(in | out) events fire after focus & blur events, -// which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order -// Related ticket - https://bugs.chromium.org/p/chromium/issues/detail?id=449857 -if ( !support.focusin ) { - jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) { - - // Attach a single capturing handler on the document while someone wants focusin/focusout - var handler = function( event ) { - jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) ); - }; - - jQuery.event.special[ fix ] = { - setup: function() { - - // Handle: regular nodes (via `this.ownerDocument`), window - // (via `this.document`) & document (via `this`). - var doc = this.ownerDocument || this.document || this, - attaches = dataPriv.access( doc, fix ); - - if ( !attaches ) { - doc.addEventListener( orig, handler, true ); - } - dataPriv.access( doc, fix, ( attaches || 0 ) + 1 ); - }, - teardown: function() { - var doc = this.ownerDocument || this.document || this, - attaches = dataPriv.access( doc, fix ) - 1; - - if ( !attaches ) { - doc.removeEventListener( orig, handler, true ); - dataPriv.remove( doc, fix ); - - } else { - dataPriv.access( doc, fix, attaches ); - } - } - }; - } ); -} -var location = window.location; - -var nonce = { guid: Date.now() }; - -var rquery = ( /\?/ ); - - - -// Cross-browser xml parsing -jQuery.parseXML = function( data ) { - var xml; - if ( !data || typeof data !== "string" ) { - return null; - } - - // Support: IE 9 - 11 only - // IE throws on parseFromString with invalid input. - try { - xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" ); - } catch ( e ) { - xml = undefined; - } - - if ( !xml || xml.getElementsByTagName( "parsererror" ).length ) { - jQuery.error( "Invalid XML: " + data ); - } - return xml; -}; - - -var - rbracket = /\[\]$/, - rCRLF = /\r?\n/g, - rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i, - rsubmittable = /^(?:input|select|textarea|keygen)/i; - -function buildParams( prefix, obj, traditional, add ) { - var name; - - if ( Array.isArray( obj ) ) { - - // Serialize array item. - jQuery.each( obj, function( i, v ) { - if ( traditional || rbracket.test( prefix ) ) { - - // Treat each array item as a scalar. - add( prefix, v ); - - } else { - - // Item is non-scalar (array or object), encode its numeric index. - buildParams( - prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]", - v, - traditional, - add - ); - } - } ); - - } else if ( !traditional && toType( obj ) === "object" ) { - - // Serialize object item. - for ( name in obj ) { - buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add ); - } - - } else { - - // Serialize scalar item. - add( prefix, obj ); - } -} - -// Serialize an array of form elements or a set of -// key/values into a query string -jQuery.param = function( a, traditional ) { - var prefix, - s = [], - add = function( key, valueOrFunction ) { - - // If value is a function, invoke it and use its return value - var value = isFunction( valueOrFunction ) ? - valueOrFunction() : - valueOrFunction; - - s[ s.length ] = encodeURIComponent( key ) + "=" + - encodeURIComponent( value == null ? "" : value ); - }; - - if ( a == null ) { - return ""; - } - - // If an array was passed in, assume that it is an array of form elements. - if ( Array.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) { - - // Serialize the form elements - jQuery.each( a, function() { - add( this.name, this.value ); - } ); - - } else { - - // If traditional, encode the "old" way (the way 1.3.2 or older - // did it), otherwise encode params recursively. - for ( prefix in a ) { - buildParams( prefix, a[ prefix ], traditional, add ); - } - } - - // Return the resulting serialization - return s.join( "&" ); -}; - -jQuery.fn.extend( { - serialize: function() { - return jQuery.param( this.serializeArray() ); - }, - serializeArray: function() { - return this.map( function() { - - // Can add propHook for "elements" to filter or add form elements - var elements = jQuery.prop( this, "elements" ); - return elements ? jQuery.makeArray( elements ) : this; - } ) - .filter( function() { - var type = this.type; - - // Use .is( ":disabled" ) so that fieldset[disabled] works - return this.name && !jQuery( this ).is( ":disabled" ) && - rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) && - ( this.checked || !rcheckableType.test( type ) ); - } ) - .map( function( _i, elem ) { - var val = jQuery( this ).val(); - - if ( val == null ) { - return null; - } - - if ( Array.isArray( val ) ) { - return jQuery.map( val, function( val ) { - return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; - } ); - } - - return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; - } ).get(); - } -} ); - - -var - r20 = /%20/g, - rhash = /#.*$/, - rantiCache = /([?&])_=[^&]*/, - rheaders = /^(.*?):[ \t]*([^\r\n]*)$/mg, - - // #7653, #8125, #8152: local protocol detection - rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/, - rnoContent = /^(?:GET|HEAD)$/, - rprotocol = /^\/\//, - - /* Prefilters - * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example) - * 2) These are called: - * - BEFORE asking for a transport - * - AFTER param serialization (s.data is a string if s.processData is true) - * 3) key is the dataType - * 4) the catchall symbol "*" can be used - * 5) execution will start with transport dataType and THEN continue down to "*" if needed - */ - prefilters = {}, - - /* Transports bindings - * 1) key is the dataType - * 2) the catchall symbol "*" can be used - * 3) selection will start with transport dataType and THEN go to "*" if needed - */ - transports = {}, - - // Avoid comment-prolog char sequence (#10098); must appease lint and evade compression - allTypes = "*/".concat( "*" ), - - // Anchor tag for parsing the document origin - originAnchor = document.createElement( "a" ); - originAnchor.href = location.href; - -// Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport -function addToPrefiltersOrTransports( structure ) { - - // dataTypeExpression is optional and defaults to "*" - return function( dataTypeExpression, func ) { - - if ( typeof dataTypeExpression !== "string" ) { - func = dataTypeExpression; - dataTypeExpression = "*"; - } - - var dataType, - i = 0, - dataTypes = dataTypeExpression.toLowerCase().match( rnothtmlwhite ) || []; - - if ( isFunction( func ) ) { - - // For each dataType in the dataTypeExpression - while ( ( dataType = dataTypes[ i++ ] ) ) { - - // Prepend if requested - if ( dataType[ 0 ] === "+" ) { - dataType = dataType.slice( 1 ) || "*"; - ( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func ); - - // Otherwise append - } else { - ( structure[ dataType ] = structure[ dataType ] || [] ).push( func ); - } - } - } - }; -} - -// Base inspection function for prefilters and transports -function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) { - - var inspected = {}, - seekingTransport = ( structure === transports ); - - function inspect( dataType ) { - var selected; - inspected[ dataType ] = true; - jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) { - var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR ); - if ( typeof dataTypeOrTransport === "string" && - !seekingTransport && !inspected[ dataTypeOrTransport ] ) { - - options.dataTypes.unshift( dataTypeOrTransport ); - inspect( dataTypeOrTransport ); - return false; - } else if ( seekingTransport ) { - return !( selected = dataTypeOrTransport ); - } - } ); - return selected; - } - - return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" ); -} - -// A special extend for ajax options -// that takes "flat" options (not to be deep extended) -// Fixes #9887 -function ajaxExtend( target, src ) { - var key, deep, - flatOptions = jQuery.ajaxSettings.flatOptions || {}; - - for ( key in src ) { - if ( src[ key ] !== undefined ) { - ( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ]; - } - } - if ( deep ) { - jQuery.extend( true, target, deep ); - } - - return target; -} - -/* Handles responses to an ajax request: - * - finds the right dataType (mediates between content-type and expected dataType) - * - returns the corresponding response - */ -function ajaxHandleResponses( s, jqXHR, responses ) { - - var ct, type, finalDataType, firstDataType, - contents = s.contents, - dataTypes = s.dataTypes; - - // Remove auto dataType and get content-type in the process - while ( dataTypes[ 0 ] === "*" ) { - dataTypes.shift(); - if ( ct === undefined ) { - ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" ); - } - } - - // Check if we're dealing with a known content-type - if ( ct ) { - for ( type in contents ) { - if ( contents[ type ] && contents[ type ].test( ct ) ) { - dataTypes.unshift( type ); - break; - } - } - } - - // Check to see if we have a response for the expected dataType - if ( dataTypes[ 0 ] in responses ) { - finalDataType = dataTypes[ 0 ]; - } else { - - // Try convertible dataTypes - for ( type in responses ) { - if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) { - finalDataType = type; - break; - } - if ( !firstDataType ) { - firstDataType = type; - } - } - - // Or just use first one - finalDataType = finalDataType || firstDataType; - } - - // If we found a dataType - // We add the dataType to the list if needed - // and return the corresponding response - if ( finalDataType ) { - if ( finalDataType !== dataTypes[ 0 ] ) { - dataTypes.unshift( finalDataType ); - } - return responses[ finalDataType ]; - } -} - -/* Chain conversions given the request and the original response - * Also sets the responseXXX fields on the jqXHR instance - */ -function ajaxConvert( s, response, jqXHR, isSuccess ) { - var conv2, current, conv, tmp, prev, - converters = {}, - - // Work with a copy of dataTypes in case we need to modify it for conversion - dataTypes = s.dataTypes.slice(); - - // Create converters map with lowercased keys - if ( dataTypes[ 1 ] ) { - for ( conv in s.converters ) { - converters[ conv.toLowerCase() ] = s.converters[ conv ]; - } - } - - current = dataTypes.shift(); - - // Convert to each sequential dataType - while ( current ) { - - if ( s.responseFields[ current ] ) { - jqXHR[ s.responseFields[ current ] ] = response; - } - - // Apply the dataFilter if provided - if ( !prev && isSuccess && s.dataFilter ) { - response = s.dataFilter( response, s.dataType ); - } - - prev = current; - current = dataTypes.shift(); - - if ( current ) { - - // There's only work to do if current dataType is non-auto - if ( current === "*" ) { - - current = prev; - - // Convert response if prev dataType is non-auto and differs from current - } else if ( prev !== "*" && prev !== current ) { - - // Seek a direct converter - conv = converters[ prev + " " + current ] || converters[ "* " + current ]; - - // If none found, seek a pair - if ( !conv ) { - for ( conv2 in converters ) { - - // If conv2 outputs current - tmp = conv2.split( " " ); - if ( tmp[ 1 ] === current ) { - - // If prev can be converted to accepted input - conv = converters[ prev + " " + tmp[ 0 ] ] || - converters[ "* " + tmp[ 0 ] ]; - if ( conv ) { - - // Condense equivalence converters - if ( conv === true ) { - conv = converters[ conv2 ]; - - // Otherwise, insert the intermediate dataType - } else if ( converters[ conv2 ] !== true ) { - current = tmp[ 0 ]; - dataTypes.unshift( tmp[ 1 ] ); - } - break; - } - } - } - } - - // Apply converter (if not an equivalence) - if ( conv !== true ) { - - // Unless errors are allowed to bubble, catch and return them - if ( conv && s.throws ) { - response = conv( response ); - } else { - try { - response = conv( response ); - } catch ( e ) { - return { - state: "parsererror", - error: conv ? e : "No conversion from " + prev + " to " + current - }; - } - } - } - } - } - } - - return { state: "success", data: response }; -} - -jQuery.extend( { - - // Counter for holding the number of active queries - active: 0, - - // Last-Modified header cache for next request - lastModified: {}, - etag: {}, - - ajaxSettings: { - url: location.href, - type: "GET", - isLocal: rlocalProtocol.test( location.protocol ), - global: true, - processData: true, - async: true, - contentType: "application/x-www-form-urlencoded; charset=UTF-8", - - /* - timeout: 0, - data: null, - dataType: null, - username: null, - password: null, - cache: null, - throws: false, - traditional: false, - headers: {}, - */ - - accepts: { - "*": allTypes, - text: "text/plain", - html: "text/html", - xml: "application/xml, text/xml", - json: "application/json, text/javascript" - }, - - contents: { - xml: /\bxml\b/, - html: /\bhtml/, - json: /\bjson\b/ - }, - - responseFields: { - xml: "responseXML", - text: "responseText", - json: "responseJSON" - }, - - // Data converters - // Keys separate source (or catchall "*") and destination types with a single space - converters: { - - // Convert anything to text - "* text": String, - - // Text to html (true = no transformation) - "text html": true, - - // Evaluate text as a json expression - "text json": JSON.parse, - - // Parse text as xml - "text xml": jQuery.parseXML - }, - - // For options that shouldn't be deep extended: - // you can add your own custom options here if - // and when you create one that shouldn't be - // deep extended (see ajaxExtend) - flatOptions: { - url: true, - context: true - } - }, - - // Creates a full fledged settings object into target - // with both ajaxSettings and settings fields. - // If target is omitted, writes into ajaxSettings. - ajaxSetup: function( target, settings ) { - return settings ? - - // Building a settings object - ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) : - - // Extending ajaxSettings - ajaxExtend( jQuery.ajaxSettings, target ); - }, - - ajaxPrefilter: addToPrefiltersOrTransports( prefilters ), - ajaxTransport: addToPrefiltersOrTransports( transports ), - - // Main method - ajax: function( url, options ) { - - // If url is an object, simulate pre-1.5 signature - if ( typeof url === "object" ) { - options = url; - url = undefined; - } - - // Force options to be an object - options = options || {}; - - var transport, - - // URL without anti-cache param - cacheURL, - - // Response headers - responseHeadersString, - responseHeaders, - - // timeout handle - timeoutTimer, - - // Url cleanup var - urlAnchor, - - // Request state (becomes false upon send and true upon completion) - completed, - - // To know if global events are to be dispatched - fireGlobals, - - // Loop variable - i, - - // uncached part of the url - uncached, - - // Create the final options object - s = jQuery.ajaxSetup( {}, options ), - - // Callbacks context - callbackContext = s.context || s, - - // Context for global events is callbackContext if it is a DOM node or jQuery collection - globalEventContext = s.context && - ( callbackContext.nodeType || callbackContext.jquery ) ? - jQuery( callbackContext ) : - jQuery.event, - - // Deferreds - deferred = jQuery.Deferred(), - completeDeferred = jQuery.Callbacks( "once memory" ), - - // Status-dependent callbacks - statusCode = s.statusCode || {}, - - // Headers (they are sent all at once) - requestHeaders = {}, - requestHeadersNames = {}, - - // Default abort message - strAbort = "canceled", - - // Fake xhr - jqXHR = { - readyState: 0, - - // Builds headers hashtable if needed - getResponseHeader: function( key ) { - var match; - if ( completed ) { - if ( !responseHeaders ) { - responseHeaders = {}; - while ( ( match = rheaders.exec( responseHeadersString ) ) ) { - responseHeaders[ match[ 1 ].toLowerCase() + " " ] = - ( responseHeaders[ match[ 1 ].toLowerCase() + " " ] || [] ) - .concat( match[ 2 ] ); - } - } - match = responseHeaders[ key.toLowerCase() + " " ]; - } - return match == null ? null : match.join( ", " ); - }, - - // Raw string - getAllResponseHeaders: function() { - return completed ? responseHeadersString : null; - }, - - // Caches the header - setRequestHeader: function( name, value ) { - if ( completed == null ) { - name = requestHeadersNames[ name.toLowerCase() ] = - requestHeadersNames[ name.toLowerCase() ] || name; - requestHeaders[ name ] = value; - } - return this; - }, - - // Overrides response content-type header - overrideMimeType: function( type ) { - if ( completed == null ) { - s.mimeType = type; - } - return this; - }, - - // Status-dependent callbacks - statusCode: function( map ) { - var code; - if ( map ) { - if ( completed ) { - - // Execute the appropriate callbacks - jqXHR.always( map[ jqXHR.status ] ); - } else { - - // Lazy-add the new callbacks in a way that preserves old ones - for ( code in map ) { - statusCode[ code ] = [ statusCode[ code ], map[ code ] ]; - } - } - } - return this; - }, - - // Cancel the request - abort: function( statusText ) { - var finalText = statusText || strAbort; - if ( transport ) { - transport.abort( finalText ); - } - done( 0, finalText ); - return this; - } - }; - - // Attach deferreds - deferred.promise( jqXHR ); - - // Add protocol if not provided (prefilters might expect it) - // Handle falsy url in the settings object (#10093: consistency with old signature) - // We also use the url parameter if available - s.url = ( ( url || s.url || location.href ) + "" ) - .replace( rprotocol, location.protocol + "//" ); - - // Alias method option to type as per ticket #12004 - s.type = options.method || options.type || s.method || s.type; - - // Extract dataTypes list - s.dataTypes = ( s.dataType || "*" ).toLowerCase().match( rnothtmlwhite ) || [ "" ]; - - // A cross-domain request is in order when the origin doesn't match the current origin. - if ( s.crossDomain == null ) { - urlAnchor = document.createElement( "a" ); - - // Support: IE <=8 - 11, Edge 12 - 15 - // IE throws exception on accessing the href property if url is malformed, - // e.g. http://example.com:80x/ - try { - urlAnchor.href = s.url; - - // Support: IE <=8 - 11 only - // Anchor's host property isn't correctly set when s.url is relative - urlAnchor.href = urlAnchor.href; - s.crossDomain = originAnchor.protocol + "//" + originAnchor.host !== - urlAnchor.protocol + "//" + urlAnchor.host; - } catch ( e ) { - - // If there is an error parsing the URL, assume it is crossDomain, - // it can be rejected by the transport if it is invalid - s.crossDomain = true; - } - } - - // Convert data if not already a string - if ( s.data && s.processData && typeof s.data !== "string" ) { - s.data = jQuery.param( s.data, s.traditional ); - } - - // Apply prefilters - inspectPrefiltersOrTransports( prefilters, s, options, jqXHR ); - - // If request was aborted inside a prefilter, stop there - if ( completed ) { - return jqXHR; - } - - // We can fire global events as of now if asked to - // Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118) - fireGlobals = jQuery.event && s.global; - - // Watch for a new set of requests - if ( fireGlobals && jQuery.active++ === 0 ) { - jQuery.event.trigger( "ajaxStart" ); - } - - // Uppercase the type - s.type = s.type.toUpperCase(); - - // Determine if request has content - s.hasContent = !rnoContent.test( s.type ); - - // Save the URL in case we're toying with the If-Modified-Since - // and/or If-None-Match header later on - // Remove hash to simplify url manipulation - cacheURL = s.url.replace( rhash, "" ); - - // More options handling for requests with no content - if ( !s.hasContent ) { - - // Remember the hash so we can put it back - uncached = s.url.slice( cacheURL.length ); - - // If data is available and should be processed, append data to url - if ( s.data && ( s.processData || typeof s.data === "string" ) ) { - cacheURL += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data; - - // #9682: remove data so that it's not used in an eventual retry - delete s.data; - } - - // Add or update anti-cache param if needed - if ( s.cache === false ) { - cacheURL = cacheURL.replace( rantiCache, "$1" ); - uncached = ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + ( nonce.guid++ ) + - uncached; - } - - // Put hash and anti-cache on the URL that will be requested (gh-1732) - s.url = cacheURL + uncached; - - // Change '%20' to '+' if this is encoded form body content (gh-2658) - } else if ( s.data && s.processData && - ( s.contentType || "" ).indexOf( "application/x-www-form-urlencoded" ) === 0 ) { - s.data = s.data.replace( r20, "+" ); - } - - // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. - if ( s.ifModified ) { - if ( jQuery.lastModified[ cacheURL ] ) { - jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] ); - } - if ( jQuery.etag[ cacheURL ] ) { - jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] ); - } - } - - // Set the correct header, if data is being sent - if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) { - jqXHR.setRequestHeader( "Content-Type", s.contentType ); - } - - // Set the Accepts header for the server, depending on the dataType - jqXHR.setRequestHeader( - "Accept", - s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ? - s.accepts[ s.dataTypes[ 0 ] ] + - ( s.dataTypes[ 0 ] !== "*" ? 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Linear Regression, basic Elements — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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1. Linear Regression, basic Elements

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Video of Lecture

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1.1. Introduction

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Our emphasis throughout this series of lectures
-is on understanding the mathematical aspects of -different algorithms used in the fields of data analysis and machine learning.

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However, where possible we will emphasize the -importance of using available software. We start thus with a hands-on -and top-down approach to machine learning. The aim is thus to start with -relevant data or data we have produced -and use these to introduce statistical data analysis -concepts and machine learning algorithms before we delve into the -algorithms themselves. The examples we will use in the beginning, start with simple -polynomials with random noise added. We will use the Python -software package Scikit-Learn and -introduce various machine learning algorithms to make fits of -the data and predictions. We move thereafter to more interesting -cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example). -These are examples where we can easily set up the data and -then use machine learning algorithms included in for example -Scikit-Learn.

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These examples will serve us the purpose of getting -started. Furthermore, they allow us to catch more than two birds with -a stone. They will allow us to bring in some programming specific -topics and tools as well as showing the power of various Python -libraries for machine learning and statistical data analysis.

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Here, we will mainly focus on two -specific Python packages for Machine Learning, Scikit-Learn and -Tensorflow (see below for links etc). Moreover, the examples we -introduce will serve as inputs to many of our discussions later, as -well as allowing you to set up models and produce your own data and -get started with programming.

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1.2. What is Machine Learning?

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Statistics, data science and machine learning form important fields of -research in modern science. They describe how to learn and make -predictions from data, as well as allowing us to extract important -correlations about physical process and the underlying laws of motion -in large data sets. The latter, big data sets, appear frequently in -essentially all disciplines, from the traditional Science, Technology, -Mathematics and Engineering fields to Life Science, Law, education -research, the Humanities and the Social Sciences.

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It has become more -and more common to see research projects on big data in for example -the Social Sciences where extracting patterns from complicated survey -data is one of many research directions. Having a solid grasp of data -analysis and machine learning is thus becoming central to scientific -computing in many fields, and competences and skills within the fields -of machine learning and scientific computing are nowadays strongly -requested by many potential employers. The latter cannot be -overstated, familiarity with machine learning has almost become a -prerequisite for many of the most exciting employment opportunities, -whether they are in bioinformatics, life science, physics or finance, -in the private or the public sector. This author has had several -students or met students who have been hired recently based on their -skills and competences in scientific computing and data science, often -with marginal knowledge of machine learning.

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Machine learning is a subfield of computer science, and is closely -related to computational statistics. It evolved from the study of -pattern recognition in artificial intelligence (AI) research, and has -made contributions to AI tasks like computer vision, natural language -processing and speech recognition. Many of the methods we will study are also -strongly rooted in basic mathematics and physics research.

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Ideally, machine learning represents the science of giving computers -the ability to learn without being explicitly programmed. The idea is -that there exist generic algorithms which can be used to find patterns -in a broad class of data sets without having to write code -specifically for each problem. The algorithm will build its own logic -based on the data. You should however always keep in mind that -machines and algorithms are to a large extent developed by humans. The -insights and knowledge we have about a specific system, play a central -role when we develop a specific machine learning algorithm.

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Machine learning is an extremely rich field, in spite of its young -age. The increases we have seen during the last three decades in -computational capabilities have been followed by developments of -methods and techniques for analyzing and handling large date sets, -relying heavily on statistics, computer science and mathematics. The -field is rather new and developing rapidly. Popular software packages -written in Python for machine learning like -Scikit-learn, -Tensorflow, -PyTorch and Keras, all -freely available at their respective GitHub sites, encompass -communities of developers in the thousands or more. And the number of -code developers and contributors keeps increasing. Not all the -algorithms and methods can be given a rigorous mathematical -justification, opening up thereby large rooms for experimenting and -trial and error and thereby exciting new developments. However, a -solid command of linear algebra, multivariate theory, probability -theory, statistical data analysis, understanding errors and Monte -Carlo methods are central elements in a proper understanding of many -of algorithms and methods we will discuss.

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The approaches to machine learning are many, but are often split into -two main categories. In supervised learning we know the answer to a -problem, and let the computer deduce the logic behind it. On the other -hand, unsupervised learning is a method for finding patterns and -relationship in data sets without any prior knowledge of the system. -Some authours also operate with a third category, namely -reinforcement learning. This is a paradigm of learning inspired by -behavioral psychology, where learning is achieved by trial-and-error, -solely from rewards and punishment.

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Another way to categorize machine learning tasks is to consider the -desired output of a system. Some of the most common tasks are:

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  • Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.

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  • Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.

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  • Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.

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The methods we cover have three main topics in common, irrespective of -whether we deal with supervised or unsupervised learning. The first -ingredient is normally our data set (which can be subdivided into -training and test data), the second item is a model which is normally a -function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model.

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The last ingredient is a so-called cost -function which allows us to present an estimate on how good our model -is in reproducing the data it is supposed to train.
-At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of gradient methods.

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1.3. Software and needed installations

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We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -Jupyter notebooks invaluable in your work. You can run R -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be -on Python.

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If you have Python installed (we strongly recommend Python3) and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via pip as

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  1. pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow

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For Python3, replace pip with pip3.

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For OSX users we recommend, after having installed Xcode, to -install brew. Brew allows for a seamless installation of additional -software via for example

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  1. brew install python3

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For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use pip as well and simply install Python as

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  1. sudo apt-get install python3 (or python for pyhton2.7)

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etc etc.

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1.4. Python installers

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If you don’t want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely

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which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system conda.

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is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license.

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Furthermore, Google’s Colab is a free Jupyter notebook environment that requires -no setup and runs entirely in the cloud. Try it out!

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1.5. Useful Python libraries

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Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)

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  • NumPy is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays

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  • The pandas library provides high-performance, easy-to-use data structures and data analysis tools

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  • Xarray is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!

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  • Scipy (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering.

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  • Matplotlib is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.

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  • Autograd can automatically differentiate native Python and Numpy code. It can handle a large subset of Python’s features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives

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  • SymPy is a Python library for symbolic mathematics.

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  • scikit-learn has simple and efficient tools for machine learning, data mining and data analysis

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  • TensorFlow is a Python library for fast numerical computing created and released by Google

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  • Keras is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano

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  • And many more such as pytorch, Theano etc

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1.6. Installing R, C++, cython or Julia

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You will also find it convenient to utilize R. We will mainly -use Python during our lectures and in various projects and exercises. -Those of you -already familiar with R should feel free to continue using R, keeping -however an eye on the parallel Python set ups. Similarly, if you are a -Python afecionado, feel free to explore R as well. Jupyter/Ipython -notebook allows you to run R codes interactively in your -browser. The software library R is really tailored for statistical data analysis -and allows for an easy usage of the tools and algorithms we will discuss in these -lectures.

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To install R with Jupyter notebook -follow the link here

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1.7. Installing R, C++, cython, Numba etc

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For the C++ aficionados, Jupyter/IPython notebook allows you also to -install C++ and run codes written in this language interactively in -the browser. Since we will emphasize writing many of the algorithms -yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming -languages.

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To add more entropy, cython can also be used when running your -notebooks. It means that Python with the jupyter notebook -setup allows you to integrate widely popular softwares and tools for -scientific computing. Similarly, the -Numba Python package delivers increased performance -capabilities with minimal rewrites of your codes. With its -versatility, including symbolic operations, Python offers a unique -computational environment. Your jupyter notebook can easily be -converted into a nicely rendered PDF file or a Latex file for -further processing. For example, convert to latex as

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    pycod jupyter nbconvert filename.ipynb --to latex 
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And to add more versatility, the Python package SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python.

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Finally, if you wish to use the light mark-up language -doconce you can convert a standard ascii text file into various HTML -formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using doconce.

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1.8. Numpy examples and Important Matrix and vector handling packages

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There are several central software libraries for linear algebra and eigenvalue problems. Several of the more -popular ones have been wrapped into ofter software packages like those from the widely used text Numerical Recipes. The original source codes in many of the available packages are often taken from the widely used -software package LAPACK, which follows two other popular packages -developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.

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  • LINPACK: package for linear equations and least square problems.

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  • LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK’s website http://www.netlib.org it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.

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  • BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from http://www.netlib.org.

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1.9. Basic Matrix Features

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Matrix properties reminder

-
-\[\begin{split} -\mathbf{A} = - \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix}\qquad -\mathbf{I} = - \begin{bmatrix} 1 & 0 & 0 & 0 \\ - 0 & 1 & 0 & 0 \\ - 0 & 0 & 1 & 0 \\ - 0 & 0 & 0 & 1 - \end{bmatrix} -\end{split}\]
-

The inverse of a matrix is defined by

-
-\[ -\mathbf{A}^{-1} \cdot \mathbf{A} = I -\]
- - - - - - - - - - - -
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \left (A^{T} \right )^{-1}$ real orthogonal $\sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \left (A^{\dagger} \right )^{-1}$ unitary $\sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij}$
-
-

1.9.1. Some famous Matrices

-
    -

  • Diagonal if \(a_{ij}=0\) for \(i\ne j\)

    • -

  • Upper triangular if \(a_{ij}=0\) for \(i > j\)

    • -

  • Lower triangular if \(a_{ij}=0\) for \(i < j\)

    • -

  • Upper Hessenberg if \(a_{ij}=0\) for \(i > j+1\)

    • -

  • Lower Hessenberg if \(a_{ij}=0\) for \(i < j+1\)

    • -

  • Tridiagonal if \(a_{ij}=0\) for \(|i -j| > 1\)

    • -

  • Lower banded with bandwidth \(p\): \(a_{ij}=0\) for \(i > j+p\)

    • -

  • Upper banded with bandwidth \(p\): \(a_{ij}=0\) for \(i < j+p\)

    • -

  • Banded, block upper triangular, block lower triangular….

    • -
-
-
-

1.9.2. More Basic Matrix Features

-

Some Equivalent Statements -For an \(N\times N\) matrix \(\mathbf{A}\) the following properties are all equivalent

-
    -

  • If the inverse of \(\mathbf{A}\) exists, \(\mathbf{A}\) is nonsingular.

    • -

  • The equation \(\mathbf{Ax}=0\) implies \(\mathbf{x}=0\).

    • -

  • The rows of \(\mathbf{A}\) form a basis of \(R^N\).

    • -

  • The columns of \(\mathbf{A}\) form a basis of \(R^N\).

    • -

  • \(\mathbf{A}\) is a product of elementary matrices.

    • -

  • \(0\) is not eigenvalue of \(\mathbf{A}\).

    • -
-
-
-
-

1.10. Numpy and arrays

-

Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as

-
-
-
import numpy as np
-
-
-
-
-

Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution,

-
-
-
n = 10
-x = np.random.normal(size=n)
-print(x)
-
-
-
-
-
[ 1.59542395e+00 -1.76635856e-01  6.09517402e-01  7.37482326e-01
-  1.23310910e+00  8.39779531e-04 -1.09487746e+00  8.32655571e-01
- -8.22808167e-01  9.71128130e-01]
-
-
-
-
-

We defined a vector \(x\) with \(n=10\) elements with its values given by the Normal distribution \(N(0,1)\). -Another alternative is to declare a vector as follows

-
-
-
import numpy as np
-x = np.array([1, 2, 3])
-print(x)
-
-
-
-
-
[1 2 3]
-
-
-
-
-

Here we have defined a vector with three elements, with \(x_0=1\), \(x_1=2\) and \(x_2=3\). Note that both Python and C++ -start numbering array elements from \(0\) and on. This means that a vector with \(n\) elements has a sequence of entities \(x_0, x_1, x_2, \dots, x_{n-1}\). We could also let (recommended) Numpy to compute the logarithms of a specific array as

-
-
-
import numpy as np
-x = np.log(np.array([4, 7, 8]))
-print(x)
-
-
-
-
-
[1.38629436 1.94591015 2.07944154]
-
-
-
-
-

In the last example we used Numpy’s unary function \(np.log\). This function is -highly tuned to compute array elements since the code is vectorized -and does not require looping. We normaly recommend that you use the -Numpy intrinsic functions instead of the corresponding log function -from Python’s math module. The looping is done explicitely by the -np.log function. The alternative, and slower way to compute the -logarithms of a vector would be to write

-
-
-
import numpy as np
-from math import log
-x = np.array([4, 7, 8])
-for i in range(0, len(x)):
-    x[i] = log(x[i])
-print(x)
-
-
-
-
-
[1 1 2]
-
-
-
-
-

We note that our code is much longer already and we need to import the log function from the math module. -The attentive reader will also notice that the output is \([1, 1, 2]\). Python interprets automagically our numbers as integers (like the automatic keyword in C++). To change this we could define our array elements to be double precision numbers as

-
-
-
import numpy as np
-x = np.log(np.array([4, 7, 8], dtype = np.float64))
-print(x)
-
-
-
-
-
[1.38629436 1.94591015 2.07944154]
-
-
-
-
-

or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is

-
-
-
import numpy as np
-x = np.log(np.array([4.0, 7.0, 8.0])
-print(x)
-
-
-
-
-
  File "<ipython-input-7-f6d7a289d493>", line 3
-    print(x)
-    ^
-SyntaxError: invalid syntax
-
-
-
-
-

To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the itemsize functionality (the array \(x\) is actually an object which inherits the functionalities defined in Numpy) as

-
-
-
import numpy as np
-x = np.log(np.array([4.0, 7.0, 8.0])
-print(x.itemsize)
-
-
-
-
-
-
-

1.11. Matrices in Python

-

Having defined vectors, we are now ready to try out matrices. We can -define a \(3 \times 3 \) real matrix \(\hat{A}\) as (recall that we user -lowercase letters for vectors and uppercase letters for matrices)

-
-
-
import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-print(A)
-
-
-
-
-

If we use the shape function we would get \((3, 3)\) as output, that is verifying that our matrix is a \(3\times 3\) matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as

-
-
-
import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-# print the first column, row-major order and elements start with 0
-print(A[:,0])
-
-
-
-
-

We can continue this was by printing out other columns or rows. The example here prints out the second column

-
-
-
import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-# print the first column, row-major order and elements start with 0
-print(A[1,:])
-
-
-
-
-

Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the Numpy website for more details. Useful functions when defining a matrix are the np.zeros function which declares a matrix of a given dimension and sets all elements to zero

-
-
-
import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to zero
-A = np.zeros( (n, n) )
-print(A)
-
-
-
-
-

or initializing all elements to

-
-
-
import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to one
-A = np.ones( (n, n) )
-print(A)
-
-
-
-
-

or as unitarily distributed random numbers (see the material on random number generators in the statistics part)

-
-
-
import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
-A = np.random.rand(n, n)
-print(A)
-
-
-
-
-

As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. -As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors -\(\hat{x}, \hat{y}, \hat{z}\) with \(n\) elements each. The covariance matrix is defined as

-
-\[\begin{split} -\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ - \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ - \sigma_{zx} & \sigma_{zy} & \sigma_{zz} - \end{bmatrix}, -\end{split}\]
-

where for example

-
-\[ -\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -\]
-

The Numpy function np.cov calculates the covariance elements using the factor \(1/(n-1)\) instead of \(1/n\) since it assumes we do not have the exact mean values. -The following simple function uses the np.vstack function which takes each vector of dimension \(1\times n\) and produces a \(3\times n\) matrix \(\hat{W}\)

-
-\[\begin{split} -\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ - x_1 & y_1 & z_1 \\ - x_2 & y_2 & z_2 \\ - \dots & \dots & \dots \\ - x_{n-2} & y_{n-2} & z_{n-2} \\ - x_{n-1} & y_{n-1} & z_{n-1} - \end{bmatrix}, -\end{split}\]
-

which in turn is converted into into the \(3\times 3\) covariance matrix -\(\hat{\Sigma}\) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \(\hat{x}\) etc using the Numpy -function np.mean(x). We can also extract the eigenvalues of the -covariance matrix through the np.linalg.eig() function.

-
-
-
# Importing various packages
-import numpy as np
-
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-z = x**3+np.random.normal(size=n)
-print(np.mean(z))
-W = np.vstack((x, y, z))
-Sigma = np.cov(W)
-print(Sigma)
-Eigvals, Eigvecs = np.linalg.eig(Sigma)
-print(Eigvals)
-
-
-
-
-
-
-
%matplotlib inline
-
-import numpy as np
-import matplotlib.pyplot as plt
-from scipy import sparse
-eye = np.eye(4)
-print(eye)
-sparse_mtx = sparse.csr_matrix(eye)
-print(sparse_mtx)
-x = np.linspace(-10,10,100)
-y = np.sin(x)
-plt.plot(x,y,marker='x')
-plt.show()
-
-
-
-
-
-
-

1.12. Meet the Pandas

- -

Another useful Python package is -pandas, which is an open source library -providing high-performance, easy-to-use data structures and data -analysis tools for Python. pandas stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data. -pandas has two major classes, the DataFrame class with two-dimensional data objects and tabular data organized in columns and the class Series with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. -pandas allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations.

-

The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of pandas, in particular in connection with classification of data.

-
-
-
import pandas as pd
-from IPython.display import display
-data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"],
-        'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"],
-        'Place of birth': ["Shire", "Shire", "Eriador", "Shire"],
-        'Date of Birth T.A.': [2968, 2890, 2931, 2980]
-        }
-data_pandas = pd.DataFrame(data)
-display(data_pandas)
-
-
-
-
-

In the above we have imported pandas with the shorthand pd, the latter has become the standard way we import pandas. We make then a list of various variables -and reorganize the aboves lists into a DataFrame and then print out a neat table with specific column labels as Name, place of birth and date of birth. -Displaying these results, we see that the indices are given by the default numbers from zero to three. -pandas is extremely flexible and we can easily change the above indices by defining a new type of indexing as

-
-
-
data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])
-display(data_pandas)
-
-
-
-
-

Thereafter we display the content of the row which begins with the index Aragorn

-
-
-
display(data_pandas.loc['Aragorn'])
-
-
-
-
-

We can easily append data to this, for example

-
-
-
new_hobbit = {'First Name': ["Peregrin"],
-              'Last Name': ["Took"],
-              'Place of birth': ["Shire"],
-              'Date of Birth T.A.': [2990]
-              }
-data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))
-display(data_pandas)
-
-
-
-
-

Here are other examples where we use the DataFrame functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix -of dimensionality \(10\times 5\) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations.

-
-
-
import numpy as np
-import pandas as pd
-from IPython.display import display
-np.random.seed(100)
-# setting up a 10 x 5 matrix
-rows = 10
-cols = 5
-a = np.random.randn(rows,cols)
-df = pd.DataFrame(a)
-display(df)
-print(df.mean())
-print(df.std())
-display(df**2)
-
-
-
-
-

Thereafter we can select specific columns only and plot final results

-
-
-
df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']
-df.index = np.arange(10)
-
-display(df)
-print(df['Second'].mean() )
-print(df.info())
-print(df.describe())
-
-from pylab import plt, mpl
-plt.style.use('seaborn')
-mpl.rcParams['font.family'] = 'serif'
-
-df.cumsum().plot(lw=2.0, figsize=(10,6))
-plt.show()
-
-
-df.plot.bar(figsize=(10,6), rot=15)
-plt.show()
-
-
-
-
-

We can produce a \(4\times 4\) matrix

-
-
-
b = np.arange(16).reshape((4,4))
-print(b)
-df1 = pd.DataFrame(b)
-print(df1)
-
-
-
-
-

and many other operations.

-

The Series class is another important class included in -pandas. You can view it as a specialization of DataFrame but where -we have just a single column of data. It shares many of the same features as _DataFrame. As with DataFrame, -most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays. -As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in. -For multidimensional arrays, we recommend strongly xarray. xarray has much of the same flexibility as pandas, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both pandas and xarray.

-

In order to study various Machine Learning algorithms, we need to -access data. Acccessing data is an essential step in all machine -learning algorithms. In particular, setting up the so-called design -matrix (to be defined below) is often the first element we need in -order to perform our calculations. To set up the design matrix means -reading (and later, when the calculations are done, writing) data -in various formats, The formats span from reading files from disk, -loading data from databases and interacting with online sources -like web application programming interfaces (APIs).

-

In handling various input formats, as discussed above, we will mainly stay with pandas, -a Python package which allows us, in a seamless and painless way, to -deal with a multitude of formats, from standard csv (comma separated -values) files, via excel, html to hdf5 formats. With pandas -and the DataFrame and Series functionalities we are able to convert text data -into the calculational formats we need for a specific algorithm. And our code is going to be -pretty close the basic mathematical expressions.

-

Our first data set is going to be a classic from nuclear physics, namely all -available data on binding energies. Don’t be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set.

-

We will show some of the -strengths of packages like Scikit-Learn in fitting nuclear binding energies to -specific functions using linear regression first. Then, as a teaser, we will show you how -you can easily implement other algorithms like decision trees and random forests and neural networks.

-

But before we really start with nuclear physics data, let’s just look at some simpler polynomial fitting cases, such as, -(don’t be offended) fitting straight lines!

-
-
-

1.13. Simple linear regression model using scikit-learn

-

We start with perhaps our simplest possible example, using Scikit-Learn to perform linear regression analysis on a data set produced by us.

-

What follows is a simple Python code where we have defined a function -\(y\) in terms of the variable \(x\). Both are defined as vectors with \(100\) entries. -The numbers in the vector \(\hat{x}\) are given -by random numbers generated with a uniform distribution with entries -\(x_i \in [0,1]\) (more about probability distribution functions -later). These values are then used to define a function \(y(x)\) -(tabulated again as a vector) with a linear dependence on \(x\) plus a -random noise added via the normal distribution.

-

The Numpy functions are imported used the import numpy as np -statement and the random number generator for the uniform distribution -is called using the function np.random.rand(), where we specificy -that we want \(100\) random variables. Using Numpy we define -automatically an array with the specified number of elements, \(100\) in -our case. With the Numpy function randn() we can compute random -numbers with the normal distribution (mean value \(\mu\) equal to zero and -variance \(\sigma^2\) set to one) and produce the values of \(y\) assuming a linear -dependence as function of \(x\)

-
-\[ -y = 2x+N(0,1), -\]
-

where \(N(0,1)\) represents random numbers generated by the normal -distribution. From Scikit-Learn we import then the -LinearRegression functionality and make a prediction \(\tilde{y} = -\alpha + \beta x\) using the function fit(x,y). We call the set of -data \((\hat{x},\hat{y})\) for our training data. The Python package -scikit-learn has also a functionality which extracts the above -fitting parameters \(\alpha\) and \(\beta\) (see below). Later we will -distinguish between training data and test data.

-

For plotting we use the Python package -matplotlib which produces publication -quality figures. Feel free to explore the extensive -gallery of examples. In -this example we plot our original values of \(x\) and \(y\) as well as the -prediction ypredict (\(\tilde{y}\)), which attempts at fitting our -data with a straight line.

-

The Python code follows here.

-
-
-
# Importing various packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression
-
-x = np.random.rand(100,1)
-y = 2*x+np.random.randn(100,1)
-linreg = LinearRegression()
-linreg.fit(x,y)
-xnew = np.array([[0],[1]])
-ypredict = linreg.predict(xnew)
-
-plt.plot(xnew, ypredict, "r-")
-plt.plot(x, y ,'ro')
-plt.axis([0,1.0,0, 5.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Simple Linear Regression')
-plt.show()
-
-
-
-
-

This example serves several aims. It allows us to demonstrate several -aspects of data analysis and later machine learning algorithms. The -immediate visualization shows that our linear fit is not -impressive. It goes through the data points, but there are many -outliers which are not reproduced by our linear regression. We could -now play around with this small program and change for example the -factor in front of \(x\) and the normal distribution. Try to change the -function \(y\) to

-
-\[ -y = 10x+0.01 \times N(0,1), -\]
-

where \(x\) is defined as before. Does the fit look better? Indeed, by -reducing the role of the noise given by the normal distribution we see immediately that -our linear prediction seemingly reproduces better the training -set. However, this testing ‘by the eye’ is obviouly not satisfactory in the -long run. Here we have only defined the training data and our model, and -have not discussed a more rigorous approach to the cost function.

-

We need more rigorous criteria in defining whether we have succeeded or -not in modeling our training data. You will be surprised to see that -many scientists seldomly venture beyond this ‘by the eye’ approach. A -standard approach for the cost function is the so-called \(\chi^2\) -function (a variant of the mean-squared error (MSE))

-
-\[ -\chi^2 = \frac{1}{n} -\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2}, -\]
-

where \(\sigma_i^2\) is the variance (to be defined later) of the entry -\(y_i\). We may not know the explicit value of \(\sigma_i^2\), it serves -however the aim of scaling the equations and make the cost function -dimensionless.

-

Minimizing the cost function is a central aspect of -our discussions to come. Finding its minima as function of the model -parameters (\(\alpha\) and \(\beta\) in our case) will be a recurring -theme in these series of lectures. Essentially all machine learning -algorithms we will discuss center around the minimization of the -chosen cost function. This depends in turn on our specific -model for describing the data, a typical situation in supervised -learning. Automatizing the search for the minima of the cost function is a -central ingredient in all algorithms. Typical methods which are -employed are various variants of gradient methods. These will be -discussed in more detail later. Again, you’ll be surprised to hear that -many practitioners minimize the above function ‘’by the eye’, popularly dubbed as -‘chi by the eye’. That is, change a parameter and see (visually and numerically) that -the \(\chi^2\) function becomes smaller.

-

There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define -the relative error (why would we prefer the MSE instead of the relative error?) as

-
-\[ -\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}. -\]
-

The squared cost function results in an arithmetic mean-unbiased -estimator, and the absolute-value cost function results in a -median-unbiased estimator (in the one-dimensional case, and a -geometric median-unbiased estimator for the multi-dimensional -case). The squared cost function has the disadvantage that it has the tendency -to be dominated by outliers.

-

We can modify easily the above Python code and plot the relative error instead

-
-
-
import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression
-
-x = np.random.rand(100,1)
-y = 5*x+0.01*np.random.randn(100,1)
-linreg = LinearRegression()
-linreg.fit(x,y)
-ypredict = linreg.predict(x)
-
-plt.plot(x, np.abs(ypredict-y)/abs(y), "ro")
-plt.axis([0,1.0,0.0, 0.5])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$\epsilon_{\mathrm{relative}}$')
-plt.title(r'Relative error')
-plt.show()
-
-
-
-
-

Depending on the parameter in front of the normal distribution, we may -have a small or larger relative error. Try to play around with -different training data sets and study (graphically) the value of the -relative error.

-

As mentioned above, Scikit-Learn has an impressive functionality. -We can for example extract the values of \(\alpha\) and \(\beta\) and -their error estimates, or the variance and standard deviation and many -other properties from the statistical data analysis.

-

Here we show an -example of the functionality of Scikit-Learn.

-
-
-
import numpy as np 
-import matplotlib.pyplot as plt 
-from sklearn.linear_model import LinearRegression 
-from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error
-
-x = np.random.rand(100,1)
-y = 2.0+ 5*x+0.5*np.random.randn(100,1)
-linreg = LinearRegression()
-linreg.fit(x,y)
-ypredict = linreg.predict(x)
-print('The intercept alpha: \n', linreg.intercept_)
-print('Coefficient beta : \n', linreg.coef_)
-# The mean squared error                               
-print("Mean squared error: %.2f" % mean_squared_error(y, ypredict))
-# Explained variance score: 1 is perfect prediction                                 
-print('Variance score: %.2f' % r2_score(y, ypredict))
-# Mean squared log error                                                        
-print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )
-# Mean absolute error                                                           
-print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))
-plt.plot(x, ypredict, "r-")
-plt.plot(x, y ,'ro')
-plt.axis([0.0,1.0,1.5, 7.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Linear Regression fit ')
-plt.show()
-
-
-
-
-

The function coef gives us the parameter \(\beta\) of our fit while intercept yields -\(\alpha\). Depending on the constant in front of the normal distribution, we get values near or far from \(alpha =2\) and \(\beta =5\). Try to play around with different parameters in front of the normal distribution. The function meansquarederror gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as

-
-\[ -MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -\]
-

The smaller the value, the better the fit. Ideally we would like to -have an MSE equal zero. The attentive reader has probably recognized -this function as being similar to the \(\chi^2\) function defined above.

-

The r2score function computes \(R^2\), the coefficient of -determination. It provides a measure of how well future samples are -likely to be predicted by the model. Best possible score is 1.0 and it -can be negative (because the model can be arbitrarily worse). A -constant model that always predicts the expected value of \(\hat{y}\), -disregarding the input features, would get a \(R^2\) score of \(0.0\).

-

If \(\tilde{\hat{y}}_i\) is the predicted value of the \(i-th\) sample and \(y_i\) is the corresponding true value, then the score \(R^2\) is defined as

-
-\[ -R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -\]
-

where we have defined the mean value of \(\hat{y}\) as

-
-\[ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -\]
-

Another quantity taht we will meet again in our discussions of regression analysis is -the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the \(l1\)-norm loss. In our discussion above we presented the relative error. -The MAE is defined as follows

-
-\[ -\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|. -\]
-

We present the -squared logarithmic (quadratic) error

-
-\[ -\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2, -\]
-

where \(\log_e (x)\) stands for the natural logarithm of \(x\). This error -estimate is best to use when targets having exponential growth, such -as population counts, average sales of a commodity over a span of -years etc.

-

Finally, another cost function is the Huber cost function used in robust regression.

-

The rationale behind this possible cost function is its reduced -sensitivity to outliers in the data set. In our discussions on -dimensionality reduction and normalization of data we will meet other -ways of dealing with outliers.

-

The Huber cost function is defined as

-
-\[\begin{split} -H_{\delta}(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}}. -\end{split}\]
-

Here \(a=\boldsymbol{y} - \boldsymbol{\tilde{y}}\). -We will discuss in more -detail these and other functions in the various lectures. We conclude this part with another example. Instead of -a linear \(x\)-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn.

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-import random
-from sklearn.linear_model import Ridge
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.pipeline import make_pipeline
-from sklearn.linear_model import LinearRegression
-
-x=np.linspace(0.02,0.98,200)
-noise = np.asarray(random.sample((range(200)),200))
-y=x**3*noise
-yn=x**3*100
-poly3 = PolynomialFeatures(degree=3)
-X = poly3.fit_transform(x[:,np.newaxis])
-clf3 = LinearRegression()
-clf3.fit(X,y)
-
-Xplot=poly3.fit_transform(x[:,np.newaxis])
-poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')
-plt.plot(x,yn, color='red', label="True Cubic")
-plt.scatter(x, y, label='Data', color='orange', s=15)
-plt.legend()
-plt.show()
-
-def error(a):
-    for i in y:
-        err=(y-yn)/yn
-    return abs(np.sum(err))/len(err)
-
-print (error(y))
-
-
-
-
-

Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding -energies. A basic quantity which can be measured for the ground -states of nuclei is the atomic mass \(M(N, Z)\) of the neutral atom with -atomic mass number \(A\) and charge \(Z\). The number of neutrons is \(N\). There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even).

-

Atomic masses are usually tabulated in terms of the mass excess defined by

-
-\[ -\Delta M(N, Z) = M(N, Z) - uA, -\]
-

where \(u\) is the Atomic Mass Unit

-
-\[ -u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2. -\]
-

The nucleon masses are

-
-\[ -m_p = 1.00727646693(9)u, -\]
-

and

-
-\[ -m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u. -\]
-

In the 2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu -there are data on masses and decays of 3437 nuclei.

-

The nuclear binding energy is defined as the energy required to break -up a given nucleus into its constituent parts of \(N\) neutrons and \(Z\) -protons. In terms of the atomic masses \(M(N, Z)\) the binding energy is -defined by

-
-\[ -BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 , -\]
-

where \(M_H\) is the mass of the hydrogen atom and \(m_n\) is the mass of the neutron. -In terms of the mass excess the binding energy is given by

-
-\[ -BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 , -\]
-

where \(\Delta_H c^2 = 7.2890\) MeV and \(\Delta_n c^2 = 8.0713\) MeV.

-

A popular and physically intuitive model which can be used to parametrize -the experimental binding energies as function of \(A\), is the so-called -liquid drop model. The ansatz is based on the following expression

-
-\[ -BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A}, -\]
-

where \(A\) stands for the number of nucleons and the \(a_i\)s are parameters which are determined by a fit -to the experimental data.

-

To arrive at the above expression we have assumed that we can make the following assumptions:

-
    -

  • There is a volume term \(a_1A\) proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.

    • -

  • There is a surface energy term \(a_2A^{2/3}\). The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.

    • -

  • There is a Coulomb energy term \(a_3\frac{Z^2}{A^{1/3}}\). The electric repulsion between each pair of protons in a nucleus yields less binding.

    • -

  • There is an asymmetry term \(a_4\frac{(N-Z)^2}{A}\). This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.

    • -
-

We could also add a so-called pairing term, which is a correction term that -arises from the tendency of proton pairs and neutron pairs to -occur. An even number of particles is more stable than an odd number.

-
-

1.13.1. Organizing our data

-

Let us start with reading and organizing our data. -We start with the compilation of masses and binding energies from 2016. -After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.

-

We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of scikit-learn.

-
-
-
# Common imports
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-import sklearn.linear_model as skl
-from sklearn.model_selection import train_test_split
-from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
-import os
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("MassEval2016.dat"),'r')
-
-
-
-
-

Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various matplotlib commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function.

-
-
-
from pylab import plt, mpl
-plt.style.use('seaborn')
-mpl.rcParams['font.family'] = 'serif'
-
-def MakePlot(x,y, styles, labels, axlabels):
-    plt.figure(figsize=(10,6))
-    for i in range(len(x)):
-        plt.plot(x[i], y[i], styles[i], label = labels[i])
-        plt.xlabel(axlabels[0])
-        plt.ylabel(axlabels[1])
-    plt.legend(loc=0)
-
-
-
-
-

Our next step is to read the data on experimental binding energies and -reorganize them as functions of the mass number \(A\), the number of -protons \(Z\) and neutrons \(N\) using pandas. Before we do this it is -always useful (unless you have a binary file or other types of compressed -data) to actually open the file and simply take a look at it!

-

In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with pandas. The file begins with some basic format information.

-
-
-
"""                                                                                                                         
-This is taken from the data file of the mass 2016 evaluation.                                                               
-All files are 3436 lines long with 124 character per line.                                                                  
-       Headers are 39 lines long.                                                                                           
-   col 1     :  Fortran character control: 1 = page feed  0 = line feed                                                     
-   format    :  a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5                     
-   These formats are reflected in the pandas widths variable below, see the statement                                       
-   widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),                                                            
-   Pandas has also a variable header, with length 39 in this case.                                                          
-"""
-
-
-
-
-

The data we are interested in are in columns 2, 3, 4 and 11, giving us -the number of neutrons, protons, mass numbers and binding energies, -respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will -covert them into the pandas DataFrame structure.

-
-
-
# Read the experimental data with Pandas
-Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
-              names=('N', 'Z', 'A', 'Element', 'Ebinding'),
-              widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
-              header=39,
-              index_col=False)
-
-# Extrapolated values are indicated by '#' in place of the decimal place, so
-# the Ebinding column won't be numeric. Coerce to float and drop these entries.
-Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
-Masses = Masses.dropna()
-# Convert from keV to MeV.
-Masses['Ebinding'] /= 1000
-
-# Group the DataFrame by nucleon number, A.
-Masses = Masses.groupby('A')
-# Find the rows of the grouped DataFrame with the maximum binding energy.
-Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
-
-
-
-
-

We have now read in the data, grouped them according to the variables we are interested in. -We see how easy it is to reorganize the data using pandas. If we -were to do these operations in C/C++ or Fortran, we would have had to -write various functions/subroutines which perform the above -reorganizations for us. Having reorganized the data, we can now start -to make some simple fits using both the functionalities in numpy and -Scikit-Learn afterwards.

-

Now we define five variables which contain -the number of nucleons \(A\), the number of protons \(Z\) and the number of neutrons \(N\), the element name and finally the energies themselves.

-
-
-
A = Masses['A']
-Z = Masses['Z']
-N = Masses['N']
-Element = Masses['Element']
-Energies = Masses['Ebinding']
-print(Masses)
-
-
-
-
-

The next step, and we will define this mathematically later, is to set up the so-called design matrix. We will throughout call this matrix \(\boldsymbol{X}\). -It has dimensionality \(p\times n\), where \(n\) is the number of data points and \(p\) are the so-called predictors. In our case here they are given by the number of polynomials in \(A\) we wish to include in the fit.

-
-
-
# Now we set up the design matrix X
-X = np.zeros((len(A),5))
-X[:,0] = 1
-X[:,1] = A
-X[:,2] = A**(2.0/3.0)
-X[:,3] = A**(-1.0/3.0)
-X[:,4] = A**(-1.0)
-
-
-
-
-

With scikitlearn we are now ready to use linear regression and fit our data.

-
-
-
clf = skl.LinearRegression().fit(X, Energies)
-fity = clf.predict(X)
-
-
-
-
-

Pretty simple!
-Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data.

-
-
-
# The mean squared error                               
-print("Mean squared error: %.2f" % mean_squared_error(Energies, fity))
-# Explained variance score: 1 is perfect prediction                                 
-print('Variance score: %.2f' % r2_score(Energies, fity))
-# Mean absolute error                                                           
-print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))
-print(clf.coef_, clf.intercept_)
-
-Masses['Eapprox']  = fity
-# Generate a plot comparing the experimental with the fitted values values.
-fig, ax = plt.subplots()
-ax.set_xlabel(r'$A = N + Z$')
-ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
-ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
-            label='Ame2016')
-ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
-            label='Fit')
-ax.legend()
-save_fig("Masses2016")
-plt.show()
-
-
-
-
-

As a teaser, let us now see how we can do this with decision trees using scikit-learn. Later we will switch to so-called random forests!

-
-
-
#Decision Tree Regression
-from sklearn.tree import DecisionTreeRegressor
-regr_1=DecisionTreeRegressor(max_depth=5)
-regr_2=DecisionTreeRegressor(max_depth=7)
-regr_3=DecisionTreeRegressor(max_depth=9)
-regr_1.fit(X, Energies)
-regr_2.fit(X, Energies)
-regr_3.fit(X, Energies)
-
-
-y_1 = regr_1.predict(X)
-y_2 = regr_2.predict(X)
-y_3=regr_3.predict(X)
-Masses['Eapprox'] = y_3
-# Plot the results
-plt.figure()
-plt.plot(A, Energies, color="blue", label="Data", linewidth=2)
-plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2)
-plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2)
-plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2)
-
-plt.xlabel("$A$")
-plt.ylabel("$E$[MeV]")
-plt.title("Decision Tree Regression")
-plt.legend()
-save_fig("Masses2016Trees")
-plt.show()
-print(Masses)
-print(np.mean( (Energies-y_1)**2))
-
-
-
-
-

The seaborn package allows us to visualize data in an efficient way. Note that we use scikit-learn’s multi-layer perceptron (or feed forward neural network) -functionality.

-
-
-
from sklearn.neural_network import MLPRegressor
-from sklearn.metrics import accuracy_score
-import seaborn as sns
-
-X_train = X
-Y_train = Energies
-n_hidden_neurons = 100
-epochs = 100
-# store models for later use
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-# store the models for later use
-DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-sns.set()
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
-                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
-        dnn.fit(X_train, Y_train)
-        DNN_scikit[i][j] = dnn
-        train_accuracy[i][j] = dnn.score(X_train, Y_train)
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-
-
-
-
-
-
-

1.14. Linear Regression, basic elements

-

Video of Lecture.

-

Fitting a continuous function with linear parameterization in terms of the parameters \(\boldsymbol{\beta}\).

-
    -

  • Method of choice for fitting a continuous function!

    • -

  • Gives an excellent introduction to central Machine Learning features with understandable pedagogical links to other methods like Neural Networks, Support Vector Machines etc

    • -

  • Analytical expression for the fitting parameters \(\boldsymbol{\beta}\)

    • -

  • Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more

    • -

  • Analytical relation with probabilistic interpretations

    • -

  • Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics

    • -

  • Easy to code! And links well with classification problems and logistic regression and neural networks

    • -

  • Allows for easy hands-on understanding of gradient descent methods

    • -

  • and many more features

    • -
-

For more discussions of Ridge and Lasso regression, Wessel van Wieringen’s article is highly recommended. -Similarly, Mehta et al’s article is also recommended.

-

Regression modeling deals with the description of the sampling distribution of a given random variable \(y\) and how it varies as function of another variable or a set of such variables \(\boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T\). -The first variable is called the dependent, the outcome or the response variable while the set of variables \(\boldsymbol{x}\) is called the independent variable, or the predictor variable or the explanatory variable.

-

A regression model aims at finding a likelihood function \(p(\boldsymbol{y}\vert \boldsymbol{x})\), that is the conditional distribution for \(\boldsymbol{y}\) with a given \(\boldsymbol{x}\). The estimation of \(p(\boldsymbol{y}\vert \boldsymbol{x})\) is made using a data set with

-
    -

  • \(n\) cases \(i = 0, 1, 2, \dots, n-1\)

    • -

  • Response (target, dependent or outcome) variable \(y_i\) with \(i = 0, 1, 2, \dots, n-1\)

    • -

  • \(p\) so-called explanatory (independent or predictor) variables \(\boldsymbol{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]\) with \(i = 0, 1, 2, \dots, n-1\) and explanatory variables running from \(0\) to \(p-1\). See below for more explicit examples.

    • -
-

The goal of the regression analysis is to extract/exploit relationship between \(\boldsymbol{y}\) and \(\boldsymbol{x}\) in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.

-

Consider an experiment in which \(p\) characteristics of \(n\) samples are -measured. The data from this experiment, for various explanatory variables \(p\) are normally represented by a matrix
-\(\mathbf{X}\).

-

The matrix \(\mathbf{X}\) is called the design -matrix. Additional information of the samples is available in the -form of \(\boldsymbol{y}\) (also as above). The variable \(\boldsymbol{y}\) is -generally referred to as the response variable. The aim of -regression analysis is to explain \(\boldsymbol{y}\) in terms of -\(\boldsymbol{X}\) through a functional relationship like \(y_i = -f(\mathbf{X}_{i,\ast})\). When no prior knowledge on the form of -\(f(\cdot)\) is available, it is common to assume a linear relationship -between \(\boldsymbol{X}\) and \(\boldsymbol{y}\). This assumption gives rise to -the linear regression model where \(\boldsymbol{\beta} = [\beta_0, \ldots, -\beta_{p-1}]^{T}\) are the regression parameters.

-

Linear regression gives us a set of analytical equations for the parameters \(\beta_j\).

-

In order to understand the relation among the predictors \(p\), the set of data \(n\) and the target (outcome, output etc) \(\boldsymbol{y}\), -consider the model we discussed for describing nuclear binding energies.

-

There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model. -Assuming

-
-\[ -BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1}, -\]
-

we have five predictors, that is the intercept, the \(A\) dependent term, the \(A^{2/3}\) term and the \(A^{-1/3}\) and \(A^{-1}\) terms. -This gives \(p=0,1,2,3,4\). Furthermore we have \(n\) entries for each predictor. It means that our design matrix is a -\(p\times n\) matrix \(\boldsymbol{X}\).

-

Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the -so-called credit card default data from Taiwan. The data set contains data on \(n=30000\) credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are \(24\) such predictors or attributes leading to a design matrix of dimensionality \(24 \times 30000\). This is however a classification problem and we will come back to it when we discuss Logistic Regression.

-

Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \(\boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}]\). We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables \(\boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}]\), that is \(y_i = y(x_i)\) with \(i=0,1,2,\dots,n-1\). The variables \(x_i\) could represent physical quantities like time, temperature, position etc. We assume that \(y(x)\) is a smooth function.

-

Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of \(y\) which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree \(n-1\) with \(n\) points, that is

-
-\[ -y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i, -\]
-

where \(\epsilon_i\) is the error in our approximation.

-

For every set of values \(y_i,x_i\) we have thus the corresponding set of equations

-
-\[\begin{split} -\begin{align*} -y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ -y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ -y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ -\dots & \dots \\ -y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\ -\end{align*} -\end{split}\]
-

Defining the vectors

-
-\[ -\boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T, -\]
-

and

-
-\[ -\boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T, -\]
-

and

-
-\[ -\boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T, -\]
-

and the design matrix

-
-\[\begin{split} -\boldsymbol{X}= -\begin{bmatrix} -1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\ -1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\ -1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\ -\dots& \dots &\dots& \dots & \dots &\dots\\ -1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\ -\end{bmatrix} -\end{split}\]
-

we can rewrite our equations as

-
-\[ -\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. -\]
-

The above design matrix is called a Vandermonde matrix.

-

We are obviously not limited to the above polynomial expansions. We -could replace the various powers of \(x\) with elements of Fourier -series or instead of \(x_i^j\) we could have \(\cos{(j x_i)}\) or \(\sin{(j -x_i)}\), or time series or other orthogonal functions. For every set -of values \(y_i,x_i\) we can then generalize the equations to

-
-\[\begin{split} -\begin{align*} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ -\dots & \dots \\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ -\dots & \dots \\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align*} -\end{split}\]
-

Note that we have \(p=n\) here. The matrix is symmetric. This is generally not the case!

-

We redefine in turn the matrix \(\boldsymbol{X}\) as

-
-\[\begin{split} -\boldsymbol{X}= -\begin{bmatrix} -x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\ -x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\ -x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\ -\dots& \dots &\dots& \dots & \dots &\dots\\ -x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\ -\end{bmatrix} -\end{split}\]
-

and without loss of generality we rewrite again our equations as

-
-\[ -\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. -\]
-

The left-hand side of this equation is kwown. Our error vector \(\boldsymbol{\epsilon}\) and the parameter vector \(\boldsymbol{\beta}\) are our unknow quantities. How can we obtain the optimal set of \(\beta_i\) values?

-

We have defined the matrix \(\boldsymbol{X}\) via the equations

-
-\[\begin{split} -\begin{align*} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ -\dots & \dots \\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ -\dots & \dots \\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align*} -\end{split}\]
-

As we noted above, we stayed with a system with the design matrix -\(\boldsymbol{X}\in {\mathbb{R}}^{n\times n}\), that is we have \(p=n\). For reasons to come later (algorithmic arguments) we will hereafter define -our matrix as \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\), with the predictors refering to the column numbers and the entries \(n\) being the row elements.

-

In our introductory notes we looked at the so-called liquid drop model. Let us remind ourselves about what we did by looking at the code.

-

We restate the parts of the code we are most interested in.

-
-
-
# Common imports
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from IPython.display import display
-import os
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("MassEval2016.dat"),'r')
-
-
-# Read the experimental data with Pandas
-Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
-              names=('N', 'Z', 'A', 'Element', 'Ebinding'),
-              widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
-              header=39,
-              index_col=False)
-
-# Extrapolated values are indicated by '#' in place of the decimal place, so
-# the Ebinding column won't be numeric. Coerce to float and drop these entries.
-Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
-Masses = Masses.dropna()
-# Convert from keV to MeV.
-Masses['Ebinding'] /= 1000
-
-# Group the DataFrame by nucleon number, A.
-Masses = Masses.groupby('A')
-# Find the rows of the grouped DataFrame with the maximum binding energy.
-Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
-A = Masses['A']
-Z = Masses['Z']
-N = Masses['N']
-Element = Masses['Element']
-Energies = Masses['Ebinding']
-
-# Now we set up the design matrix X
-X = np.zeros((len(A),5))
-X[:,0] = 1
-X[:,1] = A
-X[:,2] = A**(2.0/3.0)
-X[:,3] = A**(-1.0/3.0)
-X[:,4] = A**(-1.0)
-# Then nice printout using pandas
-DesignMatrix = pd.DataFrame(X)
-DesignMatrix.index = A
-DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']
-display(DesignMatrix)
-
-
-
-
-

With \(\boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1}\), it means that we will hereafter write our equations for the approximation as

-
-\[ -\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, -\]
-

throughout these lectures.

-

With the above we use the design matrix to define the approximation \(\boldsymbol{\tilde{y}}\) via the unknown quantity \(\boldsymbol{\beta}\) as

-
-\[ -\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, -\]
-

and in order to find the optimal parameters \(\beta_i\) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \(y_i\) (which represent hopefully the exact values) and the parameterized values \(\tilde{y}_i\), namely

-
-\[ -C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, -\]
-

or using the matrix \(\boldsymbol{X}\) and in a more compact matrix-vector notation as

-
-\[ -C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. -\]
-

This function is one possible way to define the so-called cost function.

-

It is also common to define -the function \(C\) as

-
-\[ -C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2, -\]
-

since when taking the first derivative with respect to the unknown parameters \(\beta\), the factor of \(2\) cancels out.

-

The function

-
-\[ -C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}, -\]
-

can be linked to the variance of the quantity \(y_i\) if we interpret the latter as the mean value. -When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret \(y_i\) as a mean value

-
-\[ -y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, -\]
-

where \(\langle y_i \rangle\) is the mean value. Keep in mind also that -till now we have treated \(y_i\) as the exact value. Normally, the -response (dependent or outcome) variable \(y_i\) the outcome of a -numerical experiment or another type of experiment and is thus only an -approximation to the true value. It is then always accompanied by an -error estimate, often limited to a statistical error estimate given by -the standard deviation discussed earlier. In the discussion here we -will treat \(y_i\) as our exact value for the response variable.

-

In order to find the parameters \(\beta_i\) we will then minimize the spread of \(C(\boldsymbol{\beta})\), that is we are going to solve the problem

-
-\[ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. -\]
-

In practical terms it means we will require

-
-\[ -\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0, -\]
-

which results in

-
-\[ -\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0, -\]
-

or in a matrix-vector form as

-
-\[ -\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right). -\]
-

We can rewrite

-
-\[ -\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right), -\]
-

as

-
-\[ -\boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}, -\]
-

and if the matrix \(\boldsymbol{X}^T\boldsymbol{X}\) is invertible we have the solution

-
-\[ -\boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. -\]
-

We note also that since our design matrix is defined as \(\boldsymbol{X}\in -{\mathbb{R}}^{n\times p}\), the product \(\boldsymbol{X}^T\boldsymbol{X} \in -{\mathbb{R}}^{p\times p}\). In the above case we have that \(p \ll n\), -in our case \(p=5\) meaning that we end up with inverting a small -\(5\times 5\) matrix. This is a rather common situation, in many cases we end up with low-dimensional -matrices to invert. The methods discussed here and for many other -supervised learning algorithms like classification with logistic -regression or support vector machines, exhibit dimensionalities which -allow for the usage of direct linear algebra methods such as LU decomposition or Singular Value Decomposition (SVD) for finding the inverse of the matrix -\(\boldsymbol{X}^T\boldsymbol{X}\).

-

Small question: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix \(\boldsymbol{X}^T\boldsymbol{X}\)? What kind of problems can we expect?

-

The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and -matrices as upper case boldfaced letters.

-

4 -8

-

< -< -< -! -! -M -A -T -H -_ -B -L -O -C -K

-

4 -9

-

< -< -< -! -! -M -A -T -H -_ -B -L -O -C -K

-

5 -0

-

< -< -< -! -! -M -A -T -H -_ -B -L -O -C -K

-
-\[ -\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T. -\]
-

The residuals \(\boldsymbol{\epsilon}\) are in turn given by

-
-\[ -\boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}, -\]
-

and with

-
-\[ -\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, -\]
-

we have

-
-\[ -\boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, -\]
-

meaning that the solution for \(\boldsymbol{\beta}\) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.

-

Let us now return to our nuclear binding energies and simply code the above equations.

-

It is rather straightforward to implement the matrix inversion and obtain the parameters \(\boldsymbol{\beta}\). After having defined the matrix \(\boldsymbol{X}\) we simply need to -write

-
-
-
# matrix inversion to find beta
-beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)
-# and then make the prediction
-ytilde = X @ beta
-
-
-
-
-

Alternatively, you can use the least squares functionality in Numpy as

-
-
-
fit = np.linalg.lstsq(X, Energies, rcond =None)[0]
-ytildenp = np.dot(fit,X.T)
-
-
-
-
-

And finally we plot our fit with and compare with data

-
-
-
Masses['Eapprox']  = ytilde
-# Generate a plot comparing the experimental with the fitted values values.
-fig, ax = plt.subplots()
-ax.set_xlabel(r'$A = N + Z$')
-ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
-ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
-            label='Ame2016')
-ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
-            label='Fit')
-ax.legend()
-save_fig("Masses2016OLS")
-plt.show()
-
-
-
-
-

We can easily test our fit by computing the \(R2\) score that we discussed in connection with the functionality of Scikit-Learn in the introductory slides. -Since we are not using Scikit-Learn here we can define our own \(R2\) function as

-
-
-
def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-
-
-
-
-

and we would be using it as

-
-
-
print(R2(Energies,ytilde))
-
-
-
-
-

We can easily add our MSE score as

-
-
-
def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-print(MSE(Energies,ytilde))
-
-
-
-
-

and finally the relative error as

-
-
-
def RelativeError(y_data,y_model):
-    return abs((y_data-y_model)/y_data)
-print(RelativeError(Energies, ytilde))
-
-
-
-
-
-

1.14.1. The \(\chi^2\) function

-

Normally, the response (dependent or outcome) variable \(y_i\) is the -outcome of a numerical experiment or another type of experiment and is -thus only an approximation to the true value. It is then always -accompanied by an error estimate, often limited to a statistical error -estimate given by the standard deviation discussed earlier. In the -discussion here we will treat \(y_i\) as our exact value for the -response variable.

-

Introducing the standard deviation \(\sigma_i\) for each measurement -\(y_i\), we define now the \(\chi^2\) function (omitting the \(1/n\) term) -as

-
-\[ -\chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, -\]
-

where the matrix \(\boldsymbol{\Sigma}\) is a diagonal matrix with \(\sigma_i\) as matrix elements.

-

In order to find the parameters \(\beta_i\) we will then minimize the spread of \(\chi^2(\boldsymbol{\beta})\) by requiring

-
-\[ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0, -\]
-

which results in

-
-\[ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0, -\]
-

or in a matrix-vector form as

-
-\[ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right). -\]
-

where we have defined the matrix \(\boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma}\) with matrix elements \(a_{ij} = x_{ij}/\sigma_i\) and the vector \(\boldsymbol{b}\) with elements \(b_i = y_i/\sigma_i\).

-

We can rewrite

-
-\[ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right), -\]
-

as

-
-\[ -\boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta}, -\]
-

and if the matrix \(\boldsymbol{A}^T\boldsymbol{A}\) is invertible we have the solution

-
-\[ -\boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}. -\]
-

If we then introduce the matrix

-
-\[ -\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}, -\]
-

we have then the following expression for the parameters \(\beta_j\) (the matrix elements of \(\boldsymbol{H}\) are \(h_{ij}\))

-
-\[ -\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik} -\]
-

We state without proof the expression for the uncertainty in the parameters \(\beta_j\) as (we leave this as an exercise)

-
-\[ -\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2, -\]
-

resulting in

-
-\[ -\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}! -\]
-

The first step here is to approximate the function \(y\) with a first-order polynomial, that is we write

-
-\[ -y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i. -\]
-

By computing the derivatives of \(\chi^2\) with respect to \(\beta_0\) and \(\beta_1\) show that these are given by

-
-\[ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0, -\]
-

and

-
-\[ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0. -\]
-

For a linear fit (a first-order polynomial) we don’t need to invert a matrix!!
-Defining

-
-\[ -\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2}, -\]
-
-\[ -\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2}, -\]
-
-\[ -\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right), -\]
-
-\[ -\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2}, -\]
-
-\[ -\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2}, -\]
-

we obtain

-
-\[ -\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}, -\]
-
-\[ -\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}. -\]
-

This approach (different linear and non-linear regression) suffers -often from both being underdetermined and overdetermined in the -unknown coefficients \(\beta_i\). A better approach is to use the -Singular Value Decomposition (SVD) method discussed below. Or using -Lasso and Ridge regression. See below.

-
-
-

1.14.2. Fitting an Equation of State for Dense Nuclear Matter

-

Before we continue, let us introduce yet another example. We are going to fit the -nuclear equation of state using results from many-body calculations. -The equation of state we have made available here, as function of -density, has been derived using modern nucleon-nucleon potentials with -the addition of three-body -forces. This -time the file is presented as a standard csv file.

-

The beginning of the Python code here is similar to what you have seen -before, with the same initializations and declarations. We use also -pandas again, rather extensively in order to organize our data.

-

The difference now is that we use Scikit-Learn’s regression tools -instead of our own matrix inversion implementation. Furthermore, we -sneak in Ridge regression (to be discussed below) which includes a -hyperparameter \(\lambda\), also to be explained below.

-
-
-
# Common imports
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-import matplotlib.pyplot as plt
-import sklearn.linear_model as skl
-from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("EoS.csv"),'r')
-
-# Read the EoS data as  csv file and organize the data into two arrays with density and energies
-EoS = pd.read_csv(infile, names=('Density', 'Energy'))
-EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
-EoS = EoS.dropna()
-Energies = EoS['Energy']
-Density = EoS['Density']
-#  The design matrix now as function of various polytrops
-X = np.zeros((len(Density),4))
-X[:,3] = Density**(4.0/3.0)
-X[:,2] = Density
-X[:,1] = Density**(2.0/3.0)
-X[:,0] = 1
-
-# We use now Scikit-Learn's linear regressor and ridge regressor
-# OLS part
-clf = skl.LinearRegression().fit(X, Energies)
-ytilde = clf.predict(X)
-EoS['Eols']  = ytilde
-# The mean squared error                               
-print("Mean squared error: %.2f" % mean_squared_error(Energies, ytilde))
-# Explained variance score: 1 is perfect prediction                                 
-print('Variance score: %.2f' % r2_score(Energies, ytilde))
-# Mean absolute error                                                           
-print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde))
-print(clf.coef_, clf.intercept_)
-
-# The Ridge regression with a hyperparameter lambda = 0.1
-_lambda = 0.1
-clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies)
-yridge = clf_ridge.predict(X)
-EoS['Eridge']  = yridge
-# The mean squared error                               
-print("Mean squared error: %.2f" % mean_squared_error(Energies, yridge))
-# Explained variance score: 1 is perfect prediction                                 
-print('Variance score: %.2f' % r2_score(Energies, yridge))
-# Mean absolute error                                                           
-print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge))
-print(clf_ridge.coef_, clf_ridge.intercept_)
-
-fig, ax = plt.subplots()
-ax.set_xlabel(r'$\rho[\mathrm{fm}^{-3}]$')
-ax.set_ylabel(r'Energy per particle')
-ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2,
-            label='Theoretical data')
-ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m',
-            label='OLS')
-ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g',
-            label='Ridge $\lambda = 0.1$')
-ax.legend()
-save_fig("EoSfitting")
-plt.show()
-
-
-
-
-

The above simple polynomial in density \(\rho\) gives an excellent fit -to the data.

-

We note also that there is a small deviation between the -standard OLS and the Ridge regression at higher densities. We discuss this in more detail -below.

-
-
-
-

1.15. Splitting our Data in Training and Test data

-

It is normal in essentially all Machine Learning studies to split the -data in a training set and a test set (sometimes also an additional -validation set). Scikit-Learn has an own function for this. There -is no explicit recipe for how much data should be included as training -data and say test data. An accepted rule of thumb is to use -approximately \(2/3\) to \(4/5\) of the data as training data. We will -postpone a discussion of this splitting to the end of these notes and -our discussion of the so-called bias-variance tradeoff. Here we -limit ourselves to repeat the above equation of state fitting example -but now splitting the data into a training set and a test set.

-
-
-
import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.model_selection import train_test_split
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-def R2(y_data, y_model):
-    return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
-def MSE(y_data,y_model):
-    n = np.size(y_model)
-    return np.sum((y_data-y_model)**2)/n
-
-infile = open(data_path("EoS.csv"),'r')
-
-# Read the EoS data as  csv file and organized into two arrays with density and energies
-EoS = pd.read_csv(infile, names=('Density', 'Energy'))
-EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
-EoS = EoS.dropna()
-Energies = EoS['Energy']
-Density = EoS['Density']
-#  The design matrix now as function of various polytrops
-X = np.zeros((len(Density),5))
-X[:,0] = 1
-X[:,1] = Density**(2.0/3.0)
-X[:,2] = Density
-X[:,3] = Density**(4.0/3.0)
-X[:,4] = Density**(5.0/3.0)
-# We split the data in test and training data
-X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
-# matrix inversion to find beta
-beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)
-# and then make the prediction
-ytilde = X_train @ beta
-print("Training R2")
-print(R2(y_train,ytilde))
-print("Training MSE")
-print(MSE(y_train,ytilde))
-ypredict = X_test @ beta
-print("Test R2")
-print(R2(y_test,ypredict))
-print("Test MSE")
-print(MSE(y_test,ypredict))
-
-
-
-
-
-
-

1.16. The Boston housing data example

-

The Boston housing
-data set was originally a part of UCI Machine Learning Repository -and has been removed now. The data set is now included in Scikit-Learn’s -library. There are 506 samples and 13 feature (predictor) variables -in this data set. The objective is to predict the value of prices of -the house using the features (predictors) listed here.

-

The features/predictors are

-
    -

  1. CRIM: Per capita crime rate by town

    2. -

  2. ZN: Proportion of residential land zoned for lots over 25000 square feet

    3. -

  3. INDUS: Proportion of non-retail business acres per town

    4. -

  4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)

    5. -

  5. NOX: Nitric oxide concentration (parts per 10 million)

    6. -

  6. RM: Average number of rooms per dwelling

    7. -

  7. AGE: Proportion of owner-occupied units built prior to 1940

    8. -

  8. DIS: Weighted distances to five Boston employment centers

    9. -

  9. RAD: Index of accessibility to radial highways

    10. -

  10. TAX: Full-value property tax rate per USD10000

    11. -

  11. B: \(1000(Bk - 0.63)^2\), where \(Bk\) is the proportion of [people of African American descent] by town

    12. -

  12. LSTAT: Percentage of lower status of the population

    13. -

  13. MEDV: Median value of owner-occupied homes in USD 1000s

    14. -
-
-
-

1.17. Housing data, the code

-

We start by importing the libraries

-
-
-
import numpy as np
-import matplotlib.pyplot as plt 
-
-import pandas as pd  
-import seaborn as sns
-
-
-
-
-

and load the Boston Housing DataSet from Scikit-Learn

-
-
-
from sklearn.datasets import load_boston
-
-boston_dataset = load_boston()
-
-# boston_dataset is a dictionary
-# let's check what it contains
-boston_dataset.keys()
-
-
-
-
-

Then we invoke Pandas

-
-
-
boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)
-boston.head()
-boston['MEDV'] = boston_dataset.target
-
-
-
-
-

and preprocess the data

-
-
-
# check for missing values in all the columns
-boston.isnull().sum()
-
-
-
-
-

We can then visualize the data

-
-
-
# set the size of the figure
-sns.set(rc={'figure.figsize':(11.7,8.27)})
-
-# plot a histogram showing the distribution of the target values
-sns.distplot(boston['MEDV'], bins=30)
-plt.show()
-
-
-
-
-

It is now useful to look at the correlation matrix

-
-
-
# compute the pair wise correlation for all columns  
-correlation_matrix = boston.corr().round(2)
-# use the heatmap function from seaborn to plot the correlation matrix
-# annot = True to print the values inside the square
-sns.heatmap(data=correlation_matrix, annot=True)
-
-
-
-
-

From the above coorelation plot we can see that MEDV is strongly correlated to LSTAT and RM. We see also that RAD and TAX are stronly correlated, but we don’t include this in our features together to avoid multi-colinearity

-
-
-
plt.figure(figsize=(20, 5))
-
-features = ['LSTAT', 'RM']
-target = boston['MEDV']
-
-for i, col in enumerate(features):
-    plt.subplot(1, len(features) , i+1)
-    x = boston[col]
-    y = target
-    plt.scatter(x, y, marker='o')
-    plt.title(col)
-    plt.xlabel(col)
-    plt.ylabel('MEDV')
-
-
-
-
-

Now we start training our model

-
-
-
X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])
-Y = boston['MEDV']
-
-
-
-
-

We split the data into training and test sets

-
-
-
from sklearn.model_selection import train_test_split
-
-# splits the training and test data set in 80% : 20%
-# assign random_state to any value.This ensures consistency.
-X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)
-print(X_train.shape)
-print(X_test.shape)
-print(Y_train.shape)
-print(Y_test.shape)
-
-
-
-
-

Then we use the linear regression functionality from Scikit-Learn

-
-
-
from sklearn.linear_model import LinearRegression
-from sklearn.metrics import mean_squared_error, r2_score
-
-lin_model = LinearRegression()
-lin_model.fit(X_train, Y_train)
-
-# model evaluation for training set
-
-y_train_predict = lin_model.predict(X_train)
-rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))
-r2 = r2_score(Y_train, y_train_predict)
-
-print("The model performance for training set")
-print("--------------------------------------")
-print('RMSE is {}'.format(rmse))
-print('R2 score is {}'.format(r2))
-print("\n")
-
-# model evaluation for testing set
-
-y_test_predict = lin_model.predict(X_test)
-# root mean square error of the model
-rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))
-
-# r-squared score of the model
-r2 = r2_score(Y_test, y_test_predict)
-
-print("The model performance for testing set")
-print("--------------------------------------")
-print('RMSE is {}'.format(rmse))
-print('R2 score is {}'.format(r2))
-
-
-
-
-
-
-
# plotting the y_test vs y_pred
-# ideally should have been a straight line
-plt.scatter(Y_test, y_test_predict)
-plt.show()
-
-
-
-
-
-
-

1.18. Reducing the number of degrees of freedom, overarching view

-

Many Machine Learning problems involve thousands or even millions of -features for each training instance. Not only does this make training -extremely slow, it can also make it much harder to find a good -solution, as we will see. This problem is often referred to as the -curse of dimensionality. Fortunately, in real-world problems, it is -often possible to reduce the number of features considerably, turning -an intractable problem into a tractable one.

-

Later we will discuss some of the most popular dimensionality reduction -techniques: the principal component analysis (PCA), Kernel PCA, and -Locally Linear Embedding (LLE).

-

Principal component analysis and its various variants deal with the -problem of fitting a low-dimensional affine -subspace to a set of of -data points in a high-dimensional space. With its family of methods it -is one of the most used tools in data modeling, compression and -visualization.

-

Before we proceed however, we will discuss how to preprocess our -data. Till now and in connection with our previous examples we have -not met so many cases where we are too sensitive to the scaling of our -data. Normally the data may need a rescaling and/or may be sensitive -to extreme values. Scaling the data renders our inputs much more -suitable for the algorithms we want to employ.

-

Scikit-Learn has several functions which allow us to rescale the -data, normally resulting in much better results in terms of various -accuracy scores. The StandardScaler function in Scikit-Learn -ensures that for each feature/predictor we study the mean value is -zero and the variance is one (every column in the design/feature -matrix). This scaling has the drawback that it does not ensure that -we have a particular maximum or minimum in our data set. Another -function included in Scikit-Learn is the MinMaxScaler which -ensures that all features are exactly between \(0\) and \(1\). The

-

The Normalizer scales each data -point such that the feature vector has a euclidean length of one. In other words, it -projects a data point on the circle (or sphere in the case of higher dimensions) with a -radius of 1. This means every data point is scaled by a different number (by the -inverse of it’s length). -This normalization is often used when only the direction (or angle) of the data matters, -not the length of the feature vector.

-

The RobustScaler works similarly to the StandardScaler in that it -ensures statistical properties for each feature that guarantee that -they are on the same scale. However, the RobustScaler uses the median -and quartiles, instead of mean and variance. This makes the -RobustScaler ignore data points that are very different from the rest -(like measurement errors). These odd data points are also called -outliers, and might often lead to trouble for other scaling -techniques.

-
-

1.18.1. Simple preprocessing examples, Franke function and regression

-
-
-
# Common imports
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-import sklearn.linear_model as skl
-from sklearn.metrics import mean_squared_error
-from sklearn.model_selection import  train_test_split
-from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-
-def FrankeFunction(x,y):
-	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
-	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
-	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
-	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
-	return term1 + term2 + term3 + term4
-
-
-def create_X(x, y, n ):
-	if len(x.shape) > 1:
-		x = np.ravel(x)
-		y = np.ravel(y)
-
-	N = len(x)
-	l = int((n+1)*(n+2)/2)		# Number of elements in beta
-	X = np.ones((N,l))
-
-	for i in range(1,n+1):
-		q = int((i)*(i+1)/2)
-		for k in range(i+1):
-			X[:,q+k] = (x**(i-k))*(y**k)
-
-	return X
-
-
-# Making meshgrid of datapoints and compute Franke's function
-n = 5
-N = 1000
-x = np.sort(np.random.uniform(0, 1, N))
-y = np.sort(np.random.uniform(0, 1, N))
-z = FrankeFunction(x, y)
-X = create_X(x, y, n=n)    
-# split in training and test data
-X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=0.2)
-
-
-clf = skl.LinearRegression().fit(X_train, y_train)
-
-# The mean squared error and R2 score
-print("MSE before scaling: {:.2f}".format(mean_squared_error(clf.predict(X_test), y_test)))
-print("R2 score before scaling {:.2f}".format(clf.score(X_test,y_test)))
-
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-print("Feature min values before scaling:\n {}".format(X_train.min(axis=0)))
-print("Feature max values before scaling:\n {}".format(X_train.max(axis=0)))
-
-print("Feature min values after scaling:\n {}".format(X_train_scaled.min(axis=0)))
-print("Feature max values after scaling:\n {}".format(X_train_scaled.max(axis=0)))
-
-clf = skl.LinearRegression().fit(X_train_scaled, y_train)
-
-
-print("MSE after  scaling: {:.2f}".format(mean_squared_error(clf.predict(X_test_scaled), y_test)))
-print("R2 score for  scaled data: {:.2f}".format(clf.score(X_test_scaled,y_test)))
-
-
-
-
-
-
-
- - - - -
- - -
- - 2. Linear Algebra, Handling of Arrays and more Python Features - 2. Resampling Methods - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

-
-
-
- - -
-
- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter10.html b/doc/LectureNotes/_build/html/chapter10.html deleted file mode 100644 index c0641c93b..000000000 --- a/doc/LectureNotes/_build/html/chapter10.html +++ /dev/null @@ -1,2666 +0,0 @@ - - - - - - - - 2. Building a Feed Forward Neural Network — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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- - - - - - -
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- - - - - - - - - -
- - -
- -
- - Contents -
- - -
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- -
- -
-

2. Building a Feed Forward Neural Network

-

We are now gong to develop an example based on the MNIST data -base. This is a classification problem and we need to use our -cross-entropy function we discussed in connection with logistic -regression. The cross-entropy defines our cost function for the -classificaton problems with neural networks.

-

In binary classification with two classes \((0, 1)\) we define the -logistic/sigmoid function as the probability that a particular input -is in class \(0\) or \(1\). This is possible because the logistic -function takes any input from the real numbers and inputs a number -between 0 and 1, and can therefore be interpreted as a probability. It -also has other nice properties, such as a derivative that is simple to -calculate.

-

For an input \(\boldsymbol{a}\) from the hidden layer, the probability that the input \(\boldsymbol{x}\) -is in class 0 or 1 is just. We let \(\theta\) represent the unknown weights and biases to be adjusted by our equations). The variable \(x\) -represents our activation values \(z\). We have

-
-\[ -P(y = 0 \mid \hat{x}, \hat{\theta}) = \frac{1}{1 + \exp{(- \hat{x}})} , -\]
-

and

-
-\[ -P(y = 1 \mid \hat{x}, \hat{\theta}) = 1 - P(y = 0 \mid \hat{x}, \hat{\theta}) , -\]
-

where \(y \in \{0, 1\}\) and \(\hat{\theta}\) represents the weights and biases -of our network.

-
-

2.1. Defining the cost function

-

Our cost function is given as (see the Logistic regression lectures)

-
-\[ -\mathcal{C}(\hat{\theta}) = - \ln P(\mathcal{D} \mid \hat{\theta}) = - \sum_{i=1}^n -y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\hat{\theta}) . -\]
-

This last equality means that we can interpret our cost function as a sum over the loss function -for each point in the dataset \(\mathcal{L}_i(\hat{\theta})\).
-The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather -than maximizing a negative number.

-

In multiclass classification it is common to treat each integer label as a so called one-hot vector:

-

\(y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,\) and

-

\(y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,\)

-

i.e. a binary bit string of length \(C\), where \(C = 10\) is the number of classes in the MNIST dataset (numbers from \(0\) to \(9\))..

-

If \(\hat{x}_i\) is the \(i\)-th input (image), \(y_{ic}\) refers to the \(c\)-th component of the \(i\)-th -output vector \(\hat{y}_i\).
-The probability of \(\hat{x}_i\) being in class \(c\) will be given by the softmax function:

-
-\[ -P(y_{ic} = 1 \mid \hat{x}_i, \hat{\theta}) = \frac{\exp{((\hat{a}_i^{hidden})^T \hat{w}_c)}} -{\sum_{c'=0}^{C-1} \exp{((\hat{a}_i^{hidden})^T \hat{w}_{c'})}} , -\]
-

which reduces to the logistic function in the binary case.
-The likelihood of this \(C\)-class classifier -is now given as:

-
-\[ -P(\mathcal{D} \mid \hat{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} . -\]
-

Again we take the negative log-likelihood to define our cost function:

-
-\[ -\mathcal{C}(\hat{\theta}) = - \log{P(\mathcal{D} \mid \hat{\theta})}. -\]
-

See the logistic regression lectures for a full definition of the cost function.

-

The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!

-
-

2.1.1. Example: binary classification problem

-

As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \(\beta\) as

-
-\[ -\mathcal{C}(\hat{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\hat{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\hat{\beta})}\right), -\]
-

where we had defined the logistic (sigmoid) function

-
-\[ -p(y_i =1\vert x_i,\hat{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}}, -\]
-

and

-
-\[ -p(y_i =0\vert x_i,\hat{\beta})=1-p(y_i =1\vert x_i,\hat{\beta}). -\]
-

The parameters \(\hat{\beta}\) were defined using a minimization method like gradient descent or Newton-Raphson’s method.

-

Now we replace \(x_i\) with the activation \(z_i^l\) for a given layer \(l\) and the outputs as \(y_i=a_i^l=f(z_i^l)\), with \(z_i^l\) now being a function of the weights \(w_{ij}^l\) and biases \(b_i^l\). -We have then

-
-\[ -a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}}, -\]
-

with

-
-\[ -z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l, -\]
-

where the superscript \(l-1\) indicates that these are the outputs from layer \(l-1\). -Our cost function at the final layer \(l=L\) is now

-
-\[ -\mathcal{C}(\hat{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right), -\]
-

where we have defined the targets \(t_i\). The derivatives of the cost function with respect to the output \(a_i^L\) are then easily calculated and we get

-
-\[ -\frac{\partial \mathcal{C}(\hat{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}. -\]
-

In case we use another activation function than the logistic one, we need to evaluate other derivatives.

-
-
-

2.1.2. The Softmax function

-

In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation \(z_i^l\), that is we need

-
-\[ -\frac{\partial f(z_i^l)}{\partial w_{jk}^l} = -\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}. -\]
-

For the Softmax function we have

-
-\[ -f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}. -\]
-

Its derivative with respect to \(z_j^l\) gives

-
-\[ -\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right), -\]
-

which in case of the simply binary model reduces to having \(i=j\).

-
-
-
-

2.2. Developing a code for doing neural networks with back propagation

-

One can identify a set of key steps when using neural networks to solve supervised learning problems:

-
    -

  1. Collect and pre-process data

    2. -

  2. Define model and architecture

    3. -

  3. Choose cost function and optimizer

    4. -

  4. Train the model

    5. -

  5. Evaluate model performance on test data

    6. -

  6. Adjust hyperparameters (if necessary, network architecture)

    7. -
-
-

2.2.1. Collect and pre-process data

-

Here we will be using the MNIST dataset, which is readily available through the scikit-learn -package. You may also find it for example here.
-The MNIST (Modified National Institute of Standards and Technology) database is a large database -of handwritten digits that is commonly used for training various image processing systems.
-The MNIST dataset consists of 70 000 images of size \(28\times 28\) pixels, each labeled from 0 to 9.
-The scikit-learn dataset we will use consists of a selection of 1797 images of size \(8\times 8\) collected and processed from this database.

-

To feed data into a feed-forward neural network we need to represent -the inputs as a design/feature matrix \(X = (n_{inputs}, n_{features})\). Each -row represents an input, in this case a handwritten digit, and -each column represents a feature, in this case a pixel. The -correct answers, also known as labels or targets are -represented as a 1D array of integers -\(Y = (n_{inputs}) = (5, 3, 1, 8,...)\).

-

As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from -measurements of height (in m)
-and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example:

-
-\[\begin{split} X = \begin{bmatrix} -1.85 & 81\\ -1.71 & 65\\ -1.95 & 103\\ -1.55 & 42\\ -1.63 & 56 -\end{bmatrix} ,\end{split}\]
-

and the targets would be:

-
-\[ Y = (23.7, 22.2, 27.1, 17.5, 21.1) \]
-

Since each input image is a 2D matrix, we need to flatten the image -(i.e. “unravel” the 2D matrix into a 1D array) to turn the data into a -design/feature matrix. This means we lose all spatial information in the -image, such as locality and translational invariance. More complicated -architectures such as Convolutional Neural Networks can take advantage -of such information, and are most commonly applied when analyzing -images.

-
-
-
%matplotlib inline
-
-# import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# flatten the image
-# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
-n_inputs = len(inputs)
-inputs = inputs.reshape(n_inputs, -1)
-print("X = (n_inputs, n_features) = " + str(inputs.shape))
-
-
-# choose some random images to display
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
-    plt.subplot(1, 5, i+1)
-    plt.axis('off')
-    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
-    plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-
-
-
-
-
inputs = (n_inputs, pixel_width, pixel_height) = (1797, 8, 8)
-labels = (n_inputs) = (1797,)
-X = (n_inputs, n_features) = (1797, 64)
-
-
-_images/chapter10_33_1.png -
-
-
-
-

2.2.2. Train and test datasets

-

Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.

-

We will reserve \(80 \%\) of our dataset for training and \(20 \%\) for testing.

-

It is important that the train and test datasets are drawn randomly from our dataset, to ensure -no bias in the sampling.
-Say you are taking measurements of weather data to predict the weather in the coming 5 days. -You don’t want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data -collected from 12.00 to 24.00.

-
-
-
from sklearn.model_selection import train_test_split
-
-# one-liner from scikit-learn library
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
-                                                    test_size=test_size)
-
-# equivalently in numpy
-def train_test_split_numpy(inputs, labels, train_size, test_size):
-    n_inputs = len(inputs)
-    inputs_shuffled = inputs.copy()
-    labels_shuffled = labels.copy()
-    
-    np.random.shuffle(inputs_shuffled)
-    np.random.shuffle(labels_shuffled)
-    
-    train_end = int(n_inputs*train_size)
-    X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]
-    Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]
-    
-    return X_train, X_test, Y_train, Y_test
-
-#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)
-
-print("Number of training images: " + str(len(X_train)))
-print("Number of test images: " + str(len(X_test)))
-
-
-
-
-
Number of training images: 1437
-Number of test images: 360
-
-
-
-
-
-
-

2.2.3. Define model and architecture

-

Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation \(y\) of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have

-
-\[ z = \sum_{i=1}^n w_i a_i ,\]
-
-\[ y = f(z) ,\]
-

where \(f\) is the activation function, \(a_i\) represents input from neuron \(i\) in the preceding layer -and \(w_i\) is the weight to input \(i\).
-The activation of the neurons in the input layer is just the features (e.g. a pixel value).

-

The simplest activation function for a neuron is the Heaviside function:

-
-\[\begin{split} f(z) = -\begin{cases} -1, & z > 0\\ -0, & \text{otherwise} -\end{cases} -\end{split}\]
-

A feed-forward neural network with this activation is known as a perceptron.
-For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.
-This activation can be generalized to \(k\) classes (using e.g. the one-against-all strategy), -and we call these architectures multiclass perceptrons.

-

However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and
-Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.

-

Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).
-We will be using the sigmoid function \(\sigma(x)\):

-
-\[ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,\]
-

which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.

-
-
-

2.2.4. Layers

-
    -

  • Input

    • -
-

Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.

-
    -

  • Hidden layer

    • -
-

We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.
-Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.

-
    -

  • Output

    • -
-

If we were building a binary classifier, it would be sufficient with a single neuron in the output layer, -which could output 0 or 1 according to the Heaviside function. This would be an example of a hard classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a soft classifier, which outputs the probability of being in class 0 or 1.

-

For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.

-

Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons \(j = 0,1,...,9\). The activation of each output neuron \(j\) will be according to the softmax function:

-
-\[ P(\text{class $j$} \mid \text{input $\hat{a}$}) = \frac{\exp{(\hat{a}^T \hat{w}_j)}} -{\sum_{c=0}^{9} \exp{(\hat{a}^T \hat{w}_c)}} ,\]
-

i.e. each neuron \(j\) outputs the probability of being in class \(j\) given an input from the hidden layer \(\hat{a}\), with \(\hat{w}_j\) the weights of neuron \(j\) to the inputs.
-The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.
-The exponent is just the weighted sum of inputs as before:

-
-\[ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.\]
-

Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500 -weights to the output layer.

-

Typically weights are initialized with small values distributed around zero, drawn from a uniform -or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.

-

Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range -of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron \(j\), \(b_j\):

-
-\[ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.\]
-

The bias weights \(\hat{b}\) are often initialized to zero, but a small value like \(0.01\) ensures all neurons have some output which can be backpropagated in the first training cycle.

-
-
-
# building our neural network
-
-n_inputs, n_features = X_train.shape
-n_hidden_neurons = 50
-n_categories = 10
-
-# we make the weights normally distributed using numpy.random.randn
-
-# weights and bias in the hidden layer
-hidden_weights = np.random.randn(n_features, n_hidden_neurons)
-hidden_bias = np.zeros(n_hidden_neurons) + 0.01
-
-# weights and bias in the output layer
-output_weights = np.random.randn(n_hidden_neurons, n_categories)
-output_bias = np.zeros(n_categories) + 0.01
-
-
-
-
-
-
-

2.2.5. Feed-forward pass

-

Denote \(F\) the number of features, \(H\) the number of hidden neurons and \(C\) the number of categories.
-For each input image we calculate a weighted sum of input features (pixel values) to each neuron \(j\) in the hidden layer \(l\):

-
-\[ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},\]
-

this is then passed through our activation function

-
-\[ a_{j}^{l} = f(z_{j}^{l}) .\]
-

We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron \(j\) in the output layer:

-
-\[ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.\]
-

Finally we calculate the output of neuron \(j\) in the output layer using the softmax function:

-
-\[ a_{j}^{L} = \frac{\exp{(z_j^{L})}} -{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .\]
-

Since our data has the dimensions \(X = (n_{inputs}, n_{features})\) and our weights to the hidden -layer have the dimensions
-\(W_{hidden} = (n_{features}, n_{hidden})\), -we can easily feed the network all our training data in one go by taking the matrix product

-
-\[ X W^{h} = (n_{inputs}, n_{hidden}),\]
-

and obtain a matrix that holds the weighted sum of inputs to the hidden layer -for each input image and each hidden neuron.
-We also add the bias to obtain a matrix of weighted sums to the hidden layer \(Z^{h}\):

-
-\[ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,\]
-

meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.
-This is then passed through the activation:

-
-\[ \hat{a}^{l} = f(\hat{z}^l) .\]
-

This is fed to the output layer:

-
-\[ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .\]
-

Finally we receive our output values for each image and each category by passing it through the softmax function:

-
-\[ output = softmax (\hat{z}^{L}) = (n_{inputs}, n_{categories}) .\]
-
-
-
# setup the feed-forward pass, subscript h = hidden layer
-
-def sigmoid(x):
-    return 1/(1 + np.exp(-x))
-
-def feed_forward(X):
-    # weighted sum of inputs to the hidden layer
-    z_h = np.matmul(X, hidden_weights) + hidden_bias
-    # activation in the hidden layer
-    a_h = sigmoid(z_h)
-    
-    # weighted sum of inputs to the output layer
-    z_o = np.matmul(a_h, output_weights) + output_bias
-    # softmax output
-    # axis 0 holds each input and axis 1 the probabilities of each category
-    exp_term = np.exp(z_o)
-    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-    
-    return probabilities
-
-probabilities = feed_forward(X_train)
-print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
-print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
-print("probabilities sum up to: " + str(probabilities[0].sum()))
-print()
-
-# we obtain a prediction by taking the class with the highest likelihood
-def predict(X):
-    probabilities = feed_forward(X)
-    return np.argmax(probabilities, axis=1)
-
-predictions = predict(X_train)
-print("predictions = (n_inputs) = " + str(predictions.shape))
-print("prediction for image 0: " + str(predictions[0]))
-print("correct label for image 0: " + str(Y_train[0]))
-
-
-
-
-
probabilities = (n_inputs, n_categories) = (1437, 10)
-probability that image 0 is in category 0,1,2,...,9 = 
-[5.41511965e-04 2.17174962e-03 8.84355903e-03 1.44970586e-03
- 1.10378326e-04 5.08318298e-09 2.03256632e-04 1.92507116e-03
- 9.84443254e-01 3.11507992e-04]
-probabilities sum up to: 1.0
-
-predictions = (n_inputs) = (1437,)
-prediction for image 0: 8
-correct label for image 0: 6
-
-
-
-
-
-
-

2.2.6. Choose cost function and optimizer

-

To measure how well our neural network is doing we need to introduce a cost function.
-We will call the function that gives the error of a single sample output the loss function, and the function -that gives the total error of our network across all samples the cost function. -A typical choice for multiclass classification is the cross-entropy loss, also known as the negative log likelihood.

-

In multiclass classification it is common to treat each integer label as a so called one-hot vector:

-
-\[ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,\]
-
-\[ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,\]
-

i.e. a binary bit string of length \(C\), where \(C = 10\) is the number of classes in the MNIST dataset.

-

Let \(y_{ic}\) denote the \(c\)-th component of the \(i\)-th one-hot vector.
-We define the cost function \(\mathcal{C}\) as a sum over the cross-entropy loss for each point \(\hat{x}_i\) in the dataset.

-

In the one-hot representation only one of the terms in the loss function is non-zero, namely the -probability of the correct category \(c'\)
-(i.e. the category \(c'\) such that \(y_{ic'} = 1\)). This means that the cross entropy loss only punishes you for how wrong -you got the correct label. The probability of category \(c\) is given by the softmax function. The vector \(\hat{\theta}\) represents the parameters of our network, i.e. all the weights and biases.

-
-
-

2.2.7. Optimizing the cost function

-

The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent -is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a local minimum of the cost function.
-Each parameter \(\theta\) is iteratively adjusted according to the rule

-
-\[ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,\]
-

where \(\eta\) is known as the learning rate, which controls how big a step we take towards the minimum.
-This update can be repeated for any number of iterations, or until we are satisfied with the result.

-

A simple and effective improvement is a variant called Batch Gradient Descent.
-Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient -on a subset of the data called a minibatch.
-If there are \(N\) data points and we have a minibatch size of \(M\), the total number of batches -is \(N/M\).
-We denote each minibatch \(B_k\), with \(k = 1, 2,...,N/M\). The gradient then becomes:

-
-\[ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad -\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,\]
-

i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.

-

This has two important benefits:

-
    -

  1. Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.

    2. -

  2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.

    3. -
-

The various optmization methods, with codes and algorithms, are discussed in our lectures on Gradient descent approaches.

-
-
-

2.2.8. Regularization

-

It is common to add an extra term to the cost function, proportional -to the size of the weights. This is equivalent to constraining the -size of the weights, so that they do not grow out of control. -Constraining the size of the weights means that the weights cannot -grow arbitrarily large to fit the training data, and in this way -reduces overfitting.

-

We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:

-
-\[ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad -\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2 -= \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,\]
-

i.e. we sum up all the weights squared. The factor \(\lambda\) is known as a regularization parameter.

-

In order to train the model, we need to calculate the derivative of -the cost function with respect to every bias and weight in the -network. In total our network has \((64 + 1)\times 50=3250\) weights in -the hidden layer and \((50 + 1)\times 10=510\) weights to the output -layer (\(+1\) for the bias), and the gradient must be calculated for -every parameter. We use the backpropagation algorithm discussed -above. This is a clever use of the chain rule that allows us to -calculate the gradient efficently.

-
-
-

2.2.9. Matrix multiplication

-

To more efficently train our network these equations are implemented using matrix operations.
-The error in the output layer is calculated simply as, with \(\hat{t}\) being our targets,

-
-\[ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .\]
-

The gradient for the output weights is calculated as

-
-\[ \nabla W_{L} = \hat{a}^T \delta_L = (n_{hidden}, n_{categories}) ,\]
-

where \(\hat{a} = (n_{inputs}, n_{hidden})\). This simply means that we are summing up the gradients for each input.
-Since we are going backwards we have to transpose the activation matrix.

-

The gradient with respect to the output bias is then

-
-\[ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .\]
-

The error in the hidden layer is

-
-\[ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,\]
-

where \(f'(a_{h})\) is the derivative of the activation in the hidden layer. The matrix products mean -that we are summing up the products for each neuron in the output layer. The symbol \(\circ\) denotes -the Hadamard product, meaning element-wise multiplication.

-

This again gives us the gradients in the hidden layer:

-
-\[ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,\]
-
-\[ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .\]
-
-
-
# to categorical turns our integer vector into a onehot representation
-from sklearn.metrics import accuracy_score
-
-# one-hot in numpy
-def to_categorical_numpy(integer_vector):
-    n_inputs = len(integer_vector)
-    n_categories = np.max(integer_vector) + 1
-    onehot_vector = np.zeros((n_inputs, n_categories))
-    onehot_vector[range(n_inputs), integer_vector] = 1
-    
-    return onehot_vector
-
-#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
-Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)
-
-def feed_forward_train(X):
-    # weighted sum of inputs to the hidden layer
-    z_h = np.matmul(X, hidden_weights) + hidden_bias
-    # activation in the hidden layer
-    a_h = sigmoid(z_h)
-    
-    # weighted sum of inputs to the output layer
-    z_o = np.matmul(a_h, output_weights) + output_bias
-    # softmax output
-    # axis 0 holds each input and axis 1 the probabilities of each category
-    exp_term = np.exp(z_o)
-    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-    
-    # for backpropagation need activations in hidden and output layers
-    return a_h, probabilities
-
-def backpropagation(X, Y):
-    a_h, probabilities = feed_forward_train(X)
-    
-    # error in the output layer
-    error_output = probabilities - Y
-    # error in the hidden layer
-    error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
-    
-    # gradients for the output layer
-    output_weights_gradient = np.matmul(a_h.T, error_output)
-    output_bias_gradient = np.sum(error_output, axis=0)
-    
-    # gradient for the hidden layer
-    hidden_weights_gradient = np.matmul(X.T, error_hidden)
-    hidden_bias_gradient = np.sum(error_hidden, axis=0)
-
-    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
-
-print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
-
-eta = 0.01
-lmbd = 0.01
-for i in range(1000):
-    # calculate gradients
-    dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
-    
-    # regularization term gradients
-    dWo += lmbd * output_weights
-    dWh += lmbd * hidden_weights
-    
-    # update weights and biases
-    output_weights -= eta * dWo
-    output_bias -= eta * dBo
-    hidden_weights -= eta * dWh
-    hidden_bias -= eta * dBh
-
-print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))
-
-
-
-
-
Old accuracy on training data: 0.1440501043841336
-
-
-
<ipython-input-4-16b8e3cda33a>:4: RuntimeWarning: overflow encountered in exp
-  return 1/(1 + np.exp(-x))
-
-
-
New accuracy on training data: 0.1022964509394572
-
-
-
-
-
-
-
-

2.3. Improving performance

-

As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.
-In order to obtain a network that does something useful, we will have to do a bit more work.

-

The choice of hyperparameters such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a grid-search is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \(\eta = 10^{-6}, 10^{-5},...,10^{-1}\) with different regularization parameters \(\lambda = 10^{-6},...,10^{-0}\).

-

Next, we haven’t implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an iteration, and a full training period -going through the entire dataset (\(n/M\) batches) an epoch.

-

If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.
-Andrew Ng goes through some of these considerations in this video. You can find a summary of the video here.

-

It is very natural to think of the network as an object, with specific instances of the network -being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.

-
-
-
class NeuralNetwork:
-    def __init__(
-            self,
-            X_data,
-            Y_data,
-            n_hidden_neurons=50,
-            n_categories=10,
-            epochs=10,
-            batch_size=100,
-            eta=0.1,
-            lmbd=0.0):
-
-        self.X_data_full = X_data
-        self.Y_data_full = Y_data
-
-        self.n_inputs = X_data.shape[0]
-        self.n_features = X_data.shape[1]
-        self.n_hidden_neurons = n_hidden_neurons
-        self.n_categories = n_categories
-
-        self.epochs = epochs
-        self.batch_size = batch_size
-        self.iterations = self.n_inputs // self.batch_size
-        self.eta = eta
-        self.lmbd = lmbd
-
-        self.create_biases_and_weights()
-
-    def create_biases_and_weights(self):
-        self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
-        self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01
-
-        self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
-        self.output_bias = np.zeros(self.n_categories) + 0.01
-
-    def feed_forward(self):
-        # feed-forward for training
-        self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
-        self.a_h = sigmoid(self.z_h)
-
-        self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias
-
-        exp_term = np.exp(self.z_o)
-        self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-
-    def feed_forward_out(self, X):
-        # feed-forward for output
-        z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
-        a_h = sigmoid(z_h)
-
-        z_o = np.matmul(a_h, self.output_weights) + self.output_bias
-        
-        exp_term = np.exp(z_o)
-        probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-        return probabilities
-
-    def backpropagation(self):
-        error_output = self.probabilities - self.Y_data
-        error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)
-
-        self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
-        self.output_bias_gradient = np.sum(error_output, axis=0)
-
-        self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
-        self.hidden_bias_gradient = np.sum(error_hidden, axis=0)
-
-        if self.lmbd > 0.0:
-            self.output_weights_gradient += self.lmbd * self.output_weights
-            self.hidden_weights_gradient += self.lmbd * self.hidden_weights
-
-        self.output_weights -= self.eta * self.output_weights_gradient
-        self.output_bias -= self.eta * self.output_bias_gradient
-        self.hidden_weights -= self.eta * self.hidden_weights_gradient
-        self.hidden_bias -= self.eta * self.hidden_bias_gradient
-
-    def predict(self, X):
-        probabilities = self.feed_forward_out(X)
-        return np.argmax(probabilities, axis=1)
-
-    def predict_probabilities(self, X):
-        probabilities = self.feed_forward_out(X)
-        return probabilities
-
-    def train(self):
-        data_indices = np.arange(self.n_inputs)
-
-        for i in range(self.epochs):
-            for j in range(self.iterations):
-                # pick datapoints with replacement
-                chosen_datapoints = np.random.choice(
-                    data_indices, size=self.batch_size, replace=False
-                )
-
-                # minibatch training data
-                self.X_data = self.X_data_full[chosen_datapoints]
-                self.Y_data = self.Y_data_full[chosen_datapoints]
-
-                self.feed_forward()
-                self.backpropagation()
-
-
-
-
-
-
-

2.4. Evaluate model performance on test data

-

To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.
-We measure the performance of the network using the accuracy score.
-The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of \(1\).

-
-\[ \text{Accuracy} = \frac{\sum_{i=1}^n I(\hat{y}_i = y_i)}{n} ,\]
-

where \(I\) is the indicator function, \(1\) if \(\hat{y}_i = y_i\) and \(0\) otherwise.

-
-
-
epochs = 100
-batch_size = 100
-
-dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
-                    n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
-dnn.train()
-test_predict = dnn.predict(X_test)
-
-# accuracy score from scikit library
-print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
-
-# equivalent in numpy
-def accuracy_score_numpy(Y_test, Y_pred):
-    return np.sum(Y_test == Y_pred) / len(Y_test)
-
-#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))
-
-
-
-
-
Accuracy score on test set:  0.9361111111111111
-
-
-
-
-
-
-

2.5. Adjust hyperparameters

-

We now perform a grid search to find the optimal hyperparameters for the network.
-Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around \(98\%\) (\(2\%\) error rate).

-
-
-
eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-# store the models for later use
-DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-
-# grid search
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
-                            n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
-        dnn.train()
-        
-        DNN_numpy[i][j] = dnn
-        
-        test_predict = dnn.predict(X_test)
-        
-        print("Learning rate  = ", eta)
-        print("Lambda = ", lmbd)
-        print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
-        print()
-
-
-
-
-
Learning rate  =  1e-05
-Lambda =  1e-05
-Accuracy score on test set:  0.11666666666666667
-
-
-
Learning rate  =  1e-05
-Lambda =  0.0001
-Accuracy score on test set:  0.20833333333333334
-
-
-
Learning rate  =  1e-05
-Lambda =  0.001
-Accuracy score on test set:  0.12222222222222222
-
-
-
Learning rate  =  1e-05
-Lambda =  0.01
-Accuracy score on test set:  0.14722222222222223
-
-
-
Learning rate  =  1e-05
-Lambda =  0.1
-Accuracy score on test set:  0.17777777777777778
-
-
-
Learning rate  =  1e-05
-Lambda =  1.0
-Accuracy score on test set:  0.16111111111111112
-
-
-
Learning rate  =  1e-05
-Lambda =  10.0
-Accuracy score on test set:  0.20277777777777778
-
-
-
Learning rate  =  0.0001
-Lambda =  1e-05
-Accuracy score on test set:  0.5305555555555556
-
-
-
Learning rate  =  0.0001
-Lambda =  0.0001
-Accuracy score on test set:  0.5944444444444444
-
-
-
Learning rate  =  0.0001
-Lambda =  0.001
-Accuracy score on test set:  0.5888888888888889
-
-
-
Learning rate  =  0.0001
-Lambda =  0.01
-Accuracy score on test set:  0.6111111111111112
-
-
-
Learning rate  =  0.0001
-Lambda =  0.1
-Accuracy score on test set:  0.5222222222222223
-
-
-
Learning rate  =  0.0001
-Lambda =  1.0
-Accuracy score on test set:  0.5555555555555556
-
-
-
Learning rate  =  0.0001
-Lambda =  10.0
-Accuracy score on test set:  0.8055555555555556
-
-
-
Learning rate  =  0.001
-Lambda =  1e-05
-Accuracy score on test set:  0.85
-
-
-
Learning rate  =  0.001
-Lambda =  0.0001
-Accuracy score on test set:  0.85
-
-
-
Learning rate  =  0.001
-Lambda =  0.001
-Accuracy score on test set:  0.875
-
-
-
Learning rate  =  0.001
-Lambda =  0.01
-Accuracy score on test set:  0.8666666666666667
-
-
-
Learning rate  =  0.001
-Lambda =  0.1
-Accuracy score on test set:  0.8638888888888889
-
-
-
Learning rate  =  0.001
-Lambda =  1.0
-Accuracy score on test set:  0.9555555555555556
-
-
-
Learning rate  =  0.001
-Lambda =  10.0
-Accuracy score on test set:  0.925
-
-
-
Learning rate  =  0.01
-Lambda =  1e-05
-Accuracy score on test set:  0.9583333333333334
-
-
-
Learning rate  =  0.01
-Lambda =  0.0001
-Accuracy score on test set:  0.9277777777777778
-
-
-
Learning rate  =  0.01
-Lambda =  0.001
-Accuracy score on test set:  0.9388888888888889
-
-
-
Learning rate  =  0.01
-Lambda =  0.01
-Accuracy score on test set:  0.9166666666666666
-
-
-
Learning rate  =  0.01
-Lambda =  0.1
-Accuracy score on test set:  0.9611111111111111
-
-
-
Learning rate  =  0.01
-Lambda =  1.0
-Accuracy score on test set:  0.8777777777777778
-
-
-
Learning rate  =  0.01
-Lambda =  10.0
-Accuracy score on test set:  0.11388888888888889
-
-
-
<ipython-input-4-16b8e3cda33a>:4: RuntimeWarning: overflow encountered in exp
-  return 1/(1 + np.exp(-x))
-
-
-
Learning rate  =  0.1
-Lambda =  1e-05
-Accuracy score on test set:  0.09166666666666666
-
-
-
Learning rate  =  0.1
-Lambda =  0.0001
-Accuracy score on test set:  0.10555555555555556
-
-
-
Learning rate  =  0.1
-Lambda =  0.001
-Accuracy score on test set:  0.08888888888888889
-
-
-
Learning rate  =  0.1
-Lambda =  0.01
-Accuracy score on test set:  0.10555555555555556
-
-
-
Learning rate  =  0.1
-Lambda =  0.1
-Accuracy score on test set:  0.11666666666666667
-
-
-
Learning rate  =  0.1
-Lambda =  1.0
-Accuracy score on test set:  0.12777777777777777
-
-
-
Learning rate  =  0.1
-Lambda =  10.0
-Accuracy score on test set:  0.09166666666666666
-
-
-
<ipython-input-6-2572e3a4b38d>:43: RuntimeWarning: overflow encountered in exp
-  exp_term = np.exp(self.z_o)
-<ipython-input-6-2572e3a4b38d>:44: RuntimeWarning: invalid value encountered in true_divide
-  self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
-
-
-
Learning rate  =  1.0
-Lambda =  1e-05
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  1.0
-Lambda =  0.0001
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  1.0
-Lambda =  0.001
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  1.0
-Lambda =  0.01
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  1.0
-Lambda =  0.1
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  1.0
-Lambda =  1.0
-Accuracy score on test set:  0.08888888888888889
-
-
-
Learning rate  =  1.0
-Lambda =  10.0
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  10.0
-Lambda =  1e-05
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  10.0
-Lambda =  0.0001
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  10.0
-Lambda =  0.001
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  10.0
-Lambda =  0.01
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  10.0
-Lambda =  0.1
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  10.0
-Lambda =  1.0
-Accuracy score on test set:  0.07777777777777778
-
-
-
Learning rate  =  10.0
-Lambda =  10.0
-Accuracy score on test set:  0.07777777777777778
-
-
-
-
-
-
-

2.6. Visualization

-
-
-
# visual representation of grid search
-# uses seaborn heatmap, you can also do this with matplotlib imshow
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        dnn = DNN_numpy[i][j]
-        
-        train_pred = dnn.predict(X_train) 
-        test_pred = dnn.predict(X_test)
-
-        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
-        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
-
-        
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-
-
-
-
<ipython-input-4-16b8e3cda33a>:4: RuntimeWarning: overflow encountered in exp
-  return 1/(1 + np.exp(-x))
-
-
-_images/chapter10_49_1.png -_images/chapter10_49_2.png -
-
-
-
-

2.7. scikit-learn implementation

-

scikit-learn focuses more -on traditional machine learning methods, such as regression, -clustering, decision trees, etc. As such, it has only two types of -neural networks: Multi Layer Perceptron outputting continuous values, -MPLRegressor, and Multi Layer Perceptron outputting labels, -MLPClassifier. We will see how simple it is to use these classes.

-

scikit-learn implements a few improvements from our neural network, -such as early stopping, a varying learning rate, different -optimization methods, etc. We would therefore expect a better -performance overall.

-
-
-
from sklearn.neural_network import MLPClassifier
-# store models for later use
-DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
-                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
-        dnn.fit(X_train, Y_train)
-        
-        DNN_scikit[i][j] = dnn
-        
-        print("Learning rate  = ", eta)
-        print("Lambda = ", lmbd)
-        print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
-        print()
-
-
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  1e-05
-Lambda =  1e-05
-Accuracy score on test set:  0.18333333333333332
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  1e-05
-Lambda =  0.0001
-Accuracy score on test set:  0.18611111111111112
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  1e-05
-Lambda =  0.001
-Accuracy score on test set:  0.13055555555555556
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  1e-05
-Lambda =  0.01
-Accuracy score on test set:  0.24444444444444444
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  1e-05
-Lambda =  0.1
-Accuracy score on test set:  0.23333333333333334
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  1e-05
-Lambda =  1.0
-Accuracy score on test set:  0.12777777777777777
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  1e-05
-Lambda =  10.0
-Accuracy score on test set:  0.1527777777777778
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.0001
-Lambda =  1e-05
-Accuracy score on test set:  0.9111111111111111
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.0001
-Lambda =  0.0001
-Accuracy score on test set:  0.8888888888888888
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.0001
-Lambda =  0.001
-Accuracy score on test set:  0.8722222222222222
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.0001
-Lambda =  0.01
-Accuracy score on test set:  0.8305555555555556
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.0001
-Lambda =  0.1
-Accuracy score on test set:  0.8888888888888888
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.0001
-Lambda =  1.0
-Accuracy score on test set:  0.8805555555555555
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.0001
-Lambda =  10.0
-Accuracy score on test set:  0.8944444444444445
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.001
-Lambda =  1e-05
-Accuracy score on test set:  0.975
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.001
-Lambda =  0.0001
-Accuracy score on test set:  0.9777777777777777
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.001
-Lambda =  0.001
-Accuracy score on test set:  0.9805555555555555
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.001
-Lambda =  0.01
-Accuracy score on test set:  0.9861111111111112
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.001
-Lambda =  0.1
-Accuracy score on test set:  0.9805555555555555
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.001
-Lambda =  1.0
-Accuracy score on test set:  0.9777777777777777
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.001
-Lambda =  10.0
-Accuracy score on test set:  0.9444444444444444
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.
-  warnings.warn(
-
-
-
Learning rate  =  0.01
-Lambda =  1e-05
-Accuracy score on test set:  0.9861111111111112
-
-
-
Learning rate  =  0.01
-Lambda =  0.0001
-Accuracy score on test set:  0.9888888888888889
-
-
-
Learning rate  =  0.01
-Lambda =  0.001
-Accuracy score on test set:  0.9888888888888889
-
-
-
Learning rate  =  0.01
-Lambda =  0.01
-Accuracy score on test set:  0.9861111111111112
-
-
-
Learning rate  =  0.01
-Lambda =  0.1
-Accuracy score on test set:  0.9888888888888889
-
-
-
Learning rate  =  0.01
-Lambda =  1.0
-Accuracy score on test set:  0.9722222222222222
-
-Learning rate  =  0.01
-Lambda =  10.0
-Accuracy score on test set:  0.9527777777777777
-
-
-
Learning rate  =  0.1
-Lambda =  1e-05
-Accuracy score on test set:  0.9111111111111111
-
-
-
Learning rate  =  0.1
-Lambda =  0.0001
-Accuracy score on test set:  0.9222222222222223
-
-Learning rate  =  0.1
-Lambda =  0.001
-Accuracy score on test set:  0.9111111111111111
-
-
-
Learning rate  =  0.1
-Lambda =  0.01
-Accuracy score on test set:  0.9305555555555556
-
-Learning rate  =  0.1
-Lambda =  0.1
-Accuracy score on test set:  0.8388888888888889
-
-
-
Learning rate  =  0.1
-Lambda =  1.0
-Accuracy score on test set:  0.9055555555555556
-
-Learning rate  =  0.1
-Lambda =  10.0
-Accuracy score on test set:  0.8666666666666667
-
-
-
Learning rate  =  1.0
-Lambda =  1e-05
-Accuracy score on test set:  0.09166666666666666
-
-Learning rate  =  1.0
-Lambda =  0.0001
-Accuracy score on test set:  0.11944444444444445
-
-
-
Learning rate  =  1.0
-Lambda =  0.001
-Accuracy score on test set:  0.1361111111111111
-
-Learning rate  =  1.0
-Lambda =  0.01
-Accuracy score on test set:  0.1527777777777778
-
-
-
Learning rate  =  1.0
-Lambda =  0.1
-Accuracy score on test set:  0.16666666666666666
-
-Learning rate  =  1.0
-Lambda =  1.0
-Accuracy score on test set:  0.1111111111111111
-
-
-
Learning rate  =  1.0
-Lambda =  10.0
-Accuracy score on test set:  0.05
-
-Learning rate  =  10.0
-Lambda =  1e-05
-Accuracy score on test set:  0.08888888888888889
-
-Learning rate  =  10.0
-Lambda =  0.0001
-Accuracy score on test set:  0.08611111111111111
-
-
-
Learning rate  =  10.0
-Lambda =  0.001
-Accuracy score on test set:  0.08888888888888889
-
-Learning rate  =  10.0
-Lambda =  0.01
-Accuracy score on test set:  0.08888888888888889
-
-Learning rate  =  10.0
-Lambda =  0.1
-Accuracy score on test set:  0.10555555555555556
-
-
-
Learning rate  =  10.0
-Lambda =  1.0
-Accuracy score on test set:  0.1111111111111111
-
-Learning rate  =  10.0
-Lambda =  10.0
-Accuracy score on test set:  0.1527777777777778
-
-
-
-
-
-
-

2.8. Visualization

-
-
-
# optional
-# visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        dnn = DNN_scikit[i][j]
-        
-        train_pred = dnn.predict(X_train) 
-        test_pred = dnn.predict(X_test)
-
-        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
-        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)
-
-        
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-
-
-
-_images/chapter10_53_0.png -_images/chapter10_53_1.png -
-
-
-
-

2.9. Building neural networks in Tensorflow and Keras

-

Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn -and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy -and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer.

-

In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite -clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or -NumPy arrays.

-

Tensorflow is an open source library machine learning library -developed by the Google Brain team for internal use. It was released -under the Apache 2.0 open source license in November 9, 2015.

-

Tensorflow is a computational framework that allows you to construct -machine learning models at different levels of abstraction, from -high-level, object-oriented APIs like Keras, down to the C++ kernels -that Tensorflow is built upon. The higher levels of abstraction are -simpler to use, but less flexible, and our choice of implementation -should reflect the problems we are trying to solve.

-

Tensorflow uses so-called graphs to represent your computation -in terms of the dependencies between individual operations, such that you first build a Tensorflow graph -to represent your model, and then create a Tensorflow session to run the graph.

-

In this guide we will analyze the same data as we did in our NumPy and -scikit-learn tutorial, gathered from the MNIST database of images. We -will give an introduction to the lower level Python Application -Program Interfaces (APIs), and see how we use them to build our graph. -Then we will build (effectively) the same graph in Keras, to see just -how simple solving a machine learning problem can be.

-

To install tensorflow on Unix/Linux systems, use pip as

-
-
-
pip3 install tensorflow
-
-
-
-
-
  File "<ipython-input-12-6ea927cc6e88>", line 1
-    pip3 install tensorflow
-         ^
-SyntaxError: invalid syntax
-
-
-
-
-

and/or if you use anaconda, just write (or install from the graphical user interface) -(current release of CPU-only TensorFlow)

-
-
-
conda create -n tf tensorflow
-conda activate tf
-
-
-
-
-

To install the current release of GPU TensorFlow

-
-
-
conda create -n tf-gpu tensorflow-gpu
-conda activate tf-gpu
-
-
-
-
-

Keras is a high level neural network -that supports Tensorflow, CTNK and Theano as backends.
-If you have Anaconda installed you may run the following command

-
-
-
conda install keras
-
-
-
-
-

You can look up the instructions here for more information.

-

We will to a large extent use keras in this course.

-

Let us look again at the MINST data set.

-
-
-
# import necessary packages
-import numpy as np
-import matplotlib.pyplot as plt
-import tensorflow as tf
-from sklearn import datasets
-
-
-# ensure the same random numbers appear every time
-np.random.seed(0)
-
-# display images in notebook
-%matplotlib inline
-plt.rcParams['figure.figsize'] = (12,12)
-
-
-# download MNIST dataset
-digits = datasets.load_digits()
-
-# define inputs and labels
-inputs = digits.images
-labels = digits.target
-
-print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
-print("labels = (n_inputs) = " + str(labels.shape))
-
-
-# flatten the image
-# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
-n_inputs = len(inputs)
-inputs = inputs.reshape(n_inputs, -1)
-print("X = (n_inputs, n_features) = " + str(inputs.shape))
-
-
-# choose some random images to display
-indices = np.arange(n_inputs)
-random_indices = np.random.choice(indices, size=5)
-
-for i, image in enumerate(digits.images[random_indices]):
-    plt.subplot(1, 5, i+1)
-    plt.axis('off')
-    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
-    plt.title("Label: %d" % digits.target[random_indices[i]])
-plt.show()
-
-
-
-
-
-
-
from tensorflow.keras.layers import Input
-from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
-from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
-from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
-from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
-from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
-
-from sklearn.model_selection import train_test_split
-
-# one-hot representation of labels
-labels = to_categorical(labels)
-
-# split into train and test data
-train_size = 0.8
-test_size = 1 - train_size
-X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
-                                                    test_size=test_size)
-
-
-
-
-
-
-
epochs = 100
-batch_size = 100
-n_neurons_layer1 = 100
-n_neurons_layer2 = 50
-n_categories = 10
-eta_vals = np.logspace(-5, 1, 7)
-lmbd_vals = np.logspace(-5, 1, 7)
-def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):
-    model = Sequential()
-    model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))
-    model.add(Dense(n_categories, activation='softmax'))
-    
-    sgd = optimizers.SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])
-    
-    return model
-
-
-
-
-
-
-
DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)
-        
-for i, eta in enumerate(eta_vals):
-    for j, lmbd in enumerate(lmbd_vals):
-        DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,
-                                         eta=eta, lmbd=lmbd)
-        DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)
-        scores = DNN.evaluate(X_test, Y_test)
-        
-        DNN_keras[i][j] = DNN
-        
-        print("Learning rate = ", eta)
-        print("Lambda = ", lmbd)
-        print("Test accuracy: %.3f" % scores[1])
-        print()
-
-
-
-
-
-
-
# optional
-# visual representation of grid search
-# uses seaborn heatmap, could probably do this in matplotlib
-import seaborn as sns
-
-sns.set()
-
-train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
-
-for i in range(len(eta_vals)):
-    for j in range(len(lmbd_vals)):
-        DNN = DNN_keras[i][j]
-
-        train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]
-        test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]
-
-        
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Training Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-fig, ax = plt.subplots(figsize = (10, 10))
-sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
-ax.set_title("Test Accuracy")
-ax.set_ylabel("$\eta$")
-ax.set_xlabel("$\lambda$")
-plt.show()
-
-
-
-
-
-
-

2.10. The Breast Cancer Data, now with Keras

-
-
-
import tensorflow as tf
-from tensorflow.keras.layers import Input
-from tensorflow.keras.models import Sequential      #This allows appending layers to existing models
-from tensorflow.keras.layers import Dense           #This allows defining the characteristics of a particular layer
-from tensorflow.keras import optimizers             #This allows using whichever optimiser we want (sgd,adam,RMSprop)
-from tensorflow.keras import regularizers           #This allows using whichever regularizer we want (l1,l2,l1_l2)
-from tensorflow.keras.utils import to_categorical   #This allows using categorical cross entropy as the cost function
-import numpy as np
-import matplotlib.pyplot as plt
-import seaborn as sns
-from sklearn.model_selection import train_test_split as splitter
-from sklearn.datasets import load_breast_cancer
-import pickle
-import os 
-
-
-"""Load breast cancer dataset"""
-
-np.random.seed(0)        #create same seed for random number every time
-
-cancer=load_breast_cancer()      #Download breast cancer dataset
-
-inputs=cancer.data                     #Feature matrix of 569 rows (samples) and 30 columns (parameters)
-outputs=cancer.target                  #Label array of 569 rows (0 for benign and 1 for malignant)
-labels=cancer.feature_names[0:30]
-
-print('The content of the breast cancer dataset is:')      #Print information about the datasets
-print(labels)
-print('-------------------------')
-print("inputs =  " + str(inputs.shape))
-print("outputs =  " + str(outputs.shape))
-print("labels =  "+ str(labels.shape))
-
-x=inputs      #Reassign the Feature and Label matrices to other variables
-y=outputs
-
-#%% 
-
-# Visualisation of dataset (for correlation analysis)
-
-plt.figure()
-plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean radius',fontweight='bold')
-plt.ylabel('Mean perimeter',fontweight='bold')
-plt.show()
-
-plt.figure()
-plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral)
-plt.xlabel('Mean compactness',fontweight='bold')
-plt.ylabel('Mean concavity',fontweight='bold')
-plt.show()
-
-
-plt.figure()
-plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean radius',fontweight='bold')
-plt.ylabel('Mean texture',fontweight='bold')
-plt.show()
-
-plt.figure()
-plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)
-plt.xlabel('Mean perimeter',fontweight='bold')
-plt.ylabel('Mean compactness',fontweight='bold')
-plt.show()
-
-
-# Generate training and testing datasets
-
-#Select features relevant to classification (texture,perimeter,compactness and symmetery) 
-#and add to input matrix
-
-temp1=np.reshape(x[:,1],(len(x[:,1]),1))
-temp2=np.reshape(x[:,2],(len(x[:,2]),1))
-X=np.hstack((temp1,temp2))      
-temp=np.reshape(x[:,5],(len(x[:,5]),1))
-X=np.hstack((X,temp))       
-temp=np.reshape(x[:,8],(len(x[:,8]),1))
-X=np.hstack((X,temp))       
-
-X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1)   #Split datasets into training and testing
-
-y_train=to_categorical(y_train)     #Convert labels to categorical when using categorical cross entropy
-y_test=to_categorical(y_test)
-
-del temp1,temp2,temp
-
-# %%
-
-# Define tunable parameters"
-
-eta=np.logspace(-3,-1,3)                    #Define vector of learning rates (parameter to SGD optimiser)
-lamda=0.01                                  #Define hyperparameter
-n_layers=2                                  #Define number of hidden layers in the model
-n_neuron=np.logspace(0,3,4,dtype=int)       #Define number of neurons per layer
-epochs=100                                   #Number of reiterations over the input data
-batch_size=100                              #Number of samples per gradient update
-
-# %%
-
-"""Define function to return Deep Neural Network model"""
-
-def NN_model(inputsize,n_layers,n_neuron,eta,lamda):
-    model=Sequential()      
-    for i in range(n_layers):       #Run loop to add hidden layers to the model
-        if (i==0):                  #First layer requires input dimensions
-            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize))
-        else:                       #Subsequent layers are capable of automatic shape inferencing
-            model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda)))
-    model.add(Dense(2,activation='softmax'))  #2 outputs - ordered and disordered (softmax for prob)
-    sgd=optimizers.SGD(lr=eta)
-    model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy'])
-    return model
-
-    
-Train_accuracy=np.zeros((len(n_neuron),len(eta)))      #Define matrices to store accuracy scores as a function
-Test_accuracy=np.zeros((len(n_neuron),len(eta)))       #of learning rate and number of hidden neurons for 
-
-for i in range(len(n_neuron)):     #run loops over hidden neurons and learning rates to calculate 
-    for j in range(len(eta)):      #accuracy scores 
-        DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda)
-        DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1)
-        Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1]
-        Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1]
-               
-
-def plot_data(x,y,data,title=None):
-
-    # plot results
-    fontsize=16
-
-
-    fig = plt.figure()
-    ax = fig.add_subplot(111)
-    cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1)
-    
-    cbar=fig.colorbar(cax)
-    cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize)
-    cbar.set_ticks([0,.2,.4,0.6,0.8,1.0])
-    cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%'])
-
-    # put text on matrix elements
-    for i, x_val in enumerate(np.arange(len(x))):
-        for j, y_val in enumerate(np.arange(len(y))):
-            c = "${0:.1f}\\%$".format( 100*data[j,i])  
-            ax.text(x_val, y_val, c, va='center', ha='center')
-
-    # convert axis vaues to to string labels
-    x=[str(i) for i in x]
-    y=[str(i) for i in y]
-
-
-    ax.set_xticklabels(['']+x)
-    ax.set_yticklabels(['']+y)
-
-    ax.set_xlabel('$\\mathrm{learning\\ rate}$',fontsize=fontsize)
-    ax.set_ylabel('$\\mathrm{hidden\\ neurons}$',fontsize=fontsize)
-    if title is not None:
-        ax.set_title(title)
-
-    plt.tight_layout()
-
-    plt.show()
-    
-plot_data(eta,n_neuron,Train_accuracy, 'training')
-plot_data(eta,n_neuron,Test_accuracy, 'testing')
-
-
-
-
-
-
-

2.11. Fine-tuning neural network hyperparameters

-

The flexibility of neural networks is also one of their main -drawbacks: there are many hyperparameters to tweak. Not only can you -use any imaginable network topology (how neurons/nodes are interconnected), -but even in a simple FFNN you can change the number of layers, the -number of neurons per layer, the type of activation function to use in -each layer, the weight initialization logic, the stochastic gradient optmized and much more. How do you -know what combination of hyperparameters is the best for your task?

-
    -

  • You can use grid search with cross-validation to find the right hyperparameters.

    • -
-

However,since there are many hyperparameters to tune, and since -training a neural network on a large dataset takes a lot of time, you -will only be able to explore a tiny part of the hyperparameter space.

-
    -

  • You can use randomized search.

    • -

  • Or use tools like Oscar, which implements more complex algorithms to help you find a good set of hyperparameters quickly.

    • -
-

For many problems you can start with just one or two hidden layers and it will work just fine. -For the MNIST data set you ca easily get a high accuracy using just one hidden layer with a -few hundred neurons. -You can reach for this data set above 98% accuracy using two hidden layers with the same total amount of -neurons, in roughly the same amount of training time.

-

For more complex problems, you can gradually -ramp up the number of hidden layers, until you start overfitting the training set. Very complex tasks, such -as large image classification or speech recognition, typically require networks with dozens of layers -and they need a huge amount -of training data. However, you will rarely have to train such networks from scratch: it is much more -common to reuse parts of a pretrained state-of-the-art network that performs a similar task.

-
-
-

2.12. Which activation function should I use?

-

The Back propagation algorithm we derived above works by going from -the output layer to the input layer, propagating the error gradient on -the way. Once the algorithm has computed the gradient of the cost -function with regards to each parameter in the network, it uses these -gradients to update each parameter with a Gradient Descent (GD) step.

-

Unfortunately for us, the gradients often get smaller and smaller as the -algorithm progresses down to the first hidden layers. As a result, the -GD update leaves the lower layer connection weights -virtually unchanged, and training never converges to a good -solution. This is known in the literature as -the vanishing gradients problem.

-

In other cases, the opposite can happen, namely the the gradients can grow bigger and -bigger. The result is that many of the layers get large updates of the -weights the -algorithm diverges. This is the exploding gradients problem, which is -mostly encountered in recurrent neural networks. More generally, deep -neural networks suffer from unstable gradients, different layers may -learn at widely different speeds

-

Although this unfortunate behavior has been empirically observed for -quite a while (it was one of the reasons why deep neural networks were -mostly abandoned for a long time), it is only around 2010 that -significant progress was made in understanding it.

-

A paper titled Understanding the Difficulty of Training Deep -Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio found that -the problems with the popular logistic -sigmoid activation function and the weight initialization technique -that was most popular at the time, namely random initialization using -a normal distribution with a mean of 0 and a standard deviation of -1.

-

They showed that with this activation function and this -initialization scheme, the variance of the outputs of each layer is -much greater than the variance of its inputs. Going forward in the -network, the variance keeps increasing after each layer until the -activation function saturates at the top layers. This is actually made -worse by the fact that the logistic function has a mean of 0.5, not 0 -(the hyperbolic tangent function has a mean of 0 and behaves slightly -better than the logistic function in deep networks).

-

Looking at the logistic activation function, when inputs become large -(negative or positive), the function saturates at 0 or 1, with a -derivative extremely close to 0. Thus when backpropagation kicks in, -it has virtually no gradient to propagate back through the network, -and what little gradient exists keeps getting diluted as -backpropagation progresses down through the top layers, so there is -really nothing left for the lower layers.

-

In their paper, Glorot and Bengio propose a way to significantly -alleviate this problem. We need the signal to flow properly in both -directions: in the forward direction when making predictions, and in -the reverse direction when backpropagating gradients. We don’t want -the signal to die out, nor do we want it to explode and saturate. For -the signal to flow properly, the authors argue that we need the -variance of the outputs of each layer to be equal to the variance of -its inputs, and we also need the gradients to have equal variance -before and after flowing through a layer in the reverse direction.

-

One of the insights in the 2010 paper by Glorot and Bengio was that -the vanishing/exploding gradients problems were in part due to a poor -choice of activation function. Until then most people had assumed that -if Nature had chosen to use roughly sigmoid activation functions in -biological neurons, they must be an excellent choice. But it turns out -that other activation functions behave much better in deep neural -networks, in particular the ReLU activation function, mostly because -it does not saturate for positive values (and also because it is quite -fast to compute).

-
-
-

2.13. The RELU function family

-

The ReLU activation function suffers from a problem known as the dying -ReLUs: during training, some neurons effectively die, meaning they -stop outputting anything other than 0.

-

In some cases, you may find that half of your network’s neurons are -dead, especially if you used a large learning rate. During training, -if a neuron’s weights get updated such that the weighted sum of the -neuron’s inputs is negative, it will start outputting 0. When this -happen, the neuron is unlikely to come back to life since the gradient -of the ReLU function is 0 when its input is negative.

-

To solve this problem, nowadays practitioners use a variant of the ReLU -function, such as the leaky ReLU discussed above or the so-called -exponential linear unit (ELU) function

-
-\[\begin{split} -ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\ z & z \ge 0.\end{array}\right. -\end{split}\]
-

In general it seems that the ELU activation function is better than -the leaky ReLU function (and its variants), which is better than -ReLU. ReLU performs better than \(\tanh\) which in turn performs better -than the logistic function.

-

If runtime -performance is an issue, then you may opt for the leaky ReLU function over the -ELU function If you don’t -want to tweak yet another hyperparameter, you may just use the default -\(\alpha\) of \(0.01\) for the leaky ReLU, and \(1\) for ELU. If you have -spare time and computing power, you can use cross-validation or -bootstrap to evaluate other activation functions.

-

In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).

-

It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.

-

For the output layer:

-
    -

  • For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).

    • -

  • For regression tasks, you can simply use no activation function at all.

    • -
-
-
-

2.14. Batch Normalization

-

Batch Normalization -aims to address the vanishing/exploding gradients problems, and more generally the problem that the -distribution of each layer’s inputs changes during training, as the parameters of the previous layers change.

-

The technique consists of adding an operation in the model just before the activation function of each -layer, simply zero-centering and normalizing the inputs, then scaling and shifting the result using two new -parameters per layer (one for scaling, the other for shifting). In other words, this operation lets the model -learn the optimal scale and mean of the inputs for each layer. -In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and -standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current -mini-batch, from this the name batch normalization.

-
-
-

2.15. Dropout

-

It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but -excluding the output neurons) has a probability \(p\) of being temporarily dropped out, meaning it will be -entirely ignored during this training step, but it may be active during the next step.

-

The -hyperparameter \(p\) is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore. -It is viewed as one of the most popular regularization techniques.

-
-
-

2.16. Gradient Clipping

-

A popular technique to lessen the exploding gradients problem is to simply clip the gradients during -backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural -networks).

-

This technique is called Gradient Clipping.

-

In general however, Batch -Normalization is preferred.

-
-
-

2.17. A top-down perspective on Neural networks

-

The first thing we would like to do is divide the data into two or three -parts. A training set, a validation or dev (development) set, and a -test set. The test set is the data on which we want to make -predictions. The dev set is a subset of the training data we use to -check how well we are doing out-of-sample, after training the model on -the training dataset. We use the validation error as a proxy for the -test error in order to make tweaks to our model. It is crucial that we -do not use any of the test data to train the algorithm. This is a -cardinal sin in ML. Then:

-
    -

  • Estimate optimal error rate

    • -

  • Minimize underfitting (bias) on training data set.

    • -

  • Make sure you are not overfitting.

    • -
-

If the validation and test sets are drawn from the same distributions, -then a good performance on the validation set should lead to similarly -good performance on the test set.

-

However, sometimes -the training data and test data differ in subtle ways because, for -example, they are collected using slightly different methods, or -because it is cheaper to collect data in one way versus another. In -this case, there can be a mismatch between the training and test -data. This can lead to the neural network overfitting these small -differences between the test and training sets, and a poor performance -on the test set despite having a good performance on the validation -set. To rectify this, Andrew Ng suggests making two validation or dev -sets, one constructed from the training data and one constructed from -the test data. The difference between the performance of the algorithm -on these two validation sets quantifies the train-test mismatch. This -can serve as another important diagnostic when using DNNs for -supervised learning.

-
-
-

2.18. Limitations of supervised learning with deep networks

-

Like all statistical methods, supervised learning using neural -networks has important limitations. This is especially important when -one seeks to apply these methods, especially to physics problems. Like -all tools, DNNs are not a universal solution. Often, the same or -better performance on a task can be achieved by using a few -hand-engineered features (or even a collection of random -features).

-

Here we list some of the important limitations of supervised neural network based models.

-
    -

  • Need labeled data. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).

    • -

  • Supervised neural networks are extremely data intensive. DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.

    • -

  • Homogeneous data. Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types.

    • -

  • Many problems are not about prediction. In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a wrong model. The model might or might not be useful for understanding the underlying science.

    • -
-

Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems.

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- - 1. Neural networks - 3. Solving Differential Equations with Deep Learning - -
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- - By Morten Hjorth-Jensen
- - © Copyright 2020.
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3. Solving Differential Equations with Deep Learning

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The Universal Approximation Theorem states that a neural network can -approximate any function at a single hidden layer along with one input -and output layer to any given precision.

-

An ordinary differential equation (ODE) is an equation involving functions having one variable.

-

In general, an ordinary differential equation looks like

- -
-
-\[ -\begin{equation} \label{ode} \tag{1} -f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right) = 0 -\end{equation} -\]
-

where \(g(x)\) is the function to find, and \(g^{(n)}(x)\) is the \(n\)-th derivative of \(g(x)\).

-

The \(f\left(x, g(x), g'(x), g''(x), \, \dots \, , g^{(n)}(x)\right)\) is just a way to write that there is an expression involving \(x\) and \(g(x), \ g'(x), \ g''(x), \, \dots \, , \text{ and } g^{(n)}(x)\) on the left side of the equality sign in (1). -The highest order of derivative, that is the value of \(n\), determines to the order of the equation. -The equation is referred to as a \(n\)-th order ODE. -Along with (1), some additional conditions of the function \(g(x)\) are typically given -for the solution to be unique.

-

Let the trial solution \(g_t(x)\) be

- -
-
-\[ -\begin{equation} - g_t(x) = h_1(x) + h_2(x,N(x,P)) -\label{_auto1} \tag{2} -\end{equation} -\]
-

where \(h_1(x)\) is a function that makes \(g_t(x)\) satisfy a given set -of conditions, \(N(x,P)\) a neural network with weights and biases -described by \(P\) and \(h_2(x, N(x,P))\) some expression involving the -neural network. The role of the function \(h_2(x, N(x,P))\), is to -ensure that the output from \(N(x,P)\) is zero when \(g_t(x)\) is -evaluated at the values of \(x\) where the given conditions must be -satisfied. The function \(h_1(x)\) should alone make \(g_t(x)\) satisfy -the conditions.

-

But what about the network \(N(x,P)\)?

-

As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.

-

For the minimization to be defined, we need to have a cost function at hand to minimize.

-

It is given that \(f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right)\) should be equal to zero in (1). -We can choose to consider the mean squared error as the cost function for an input \(x\). -Since we are looking at one input, the cost function is just \(f\) squared. -The cost function \(c\left(x, P \right)\) can therefore be expressed as

-
-\[ -C\left(x, P\right) = \big(f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right)\big)^2 -\]
-

If \(N\) inputs are given as a vector \(\boldsymbol{x}\) with elements \(x_i\) for \(i = 1,\dots,N\), -the cost function becomes

- -
-
-\[ -\begin{equation} \label{cost} \tag{3} - C\left(\boldsymbol{x}, P\right) = \frac{1}{N} \sum_{i=1}^N \big(f\left(x_i, \, g(x_i), \, g'(x_i), \, g''(x_i), \, \dots \, , \, g^{(n)}(x_i)\right)\big)^2 -\end{equation} -\]
-

The neural net should then find the parameters \(P\) that minimizes the cost function in -(3) for a set of \(N\) training samples \(x_i\).

-

To perform the minimization using gradient descent, the gradient of \(C\left(\boldsymbol{x}, P\right)\) is needed. -It might happen so that finding an analytical expression of the gradient of \(C(\boldsymbol{x}, P)\) from (3) gets too messy, depending on which cost function one desires to use.

-

Luckily, there exists libraries that makes the job for us through automatic differentiation. -Automatic differentiation is a method of finding the derivatives numerically with very high precision.

-
-

3.1. Example: Exponential decay

-

An exponential decay of a quantity \(g(x)\) is described by the equation

- -
-
-\[ -\begin{equation} \label{solve_expdec} \tag{4} - g'(x) = -\gamma g(x) -\end{equation} -\]
-

with \(g(0) = g_0\) for some chosen initial value \(g_0\).

-

The analytical solution of (4) is

- -
-
-\[ -\begin{equation} - g(x) = g_0 \exp\left(-\gamma x\right) -\label{_auto2} \tag{5} -\end{equation} -\]
-

Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of (4).

-

The program will use a neural network to solve

- -
-
-\[ -\begin{equation} \label{solveode} \tag{6} -g'(x) = -\gamma g(x) -\end{equation} -\]
-

where \(g(0) = g_0\) with \(\gamma\) and \(g_0\) being some chosen values.

-

In this example, \(\gamma = 2\) and \(g_0 = 10\).

-

To begin with, a trial solution \(g_t(t)\) must be chosen. A general trial solution for ordinary differential equations could be

-
-\[ -g_t(x, P) = h_1(x) + h_2(x, N(x, P)) -\]
-

with \(h_1(x)\) ensuring that \(g_t(x)\) satisfies some conditions and \(h_2(x,N(x, P))\) an expression involving \(x\) and the output from the neural network \(N(x,P)\) with \(P \) being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.

-

In this network, there are no weights and bias at the input layer, so \(P = \{ P_{\text{hidden}}, P_{\text{output}} \}\). -If there are \(N_{\text{hidden} }\) neurons in the hidden layer, then \(P_{\text{hidden}}\) is a \(N_{\text{hidden} } \times (1 + N_{\text{input}})\) matrix, given that there are \(N_{\text{input}}\) neurons in the input layer.

-

The first column in \(P_{\text{hidden} }\) represents the bias for each neuron in the hidden layer and the second column represents the weights for each neuron in the hidden layer from the input layer. -If there are \(N_{\text{output} }\) neurons in the output layer, then \(P_{\text{output}} \) is a \(N_{\text{output} } \times (1 + N_{\text{hidden} })\) matrix.

-

Its first column represents the bias of each neuron and the remaining columns represents the weights to each neuron.

-

It is given that \(g(0) = g_0\). The trial solution must fulfill this condition to be a proper solution of (6). A possible way to ensure that \(g_t(0, P) = g_0\), is to let \(F(N(x,P)) = x \cdot N(x,P)\) and \(A(x) = g_0\). This gives the following trial solution:

- -
-
-\[ -\begin{equation} \label{trial} \tag{7} -g_t(x, P) = g_0 + x \cdot N(x, P) -\end{equation} -\]
-
-
-

3.2. Reformulating the problem

-

We wish that our neural network manages to minimize a given cost function.

-

A reformulation of out equation, (6), must therefore be done, -such that it describes the problem a neural network can solve for.

-

The neural network must find the set of weights and biases \(P\) such that the trial solution in (7) satisfies (6).

-

The trial solution

-
-\[ -g_t(x, P) = g_0 + x \cdot N(x, P) -\]
-

has been chosen such that it already solves the condition \(g(0) = g_0\). What remains, is to find \(P\) such that

- -
-
-\[ -\begin{equation} \label{nnmin} \tag{8} -g_t'(x, P) = - \gamma g_t(x, P) -\end{equation} -\]
-

is fulfilled as best as possible.

-

The left hand side and right hand side of (8) must be computed separately, and then the neural network must choose weights and biases, contained in \(P\), such that the sides are equal as best as possible. -This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero. -In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to \(P\) of the neural network.

-

This gives the following cost function our neural network must solve for:

-
-\[ -\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\} -\]
-

(the notation \(\min_{P}\{ f(x, P) \}\) means that we desire to find \(P\) that yields the minimum of \(f(x, P)\))

-

or, in terms of weights and biases for the hidden and output layer in our network:

-
-\[ -\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\} -\]
-

for an input value \(x\).

-

If the neural network evaluates \(g_t(x, P)\) at more values for \(x\), say \(N\) values \(x_i\) for \(i = 1, \dots, N\), then the total error to minimize becomes

- -
-
-\[ -\begin{equation} \label{min} \tag{9} -\min_{P}\Big\{\frac{1}{N} \sum_{i=1}^N \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\} -\end{equation} -\]
-

Letting \(\boldsymbol{x}\) be a vector with elements \(x_i\) and \(C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2\) denote the cost function, the minimization problem that our network must solve, becomes

-
-\[ -\min_{P} C(\boldsymbol{x}, P) -\]
-

In terms of \(P_{\text{hidden} }\) and \(P_{\text{output} }\), this could also be expressed as

-
-\[ -\min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) -\]
-

For simplicity, it is assumed that the input is an array \(\boldsymbol{x} = (x_1, \dots, x_N)\) with \(N\) elements. It is at these points the neural network should find \(P\) such that it fulfills (9).

-

First, the neural network must feed forward the inputs. -This means that \(\boldsymbol{x}s\) must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further. -The input layer will consist of \(N_{\text{input} }\) neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be \(N_{\text{hidden} }\).

-

For the \(i\)-th in the hidden layer with weight \(w_i^{\text{hidden} }\) and bias \(b_i^{\text{hidden} }\), the weighting from the \(j\)-th neuron at the input layer is:

-
-\[\begin{split} -\begin{aligned} -z_{i,j}^{\text{hidden}} &= b_i^{\text{hidden}} + w_i^{\text{hidden}}x_j \\ -&= -\begin{pmatrix} -b_i^{\text{hidden}} & w_i^{\text{hidden}} -\end{pmatrix} -\begin{pmatrix} -1 \\ -x_j -\end{pmatrix} -\end{aligned} -\end{split}\]
-

The result after weighting the inputs at the \(i\)-th hidden neuron can be written as a vector:

-
-\[\begin{split} -\begin{aligned} -\boldsymbol{z}_{i}^{\text{hidden}} &= \Big( b_i^{\text{hidden}} + w_i^{\text{hidden}}x_1 , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_2, \ \dots \, , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_N\Big) \\ -&= -\begin{pmatrix} - b_i^{\text{hidden}} & w_i^{\text{hidden}} -\end{pmatrix} -\begin{pmatrix} -1 & 1 & \dots & 1 \\ -x_1 & x_2 & \dots & x_N -\end{pmatrix} \\ -&= \boldsymbol{p}_{i, \text{hidden}}^T X -\end{aligned} -\end{split}\]
-

The vector \(\boldsymbol{p}_{i, \text{hidden}}^T\) constitutes each row in \(P_{\text{hidden} }\), which contains the weights for the neural network to minimize according to (9).

-

After having found \(\boldsymbol{z}_{i}^{\text{hidden}} \) for every \(i\)-th neuron within the hidden layer, the vector will be sent to an activation function \(a_i(\boldsymbol{z})\).

-

In this example, the sigmoid function has been chosen to be the activation function for each hidden neuron:

-
-\[ -f(z) = \frac{1}{1 + \exp{(-z)}} -\]
-

It is possible to use other activations functions for the hidden layer also.

-

The output \(\boldsymbol{x}_i^{\text{hidden}}\) from each \(i\)-th hidden neuron is:

-
-\[ -\boldsymbol{x}_i^{\text{hidden} } = f\big( \boldsymbol{z}_{i}^{\text{hidden}} \big) -\]
-

The outputs \(\boldsymbol{x}_i^{\text{hidden} } \) are then sent to the output layer.

-

The output layer consists of one neuron in this case, and combines the -output from each of the neurons in the hidden layers. The output layer -combines the results from the hidden layer using some weights \(w_i^{\text{output}}\) -and biases \(b_i^{\text{output}}\). In this case, -it is assumes that the number of neurons in the output layer is one.

-

The procedure of weighting the output neuron \(j\) in the hidden layer to the \(i\)-th neuron in the output layer is similar as for the hidden layer described previously.

-
-\[\begin{split} -\begin{aligned} -z_{1,j}^{\text{output}} & = -\begin{pmatrix} -b_1^{\text{output}} & \boldsymbol{w}_1^{\text{output}} -\end{pmatrix} -\begin{pmatrix} -1 \\ -\boldsymbol{x}_j^{\text{hidden}} -\end{pmatrix} -\end{aligned} -\end{split}\]
-

Expressing \(z_{1,j}^{\text{output}}\) as a vector gives the following way of weighting the inputs from the hidden layer:

-
-\[\begin{split} -\boldsymbol{z}_{1}^{\text{output}} = -\begin{pmatrix} -b_1^{\text{output}} & \boldsymbol{w}_1^{\text{output}} -\end{pmatrix} -\begin{pmatrix} -1 & 1 & \dots & 1 \\ -\boldsymbol{x}_1^{\text{hidden}} & \boldsymbol{x}_2^{\text{hidden}} & \dots & \boldsymbol{x}_N^{\text{hidden}} -\end{pmatrix} -\end{split}\]
-

In this case we seek a continuous range of values since we are approximating a function. This means that after computing \(\boldsymbol{z}_{1}^{\text{output}}\) the neural network has finished its feed forward step, and \(\boldsymbol{z}_{1}^{\text{output}}\) is the final output of the network.

-

The next step is to decide how the parameters should be changed such that they minimize the cost function.

-

The chosen cost function for this problem is

-
-\[ -C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 -\]
-

In order to minimize the cost function, an optimization method must be chosen.

-

Here, gradient descent with a constant step size has been chosen.

-
-
-

3.3. Gradient descent

-

The idea of the gradient descent algorithm is to update parameters in -a direction where the cost function decreases goes to a minimum.

-

In general, the update of some parameters \(\boldsymbol{\omega}\) given a cost -function defined by some weights \(\boldsymbol{\omega}\), \(C(\boldsymbol{x}, -\boldsymbol{\omega})\), goes as follows:

-
-\[ -\boldsymbol{\omega}_{\text{new} } = \boldsymbol{\omega} - \lambda \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega}) -\]
-

for a number of iterations or until \( \big|\big| \boldsymbol{\omega}_{\text{new} } - \boldsymbol{\omega} \big|\big|\) becomes smaller than some given tolerance.

-

The value of \(\lambda\) decides how large steps the algorithm must take -in the direction of \( \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})\). -The notation \(\nabla_{\boldsymbol{\omega}}\) express the gradient with respect -to the elements in \(\boldsymbol{\omega}\).

-

In our case, we have to minimize the cost function \(C(\boldsymbol{x}, P)\) with -respect to the two sets of weights and biases, that is for the hidden -layer \(P_{\text{hidden} }\) and for the output layer \(P_{\text{output} -}\) .

-

This means that \(P_{\text{hidden} }\) and \(P_{\text{output} }\) is updated by

-
-\[\begin{split} -\begin{aligned} -P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P) \\ -P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P) -\end{aligned} -\end{split}\]
-
-
-

3.4. The code for solving the ODE

-
-
-
%matplotlib inline
-
-import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-# Assuming one input, hidden, and output layer
-def neural_network(params, x):
-
-    # Find the weights (including and biases) for the hidden and output layer.
-    # Assume that params is a list of parameters for each layer.
-    # The biases are the first element for each array in params,
-    # and the weights are the remaning elements in each array in params.
-
-    w_hidden = params[0]
-    w_output = params[1]
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    ## Hidden layer:
-
-    # Add a row of ones to include bias
-    x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)
-
-    z_hidden = np.matmul(w_hidden, x_input)
-    x_hidden = sigmoid(z_hidden)
-
-    ## Output layer:
-
-    # Include bias:
-    x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)
-
-    z_output = np.matmul(w_output, x_hidden)
-    x_output = z_output
-
-    return x_output
-
-# The trial solution using the deep neural network:
-def g_trial(x,params, g0 = 10):
-    return g0 + x*neural_network(params,x)
-
-# The right side of the ODE:
-def g(x, g_trial, gamma = 2):
-    return -gamma*g_trial
-
-# The cost function:
-def cost_function(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial(x,P)
-
-    # Find the derivative w.r.t x of the neural network
-    d_net_out = elementwise_grad(neural_network,1)(P,x)
-
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial,0)(x,P)
-
-    # The right side of the ODE
-    func = g(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum / np.size(err_sqr)
-
-# Solve the exponential decay ODE using neural network with one input, hidden, and output layer
-def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):
-    ## Set up initial weights and biases
-
-    # For the hidden layer
-    p0 = npr.randn(num_neurons_hidden, 2 )
-
-    # For the output layer
-    p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included
-
-    P = [p0, p1]
-
-    print('Initial cost: %g'%cost_function(P, x))
-
-    ## Start finding the optimal weights using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_grad = grad(cost_function,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of two arrays;
-        # one for the gradient w.r.t P_hidden and
-        # one for the gradient w.r.t P_output
-        cost_grad =  cost_function_grad(P, x)
-
-        P[0] = P[0] - lmb * cost_grad[0]
-        P[1] = P[1] - lmb * cost_grad[1]
-
-    print('Final cost: %g'%cost_function(P, x))
-
-    return P
-
-def g_analytic(x, gamma = 2, g0 = 10):
-    return g0*np.exp(-gamma*x)
-
-# Solve the given problem
-if __name__ == '__main__':
-    # Set seed such that the weight are initialized
-    # with same weights and biases for every run.
-    npr.seed(15)
-
-    ## Decide the vales of arguments to the function to solve
-    N = 10
-    x = np.linspace(0, 1, N)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = 10
-    num_iter = 10000
-    lmb = 0.001
-
-    # Use the network
-    P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-    # Print the deviation from the trial solution and true solution
-    res = g_trial(x,P)
-    res_analytical = g_analytic(x)
-
-    print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))
-
-    # Plot the results
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(x, res_analytical)
-    plt.plot(x, res[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('x')
-    plt.ylabel('g(x)')
-    plt.show()
-
-
-
-
-
Initial cost: 367.01
-
-
-
Final cost: 0.0666807
-Max absolute difference: 0.0437499
-
-
-_images/chapter11_47_2.png -
-
-
-
-

3.5. The network with one input layer, specified number of hidden layers, and one output layer

-

It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.

-

The number of neurons within each hidden layer are given as a list of integers in the program below.

-
-
-
import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-# The neural network with one input layer and one output layer,
-# but with number of hidden layers specified by the user.
-def deep_neural_network(deep_params, x):
-    # N_hidden is the number of hidden layers
-
-    N_hidden = np.size(deep_params) - 1 # -1 since params consists of
-                                        # parameters to all the hidden
-                                        # layers AND the output layer.
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
-# The trial solution using the deep neural network:
-def g_trial_deep(x,params, g0 = 10):
-    return g0 + x*deep_neural_network(params, x)
-
-# The right side of the ODE:
-def g(x, g_trial, gamma = 2):
-    return -gamma*g_trial
-
-# The same cost function as before, but calls deep_neural_network instead.
-def cost_function_deep(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-
-    # Find the derivative w.r.t x of the neural network
-    d_net_out = elementwise_grad(deep_neural_network,1)(P,x)
-
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)
-
-    # The right side of the ODE
-    func = g(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum / np.size(err_sqr)
-
-# Solve the exponential decay ODE using neural network with one input and one output layer,
-# but with specified number of hidden layers from the user.
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # The number of elements in the list num_hidden_neurons thus represents
-    # the number of hidden layers.
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weights and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 )
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-
-    ## Start finding the optimal weights using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-
-    return P
-
-def g_analytic(x, gamma = 2, g0 = 10):
-    return g0*np.exp(-gamma*x)
-
-# Solve the given problem
-if __name__ == '__main__':
-    npr.seed(15)
-
-    ## Decide the vales of arguments to the function to solve
-    N = 10
-    x = np.linspace(0, 1, N)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = np.array([10,10])
-    num_iter = 10000
-    lmb = 0.001
-
-    P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-    res = g_trial_deep(x,P)
-    res_analytical = g_analytic(x)
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')
-    plt.plot(x, res_analytical)
-    plt.plot(x, res[0,:])
-    plt.legend(['analytical','dnn'])
-    plt.ylabel('g(x)')
-    plt.show()
-
-
-
-
-
Initial cost: 324.246
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
-  return array(a, dtype, copy=False, order=order)
-
-
-
Final cost: 0.119936
-
-
-_images/chapter11_49_3.png -
-
-
-

3.5.1. Example: Population growth

-

A logistic model of population growth assumes that a population converges toward an equilibrium. -The population growth can be modeled by

- -
-
-\[ -\begin{equation} \label{log} \tag{10} - g'(t) = \alpha g(t)(A - g(t)) -\end{equation} -\]
-

where \(g(t)\) is the population density at time \(t\), \(\alpha > 0\) the growth rate and \(A > 0\) is the maximum population number in the environment. -Also, at \(t = 0\) the population has the size \(g(0) = g_0\), where \(g_0\) is some chosen constant.

-

In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability -and high execution time (this might be more apparent in the examples solving PDEs), -using a library like TensorFlow is recommended. -Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.

-

Here, we will model a population \(g(t)\) in an environment having carrying capacity \(A\). -The population follows the model

- -
-
-\[ -\begin{equation} \label{solveode_population} \tag{11} -g'(t) = \alpha g(t)(A - g(t)) -\end{equation} -\]
-

where \(g(0) = g_0\).

-

In this example, we let \(\alpha = 2\), \(A = 1\), and \(g_0 = 1.2\).

-

We will get a slightly different trial solution, as the boundary conditions are different -compared to the case for exponential decay.

-

A possible trial solution satisfying the condition \(g(0) = g_0\) could be

-
-\[ -h_1(t) = g_0 + t \cdot N(t,P) -\]
-

with \(N(t,P)\) being the output from the neural network with weights and biases for each layer collected in the set \(P\).

-

The analytical solution is

-
-\[ -g(t) = \frac{Ag_0}{g_0 + (A - g_0)\exp(-\alpha A t)} -\]
-

The network will be the similar as for the exponential decay example, but with some small modifications for our problem.

-
-
-
import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-# Function to get the parameters.
-# Done such that one can easily change the paramaters after one's liking.
-def get_parameters():
-    alpha = 2
-    A = 1
-    g0 = 1.2
-    return alpha, A, g0
-
-def deep_neural_network(P, x):
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(P) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = P[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = P[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
-
-def cost_function_deep(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-
-    # Find the derivative w.r.t x of the trial function
-    d_g_t = elementwise_grad(g_trial_deep,0)(x,P)
-
-    # The right side of the ODE
-    func = f(x, g_t)
-
-    err_sqr = (d_g_t - func)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum / np.size(err_sqr)
-
-# The right side of the ODE:
-def f(x, g_trial):
-    alpha,A, g0 = get_parameters()
-    return alpha*g_trial*(A - g_trial)
-
-# The trial solution using the deep neural network:
-def g_trial_deep(x, params):
-    alpha,A, g0 = get_parameters()
-    return g0 + x*deep_neural_network(params,x)
-
-# The analytical solution:
-def g_analytic(t):
-    alpha,A, g0 = get_parameters()
-    return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))
-
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 )
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-
-    ## Start finding the optimal weigths using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-
-    return P
-
-if __name__ == '__main__':
-    npr.seed(4155)
-
-    ## Decide the vales of arguments to the function to solve
-    Nt = 10
-    T = 1
-    t = np.linspace(0,T, Nt)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = [100, 50, 25]
-    num_iter = 1000
-    lmb = 1e-3
-
-    P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)
-
-    g_dnn_ag = g_trial_deep(t,P)
-    g_analytical = g_analytic(t)
-
-    # Find the maximum absolute difference between the solutons:
-    diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))
-    print("The max absolute difference between the solutions is: %g"%diff_ag)
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(t, g_analytical)
-    plt.plot(t, g_dnn_ag[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('t')
-    plt.ylabel('g(t)')
-
-    plt.show()
-
-
-
-
-
Initial cost: 0.221805
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
-  return array(a, dtype, copy=False, order=order)
-
-
-
Final cost: 0.000417932
-The max absolute difference between the solutions is: 0.00424909
-
-
-_images/chapter11_55_3.png -
-
-
-
-
-

3.6. Using forward Euler to solve the ODE

-

A straightforward way of solving an ODE numerically, is to use Euler’s method.

-

Euler’s method uses Taylor series to approximate the value at a function \(f\) at a step \(\Delta x\) from \(x\):

-
-\[ -f(x + \Delta x) \approx f(x) + \Delta x f'(x) -\]
-

In our case, using Euler’s method to approximate the value of \(g\) at a step \(\Delta t\) from \(t\) yields

-
-\[\begin{split} -\begin{aligned} - g(t + \Delta t) &\approx g(t) + \Delta t g'(t) \\ - &= g(t) + \Delta t \big(\alpha g(t)(A - g(t))\big) -\end{aligned} -\end{split}\]
-

along with the condition that \(g(0) = g_0\).

-

Let \(t_i = i \cdot \Delta t\) where \(\Delta t = \frac{T}{N_t-1}\) where \(T\) is the final time our solver must solve for and \(N_t\) the number of values for \(t \in [0, T]\) for \(i = 0, \dots, N_t-1\).

-

For \(i \geq 1\), we have that

-
-\[\begin{split} -\begin{aligned} -t_i &= i\Delta t \\ -&= (i - 1)\Delta t + \Delta t \\ -&= t_{i-1} + \Delta t -\end{aligned} -\end{split}\]
-

Now, if \(g_i = g(t_i)\) then

- -
-
-\[\begin{split} -\begin{equation} - \begin{aligned} - g_i &= g(t_i) \\ - &= g(t_{i-1} + \Delta t) \\ - &\approx g(t_{i-1}) + \Delta t \big(\alpha g(t_{i-1})(A - g(t_{i-1}))\big) \\ - &= g_{i-1} + \Delta t \big(\alpha g_{i-1}(A - g_{i-1})\big) - \end{aligned} -\end{equation} \label{odenum} \tag{12} -\end{split}\]
-

for \(i \geq 1\) and \(g_0 = g(t_0) = g(0) = g_0\).

-

Equation (12) could be implemented in the following way, -extending the program that uses the network using Autograd:

-
-
-
# Assume that all function definitions from the example program using Autograd
-# are located here.
-
-if __name__ == '__main__':
-    npr.seed(4155)
-
-    ## Decide the vales of arguments to the function to solve
-    Nt = 10
-    T = 1
-    t = np.linspace(0,T, Nt)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = [100,50,25]
-    num_iter = 1000
-    lmb = 1e-3
-
-    P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)
-
-    g_dnn_ag = g_trial_deep(t,P)
-    g_analytical = g_analytic(t)
-
-    # Find the maximum absolute difference between the solutons:
-    diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))
-    print("The max absolute difference between the solutions is: %g"%diff_ag)
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(t, g_analytical)
-    plt.plot(t, g_dnn_ag[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('t')
-    plt.ylabel('g(t)')
-
-    ## Find an approximation to the funtion using forward Euler
-
-    alpha, A, g0 = get_parameters()
-    dt = T/(Nt - 1)
-
-    # Perform forward Euler to solve the ODE
-    g_euler = np.zeros(Nt)
-    g_euler[0] = g0
-
-    for i in range(1,Nt):
-        g_euler[i] = g_euler[i-1] + dt*(alpha*g_euler[i-1]*(A - g_euler[i-1]))
-
-    # Print the errors done by each method
-    diff1 = np.max(np.abs(g_euler - g_analytical))
-    diff2 = np.max(np.abs(g_dnn_ag[0,:] - g_analytical))
-
-    print('Max absolute difference between Euler method and analytical: %g'%diff1)
-    print('Max absolute difference between deep neural network and analytical: %g'%diff2)
-
-    # Plot results
-    plt.figure(figsize=(10,10))
-
-    plt.plot(t,g_euler)
-    plt.plot(t,g_analytical)
-    plt.plot(t,g_dnn_ag[0,:])
-
-    plt.legend(['euler','analytical','dnn'])
-    plt.xlabel('Time t')
-    plt.ylabel('g(t)')
-
-    plt.show()
-
-
-
-
-
Initial cost: 0.221805
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
-  return array(a, dtype, copy=False, order=order)
-
-
-
Final cost: 0.000417932
-The max absolute difference between the solutions is: 0.00424909
-Max absolute difference between Euler method and analytical: 0.011225
-Max absolute difference between deep neural network and analytical: 0.00424909
-
-
-_images/chapter11_63_3.png -_images/chapter11_63_4.png -
-
-
-
-

3.7. Solving the one dimensional Poisson equation

-

The Poisson equation for \(g(x)\) in one dimension is

- -
-
-\[ -\begin{equation} \label{poisson} \tag{13} - -g''(x) = f(x) -\end{equation} -\]
-

where \(f(x)\) is a given function for \(x \in (0,1)\).

-

The conditions that \(g(x)\) is chosen to fulfill, are

-
-\[\begin{split} -\begin{align*} - g(0) &= 0 \\ - g(1) &= 0 -\end{align*} -\end{split}\]
-

This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used. -The results from the networks can then be compared to the analytical solution. -In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.

-

Here, the function \(g(x)\) to solve for follows the equation

-
-\[ --g''(x) = f(x),\qquad x \in (0,1) -\]
-

where \(f(x)\) is a given function, along with the chosen conditions

- -
-
-\[ -\begin{aligned} -g(0) = g(1) = 0 -\end{aligned}\label{cond} \tag{14} -\]
-

In this example, we consider the case when \(f(x) = (3x + x^2)\exp(x)\).

-

For this case, a possible trial solution satisfying the conditions could be

-
-\[ -g_t(x) = x \cdot (1-x) \cdot N(P,x) -\]
-

The analytical solution for this problem is

-
-\[ -g(x) = x(1 - x)\exp(x) -\]
-
-
-
import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 )
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-
-    ## Start finding the optimal weigths using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-
-    return P
-
-## Set up the cost function specified for this Poisson equation:
-
-# The right side of the ODE
-def f(x):
-    return (3*x + x**2)*np.exp(x)
-
-def cost_function_deep(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-
-    # Find the derivative w.r.t x of the trial function
-    d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)
-
-    right_side = f(x)
-
-    err_sqr = (-d2_g_t - right_side)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum/np.size(err_sqr)
-
-# The trial solution:
-def g_trial_deep(x,P):
-    return x*(1-x)*deep_neural_network(P,x)
-
-# The analytic solution;
-def g_analytic(x):
-    return x*(1-x)*np.exp(x)
-
-if __name__ == '__main__':
-    npr.seed(4155)
-
-    ## Decide the vales of arguments to the function to solve
-    Nx = 10
-    x = np.linspace(0,1, Nx)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = [200,100]
-    num_iter = 1000
-    lmb = 1e-3
-
-    P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-    g_dnn_ag = g_trial_deep(x,P)
-    g_analytical = g_analytic(x)
-
-    # Find the maximum absolute difference between the solutons:
-    max_diff = np.max(np.abs(g_dnn_ag - g_analytical))
-    print("The max absolute difference between the solutions is: %g"%max_diff)
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(x, g_analytical)
-    plt.plot(x, g_dnn_ag[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('x')
-    plt.ylabel('g(x)')
-    plt.show()
-
-
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
-  return array(a, dtype, copy=False, order=order)
-
-
-
Initial cost: 457.256
-
-
-
Final cost: 0.00310113
-The max absolute difference between the solutions is: 0.000464088
-
-
-_images/chapter11_76_3.png -
-
-
-

3.7.1. Comparing with a numerical scheme

-

The Poisson equation is possible to solve using Taylor series to approximate the second derivative.

-

Using Taylor series, the second derivative can be expressed as

-
-\[ -g''(x) = \frac{g(x + \Delta x) - 2g(x) + g(x-\Delta x)}{\Delta x^2} + E_{\Delta x}(x) -\]
-

where \(\Delta x\) is a small step size and \(E_{\Delta x}(x)\) being the error term.

-

Looking away from the error terms gives an approximation to the second derivative:

- -
-
-\[ -\begin{equation} \label{approx} \tag{15} -g''(x) \approx \frac{g(x + \Delta x) - 2g(x) + g(x-\Delta x)}{\Delta x^2} -\end{equation} -\]
-

If \(x_i = i \Delta x = x_{i-1} + \Delta x\) and \(g_i = g(x_i)\) for \(i = 1,\dots N_x - 2\) with \(N_x\) being the number of values for \(x\), (15) becomes

-
-\[\begin{split} -\begin{aligned} -g''(x_i) &\approx \frac{g(x_i + \Delta x) - 2g(x_i) + g(x_i -\Delta x)}{\Delta x^2} \\ -&= \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2} -\end{aligned} -\end{split}\]
-

Since we know from our problem that

-
-\[\begin{split} -\begin{aligned} --g''(x) &= f(x) \\ -&= (3x + x^2)\exp(x) -\end{aligned} -\end{split}\]
-

along with the conditions \(g(0) = g(1) = 0\), -the following scheme can be used to find an approximate solution for \(g(x)\) numerically:

- -
-
-\[\begin{split} -\begin{equation} - \begin{aligned} - -\Big( \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2} \Big) &= f(x_i) \\ - -g_{i+1} + 2g_i - g_{i-1} &= \Delta x^2 f(x_i) - \end{aligned} -\end{equation} \label{odesys} \tag{16} -\end{split}\]
-

for \(i = 1, \dots, N_x - 2\) where \(g_0 = g_{N_x - 1} = 0\) and \(f(x_i) = (3x_i + x_i^2)\exp(x_i)\), which is given for our specific problem.

-

The equation can be rewritten into a matrix equation:

-
-\[\begin{split} -\begin{aligned} -\begin{pmatrix} -2 & -1 & 0 & \dots & 0 \\ --1 & 2 & -1 & \dots & 0 \\ -\vdots & & \ddots & & \vdots \\ -0 & \dots & -1 & 2 & -1 \\ -0 & \dots & 0 & -1 & 2\\ -\end{pmatrix} -\begin{pmatrix} -g_1 \\ -g_2 \\ -\vdots \\ -g_{N_x - 3} \\ -g_{N_x - 2} -\end{pmatrix} -&= -\Delta x^2 -\begin{pmatrix} -f(x_1) \\ -f(x_2) \\ -\vdots \\ -f(x_{N_x - 3}) \\ -f(x_{N_x - 2}) -\end{pmatrix} \\ -\boldsymbol{A}\boldsymbol{g} &= \boldsymbol{f}, -\end{aligned} -\end{split}\]
-

which makes it possible to solve for the vector \(\boldsymbol{g}\).

-

We can then compare the result from this numerical scheme with the output from our network using Autograd:

-
-
-
import autograd.numpy as np
-from autograd import grad, elementwise_grad
-import autograd.numpy.random as npr
-from matplotlib import pyplot as plt
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assumes input x being an one-dimensional array
-    num_values = np.size(x)
-    x = x.reshape(-1, num_values)
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-
-    # Due to multiple hidden layers, define a variable referencing to the
-    # output of the previous layer:
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output
-
-def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):
-    # num_hidden_neurons is now a list of number of neurons within each hidden layer
-
-    # Find the number of hidden layers:
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 )
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: %g'%cost_function_deep(P, x))
-
-    ## Start finding the optimal weigths using gradient descent
-
-    # Find the Python function that represents the gradient of the cost function
-    # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer
-    cost_function_deep_grad = grad(cost_function_deep,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        # Evaluate the gradient at the current weights and biases in P.
-        # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases
-        # in the hidden layers and output layers evaluated at x.
-        cost_deep_grad =  cost_function_deep_grad(P, x)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_deep_grad[l]
-
-    print('Final cost: %g'%cost_function_deep(P, x))
-
-    return P
-
-## Set up the cost function specified for this Poisson equation:
-
-# The right side of the ODE
-def f(x):
-    return (3*x + x**2)*np.exp(x)
-
-def cost_function_deep(P, x):
-
-    # Evaluate the trial function with the current parameters P
-    g_t = g_trial_deep(x,P)
-
-    # Find the derivative w.r.t x of the trial function
-    d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)
-
-    right_side = f(x)
-
-    err_sqr = (-d2_g_t - right_side)**2
-    cost_sum = np.sum(err_sqr)
-
-    return cost_sum/np.size(err_sqr)
-
-# The trial solution:
-def g_trial_deep(x,P):
-    return x*(1-x)*deep_neural_network(P,x)
-
-# The analytic solution;
-def g_analytic(x):
-    return x*(1-x)*np.exp(x)
-
-if __name__ == '__main__':
-    npr.seed(4155)
-
-    ## Decide the vales of arguments to the function to solve
-    Nx = 10
-    x = np.linspace(0,1, Nx)
-
-    ## Set up the initial parameters
-    num_hidden_neurons = [200,100]
-    num_iter = 1000
-    lmb = 1e-3
-
-    P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)
-
-    g_dnn_ag = g_trial_deep(x,P)
-    g_analytical = g_analytic(x)
-
-    # Find the maximum absolute difference between the solutons:
-
-    plt.figure(figsize=(10,10))
-
-    plt.title('Performance of neural network solving an ODE compared to the analytical solution')
-    plt.plot(x, g_analytical)
-    plt.plot(x, g_dnn_ag[0,:])
-    plt.legend(['analytical','nn'])
-    plt.xlabel('x')
-    plt.ylabel('g(x)')
-
-    ## Perform the computation using the numerical scheme
-
-    dx = 1/(Nx - 1)
-
-    # Set up the matrix A
-    A = np.zeros((Nx-2,Nx-2))
-
-    A[0,0] = 2
-    A[0,1] = -1
-
-    for i in range(1,Nx-3):
-        A[i,i-1] = -1
-        A[i,i] = 2
-        A[i,i+1] = -1
-
-    A[Nx - 3, Nx - 4] = -1
-    A[Nx - 3, Nx - 3] = 2
-
-    # Set up the vector f
-    f_vec = dx**2 * f(x[1:-1])
-
-    # Solve the equation
-    g_res = np.linalg.solve(A,f_vec)
-
-    g_vec = np.zeros(Nx)
-    g_vec[1:-1] = g_res
-
-    # Print the differences between each method
-    max_diff1 = np.max(np.abs(g_dnn_ag - g_analytical))
-    max_diff2 = np.max(np.abs(g_vec - g_analytical))
-    print("The max absolute difference between the analytical solution and DNN Autograd: %g"%max_diff1)
-    print("The max absolute difference between the analytical solution and numerical scheme: %g"%max_diff2)
-
-    # Plot the results
-    plt.figure(figsize=(10,10))
-
-    plt.plot(x,g_vec)
-    plt.plot(x,g_analytical)
-    plt.plot(x,g_dnn_ag[0,:])
-
-    plt.legend(['numerical scheme','analytical','dnn'])
-    plt.show()
-
-
-
-
-
Initial cost: 457.256
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
-  return array(a, dtype, copy=False, order=order)
-
-
-
Final cost: 0.00310113
-The max absolute difference between the analytical solution and DNN Autograd: 0.000464088
-The max absolute difference between the analytical solution and numerical scheme: 0.00266858
-
-
-_images/chapter11_88_3.png -_images/chapter11_88_4.png -
-
-
-
-
-

3.8. Partial Differential Equations

-

A partial differential equation (PDE) has a solution here the function -is defined by multiple variables. The equation may involve all kinds -of combinations of which variables the function is differentiated with -respect to.

-

In general, a partial differential equation for a function \(g(x_1,\dots,x_N)\) with \(N\) variables may be expressed as

- -
-
-\[ -\begin{equation} \label{PDE} \tag{17} - f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) = 0 -\end{equation} -\]
-

where \(f\) is an expression involving all kinds of possible mixed derivatives of \(g(x_1,\dots,x_N)\) up to an order \(n\). In order for the solution to be unique, some additional conditions must also be given.

-
-

3.8.1. Type of problem

-

The problem our network must solve for, is similar to the ODE case. -We must have a trial solution \(g_t\) at hand.

-

For instance, the trial solution could be expressed as

-
-\[ -\begin{align*} - g_t(x_1,\dots,x_N) = h_1(x_1,\dots,x_N) + h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) -\end{align*} -\]
-

where \(h_1(x_1,\dots,x_N)\) is a function that ensures \(g_t(x_1,\dots,x_N)\) satisfies some given conditions. -The neural network \(N(x_1,\dots,x_N,P)\) has weights and biases described by \(P\) and \(h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))\) is an expression using the output from the neural network in some way.

-

The role of the function \(h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))\), is to ensure that the output of \(N(x_1,\dots,x_N,P)\) is zero when \(g_t(x_1,\dots,x_N)\) is evaluated at the values of \(x_1,\dots,x_N\) where the given conditions must be satisfied. The function \(h_1(x_1,\dots,x_N)\) should alone make \(g_t(x_1,\dots,x_N)\) satisfy the conditions.

-
-
-

3.8.2. Network requirements

-

The network tries then the minimize the cost function following the -same ideas as described for the ODE case, but now with more than one -variables to consider. The concept still remains the same; find a set -of parameters \(P\) such that the expression \(f\) in (17) is as -close to zero as possible.

-

As for the ODE case, the cost function is the mean squared error that -the network must try to minimize. The cost function for the network to -minimize is

-
-\[ -C\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2 -\]
-

If we let \(\boldsymbol{x} = \big( x_1, \dots, x_N \big)\) be an array containing the values for \(x_1, \dots, x_N\) respectively, the cost function can be reformulated into the following:

-
-\[ -C\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}) }{\partial x_N^n} \right) \right)^2 -\]
-

If we also have \(M\) different sets of values for \(x_1, \dots, x_N\), that is \(\boldsymbol{x}_i = \big(x_1^{(i)}, \dots, x_N^{(i)}\big)\) for \(i = 1,\dots,M\) being the rows in matrix \(X\), the cost function can be generalized into

-
-\[ -C\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2. -\]
-
-
-
-

3.9. Example: The diffusion equation

-

In one spatial dimension, the equation reads

-
-\[ -\frac{\partial g(x,t)}{\partial t} = \frac{\partial^2 g(x,t)}{\partial x^2} -\]
-

where a possible choice of conditions are

-
-\[\begin{split} -\begin{align*} -g(0,t) &= 0 ,\qquad t \geq 0 \\ -g(1,t) &= 0, \qquad t \geq 0 \\ -g(x,0) &= u(x),\qquad x\in [0,1] -\end{align*} -\end{split}\]
-

with \(u(x)\) being some given function.

-

For this case, we want to find \(g(x,t)\) such that

- -
-
-\[ -\begin{equation} - \frac{\partial g(x,t)}{\partial t} = \frac{\partial^2 g(x,t)}{\partial x^2} -\end{equation} \label{diffonedim} \tag{18} -\]
-

and

-
-\[\begin{split} -\begin{align*} -g(0,t) &= 0 ,\qquad t \geq 0 \\ -g(1,t) &= 0, \qquad t \geq 0 \\ -g(x,0) &= u(x),\qquad x\in [0,1] -\end{align*} -\end{split}\]
-

with \(u(x) = \sin(\pi x)\).

-

First, let us set up the deep neural network. -The deep neural network will follow the same structure as discussed in the examples solving the ODEs. -First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions.

-

The only change to do here, is to extend our network such that -functions of multiple parameters are correctly handled. In this case -we have two variables in our function to solve for, that is time \(t\) -and position \(x\). The variables will be represented by a -one-dimensional array in the program. The program will evaluate the -network at each possible pair \((x,t)\), given an array for the desired -\(x\)-values and \(t\)-values to approximate the solution at.

-
-
-
def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # x is now a point and a 1D numpy array; make it a column vector
-    num_coordinates = np.size(x,0)
-    x = x.reshape(num_coordinates,-1)
-
-    num_points = np.size(x,1)
-
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output[0][0]
-
-
-
-
-

The cost function must then iterate through the given arrays -containing values for \(x\) and \(t\), defines a point \((x,t)\) the deep -neural network and the trial solution is evaluated at, and then finds -the Jacobian of the trial solution.

-

A possible trial solution for this PDE is

-
-\[ -g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P) -\]
-

with \(A(x,t)\) being a function ensuring that \(g_t(x,t)\) satisfies our given conditions, and \(N(x,t,P)\) being the output from the deep neural network using weights and biases for each layer from \(P\).

-

To fulfill the conditions, \(A(x,t)\) could be:

-
-\[ -h_1(x,t) = (1-t)\Big(u(x) - \big((1-x)u(0) + x u(1)\big)\Big) = (1-t)u(x) = (1-t)\sin(\pi x) -\]
-

since \((0) = u(1) = 0\) and \(u(x) = \sin(\pi x)\).

-

The Jacobian is used because the program must find the derivative of -the trial solution with respect to \(x\) and \(t\).

-

This gives the necessity of computing the Jacobian matrix, as we want -to evaluate the gradient with respect to \(x\) and \(t\) (note that the -Jacobian of a scalar-valued multivariate function is simply its -gradient).

-

In Autograd, the differentiation is by default done with respect to -the first input argument of your Python function. Since the points is -an array representing \(x\) and \(t\), the Jacobian is calculated using -the values of \(x\) and \(t\).

-

To find the second derivative with respect to \(x\) and \(t\), the -Jacobian can be found for the second time. The result is a Hessian -matrix, which is the matrix containing all the possible second order -mixed derivatives of \(g(x,t)\).

-
-
-
# Set up the trial function:
-def u(x):
-    return np.sin(np.pi*x)
-
-def g_trial(point,P):
-    x,t = point
-    return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)
-
-# The right side of the ODE:
-def f(point):
-    return 0.
-
-# The cost function:
-def cost_function(P, x, t):
-    cost_sum = 0
-
-    g_t_jacobian_func = jacobian(g_trial)
-    g_t_hessian_func = hessian(g_trial)
-
-    for x_ in x:
-        for t_ in t:
-            point = np.array([x_,t_])
-
-            g_t = g_trial(point,P)
-            g_t_jacobian = g_t_jacobian_func(point,P)
-            g_t_hessian = g_t_hessian_func(point,P)
-
-            g_t_dt = g_t_jacobian[1]
-            g_t_d2x = g_t_hessian[0][0]
-
-            func = f(point)
-
-            err_sqr = ( (g_t_dt - g_t_d2x) - func)**2
-            cost_sum += err_sqr
-
-    return cost_sum
-
-
-
-
-
-

3.9.1. Setting up the network using Autograd; The full program

-

Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.

-

The analytical solution of our problem is

-
-\[ -g(x,t) = \exp(-\pi^2 t)\sin(\pi x) -\]
-

A possible way to implement a neural network solving the PDE, is given below. -Be aware, though, that it is fairly slow for the parameters used. -A better result is possible, but requires more iterations, and thus longer time to complete.

-

Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE. -Using TensorFlow results in a much better execution time. Try it!

-
-
-
import autograd.numpy as np
-from autograd import jacobian,hessian,grad
-import autograd.numpy.random as npr
-from matplotlib import cm
-from matplotlib import pyplot as plt
-from mpl_toolkits.mplot3d import axes3d
-
-## Set up the network
-
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # x is now a point and a 1D numpy array; make it a column vector
-    num_coordinates = np.size(x,0)
-    x = x.reshape(num_coordinates,-1)
-
-    num_points = np.size(x,1)
-
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output[0][0]
-
-## Define the trial solution and cost function
-def u(x):
-    return np.sin(np.pi*x)
-
-def g_trial(point,P):
-    x,t = point
-    return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)
-
-# The right side of the ODE:
-def f(point):
-    return 0.
-
-# The cost function:
-def cost_function(P, x, t):
-    cost_sum = 0
-
-    g_t_jacobian_func = jacobian(g_trial)
-    g_t_hessian_func = hessian(g_trial)
-
-    for x_ in x:
-        for t_ in t:
-            point = np.array([x_,t_])
-
-            g_t = g_trial(point,P)
-            g_t_jacobian = g_t_jacobian_func(point,P)
-            g_t_hessian = g_t_hessian_func(point,P)
-
-            g_t_dt = g_t_jacobian[1]
-            g_t_d2x = g_t_hessian[0][0]
-
-            func = f(point)
-
-            err_sqr = ( (g_t_dt - g_t_d2x) - func)**2
-            cost_sum += err_sqr
-
-    return cost_sum /( np.size(x)*np.size(t) )
-
-## For comparison, define the analytical solution
-def g_analytic(point):
-    x,t = point
-    return np.exp(-np.pi**2*t)*np.sin(np.pi*x)
-
-## Set up a function for training the network to solve for the equation
-def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):
-    ## Set up initial weigths and biases
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: ',cost_function(P, x, t))
-
-    cost_function_grad = grad(cost_function,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        cost_grad =  cost_function_grad(P, x , t)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_grad[l]
-
-    print('Final cost: ',cost_function(P, x, t))
-
-    return P
-
-if __name__ == '__main__':
-    ### Use the neural network:
-    npr.seed(15)
-
-    ## Decide the vales of arguments to the function to solve
-    Nx = 10; Nt = 10
-    x = np.linspace(0, 1, Nx)
-    t = np.linspace(0,1,Nt)
-
-    ## Set up the parameters for the network
-    num_hidden_neurons = [100, 25]
-    num_iter = 250
-    lmb = 0.01
-
-    P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
-
-    ## Store the results
-    g_dnn_ag = np.zeros((Nx, Nt))
-    G_analytical = np.zeros((Nx, Nt))
-    for i,x_ in enumerate(x):
-        for j, t_ in enumerate(t):
-            point = np.array([x_, t_])
-            g_dnn_ag[i,j] = g_trial(point,P)
-
-            G_analytical[i,j] = g_analytic(point)
-
-    # Find the map difference between the analytical and the computed solution
-    diff_ag = np.abs(g_dnn_ag - G_analytical)
-    print('Max absolute difference between the analytical solution and the network: %g'%np.max(diff_ag))
-
-    ## Plot the solutions in two dimensions, that being in position and time
-
-    T,X = np.meshgrid(t,x)
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))
-    s = ax.plot_surface(T,X,g_dnn_ag,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Analytical solution')
-    s = ax.plot_surface(T,X,G_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Difference')
-    s = ax.plot_surface(T,X,diff_ag,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-    ## Take some slices of the 3D plots just to see the solutions at particular times
-    indx1 = 0
-    indx2 = int(Nt/2)
-    indx3 = Nt-1
-
-    t1 = t[indx1]
-    t2 = t[indx2]
-    t3 = t[indx3]
-
-    # Slice the results from the DNN
-    res1 = g_dnn_ag[:,indx1]
-    res2 = g_dnn_ag[:,indx2]
-    res3 = g_dnn_ag[:,indx3]
-
-    # Slice the analytical results
-    res_analytical1 = G_analytical[:,indx1]
-    res_analytical2 = G_analytical[:,indx2]
-    res_analytical3 = G_analytical[:,indx3]
-
-    # Plot the slices
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t1)
-    plt.plot(x, res1)
-    plt.plot(x,res_analytical1)
-    plt.legend(['dnn','analytical'])
-
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t2)
-    plt.plot(x, res2)
-    plt.plot(x,res_analytical2)
-    plt.legend(['dnn','analytical'])
-
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t3)
-    plt.plot(x, res3)
-    plt.plot(x,res_analytical3)
-    plt.legend(['dnn','analytical'])
-
-    plt.show()
-
-
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray
-  return array(a, dtype, copy=False, order=order)
-
-
-
Initial cost:  41.05505310046363
-
-
-
-
-
-
-
-

3.10. Solving the wave equation with Neural Networks

-

The wave equation is

-
-\[ -\frac{\partial^2 g(x,t)}{\partial t^2} = c^2\frac{\partial^2 g(x,t)}{\partial x^2} -\]
-

with \(c\) being the specified wave speed.

-

Here, the chosen conditions are

-
-\[\begin{split} -\begin{align*} - g(0,t) &= 0 \\ - g(1,t) &= 0 \\ - g(x,0) &= u(x) \\ - \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} &= v(x) -\end{align*} -\end{split}\]
-

where \(\frac{\partial g(x,t)}{\partial t} \Big |_{t = 0}\) means the derivative of \(g(x,t)\) with respect to \(t\) is evaluated at \(t = 0\), and \(u(x)\) and \(v(x)\) being given functions.

-

The wave equation to solve for, is

- -
-
-\[ -\begin{equation} \label{wave} \tag{19} -\frac{\partial^2 g(x,t)}{\partial t^2} = c^2 \frac{\partial^2 g(x,t)}{\partial x^2} -\end{equation} -\]
-

where \(c\) is the given wave speed. -The chosen conditions for this equation are

- -
-
-\[\begin{split} -\begin{aligned} -g(0,t) &= 0, &t \geq 0 \\ -g(1,t) &= 0, &t \geq 0 \\ -g(x,0) &= u(x), &x\in[0,1] \\ -\frac{\partial g(x,t)}{\partial t}\Big |_{t = 0} &= v(x), &x \in [0,1] -\end{aligned} \label{condwave} \tag{20} -\end{split}\]
-

In this example, let \(c = 1\) and \(u(x) = \sin(\pi x)\) and \(v(x) = -\pi\sin(\pi x)\).

-

Setting up the network is done in similar matter as for the example of solving the diffusion equation. -The only things we have to change, is the trial solution such that it satisfies the conditions from (20) and the cost function.

-

The trial solution becomes slightly different since we have other conditions than in the example of solving the diffusion equation. Here, a possible trial solution \(g_t(x,t)\) is

-
-\[ -g_t(x,t) = h_1(x,t) + x(1-x)t^2N(x,t,P) -\]
-

where

-
-\[ -h_1(x,t) = (1-t^2)u(x) + tv(x) -\]
-

Note that this trial solution satisfies the conditions only if \(u(0) = v(0) = u(1) = v(1) = 0\), which is the case in this example.

-

The analytical solution for our specific problem, is

-
-\[ -g(x,t) = \sin(\pi x)\cos(\pi t) - \sin(\pi x)\sin(\pi t) -\]
-
-
-
import autograd.numpy as np
-from autograd import hessian,grad
-import autograd.numpy.random as npr
-from matplotlib import cm
-from matplotlib import pyplot as plt
-from mpl_toolkits.mplot3d import axes3d
-
-## Set up the trial function:
-def u(x):
-    return np.sin(np.pi*x)
-
-def v(x):
-    return -np.pi*np.sin(np.pi*x)
-
-def h1(point):
-    x,t = point
-    return (1 - t**2)*u(x) + t*v(x)
-
-def g_trial(point,P):
-    x,t = point
-    return h1(point) + x*(1-x)*t**2*deep_neural_network(P,point)
-
-## Define the cost function
-def cost_function(P, x, t):
-    cost_sum = 0
-
-    g_t_hessian_func = hessian(g_trial)
-
-    for x_ in x:
-        for t_ in t:
-            point = np.array([x_,t_])
-
-            g_t_hessian = g_t_hessian_func(point,P)
-
-            g_t_d2x = g_t_hessian[0][0]
-            g_t_d2t = g_t_hessian[1][1]
-
-            err_sqr = ( (g_t_d2t - g_t_d2x) )**2
-            cost_sum += err_sqr
-
-    return cost_sum / (np.size(t) * np.size(x))
-
-## The neural network
-def sigmoid(z):
-    return 1/(1 + np.exp(-z))
-
-def deep_neural_network(deep_params, x):
-    # x is now a point and a 1D numpy array; make it a column vector
-    num_coordinates = np.size(x,0)
-    x = x.reshape(num_coordinates,-1)
-
-    num_points = np.size(x,1)
-
-    # N_hidden is the number of hidden layers
-    N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer
-
-    # Assume that the input layer does nothing to the input x
-    x_input = x
-    x_prev = x_input
-
-    ## Hidden layers:
-
-    for l in range(N_hidden):
-        # From the list of parameters P; find the correct weigths and bias for this layer
-        w_hidden = deep_params[l]
-
-        # Add a row of ones to include bias
-        x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)
-
-        z_hidden = np.matmul(w_hidden, x_prev)
-        x_hidden = sigmoid(z_hidden)
-
-        # Update x_prev such that next layer can use the output from this layer
-        x_prev = x_hidden
-
-    ## Output layer:
-
-    # Get the weights and bias for this layer
-    w_output = deep_params[-1]
-
-    # Include bias:
-    x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)
-
-    z_output = np.matmul(w_output, x_prev)
-    x_output = z_output
-
-    return x_output[0][0]
-
-## The analytical solution
-def g_analytic(point):
-    x,t = point
-    return np.sin(np.pi*x)*np.cos(np.pi*t) - np.sin(np.pi*x)*np.sin(np.pi*t)
-
-def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):
-    ## Set up initial weigths and biases
-    N_hidden = np.size(num_neurons)
-
-    ## Set up initial weigths and biases
-
-    # Initialize the list of parameters:
-    P = [None]*(N_hidden + 1) # + 1 to include the output layer
-
-    P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias
-    for l in range(1,N_hidden):
-        P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias
-
-    # For the output layer
-    P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included
-
-    print('Initial cost: ',cost_function(P, x, t))
-
-    cost_function_grad = grad(cost_function,0)
-
-    # Let the update be done num_iter times
-    for i in range(num_iter):
-        cost_grad =  cost_function_grad(P, x , t)
-
-        for l in range(N_hidden+1):
-            P[l] = P[l] - lmb * cost_grad[l]
-
-
-    print('Final cost: ',cost_function(P, x, t))
-
-    return P
-
-if __name__ == '__main__':
-    ### Use the neural network:
-    npr.seed(15)
-
-    ## Decide the vales of arguments to the function to solve
-    Nx = 10; Nt = 10
-    x = np.linspace(0, 1, Nx)
-    t = np.linspace(0,1,Nt)
-
-    ## Set up the parameters for the network
-    num_hidden_neurons = [50,20]
-    num_iter = 1000
-    lmb = 0.01
-
-    P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)
-
-    ## Store the results
-    res = np.zeros((Nx, Nt))
-    res_analytical = np.zeros((Nx, Nt))
-    for i,x_ in enumerate(x):
-        for j, t_ in enumerate(t):
-            point = np.array([x_, t_])
-            res[i,j] = g_trial(point,P)
-
-            res_analytical[i,j] = g_analytic(point)
-
-    diff = np.abs(res - res_analytical)
-    print("Max difference between analytical and solution from nn: %g"%np.max(diff))
-
-    ## Plot the solutions in two dimensions, that being in position and time
-
-    T,X = np.meshgrid(t,x)
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))
-    s = ax.plot_surface(T,X,res,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Analytical solution')
-    s = ax.plot_surface(T,X,res_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-
-    fig = plt.figure(figsize=(10,10))
-    ax = fig.gca(projection='3d')
-    ax.set_title('Difference')
-    s = ax.plot_surface(T,X,diff,linewidth=0,antialiased=False,cmap=cm.viridis)
-    ax.set_xlabel('Time $t$')
-    ax.set_ylabel('Position $x$');
-
-    ## Take some slices of the 3D plots just to see the solutions at particular times
-    indx1 = 0
-    indx2 = int(Nt/2)
-    indx3 = Nt-1
-
-    t1 = t[indx1]
-    t2 = t[indx2]
-    t3 = t[indx3]
-
-    # Slice the results from the DNN
-    res1 = res[:,indx1]
-    res2 = res[:,indx2]
-    res3 = res[:,indx3]
-
-    # Slice the analytical results
-    res_analytical1 = res_analytical[:,indx1]
-    res_analytical2 = res_analytical[:,indx2]
-    res_analytical3 = res_analytical[:,indx3]
-
-    # Plot the slices
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t1)
-    plt.plot(x, res1)
-    plt.plot(x,res_analytical1)
-    plt.legend(['dnn','analytical'])
-
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t2)
-    plt.plot(x, res2)
-    plt.plot(x,res_analytical2)
-    plt.legend(['dnn','analytical'])
-
-    plt.figure(figsize=(10,10))
-    plt.title("Computed solutions at time = %g"%t3)
-    plt.plot(x, res3)
-    plt.plot(x,res_analytical3)
-    plt.legend(['dnn','analytical'])
-
-    plt.show()
-
-
-
-
-
- -
- - - - -
- - -
- - 2. Building a Feed Forward Neural Network - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

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- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter2.html b/doc/LectureNotes/_build/html/chapter2.html deleted file mode 100644 index 08cc51291..000000000 --- a/doc/LectureNotes/_build/html/chapter2.html +++ /dev/null @@ -1,1444 +0,0 @@ - - - - - - - - 2. Resampling Methods — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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2. Resampling Methods

-

Video of Lecture

-
-

2.1. Introduction

-

Resampling methods are an indispensable tool in modern -statistics. They involve repeatedly drawing samples from a training -set and refitting a model of interest on each sample in order to -obtain additional information about the fitted model. For example, in -order to estimate the variability of a linear regression fit, we can -repeatedly draw different samples from the training data, fit a linear -regression to each new sample, and then examine the extent to which -the resulting fits differ. Such an approach may allow us to obtain -information that would not be available from fitting the model only -once using the original training sample.

-

Two resampling methods are often used in Machine Learning analyses,

-
    -

  1. The bootstrap method

    2. -

  2. and Cross-Validation

    3. -
-

In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular -cross-validation and the bootstrap method.

-

Resampling approaches can be computationally expensive, because they -involve fitting the same statistical method multiple times using -different subsets of the training data. However, due to recent -advances in computing power, the computational requirements of -resampling methods generally are not prohibitive. In this chapter, we -discuss two of the most commonly used resampling methods, -cross-validation and the bootstrap. Both methods are important tools -in the practical application of many statistical learning -procedures. For example, cross-validation can be used to estimate the -test error associated with a given statistical learning method in -order to evaluate its performance, or to select the appropriate level -of flexibility. The process of evaluating a model’s performance is -known as model assessment, whereas the process of selecting the proper -level of flexibility for a model is known as model selection. The -bootstrap is widely used.

-
    -

  • Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods

    • -

  • The results can be analysed with the same statistical tools as we would use analysing experimental data.

    • -

  • As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.

    • -
-
-
-

2.2. Reminder on Statistics

-
    -

  • As in other experiments, many numerical experiments have two classes of errors:

    -
      -

    • Statistical errors

      • -

    • Systematical errors

      • -
    -
    • -

  • Statistical errors can be estimated using standard tools from statistics

    • -

  • Systematical errors are method specific and must be treated differently from case to case.

    • -
-

The -advantage of doing linear regression is that we actually end up with -analytical expressions for several statistical quantities.
-Standard least squares and Ridge regression allow us to -derive quantities like the variance and other expectation values in a -rather straightforward way.

-

It is assumed that \(\varepsilon_i -\sim \mathcal{N}(0, \sigma^2)\) and the \(\varepsilon_{i}\) are -independent, i.e.:

-
-\[\begin{split} -\begin{align*} -\mbox{Cov}(\varepsilon_{i_1}, -\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if} -& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right. -\end{align*} -\end{split}\]
-

The randomness of \(\varepsilon_i\) implies that -\(\mathbf{y}_i\) is also a random variable. In particular, -\(\mathbf{y}_i\) is normally distributed, because \(\varepsilon_i \sim -\mathcal{N}(0, \sigma^2)\) and \(\mathbf{X}_{i,\ast} \, \boldsymbol{\beta}\) is a -non-random scalar. To specify the parameters of the distribution of -\(\mathbf{y}_i\) we need to calculate its first two moments.

-

Recall that \(\boldsymbol{X}\) is a matrix of dimensionality \(n\times p\). The -notation above \(\mathbf{X}_{i,\ast}\) means that we are looking at the -row number \(i\) and perform a sum over all values \(p\).

-

The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off) -that there exists a function \(f(\boldsymbol{x})\) and a normal distributed error \(\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)\) -which describe our data

-
-\[ -\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} -\]
-

We approximate this function with our model from the solution of the linear regression equations, that is our -function \(f\) is approximated by \(\boldsymbol{\tilde{y}}\) where we want to minimize \((\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\), our MSE, with

-
-\[ -\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. -\]
-

We can calculate the expectation value of \(\boldsymbol{y}\) for a given element \(i\)

-
-\[ -\begin{align*} -\mathbb{E}(y_i) & = -\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i) -\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, -\end{align*} -\]
-

while -its variance is

-
-\[\begin{split} -\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i -- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - -[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, -\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ & -= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i -\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, -\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 -\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + -\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 -\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, -\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. -\end{align*} -\end{split}\]
-

Hence, \(y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)\), that is \(\boldsymbol{y}\) follows a normal distribution with -mean value \(\boldsymbol{X}\boldsymbol{\beta}\) and variance \(\sigma^2\) (not be confused with the singular values of the SVD).

-

With the OLS expressions for the parameters \(\boldsymbol{\beta}\) we can evaluate the expectation value

-
-\[ -\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}. -\]
-

This means that the estimator of the regression parameters is unbiased.

-

We can also calculate the variance

-

The variance of \(\boldsymbol{\beta}\) is

-
-\[\begin{split} -\begin{eqnarray*} -\mbox{Var}(\boldsymbol{\beta}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \} -\\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \} -\\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} -% \\ -% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T -\\ -& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, -\end{eqnarray*} -\end{split}\]
-

where we have used that \(\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = -\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + -\sigma^2 \, \mathbf{I}_{nn}\). From \(\mbox{Var}(\boldsymbol{\beta}) = \sigma^2 -\, (\mathbf{X}^{T} \mathbf{X})^{-1}\), one obtains an estimate of the -variance of the estimate of the \(j\)-th regression coefficient: -\(\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 \sqrt{ -[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} }\). This may be used to -construct a confidence interval for the estimates.

-

In a similar way, we can obtain analytical expressions for say the -expectation values of the parameters \(\boldsymbol{\beta}\) and their variance -when we employ Ridge regression, allowing us again to define a confidence interval.

-

It is rather straightforward to show that

-
-\[ -\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}. -\]
-

We see clearly that -\(\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}}\) for any \(\lambda > 0\). We say then that the ridge estimator is biased.

-

We can also compute the variance as

-
-\[ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}, -\]
-

and it is easy to see that if the parameter \(\lambda\) goes to infinity then the variance of Ridge parameters \(\boldsymbol{\beta}\) goes to zero.

-

With this, we can compute the difference

-
-\[ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}. -\]
-

The difference is non-negative definite since each component of the -matrix product is non-negative definite. -This means the variance we obtain with the standard OLS will always for \(\lambda > 0\) be larger than the variance of \(\boldsymbol{\beta}\) obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below.

-
-
-

2.3. Resampling methods

-

With all these analytical equations for both the OLS and Ridge -regression, we will now outline how to assess a given model. This will -lead us to a discussion of the so-called bias-variance tradeoff (see -below) and so-called resampling methods.

-

One of the quantities we have discussed as a way to measure errors is -the mean-squared error (MSE), mainly used for fitting of continuous -functions. Another choice is the absolute error.

-

In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data, -we discuss the

-
    -

  1. prediction error or simply the test error \(\mathrm{Err_{Test}}\), where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the

    2. -

  2. training error \(\mathrm{Err_{Train}}\), which is the average loss over the training data.

    3. -
-

As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error. -For a certain level of complexity the test error will reach minimum, before starting to increase again. The -training error reaches a saturation.

-

Two famous -resampling methods are the independent bootstrap and the jackknife.

-

The jackknife is a special case of the independent bootstrap. Still, the jackknife was made -popular prior to the independent bootstrap. And as the popularity of -the independent bootstrap soared, new variants, such as the dependent bootstrap.

-

The Jackknife and independent bootstrap work for -independent, identically distributed random variables. -If these conditions are not -satisfied, the methods will fail. Yet, it should be said that if the data are -independent, identically distributed, and we only want to estimate the -variance of \(\overline{X}\) (which often is the case), then there is no -need for bootstrapping.

-

The Jackknife works by making many replicas of the estimator \(\widehat{\theta}\). -The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values \(\boldsymbol{x} = (x_1,x_2,\cdots,X_n)\). -Let \(\boldsymbol{x}_i\) denote the vector

-
-\[ -\boldsymbol{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n), -\]
-

which equals the vector \(\boldsymbol{x}\) with the exception that observation -number \(i\) is left out. Using this notation, define -\(\widehat{\theta}_i\) to be the estimator -\(\widehat{\theta}\) computed using \(\vec{X}_i\).

-
-
-
from numpy import *
-from numpy.random import randint, randn
-from time import time
-
-def jackknife(data, stat):
-    n = len(data);t = zeros(n); inds = arange(n); t0 = time()
-    ## 'jackknifing' by leaving out an observation for each i                                                                                                                      
-    for i in range(n):
-        t[i] = stat(delete(data,i) )
-
-    # analysis                                                                                                                                                                     
-    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
-    print("original           bias      std. error")
-    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))
-
-    return t
-
-
-# Returns mean of data samples                                                                                                                                                     
-def stat(data):
-    return mean(data)
-
-
-mu, sigma = 100, 15
-datapoints = 10000
-x = mu + sigma*random.randn(datapoints)
-# jackknife returns the data sample                                                                                                                                                
-t = jackknife(x, stat)
-
-
-
-
-
Runtime: 0.139172 sec
-Jackknife Statistics :
-original           bias      std. error
- 100.048        100.038        0.150004
-
-
-
-
-
-

2.3.1. Bootstrap

-

Bootstrapping is a nonparametric approach to statistical inference -that substitutes computation for more traditional distributional -assumptions and asymptotic results. Bootstrapping offers a number of -advantages:

-
    -

  1. The bootstrap is quite general, although there are some cases in which it fails.

    2. -

  2. Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.

    3. -

  3. It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.

    4. -

  4. It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).

    5. -
-

Since \(\widehat{\theta} = \widehat{\theta}(\boldsymbol{X})\) is a function of random variables, -\(\widehat{\theta}\) itself must be a random variable. Thus it has -a pdf, call this function \(p(\boldsymbol{t})\). The aim of the bootstrap is to -estimate \(p(\boldsymbol{t})\) by the relative frequency of -\(\widehat{\theta}\). You can think of this as using a histogram -in the place of \(p(\boldsymbol{t})\). If the relative frequency closely -resembles \(p(\vec{t})\), then using numerics, it is straight forward to -estimate all the interesting parameters of \(p(\boldsymbol{t})\) using point -estimators.

-

In the case that \(\widehat{\theta}\) has -more than one component, and the components are independent, we use the -same estimator on each component separately. If the probability -density function of \(X_i\), \(p(x)\), had been known, then it would have -been straight forward to do this by:

-
    -

  1. Drawing lots of numbers from \(p(x)\), suppose we call one such set of numbers \((X_1^*, X_2^*, \cdots, X_n^*)\).

    2. -

  2. Then using these numbers, we could compute a replica of \(\widehat{\theta}\) called \(\widehat{\theta}^*\).

    3. -
-

By repeated use of (1) and (2), many -estimates of \(\widehat{\theta}\) could have been obtained. The -idea is to use the relative frequency of \(\widehat{\theta}^*\) -(think of a histogram) as an estimate of \(p(\boldsymbol{t})\).

-

But -unless there is enough information available about the process that -generated \(X_1,X_2,\cdots,X_n\), \(p(x)\) is in general -unknown. Therefore, Efron in 1979 asked the -question: What if we replace \(p(x)\) by the relative frequency -of the observation \(X_i\); if we draw observations in accordance with -the relative frequency of the observations, will we obtain the same -result in some asymptotic sense? The answer is yes.

-

Instead of generating the histogram for the relative -frequency of the observation \(X_i\), just draw the values -\((X_1^*,X_2^*,\cdots,X_n^*)\) with replacement from the vector -\(\boldsymbol{X}\).

-

The independent bootstrap works like this:

-
    -

  1. Draw with replacement \(n\) numbers for the observed variables \(\boldsymbol{x} = (x_1,x_2,\cdots,x_n)\).

    2. -

  2. Define a vector \(\boldsymbol{x}^*\) containing the values which were drawn from \(\boldsymbol{x}\).

    3. -

  3. Using the vector \(\boldsymbol{x}^*\) compute \(\widehat{\theta}^*\) by evaluating \(\widehat \theta\) under the observations \(\boldsymbol{x}^*\).

    4. -

  4. Repeat this process \(k\) times.

    5. -
-

When you are done, you can draw a histogram of the relative frequency -of \(\widehat \theta^*\). This is your estimate of the probability -distribution \(p(t)\). Using this probability distribution you can -estimate any statistics thereof. In principle you never draw the -histogram of the relative frequency of \(\widehat{\theta}^*\). Instead -you use the estimators corresponding to the statistic of interest. For -example, if you are interested in estimating the variance of \(\widehat -\theta\), apply the etsimator \(\widehat \sigma^2\) to the values -\(\widehat \theta ^*\).

-

The following code starts with a Gaussian distribution with mean value -\(\mu =100\) and variance \(\sigma=15\). We use this to generate the data -used in the bootstrap analysis. The bootstrap analysis returns a data -set after a given number of bootstrap operations (as many as we have -data points). This data set consists of estimated mean values for each -bootstrap operation. The histogram generated by the bootstrap method -shows that the distribution for these mean values is also a Gaussian, -centered around the mean value \(\mu=100\) but with standard deviation -\(\sigma/\sqrt{n}\), where \(n\) is the number of bootstrap samples (in -this case the same as the number of original data points). The value -of the standard deviation is what we expect from the central limit -theorem.

-
-
-
%matplotlib inline
-
-from numpy import *
-from numpy.random import randint, randn
-from time import time
-import matplotlib.mlab as mlab
-import matplotlib.pyplot as plt
-
-# Returns mean of bootstrap samples                                                                                                                                                
-def stat(data):
-    return mean(data)
-
-# Bootstrap algorithm
-def bootstrap(data, statistic, R):
-    t = zeros(R); n = len(data); inds = arange(n); t0 = time()
-    # non-parametric bootstrap         
-    for i in range(R):
-        t[i] = statistic(data[randint(0,n,n)])
-
-    # analysis    
-    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
-    print("original           bias      std. error")
-    print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t)))
-    return t
-
-
-mu, sigma = 100, 15
-datapoints = 10000
-x = mu + sigma*random.randn(datapoints)
-# bootstrap returns the data sample                                    
-t = bootstrap(x, stat, datapoints)
-# the histogram of the bootstrapped  data                                                                                                    
-n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
-
-# add a 'best fit' line  
-y = mlab.normpdf( binsboot, mean(t), std(t))
-lt = plt.plot(binsboot, y, 'r--', linewidth=1)
-plt.xlabel('Smarts')
-plt.ylabel('Probability')
-plt.axis([99.5, 100.6, 0, 3.0])
-plt.grid(True)
-
-plt.show()
-
-
-
-
-
Runtime: 1.73359 sec
-Bootstrap Statistics :
-original           bias      std. error
- 99.9929  15.0315         99.994        0.150978
-
-
-
---------------------------------------------------------------------------
-AttributeError                            Traceback (most recent call last)
-<ipython-input-2-772b904ae9cb> in <module>
-     31 t = bootstrap(x, stat, datapoints)
-     32 # the histogram of the bootstrapped  data
----> 33 n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)
-     34 
-     35 # add a 'best fit' line
-
-~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/pyplot.py in hist(x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, data, **kwargs)
-   2683         orientation='vertical', rwidth=None, log=False, color=None,
-   2684         label=None, stacked=False, *, data=None, **kwargs):
--> 2685     return gca().hist(
-   2686         x, bins=bins, range=range, density=density, weights=weights,
-   2687         cumulative=cumulative, bottom=bottom, histtype=histtype,
-
-~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/__init__.py in inner(ax, data, *args, **kwargs)
-   1445     def inner(ax, *args, data=None, **kwargs):
-   1446         if data is None:
--> 1447             return func(ax, *map(sanitize_sequence, args), **kwargs)
-   1448 
-   1449         bound = new_sig.bind(ax, *args, **kwargs)
-
-~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/axes/_axes.py in hist(self, x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, **kwargs)
-   6813             if patch:
-   6814                 p = patch[0]
--> 6815                 p.update(kwargs)
-   6816                 if lbl is not None:
-   6817                     p.set_label(lbl)
-
-~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/artist.py in update(self, props)
-    994                     func = getattr(self, f"set_{k}", None)
-    995                     if not callable(func):
---> 996                         raise AttributeError(f"{type(self).__name__!r} object "
-    997                                              f"has no property {k!r}")
-    998                     ret.append(func(v))
-
-AttributeError: 'Rectangle' object has no property 'normed'
-
-
-_images/chapter2_25_2.png -
-
-
-
-
-

2.4. Various steps in cross-validation

-

When the repetitive splitting of the data set is done randomly, -samples may accidently end up in a fast majority of the splits in -either training or test set. Such samples may have an unbalanced -influence on either model building or prediction evaluation. To avoid -this \(k\)-fold cross-validation structures the data splitting. The -samples are divided into \(k\) more or less equally sized exhaustive and -mutually exclusive subsets. In turn (at each split) one of these -subsets plays the role of the test set while the union of the -remaining subsets constitutes the training set. Such a splitting -warrants a balanced representation of each sample in both training and -test set over the splits. Still the division into the \(k\) subsets -involves a degree of randomness. This may be fully excluded when -choosing \(k=n\). This particular case is referred to as leave-one-out -cross-validation (LOOCV).

-
    -

  • Define a range of interest for the penalty parameter.

    • -

  • Divide the data set into training and test set comprising samples \(\{1, \ldots, n\} \setminus i\) and \(\{ i \}\), respectively.

    • -

  • Fit the linear regression model by means of ridge estimation for each \(\lambda\) in the grid using the training set, and the corresponding estimate of the error variance \(\boldsymbol{\sigma}_{-i}^2(\lambda)\), as

    • -
-
-\[ -\begin{align*} -\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T} -\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1} -\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i} -\end{align*} -\]
-
    -

  • Evaluate the prediction performance of these models on the test set by \(\log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}\). Or, by the prediction error \(|y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)|\), the relative error, the error squared or the R2 score function.

    • -

  • Repeat the first three steps such that each sample plays the role of the test set once.

    • -

  • Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as

    • -
-
-\[ -\begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}. -\end{align*} -\]
-

For the various values of \(k\)

-
    -

  1. shuffle the dataset randomly.

    2. -

  2. Split the dataset into \(k\) groups.

    3. -

  3. For each unique group:

    4. -
-

a. Decide which group to use as set for test data

-

b. Take the remaining groups as a training data set

-

c. Fit a model on the training set and evaluate it on the test set

-

d. Retain the evaluation score and discard the model

-
    -

  1. Summarize the model using the sample of model evaluation scores

    2. -
-

The code here uses Ridge regression with cross-validation (CV) resampling and \(k\)-fold CV in order to fit a specific polynomial.

-
-
-
import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.model_selection import KFold
-from sklearn.linear_model import Ridge
-from sklearn.model_selection import cross_val_score
-from sklearn.preprocessing import PolynomialFeatures
-
-# A seed just to ensure that the random numbers are the same for every run.
-# Useful for eventual debugging.
-np.random.seed(3155)
-
-# Generate the data.
-nsamples = 100
-x = np.random.randn(nsamples)
-y = 3*x**2 + np.random.randn(nsamples)
-
-## Cross-validation on Ridge regression using KFold only
-
-# Decide degree on polynomial to fit
-poly = PolynomialFeatures(degree = 6)
-
-# Decide which values of lambda to use
-nlambdas = 500
-lambdas = np.logspace(-3, 5, nlambdas)
-
-# Initialize a KFold instance
-k = 5
-kfold = KFold(n_splits = k)
-
-# Perform the cross-validation to estimate MSE
-scores_KFold = np.zeros((nlambdas, k))
-
-i = 0
-for lmb in lambdas:
-    ridge = Ridge(alpha = lmb)
-    j = 0
-    for train_inds, test_inds in kfold.split(x):
-        xtrain = x[train_inds]
-        ytrain = y[train_inds]
-
-        xtest = x[test_inds]
-        ytest = y[test_inds]
-
-        Xtrain = poly.fit_transform(xtrain[:, np.newaxis])
-        ridge.fit(Xtrain, ytrain[:, np.newaxis])
-
-        Xtest = poly.fit_transform(xtest[:, np.newaxis])
-        ypred = ridge.predict(Xtest)
-
-        scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)
-
-        j += 1
-    i += 1
-
-
-estimated_mse_KFold = np.mean(scores_KFold, axis = 1)
-
-## Cross-validation using cross_val_score from sklearn along with KFold
-
-# kfold is an instance initialized above as:
-# kfold = KFold(n_splits = k)
-
-estimated_mse_sklearn = np.zeros(nlambdas)
-i = 0
-for lmb in lambdas:
-    ridge = Ridge(alpha = lmb)
-
-    X = poly.fit_transform(x[:, np.newaxis])
-    estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)
-
-    # cross_val_score return an array containing the estimated negative mse for every fold.
-    # we have to the the mean of every array in order to get an estimate of the mse of the model
-    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
-
-    i += 1
-
-## Plot and compare the slightly different ways to perform cross-validation
-
-plt.figure()
-
-plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
-plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')
-
-plt.xlabel('log10(lambda)')
-plt.ylabel('mse')
-
-plt.legend()
-
-plt.show()
-
-
-
-
-
-
-

2.5. The bias-variance tradeoff

-

We will discuss the bias-variance tradeoff in the context of -continuous predictions such as regression. However, many of the -intuitions and ideas discussed here also carry over to classification -tasks. Consider a dataset \(\mathcal{L}\) consisting of the data -\(\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\).

-

Let us assume that the true data is generated from a noisy model

-
-\[ -\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon} -\]
-

where \(\epsilon\) is normally distributed with mean zero and standard deviation \(\sigma^2\).

-

In our derivation of the ordinary least squares method we defined then -an approximation to the function \(f\) in terms of the parameters -\(\boldsymbol{\beta}\) and the design matrix \(\boldsymbol{X}\) which embody our model, -that is \(\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta}\).

-

Thereafter we found the parameters \(\boldsymbol{\beta}\) by optimizing the means squared error via the so-called cost function

-
-\[ -C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. -\]
-

We can rewrite this as

-
-\[ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2. -\]
-

The three terms represent the square of the bias of the learning -method, which can be thought of as the error caused by the simplifying -assumptions built into the method. The second term represents the -variance of the chosen model and finally the last terms is variance of -the error \(\boldsymbol{\epsilon}\).

-

To derive this equation, we need to recall that the variance of \(\boldsymbol{y}\) and \(\boldsymbol{\epsilon}\) are both equal to \(\sigma^2\). The mean value of \(\boldsymbol{\epsilon}\) is by definition equal to zero. Furthermore, the function \(f\) is not a stochastics variable, idem for \(\boldsymbol{\tilde{y}}\). -We use a more compact notation in terms of the expectation value

-
-\[ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right], -\]
-

and adding and subtracting \(\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]\) we get

-
-\[ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right], -\]
-

which, using the abovementioned expectation values can be rewritten as

-
-\[ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2, -\]
-

that is the rewriting in terms of the so-called bias, the variance of the model \(\boldsymbol{\tilde{y}}\) and the variance of \(\boldsymbol{\epsilon}\).

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.model_selection import train_test_split
-from sklearn.pipeline import make_pipeline
-from sklearn.utils import resample
-
-np.random.seed(2018)
-
-n = 500
-n_boostraps = 100
-degree = 18  # A quite high value, just to show.
-noise = 0.1
-
-# Make data set.
-x = np.linspace(-1, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape)
-
-# Hold out some test data that is never used in training.
-x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
-
-# Combine x transformation and model into one operation.
-# Not neccesary, but convenient.
-model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
-
-# The following (m x n_bootstraps) matrix holds the column vectors y_pred
-# for each bootstrap iteration.
-y_pred = np.empty((y_test.shape[0], n_boostraps))
-for i in range(n_boostraps):
-    x_, y_ = resample(x_train, y_train)
-
-    # Evaluate the new model on the same test data each time.
-    y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
-
-# Note: Expectations and variances taken w.r.t. different training
-# data sets, hence the axis=1. Subsequent means are taken across the test data
-# set in order to obtain a total value, but before this we have error/bias/variance
-# calculated per data point in the test set.
-# Note 2: The use of keepdims=True is important in the calculation of bias as this 
-# maintains the column vector form. Dropping this yields very unexpected results.
-error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-print('Error:', error)
-print('Bias^2:', bias)
-print('Var:', variance)
-print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))
-
-plt.plot(x[::5, :], y[::5, :], label='f(x)')
-plt.scatter(x_test, y_test, label='Data points')
-plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred')
-plt.legend()
-plt.show()
-
-
-
-
-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.model_selection import train_test_split
-from sklearn.pipeline import make_pipeline
-from sklearn.utils import resample
-
-np.random.seed(2018)
-
-n = 40
-n_boostraps = 100
-maxdegree = 14
-
-
-# Make data set.
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-error = np.zeros(maxdegree)
-bias = np.zeros(maxdegree)
-variance = np.zeros(maxdegree)
-polydegree = np.zeros(maxdegree)
-x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
-
-for degree in range(maxdegree):
-    model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
-    y_pred = np.empty((y_test.shape[0], n_boostraps))
-    for i in range(n_boostraps):
-        x_, y_ = resample(x_train, y_train)
-        y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()
-
-    polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-    print('Polynomial degree:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
-
-plt.plot(polydegree, error, label='Error')
-plt.plot(polydegree, bias, label='bias')
-plt.plot(polydegree, variance, label='Variance')
-plt.legend()
-plt.show()
-
-
-
-
-

The bias-variance tradeoff summarizes the fundamental tension in -machine learning, particularly supervised learning, between the -complexity of a model and the amount of training data needed to train -it. Since data is often limited, in practice it is often useful to -use a less-complex model with higher bias, that is a model whose asymptotic -performance is worse than another model because it is easier to -train and less sensitive to sampling noise arising from having a -finite-sized training dataset (smaller variance).

-

The above equations tell us that in -order to minimize the expected test error, we need to select a -statistical learning method that simultaneously achieves low variance -and low bias. Note that variance is inherently a nonnegative quantity, -and squared bias is also nonnegative. Hence, we see that the expected -test MSE can never lie below \(Var(\epsilon)\), the irreducible error.

-

What do we mean by the variance and bias of a statistical learning -method? The variance refers to the amount by which our model would change if we -estimated it using a different training data set. Since the training -data are used to fit the statistical learning method, different -training data sets will result in a different estimate. But ideally the -estimate for our model should not vary too much between training -sets. However, if a method has high variance then small changes in -the training data can result in large changes in the model. In general, more -flexible statistical methods have higher variance.

-

You may also find this recent article of interest.

-
-
-
"""
-============================
-Underfitting vs. Overfitting
-============================
-
-This example demonstrates the problems of underfitting and overfitting and
-how we can use linear regression with polynomial features to approximate
-nonlinear functions. The plot shows the function that we want to approximate,
-which is a part of the cosine function. In addition, the samples from the
-real function and the approximations of different models are displayed. The
-models have polynomial features of different degrees. We can see that a
-linear function (polynomial with degree 1) is not sufficient to fit the
-training samples. This is called **underfitting**. A polynomial of degree 4
-approximates the true function almost perfectly. However, for higher degrees
-the model will **overfit** the training data, i.e. it learns the noise of the
-training data.
-We evaluate quantitatively **overfitting** / **underfitting** by using
-cross-validation. We calculate the mean squared error (MSE) on the validation
-set, the higher, the less likely the model generalizes correctly from the
-training data.
-"""
-
-print(__doc__)
-
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.linear_model import LinearRegression
-from sklearn.model_selection import cross_val_score
-
-
-def true_fun(X):
-    return np.cos(1.5 * np.pi * X)
-
-np.random.seed(0)
-
-n_samples = 30
-degrees = [1, 4, 15]
-
-X = np.sort(np.random.rand(n_samples))
-y = true_fun(X) + np.random.randn(n_samples) * 0.1
-
-plt.figure(figsize=(14, 5))
-for i in range(len(degrees)):
-    ax = plt.subplot(1, len(degrees), i + 1)
-    plt.setp(ax, xticks=(), yticks=())
-
-    polynomial_features = PolynomialFeatures(degree=degrees[i],
-                                             include_bias=False)
-    linear_regression = LinearRegression()
-    pipeline = Pipeline([("polynomial_features", polynomial_features),
-                         ("linear_regression", linear_regression)])
-    pipeline.fit(X[:, np.newaxis], y)
-
-    # Evaluate the models using crossvalidation
-    scores = cross_val_score(pipeline, X[:, np.newaxis], y,
-                             scoring="neg_mean_squared_error", cv=10)
-
-    X_test = np.linspace(0, 1, 100)
-    plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model")
-    plt.plot(X_test, true_fun(X_test), label="True function")
-    plt.scatter(X, y, edgecolor='b', s=20, label="Samples")
-    plt.xlabel("x")
-    plt.ylabel("y")
-    plt.xlim((0, 1))
-    plt.ylim((-2, 2))
-    plt.legend(loc="best")
-    plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
-        degrees[i], -scores.mean(), scores.std()))
-plt.show()
-
-
-
-
-
-
-
# Common imports
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.model_selection import train_test_split
-from sklearn.utils import resample
-from sklearn.metrics import mean_squared_error
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("EoS.csv"),'r')
-
-# Read the EoS data as  csv file and organize the data into two arrays with density and energies
-EoS = pd.read_csv(infile, names=('Density', 'Energy'))
-EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
-EoS = EoS.dropna()
-Energies = EoS['Energy']
-Density = EoS['Density']
-#  The design matrix now as function of various polytrops
-
-Maxpolydegree = 30
-X = np.zeros((len(Density),Maxpolydegree))
-X[:,0] = 1.0
-testerror = np.zeros(Maxpolydegree)
-trainingerror = np.zeros(Maxpolydegree)
-polynomial = np.zeros(Maxpolydegree)
-
-trials = 100
-for polydegree in range(1, Maxpolydegree):
-    polynomial[polydegree] = polydegree
-    for degree in range(polydegree):
-        X[:,degree] = Density**(degree/3.0)
-
-# loop over trials in order to estimate the expectation value of the MSE
-    testerror[polydegree] = 0.0
-    trainingerror[polydegree] = 0.0
-    for samples in range(trials):
-        x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
-        model = LinearRegression(fit_intercept=True).fit(x_train, y_train)
-        ypred = model.predict(x_train)
-        ytilde = model.predict(x_test)
-        testerror[polydegree] += mean_squared_error(y_test, ytilde)
-        trainingerror[polydegree] += mean_squared_error(y_train, ypred) 
-
-    testerror[polydegree] /= trials
-    trainingerror[polydegree] /= trials
-    print("Degree of polynomial: %3d"% polynomial[polydegree])
-    print("Mean squared error on training data: %.8f" % trainingerror[polydegree])
-    print("Mean squared error on test data: %.8f" % testerror[polydegree])
-
-plt.plot(polynomial, np.log10(trainingerror), label='Training Error')
-plt.plot(polynomial, np.log10(testerror), label='Test Error')
-plt.xlabel('Polynomial degree')
-plt.ylabel('log10[MSE]')
-plt.legend()
-plt.show()
-
-
-
-
-
-
-
# Common imports
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.metrics import mean_squared_error
-from sklearn.model_selection import KFold
-from sklearn.model_selection import cross_val_score
-
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("EoS.csv"),'r')
-
-# Read the EoS data as  csv file and organize the data into two arrays with density and energies
-EoS = pd.read_csv(infile, names=('Density', 'Energy'))
-EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
-EoS = EoS.dropna()
-Energies = EoS['Energy']
-Density = EoS['Density']
-#  The design matrix now as function of various polytrops
-
-Maxpolydegree = 30
-X = np.zeros((len(Density),Maxpolydegree))
-X[:,0] = 1.0
-estimated_mse_sklearn = np.zeros(Maxpolydegree)
-polynomial = np.zeros(Maxpolydegree)
-k =5
-kfold = KFold(n_splits = k)
-
-for polydegree in range(1, Maxpolydegree):
-    polynomial[polydegree] = polydegree
-    for degree in range(polydegree):
-        X[:,degree] = Density**(degree/3.0)
-        OLS = LinearRegression()
-# loop over trials in order to estimate the expectation value of the MSE
-    estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)
-#[:, np.newaxis]
-    estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)
-
-plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')
-plt.xlabel('Polynomial degree')
-plt.ylabel('log10[MSE]')
-plt.legend()
-plt.show()
-
-
-
-
-
-
-
import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.model_selection import KFold
-from sklearn.linear_model import Ridge
-from sklearn.model_selection import cross_val_score
-from sklearn.preprocessing import PolynomialFeatures
-
-# A seed just to ensure that the random numbers are the same for every run.
-np.random.seed(3155)
-# Generate the data.
-n = 100
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-# Decide degree on polynomial to fit
-poly = PolynomialFeatures(degree = 10)
-
-# Decide which values of lambda to use
-nlambdas = 500
-lambdas = np.logspace(-3, 5, nlambdas)
-# Initialize a KFold instance
-k = 5
-kfold = KFold(n_splits = k)
-estimated_mse_sklearn = np.zeros(nlambdas)
-i = 0
-for lmb in lambdas:
-    ridge = Ridge(alpha = lmb)
-    estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)
-    estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)
-    i += 1
-plt.figure()
-plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')
-plt.xlabel('log10(lambda)')
-plt.ylabel('MSE')
-plt.legend()
-plt.show()
-
-
-
-
-
-
- - - - -
- - -
- - 1. Linear Regression, basic Elements - 3. Ridge and Lasso Regression - -
- -
-
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-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

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- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter3.html b/doc/LectureNotes/_build/html/chapter3.html deleted file mode 100644 index 5f58e6c57..000000000 --- a/doc/LectureNotes/_build/html/chapter3.html +++ /dev/null @@ -1,1175 +0,0 @@ - - - - - - - - 3. Ridge and Lasso Regression — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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3. Ridge and Lasso Regression

-

Video of Lecture

-
-

3.1. The singular value decomposition

-

The examples we have looked at so far are cases where we normally can -invert the matrix \(\boldsymbol{X}^T\boldsymbol{X}\). Using a polynomial expansion as we -did both for the masses and the fitting of the equation of state, -leads to row vectors of the design matrix which are essentially -orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition.

-

This may -however not the be case in general and a standard matrix inversion -algorithm based on say LU, QR or Cholesky decomposition may lead to singularities. We will see examples of this below.

-

There is however a way to partially circumvent this problem and also gain some insights about the ordinary least squares approach, and later shrinkage methods like Ridge and Lasso regressions.

-

This is given by the Singular Value Decomposition algorithm, perhaps -the most powerful linear algebra algorithm. Let us look at a -different example where we may have problems with the standard matrix -inversion algorithm. Thereafter we dive into the math of the SVD.

-

One of the typical problems we encounter with linear regression, in particular -when the matrix \(\boldsymbol{X}\) (our so-called design matrix) is high-dimensional, -are problems with near singular or singular matrices. The column vectors of \(\boldsymbol{X}\) -may be linearly dependent, normally referred to as super-collinearity.
-This means that the matrix may be rank deficient and it is basically impossible to -to model the data using linear regression. As an example, consider the matrix

-
-\[\begin{split} -\begin{align*} -\mathbf{X} & = \left[ -\begin{array}{rrr} -1 & -1 & 2 -\\ -1 & 0 & 1 -\\ -1 & 2 & -1 -\\ -1 & 1 & 0 -\end{array} \right] -\end{align*} -\end{split}\]
-

The columns of \(\boldsymbol{X}\) are linearly dependent. We see this easily since the -the first column is the row-wise sum of the other two columns. The rank (more correct, -the column rank) of a matrix is the dimension of the space spanned by the -column vectors. Hence, the rank of \(\mathbf{X}\) is equal to the number -of linearly independent columns. In this particular case the matrix has rank 2.

-

Super-collinearity of an \((n \times p)\)-dimensional design matrix \(\mathbf{X}\) implies -that the inverse of the matrix \(\boldsymbol{X}^T\boldsymbol{X}\) (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this

-
-\[\begin{split} -\begin{align*} -\boldsymbol{X} & = \left[ -\begin{array}{rr} -1 & -1 -\\ -1 & -1 -\end{array} \right]. -\end{align*} -\end{split}\]
-

We see easily that \(\mbox{det}(\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0\). Hence, \(\mathbf{X}\) is singular and its inverse is undefined. -This is equivalent to saying that the matrix \(\boldsymbol{X}\) has at least an eigenvalue which is zero.

-

If our design matrix \(\boldsymbol{X}\) which enters the linear regression problem

- -
-
-\[ -\begin{equation} -\boldsymbol{\beta} = (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y}, -\label{_auto1} \tag{1} -\end{equation} -\]
-

has linearly dependent column vectors, we will not be able to compute the inverse -of \(\boldsymbol{X}^T\boldsymbol{X}\) and we cannot find the parameters (estimators) \(\beta_i\). -The estimators are only well-defined if \((\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\) exits. -This is more likely to happen when the matrix \(\boldsymbol{X}\) is high-dimensional. In this case it is likely to encounter a situation where -the regression parameters \(\beta_i\) cannot be estimated.

-

A cheap ad hoc approach is simply to add a small diagonal component to the matrix to invert, that is we change

-
-\[ -\boldsymbol{X}^{T} \boldsymbol{X} \rightarrow \boldsymbol{X}^{T} \boldsymbol{X}+\lambda \boldsymbol{I}, -\]
-

where \(\boldsymbol{I}\) is the identity matrix. When we discuss Ridge regression this is actually what we end up evaluating. The parameter \(\lambda\) is called a hyperparameter. More about this later.

-

From standard linear algebra we know that a square matrix \(\boldsymbol{X}\) can be diagonalized if and only it is -a so-called normal matrix, that is if \(\boldsymbol{X}\in {\mathbb{R}}^{n\times n}\) -we have \(\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X}\) or if \(\boldsymbol{X}\in {\mathbb{C}}^{n\times n}\) we have \(\boldsymbol{X}\boldsymbol{X}^{\dagger}=\boldsymbol{X}^{\dagger}\boldsymbol{X}\). -The matrix has then a set of eigenpairs

-
-\[ -(\lambda_1,\boldsymbol{u}_1),\dots, (\lambda_n,\boldsymbol{u}_n), -\]
-

and the eigenvalues are given by the diagonal matrix

-
-\[ -\boldsymbol{\Sigma}=\mathrm{Diag}(\lambda_1, \dots,\lambda_n). -\]
-

The matrix \(\boldsymbol{X}\) can be written in terms of an orthogonal/unitary transformation \(\boldsymbol{U}\)

-
-\[ -\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, -\]
-

with \(\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I}\) or \(\boldsymbol{U}\boldsymbol{U}^{\dagger}=\boldsymbol{I}\).

-

Not all square matrices are diagonalizable. A matrix like the one discussed above

-
-\[\begin{split} -\boldsymbol{X} = \begin{bmatrix} -1& -1 \\ -1& -1\\ -\end{bmatrix} -\end{split}\]
-

is not diagonalizable, it is a so-called defective matrix. It is easy to see that the condition -\(\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X}\) is not fulfilled.

-
-
-

3.2. The SVD, a Fantastic Algorithm

-

However, and this is the strength of the SVD algorithm, any general -matrix \(\boldsymbol{X}\) can be decomposed in terms of a diagonal matrix and -two orthogonal/unitary matrices. The Singular Value Decompostion -(SVD) theorem -states that a general \(m\times n\) matrix \(\boldsymbol{X}\) can be written in -terms of a diagonal matrix \(\boldsymbol{\Sigma}\) of dimensionality \(m\times n\) -and two orthognal matrices \(\boldsymbol{U}\) and \(\boldsymbol{V}\), where the first has -dimensionality \(m \times m\) and the last dimensionality \(n\times n\). -We have then

-
-\[ -\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T -\]
-

As an example, the above defective matrix can be decomposed as

-
-\[\begin{split} -\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, -\end{split}\]
-

with eigenvalues \(\sigma_1=2\) and \(\sigma_2=0\). -The SVD exits always!

-

The SVD -decomposition (singular values) gives eigenvalues -\(\sigma_i\geq\sigma_{i+1}\) for all \(i\) and for dimensions larger than \(i=p\), the -eigenvalues (singular values) are zero.

-

In the general case, where our design matrix \(\boldsymbol{X}\) has dimension -\(n\times p\), the matrix is thus decomposed into an \(n\times n\) -orthogonal matrix \(\boldsymbol{U}\), a \(p\times p\) orthogonal matrix \(\boldsymbol{V}\) -and a diagonal matrix \(\boldsymbol{\Sigma}\) with \(r=\mathrm{min}(n,p)\) -singular values \(\sigma_i\geq 0\) on the main diagonal and zeros filling -the rest of the matrix. There are at most \(p\) singular values -assuming that \(n > p\). In our regression examples for the nuclear -masses and the equation of state this is indeed the case, while for -the Ising model we have \(p > n\). These are often cases that lead to -near singular or singular matrices.

-

The columns of \(\boldsymbol{U}\) are called the left singular vectors while the columns of \(\boldsymbol{V}\) are the right singular vectors.

-
-
-

3.3. Economy-size SVD

-

If we assume that \(n > p\), then our matrix \(\boldsymbol{U}\) has dimension \(n -\times n\). The last \(n-p\) columns of \(\boldsymbol{U}\) become however -irrelevant in our calculations since they are multiplied with the -zeros in \(\boldsymbol{\Sigma}\).

-

The economy-size decomposition removes extra rows or columns of zeros -from the diagonal matrix of singular values, \(\boldsymbol{\Sigma}\), along with the columns -in either \(\boldsymbol{U}\) or \(\boldsymbol{V}\) that multiply those zeros in the expression. -Removing these zeros and columns can improve execution time -and reduce storage requirements without compromising the accuracy of -the decomposition.

-

If \(n > p\), we keep only the first \(p\) columns of \(\boldsymbol{U}\) and \(\boldsymbol{\Sigma}\) has dimension \(p\times p\). -If \(p > n\), then only the first \(n\) columns of \(\boldsymbol{V}\) are computed and \(\boldsymbol{\Sigma}\) has dimension \(n\times n\). -The \(n=p\) case is obvious, we retain the full SVD. -In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy.

-
-
-
import numpy as np
-# SVD inversion
-def SVDinv(A):
-    ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).
-    SVD is numerically more stable than the inversion algorithms provided by
-    numpy and scipy.linalg at the cost of being slower.
-    '''
-    U, s, VT = np.linalg.svd(A)
-#    print('test U')
-#    print( (np.transpose(U) @ U - U @np.transpose(U)))
-#    print('test VT')
-#    print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))
-    print(U)
-    print(s)
-    print(VT)
-
-    D = np.zeros((len(U),len(VT)))
-    for i in range(0,len(VT)):
-        D[i,i]=s[i]
-    UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
-    return np.matmul(V,np.matmul(invD,UT))
-
-
-X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
-print(X)
-A = np.transpose(X) @ X
-print(A)
-# Brute force inversion of super-collinear matrix
-#B = np.linalg.inv(A)
-#print(B)
-C = SVDinv(A)
-print(C)
-
-
-
-
-
[[ 1. -1.  2.]
- [ 1.  0.  1.]
- [ 1.  2. -1.]
- [ 1.  1.  0.]]
-[[ 4.  2.  2.]
- [ 2.  6. -4.]
- [ 2. -4.  6.]]
-[[-1.18404906e-16  8.16496581e-01 -5.77350269e-01]
- [-7.07106781e-01  4.08248290e-01  5.77350269e-01]
- [ 7.07106781e-01  4.08248290e-01  5.77350269e-01]]
-[1.00000000e+01 6.00000000e+00 9.10898112e-32]
-[[ 3.33066907e-17 -7.07106781e-01  7.07106781e-01]
- [ 8.16496581e-01  4.08248290e-01  4.08248290e-01]
- [ 5.77350269e-01 -5.77350269e-01 -5.77350269e-01]]
-[[-3.65939208e+30  3.65939208e+30  3.65939208e+30]
- [ 3.65939208e+30 -3.65939208e+30 -3.65939208e+30]
- [ 3.65939208e+30 -3.65939208e+30 -3.65939208e+30]]
-
-
-
-
-

The matrix \(\boldsymbol{X}\) has columns that are linearly dependent. The first -column is the row-wise sum of the other two columns. The rank of a -matrix (the column rank) is the dimension of space spanned by the -column vectors. The rank of the matrix is the number of linearly -independent columns, in this case just \(2\). We see this from the -singular values when running the above code. Running the standard -inversion algorithm for matrix inversion with \(\boldsymbol{X}^T\boldsymbol{X}\) results -in the program terminating due to a singular matrix.

-

There are several interesting mathematical properties which will be -relevant when we are going to discuss the differences between say -ordinary least squares (OLS) and Ridge regression.

-

We have from OLS that the parameters of the linear approximation are given by

-
-\[ -\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. -\]
-

The matrix to invert can be rewritten in terms of our SVD decomposition as

-
-\[ -\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T. -\]
-

Using the orthogonality properties of \(\boldsymbol{U}\) we have

-
-\[ -\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T = \boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T, -\]
-

with \(\boldsymbol{D}\) being a diagonal matrix with values along the diagonal given by the singular values squared.

-

This means that

-
-\[ -(\boldsymbol{X}^T\boldsymbol{X})\boldsymbol{V} = \boldsymbol{V}\boldsymbol{D}, -\]
-

that is the eigenvectors of \((\boldsymbol{X}^T\boldsymbol{X})\) are given by the columns of the right singular matrix of \(\boldsymbol{X}\) and the eigenvalues are the squared singular values. It is easy to show (show this) that

-
-\[ -(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}, -\]
-

that is, the eigenvectors of \((\boldsymbol{X}\boldsymbol{X})^T\) are the columns of the left singular matrix and the eigenvalues are the same.

-

Going back to our OLS equation we have

-
-\[ -\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}. -\]
-

We will come back to this expression when we discuss Ridge regression.

-

$\( \tilde{y}^{OLS}=\boldsymbol{X}\hat{\beta}^{OLS}=\sum_{j=1}^p \boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}\)$ and for Ridge we have

-

$\( \tilde{y}^{Ridge}=\boldsymbol{X}\hat{\beta}^{Ridge}=\sum_{j=1}^p \boldsymbol{u}_j\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{u}_j^T\boldsymbol{y}\)$ .

-

It is indeed the economy-sized SVD, note the summation runs up tp $\(p\)\( only and not \)\(n\)$.

-

Here we have that $\(\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\)\(, with \)\(\Sigma\)\( being an \)\( n\times p\)\( matrix and \)\(\boldsymbol{V}\)\( being a \)\( p\times p\)\( matrix. We also have assumed here that \)\( n > p\)$.

-
-
-

3.4. Ridge and LASSO Regression

-

Video of Lecture

-

Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is -our optimization problem is

-
-\[ -{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. -\]
-

or we can state it as

-
-\[ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2, -\]
-

where we have used the definition of a norm-2 vector, that is

-
-\[ -\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. -\]
-

By minimizing the above equation with respect to the parameters -\(\boldsymbol{\beta}\) we could then obtain an analytical expression for the -parameters \(\boldsymbol{\beta}\). We can add a regularization parameter \(\lambda\) by -defining a new cost function to be optimized, that is

-
-\[ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2 -\]
-

which leads to the Ridge regression minimization problem where we -require that \(\vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t\), where \(t\) is -a finite number larger than zero. By defining

-
-\[ -C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1, -\]
-

we have a new optimization equation

-
-\[ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1 -\]
-

which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator.

-

Here we have defined the norm-1 as

-
-\[ -\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert. -\]
-

Using the matrix-vector expression for Ridge regression,

-
-\[ -C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta}, -\]
-

by taking the derivatives with respect to \(\boldsymbol{\beta}\) we obtain then -a slightly modified matrix inversion problem which for finite values -of \(\lambda\) does not suffer from singularity problems. We obtain

-
-\[ -\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, -\]
-

with \(\boldsymbol{I}\) being a \(p\times p\) identity matrix with the constraint that

-
-\[ -\sum_{i=0}^{p-1} \beta_i^2 \leq t, -\]
-

with \(t\) a finite positive number.

-

We see that Ridge regression is nothing but the standard -OLS with a modified diagonal term added to \(\boldsymbol{X}^T\boldsymbol{X}\). The -consequences, in particular for our discussion of the bias-variance tradeoff -are rather interesting.

-

Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had

-
-\[ -(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}. -\]
-

We can analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix \(\boldsymbol{U}\) as

-
-\[ -\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y} -\]
-

For Ridge regression this becomes

-
-\[ -\boldsymbol{X}\boldsymbol{\beta}^{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y}, -\]
-

with the vectors \(\boldsymbol{u}_j\) being the columns of \(\boldsymbol{U}\).

-

Since \(\lambda \geq 0\), it means that compared to OLS, we have

-
-\[ -\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1. -\]
-

Ridge regression finds the coordinates of \(\boldsymbol{y}\) with respect to the -orthonormal basis \(\boldsymbol{U}\), it then shrinks the coordinates by -\(\frac{\sigma_j^2}{\sigma_j^2+\lambda}\). Recall that the SVD has -eigenvalues ordered in a descending way, that is \(\sigma_i \geq -\sigma_{i+1}\).

-

For small eigenvalues \(\sigma_i\) it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. -Actually, calculating the variance of \(\boldsymbol{X}\boldsymbol{v}_j\) shows that this quantity is equal to \(\sigma_j^2/n\). -With a parameter \(\lambda\) we can thus shrink the role of specific parameters.

-

For the sake of simplicity, let us assume that the design matrix is orthonormal, that is

-
-\[ -\boldsymbol{X}^T\boldsymbol{X}=(\boldsymbol{X}^T\boldsymbol{X})^{-1} =\boldsymbol{I}. -\]
-

In this case the standard OLS results in

-
-\[ -\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}, -\]
-

and

-
-\[ -\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{I}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\left(1+\lambda\right)^{-1}\boldsymbol{\beta}^{\mathrm{OLS}}, -\]
-

that is the Ridge estimator scales the OLS estimator by the inverse of a factor \(1+\lambda\), and -the Ridge estimator converges to zero when the hyperparameter goes to -infinity.

-

We will come back to more interpreations after we have gone through some of the statistical analysis part.

-

For more discussions of Ridge and Lasso regression, Wessel van Wieringen’s article is highly recommended. -Similarly, Mehta et al’s article is also recommended.

-
-
-

3.5. A better understanding of regularization

-

The parameter \(\lambda\) that we have introduced in the Ridge (and -Lasso as well) regression is often called a regularization parameter -or shrinkage parameter. It is common to call it a hyperparameter. What does it mean mathemtically?

-

Here we will first look at how to analyze the difference between the -standard OLS equations and the Ridge expressions in terms of a linear -algebra analysis using the SVD algorithm. Thereafter, we will link -(see the material on the bias-variance tradeoff below) these -observation to the statisical analysis of the results. In particular -we consider how the variance of the parameters \(\boldsymbol{\beta}\) is -affected by changing the parameter \(\lambda\).

-

We have our design matrix -\(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\). With the SVD we decompose it as

-
-\[ -\boldsymbol{X} = \boldsymbol{U\Sigma V^T}, -\]
-

with \(\boldsymbol{U}\in {\mathbb{R}}^{n\times n}\), \(\boldsymbol{\Sigma}\in {\mathbb{R}}^{n\times p}\) -and \(\boldsymbol{V}\in {\mathbb{R}}^{p\times p}\).

-

The matrices \(\boldsymbol{U}\) and \(\boldsymbol{V}\) are unitary/orthonormal matrices, that is in case the matrices are real we have \(\boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I}\) and \(\boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I}\).

-
-
-

3.6. Introducing the Covariance and Correlation functions

-

Before we discuss the link between for example Ridge regression and the singular value decomposition, we need to remind ourselves about -the definition of the covariance and the correlation function. These are quantities

-

Suppose we have defined two vectors -\(\hat{x}\) and \(\hat{y}\) with \(n\) elements each. The covariance matrix \(\boldsymbol{C}\) is defined as

-
-\[\begin{split} -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ - \end{bmatrix}, -\end{split}\]
-

where for example

-
-\[ -\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -\]
-

With this definition and recalling that the variance is defined as

-
-\[ -\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, -\]
-

we can rewrite the covariance matrix as

-
-\[\begin{split} -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ - \end{bmatrix}. -\end{split}\]
-

The covariance takes values between zero and infinity and may thus -lead to problems with loss of numerical precision for particularly -large values. It is common to scale the covariance matrix by -introducing instead the correlation matrix defined via the so-called -correlation function

-
-\[ -\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. -\]
-

The correlation function is then given by values \(\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] -\in [-1,1]\). This avoids eventual problems with too large values. We -can then define the correlation matrix for the two vectors \(\boldsymbol{x}\) -and \(\boldsymbol{y}\) as

-
-\[\begin{split} -\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\ - \end{bmatrix}, -\end{split}\]
-

In the above example this is the function we constructed using pandas.

-

In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression -we defined the design/feature matrix \(\boldsymbol{X}\) as

-
-\[\begin{split} -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -\end{split}\]
-

with \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\), with the predictors/features \(p\) refering to the column numbers and the -entries \(n\) being the row elements. -We can rewrite the design/feature matrix in terms of its column vectors as

-
-\[ -\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, -\]
-

with a given vector

-
-\[ -\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. -\]
-

With these definitions, we can now rewrite our \(2\times 2\) -correaltion/covariance matrix in terms of a moe general design/feature -matrix \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\). This leads to a \(p\times p\) -covariance matrix for the vectors \(\boldsymbol{x}_i\) with \(i=0,1,\dots,p-1\)

-
-\[\begin{split} -\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} -\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ -\end{bmatrix}, -\end{split}\]
-

and the correlation matrix

-
-\[\begin{split} -\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} -1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ -\end{bmatrix}, -\end{split}\]
-

The Numpy function np.cov calculates the covariance elements using -the factor \(1/(n-1)\) instead of \(1/n\) since it assumes we do not have -the exact mean values. The following simple function uses the -np.vstack function which takes each vector of dimension \(1\times n\) -and produces a \(2\times n\) matrix \(\boldsymbol{W}\)

-
-\[\begin{split} -\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ - x_1 & y_1 \\ - x_2 & y_2\\ - \dots & \dots \\ - x_{n-2} & y_{n-2}\\ - x_{n-1} & y_{n-1} & - \end{bmatrix}, -\end{split}\]
-

which in turn is converted into into the \(2\times 2\) covariance matrix -\(\boldsymbol{C}\) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \(\boldsymbol{x}\) etc using the Numpy -function np.mean(x). We can also extract the eigenvalues of the -covariance matrix through the np.linalg.eig() function.

-
-
-
# Importing various packages
-import numpy as np
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-W = np.vstack((x, y))
-C = np.cov(W)
-print(C)
-
-
-
-
-
0.062127739929490035
-4.226441441217558
-[[0.95977893 2.78652875]
- [2.78652875 9.09409124]]
-
-
-
-
-

The previous example can be converted into the correlation matrix by -simply scaling the matrix elements with the variances. We should also -subtract the mean values for each column. This leads to the following -code which sets up the correlations matrix for the previous example in -a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \(2\times 2\) correlation matrix (since we have only two vectors).

-
-
-
import numpy as np
-n = 100
-# define two vectors                                                                                           
-x = np.random.random(size=n)
-y = 4+3*x+np.random.normal(size=n)
-#scaling the x and y vectors                                                                                   
-x = x - np.mean(x)
-y = y - np.mean(y)
-variance_x = np.sum(x@x)/n
-variance_y = np.sum(y@y)/n
-print(variance_x)
-print(variance_y)
-cov_xy = np.sum(x@y)/n
-cov_xx = np.sum(x@x)/n
-cov_yy = np.sum(y@y)/n
-C = np.zeros((2,2))
-C[0,0]= cov_xx/variance_x
-C[1,1]= cov_yy/variance_y
-C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
-C[1,0]= C[0,1]
-print(C)
-
-
-
-
-
0.09053391104887817
-1.9755272664385481
-[[1.         0.64723729]
- [0.64723729 1.        ]]
-
-
-
-
-

We see that the matrix elements along the diagonal are one as they -should be and that the matrix is symmetric. Furthermore, diagonalizing -this matrix we easily see that it is a positive definite matrix.

-

The above procedure with numpy can be made more compact if we use pandas.

-

We whow here how we can set up the correlation matrix using pandas, as done in this simple code

-
-
-
import numpy as np
-import pandas as pd
-n = 10
-x = np.random.normal(size=n)
-x = x - np.mean(x)
-y = 4+3*x+np.random.normal(size=n)
-y = y - np.mean(y)
-X = (np.vstack((x, y))).T
-print(X)
-Xpd = pd.DataFrame(X)
-print(Xpd)
-correlation_matrix = Xpd.corr()
-print(correlation_matrix)
-
-
-
-
-
[[ 0.52374318  0.7528421 ]
- [-1.07892554 -3.5697027 ]
- [ 0.65536057  2.84854   ]
- [-0.9936011  -1.75368597]
- [-0.22233456 -1.63866932]
- [ 1.10799046  4.51410028]
- [ 1.30401938  5.04686521]
- [ 0.70055962  1.28566384]
- [-1.69423925 -6.23684061]
- [-0.30257276 -1.24911284]]
-          0         1
-0  0.523743  0.752842
-1 -1.078926 -3.569703
-2  0.655361  2.848540
-3 -0.993601 -1.753686
-4 -0.222335 -1.638669
-5  1.107990  4.514100
-6  1.304019  5.046865
-7  0.700560  1.285664
-8 -1.694239 -6.236841
-9 -0.302573 -1.249113
-          0         1
-0  1.000000  0.967871
-1  0.967871  1.000000
-
-
-
-
-

We expand this model to the Franke function discussed above.

-
-
-
# Common imports
-import numpy as np
-import pandas as pd
-
-
-def FrankeFunction(x,y):
-	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
-	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
-	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
-	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
-	return term1 + term2 + term3 + term4
-
-
-def create_X(x, y, n ):
-	if len(x.shape) > 1:
-		x = np.ravel(x)
-		y = np.ravel(y)
-
-	N = len(x)
-	l = int((n+1)*(n+2)/2)		# Number of elements in beta
-	X = np.ones((N,l))
-
-	for i in range(1,n+1):
-		q = int((i)*(i+1)/2)
-		for k in range(i+1):
-			X[:,q+k] = (x**(i-k))*(y**k)
-
-	return X
-
-
-# Making meshgrid of datapoints and compute Franke's function
-n = 4
-N = 100
-x = np.sort(np.random.uniform(0, 1, N))
-y = np.sort(np.random.uniform(0, 1, N))
-z = FrankeFunction(x, y)
-X = create_X(x, y, n=n)    
-
-Xpd = pd.DataFrame(X)
-# subtract the mean values and set up the covariance matrix
-Xpd = Xpd - Xpd.mean()
-covariance_matrix = Xpd.cov()
-print(covariance_matrix)
-
-
-
-
-
     0         1         2         3         4         5         6         7   \
-0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   
-1   0.0  0.076075  0.081429  0.075275  0.076780  0.077999  0.067453  0.067971   
-2   0.0  0.081429  0.088214  0.081300  0.083371  0.085063  0.072811  0.073594   
-3   0.0  0.075275  0.081300  0.080335  0.082127  0.083567  0.075400  0.075990   
-4   0.0  0.076780  0.083371  0.082127  0.084184  0.085857  0.076996  0.077729   
-5   0.0  0.077999  0.085063  0.083567  0.085857  0.087738  0.078264  0.079128   
-6   0.0  0.067453  0.072811  0.075400  0.076996  0.078264  0.072961  0.073444   
-7   0.0  0.067971  0.073594  0.075990  0.077729  0.079128  0.073444  0.074016   
-8   0.0  0.068431  0.074291  0.076495  0.078367  0.079889  0.073843  0.074498   
-9   0.0  0.068860  0.074936  0.076947  0.078943  0.080582  0.074186  0.074922   
-10  0.0  0.059693  0.064192  0.068842  0.070144  0.071159  0.068084  0.068427   
-11  0.0  0.059875  0.064519  0.069009  0.070400  0.071499  0.068172  0.068575   
-12  0.0  0.060056  0.064837  0.069164  0.070641  0.071822  0.068246  0.068709   
-13  0.0  0.060243  0.065156  0.069319  0.070878  0.072139  0.068315  0.068837   
-14  0.0  0.060442  0.065483  0.069478  0.071119  0.072459  0.068387  0.068966   
-
-          8         9         10        11        12        13        14  
-0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  
-1   0.068431  0.068860  0.059693  0.059875  0.060056  0.060243  0.060442  
-2   0.074291  0.074936  0.064192  0.064519  0.064837  0.065156  0.065483  
-3   0.076495  0.076947  0.068842  0.069009  0.069164  0.069319  0.069478  
-4   0.078367  0.078943  0.070144  0.070400  0.070641  0.070878  0.071119  
-5   0.079889  0.080582  0.071159  0.071499  0.071822  0.072139  0.072459  
-6   0.073843  0.074186  0.068084  0.068172  0.068246  0.068315  0.068387  
-7   0.074498  0.074922  0.068427  0.068575  0.068709  0.068837  0.068966  
-8   0.075062  0.075564  0.068693  0.068901  0.069093  0.069278  0.069465  
-9   0.075564  0.076143  0.068909  0.069174  0.069423  0.069665  0.069908  
-10  0.068693  0.068909  0.064578  0.064582  0.064574  0.064559  0.064545  
-11  0.068901  0.069174  0.064582  0.064632  0.064668  0.064698  0.064728  
-12  0.069093  0.069423  0.064574  0.064668  0.064748  0.064822  0.064896  
-13  0.069278  0.069665  0.064559  0.064698  0.064822  0.064940  0.065058  
-14  0.069465  0.069908  0.064545  0.064728  0.064896  0.065058  0.065220  
-
-
-
-
-

We note here that the covariance is zero for the first rows and -columns since all matrix elements in the design matrix were set to one -(we are fitting the function in terms of a polynomial of degree \(n\)).

-

This means that the variance for these elements will be zero and will -cause problems when we set up the correlation matrix. We can simply -drop these elements and construct a correlation -matrix without these elements.

-

We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \(\boldsymbol{X}\) as

-
-\[ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}]. -\]
-

To see this let us simply look at a design matrix \(\boldsymbol{X}\in {\mathbb{R}}^{2\times 2}\)

-
-\[\begin{split} -\boldsymbol{X}=\begin{bmatrix} -x_{00} & x_{01}\\ -x_{10} & x_{11}\\ -\end{bmatrix}=\begin{bmatrix} -\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\ -\end{bmatrix}. -\end{split}\]
-

If we then compute the expectation value

-
-\[\begin{split} -\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix} -x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ -x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ -\end{bmatrix}, -\end{split}\]
-

which is just

-
-\[\begin{split} -\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\ - \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\ - \end{bmatrix}, -\end{split}\]
-

where we wrote $\(\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]\)\( to indicate that this the covariance of the vectors \)\boldsymbol{x}\( of the design/feature matrix \)\boldsymbol{X}$.

-

It is easy to generalize this to a matrix \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\).

-
-
-

3.7. Linking with SVD

-
-
- - - - -
- - -
- - 2. Resampling Methods - 4. Logistic Regression - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

-
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-
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- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter4.html b/doc/LectureNotes/_build/html/chapter4.html deleted file mode 100644 index d0e056c7f..000000000 --- a/doc/LectureNotes/_build/html/chapter4.html +++ /dev/null @@ -1,2292 +0,0 @@ - - - - - - - - 4. Logistic Regression — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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4. Logistic Regression

-

Video of Lecture

-
-

4.1. Logistic Regression

-

In linear regression our main interest was centered on learning the -coefficients of a functional fit (say a polynomial) in order to be -able to predict the response of a continuous variable on some unseen -data. The fit to the continuous variable \(y_i\) is based on some -independent variables \(\hat{x}_i\). Linear regression resulted in -analytical expressions for standard ordinary Least Squares or Ridge -regression (in terms of matrices to invert) for several quantities, -ranging from the variance and thereby the confidence intervals of the -parameters \(\hat{\beta}\) to the mean squared error. If we can invert -the product of the design matrices, linear regression gives then a -simple recipe for fitting our data.

-

Classification problems, however, are concerned with outcomes taking -the form of discrete variables (i.e. categories). We may for example, -on the basis of DNA sequencing for a number of patients, like to find -out which mutations are important for a certain disease; or based on -scans of various patients’ brains, figure out if there is a tumor or -not; or given a specific physical system, we’d like to identify its -state, say whether it is an ordered or disordered system (typical -situation in solid state physics); or classify the status of a -patient, whether she/he has a stroke or not and many other similar -situations.

-

The most common situation we encounter when we apply logistic -regression is that of two possible outcomes, normally denoted as a -binary outcome, true or false, positive or negative, success or -failure etc.

-

Logistic regression will also serve as our stepping stone towards -neural network algorithms and supervised deep learning. For logistic -learning, the minimization of the cost function leads to a non-linear -equation in the parameters \(\hat{\beta}\). The optimization of the -problem calls therefore for minimization algorithms. This forms the -bottle neck of all machine learning algorithms, namely how to find -reliable minima of a multi-variable function. This leads us to the -family of gradient descent methods. The latter are the working horses -of basically all modern machine learning algorithms.

-

We note also that many of the topics discussed here on logistic -regression are also commonly used in modern supervised Deep Learning -models, as we will see later.

-
-
-

4.2. Basics

-

We consider the case where the dependent variables, also called the -responses or the outcomes, \(y_i\) are discrete and only take values -from \(k=0,\dots,K-1\) (i.e. \(K\) classes).

-

The goal is to predict the -output classes from the design matrix \(\hat{X}\in\mathbb{R}^{n\times p}\) -made of \(n\) samples, each of which carries \(p\) features or predictors. The -primary goal is to identify the classes to which new unseen samples -belong.

-

Let us specialize to the case of two classes only, with outputs -\(y_i=0\) and \(y_i=1\). Our outcomes could represent the status of a -credit card user that could default or not on her/his credit card -debt. That is

-
-\[\begin{split} -y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. -\end{split}\]
-

Before moving to the logistic model, let us try to use our linear -regression model to classify these two outcomes. We could for example -fit a linear model to the default case if \(y_i > 0.5\) and the no -default case \(y_i \leq 0.5\).

-

We would then have our -weighted linear combination, namely

- -
-
-\[ -\begin{equation} -\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, -\label{_auto1} \tag{1} -\end{equation} -\]
-

where \(\hat{y}\) is a vector representing the possible outcomes, \(\hat{X}\) is our -\(n\times p\) design matrix and \(\hat{\beta}\) represents our estimators/predictors.

-

The main problem with our function is that it takes values on the -entire real axis. In the case of logistic regression, however, the -labels \(y_i\) are discrete variables. A typical example is the credit -card data discussed below here, where we can set the state of -defaulting the debt to \(y_i=1\) and not to \(y_i=0\) for one the persons -in the data set (see the full example below).

-

One simple way to get a discrete output is to have sign -functions that map the output of a linear regressor to values \(\{0,1\}\), -\(f(s_i)=sign(s_i)=1\) if \(s_i\ge 0\) and 0 if otherwise. -We will encounter this model in our first demonstration of neural networks. Historically it is called the perceptron" model in the machine learning literature. This model is extremely simple. However, in many cases it is more favorable to use a soft” classifier that outputs -the probability of a given category. This leads us to the logistic function.

-

The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person’s against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful.

-
-
-
%matplotlib inline
-
-# Common imports
-import os
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.linear_model import LinearRegression, Ridge, Lasso
-from sklearn.model_selection import train_test_split
-from sklearn.utils import resample
-from sklearn.metrics import mean_squared_error
-from IPython.display import display
-from pylab import plt, mpl
-plt.style.use('seaborn')
-mpl.rcParams['font.family'] = 'serif'
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("chddata.csv"),'r')
-
-# Read the chd data as  csv file and organize the data into arrays with age group, age, and chd
-chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))
-chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']
-output = chd['CHD']
-age = chd['Age']
-agegroup = chd['Agegroup']
-numberID  = chd['ID'] 
-display(chd)
-
-plt.scatter(age, output, marker='o')
-plt.axis([18,70.0,-0.1, 1.2])
-plt.xlabel(r'Age')
-plt.ylabel(r'CHD')
-plt.title(r'Age distribution and Coronary heart disease')
-plt.show()
-
-
-
-
-
---------------------------------------------------------------------------
-FileNotFoundError                         Traceback (most recent call last)
-<ipython-input-1-a77d5ac269b2> in <module>
-     38     plt.savefig(image_path(fig_id) + ".png", format='png')
-     39 
----> 40 infile = open(data_path("chddata.csv"),'r')
-     41 
-     42 # Read the chd data as  csv file and organize the data into arrays with age group, age, and chd
-
-FileNotFoundError: [Errno 2] No such file or directory: 'DataFiles/chddata.csv'
-
-
-
-
-

What we could attempt however is to plot the mean value for each group.

-
-
-
agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])
-group = np.array([1, 2, 3, 4, 5, 6, 7, 8])
-plt.plot(group, agegroupmean, "r-")
-plt.axis([0,9,0, 1.0])
-plt.xlabel(r'Age group')
-plt.ylabel(r'CHD mean values')
-plt.title(r'Mean values for each age group')
-plt.show()
-
-
-
-
-

We are now trying to find a function \(f(y\vert x)\), that is a function which gives us an expected value for the output \(y\) with a given input \(x\). -In standard linear regression with a linear dependence on \(x\), we would write this in terms of our model

-
-\[ -f(y_i\vert x_i)=\beta_0+\beta_1 x_i. -\]
-

This expression implies however that \(f(y_i\vert x_i)\) could take any -value from minus infinity to plus infinity. If we however let -\(f(y\vert y)\) be represented by the mean value, the above example -shows us that we can constrain the function to take values between -zero and one, that is we have \(0 \le f(y_i\vert x_i) \le 1\). Looking -at our last curve we see also that it has an S-shaped form. This leads -us to a very popular model for the function \(f\), namely the so-called -Sigmoid function or logistic model. We will consider this function as -representing the probability for finding a value of \(y_i\) with a given -\(x_i\).

-
-
-

4.3. The logistic function

-

Another widely studied model, is the so-called -perceptron model, which is an example of a “hard classification” model. We -will encounter this model when we discuss neural networks as -well. Each datapoint is deterministically assigned to a category (i.e -\(y_i=0\) or \(y_i=1\)). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a “soft” -classifier that outputs the probability of a given category rather -than a single value. For example, given \(x_i\), the classifier -outputs the probability of being in a category \(k\). Logistic regression -is the most common example of a so-called soft classifier. In logistic -regression, the probability that a data point \(x_i\) -belongs to a category \(y_i=\{0,1\}\) is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,

-
-\[ -p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. -\]
-

Note that \(1-p(t)= p(-t)\).

-
-
-

4.4. Examples of likelihood functions used in logistic regression and nueral networks

-

The following code plots the logistic function, the step function and other functions we will encounter from here and on.

-
-
-
"""The sigmoid function (or the logistic curve) is a
-function that takes any real number, z, and outputs a number (0,1).
-It is useful in neural networks for assigning weights on a relative scale.
-The value z is the weighted sum of parameters involved in the learning algorithm."""
-
-import numpy
-import matplotlib.pyplot as plt
-import math as mt
-
-z = numpy.arange(-5, 5, .1)
-sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
-sigma = sigma_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, sigma)
-ax.set_ylim([-0.1, 1.1])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sigmoid function')
-
-plt.show()
-
-"""Step Function"""
-z = numpy.arange(-5, 5, .02)
-step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
-step = step_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, step)
-ax.set_ylim([-0.5, 1.5])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('step function')
-
-plt.show()
-
-"""tanh Function"""
-z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
-t = numpy.tanh(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, t)
-ax.set_ylim([-1.0, 1.0])
-ax.set_xlim([-2*mt.pi,2*mt.pi])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('tanh function')
-
-plt.show()
-
-
-
-
-

We assume now that we have two classes with \(y_i\) either \(0\) or \(1\). Furthermore we assume also that we have only two parameters \(\beta\) in our fitting of the Sigmoid function, that is we define probabilities

-
-\[\begin{split} -\begin{align*} -p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ -p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), -\end{align*} -\end{split}\]
-

where \(\hat{\beta}\) are the weights we wish to extract from data, in our case \(\beta_0\) and \(\beta_1\).

-

Note that we used

-
-\[ -p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}). -\]
-

In order to define the total likelihood for all possible outcomes from a
-dataset \(\mathcal{D}=\{(y_i,x_i)\}\), with the binary labels -\(y_i\in\{0,1\}\) and where the data points are drawn independently, we use the so-called Maximum Likelihood Estimation (MLE) principle. -We aim thus at maximizing -the probability of seeing the observed data. We can then approximate the -likelihood in terms of the product of the individual probabilities of a specific outcome \(y_i\), that is

-
-\[\begin{split} -\begin{align*} -P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ -\end{align*} -\end{split}\]
-

from which we obtain the log-likelihood and our cost/loss function

-
-\[ -\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right). -\]
-

Reordering the logarithms, we can rewrite the cost/loss function as

-
-\[ -\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). -\]
-

The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to \(\beta\). -Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that

-
-\[ -\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). -\]
-

This equation is known in statistics as the cross entropy. Finally, we note that just as in linear regression, -in practice we often supplement the cross-entropy with additional regularization terms, usually \(L_1\) and \(L_2\) regularization as we did for Ridge and Lasso regression.

-

The cross entropy is a convex function of the weights \(\hat{\beta}\) and, -therefore, any local minimizer is a global minimizer.

-

Minimizing this -cost function with respect to the two parameters \(\beta_0\) and \(\beta_1\) we obtain

-
-\[ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), -\]
-

and

-
-\[ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). -\]
-

Let us now define a vector \(\hat{y}\) with \(n\) elements \(y_i\), an -\(n\times p\) matrix \(\hat{X}\) which contains the \(x_i\) values and a -vector \(\hat{p}\) of fitted probabilities \(p(y_i\vert x_i,\hat{\beta})\). We can rewrite in a more compact form the first -derivative of cost function as

-
-\[ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). -\]
-

If we in addition define a diagonal matrix \(\hat{W}\) with elements -\(p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta})\), we can obtain a compact expression of the second derivative as

-
-\[ -\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. -\]
-

Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with \(p\) predictors

-
-\[ -\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. -\]
-

Here we defined \(\hat{x}=[1,x_1,x_2,\dots,x_p]\) and \(\hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p]\) leading to

-
-\[ -p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. -\]
-

Till now we have mainly focused on two classes, the so-called binary -system. Suppose we wish to extend to \(K\) classes. Let us for the sake -of simplicity assume we have only two predictors. We have then following model

-
-\[ -\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, -\]
-

and

-
-\[ -\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, -\]
-

and so on till the class \(C=K-1\) class

-
-\[ -\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, -\]
-

and the model is specified in term of \(K-1\) so-called log-odds or -logit transformations.

-

In our discussion of neural networks we will encounter the above again -in terms of a slightly modified function, the so-called Softmax function.

-

The softmax function is used in various multiclass classification -methods, such as multinomial logistic regression (also known as -softmax regression), multiclass linear discriminant analysis, naive -Bayes classifiers, and artificial neural networks. Specifically, in -multinomial logistic regression and linear discriminant analysis, the -input to the function is the result of \(K\) distinct linear functions, -and the predicted probability for the \(k\)-th class given a sample -vector \(\hat{x}\) and a weighting vector \(\hat{\beta}\) is (with two -predictors):

-
-\[ -p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. -\]
-

It is easy to extend to more predictors. The final class is

-
-\[ -p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, -\]
-

and they sum to one. Our earlier discussions were all specialized to -the case with two classes only. It is easy to see from the above that -what we derived earlier is compatible with these equations.

-

To find the optimal parameters we would typically use a gradient -descent method. Newton’s method and gradient descent methods are -discussed in the material on optimization -methods.

-
-
-

4.5. Wisconsin Cancer Data

-

We show here how we can use a simple regression case on the breast -cancer data using Logistic regression as our algorithm for -classification.

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-# Logistic Regression
-logreg = LogisticRegression(solver='lbfgs')
-logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Logistic Regression
-logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-
-
-
-
-

In addition to the above scores, we could also study the covariance (and the correlation matrix). -We use Pandas to compute the correlation matrix.

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-cancer = load_breast_cancer()
-import pandas as pd
-# Making a data frame
-cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
-
-fig, axes = plt.subplots(15,2,figsize=(10,20))
-malignant = cancer.data[cancer.target == 0]
-benign = cancer.data[cancer.target == 1]
-ax = axes.ravel()
-
-for i in range(30):
-    _, bins = np.histogram(cancer.data[:,i], bins =50)
-    ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5)
-    ax[i].hist(benign[:,i], bins = bins, alpha = 0.5)
-    ax[i].set_title(cancer.feature_names[i])
-    ax[i].set_yticks(())
-ax[0].set_xlabel("Feature magnitude")
-ax[0].set_ylabel("Frequency")
-ax[0].legend(["Malignant", "Benign"], loc ="best")
-fig.tight_layout()
-plt.show()
-
-import seaborn as sns
-correlation_matrix = cancerpd.corr().round(1)
-# use the heatmap function from seaborn to plot the correlation matrix
-# annot = True to print the values inside the square
-plt.figure(figsize=(15,8))
-sns.heatmap(data=correlation_matrix, annot=True)
-plt.show()
-
-
-
-
-

In the above example we note two things. In the first plot we display -the overlap of benign and malignant tumors as functions of the various -features in the Wisconsing breast cancer data set. We see that for -some of the features we can distinguish clearly the benign and -malignant cases while for other features we cannot. This can point to -us which features may be of greater interest when we wish to classify -a benign or not benign tumour.

-

In the second figure we have computed the so-called correlation -matrix, which in our case with thirty features becomes a \(30\times 30\) -matrix.

-

We constructed this matrix using pandas via the statements

-
-
-
cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
-
-
-
-
-

and then

-
-
-
correlation_matrix = cancerpd.corr().round(1)
-
-
-
-
-

Diagonalizing this matrix we can in turn say something about which -features are of relevance and which are not. This leads us to -the classical Principal Component Analysis (PCA) theorem with -applications. This will be discussed later this semester (week 43).

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-# Logistic Regression
-logreg = LogisticRegression(solver='lbfgs')
-logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Logistic Regression
-logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-
-
-from sklearn.preprocessing import LabelEncoder
-from sklearn.model_selection import cross_validate
-#Cross validation
-accuracy = cross_validate(logreg,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Logistic Regression  and scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-
-
-import scikitplot as skplt
-y_pred = logreg.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-plt.show()
-y_probas = logreg.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-plt.show()
-
-
-
-
-
-
-

4.6. Optimization, the central part of any Machine Learning algortithm

-

Almost every problem in machine learning and data science starts with -a dataset \(X\), a model \(g(\beta)\), which is a function of the -parameters \(\beta\) and a cost function \(C(X, g(\beta))\) that allows -us to judge how well the model \(g(\beta)\) explains the observations -\(X\). The model is fit by finding the values of \(\beta\) that minimize -the cost function. Ideally we would be able to solve for \(\beta\) -analytically, however this is not possible in general and we must use -some approximative/numerical method to compute the minimum.

-
-
-

4.7. Revisiting our Logistic Regression case

-

In our discussion on Logistic Regression we studied the -case of -two classes, with \(y_i\) either -\(0\) or \(1\). Furthermore we assumed also that we have only two -parameters \(\beta\) in our fitting, that is we -defined probabilities

-
-\[\begin{split} -\begin{align*} -p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ -p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}), -\end{align*} -\end{split}\]
-

where \(\boldsymbol{\beta}\) are the weights we wish to extract from data, in our case \(\beta_0\) and \(\beta_1\).

-
-
-

4.8. The equations to solve

-

Our compact equations used a definition of a vector \(\boldsymbol{y}\) with \(n\) -elements \(y_i\), an \(n\times p\) matrix \(\boldsymbol{X}\) which contains the -\(x_i\) values and a vector \(\boldsymbol{p}\) of fitted probabilities -\(p(y_i\vert x_i,\boldsymbol{\beta})\). We rewrote in a more compact form -the first derivative of the cost function as

-
-\[ -\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right). -\]
-

If we in addition define a diagonal matrix \(\boldsymbol{W}\) with elements -\(p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta})\), we can obtain a compact expression of the second derivative as

-
-\[ -\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}. -\]
-

This defines what is called the Hessian matrix.

-
-
-

4.9. Solving using Newton-Raphson’s method

-

If we can set up these equations, Newton-Raphson’s iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives.

-

Our iterative scheme is then given by

-
-\[ -\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T}\right)^{-1}_{\boldsymbol{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}\right)_{\boldsymbol{\beta}^{\mathrm{old}}}, -\]
-

or in matrix form as

-
-\[ -\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X} \right)^{-1}\times \left(-\boldsymbol{X}^T(\boldsymbol{y}-\boldsymbol{p}) \right)_{\boldsymbol{\beta}^{\mathrm{old}}}. -\]
-

The right-hand side is computed with the old values of \(\beta\).

-

If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement.

-
-
-

4.10. Brief reminder on Newton-Raphson’s method

-

Let us quickly remind ourselves how we derive the above method.

-

Perhaps the most celebrated of all one-dimensional root-finding -routines is Newton’s method, also called the Newton-Raphson -method. This method requires the evaluation of both the -function \(f\) and its derivative \(f'\) at arbitrary points. -If you can only calculate the derivative -numerically and/or your function is not of the smooth type, we -normally discourage the use of this method.

-
-
-

4.11. The equations

-

The Newton-Raphson formula consists geometrically of extending the -tangent line at a current point until it crosses zero, then setting -the next guess to the abscissa of that zero-crossing. The mathematics -behind this method is rather simple. Employing a Taylor expansion for -\(x\) sufficiently close to the solution \(s\), we have

- -
-
-\[ -f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots. - \label{eq:taylornr} \tag{2} -\]
-

For small enough values of the function and for well-behaved -functions, the terms beyond linear are unimportant, hence we obtain

-
-\[ -f(x)+(s-x)f'(x)\approx 0, -\]
-

yielding

-
-\[ -s\approx x-\frac{f(x)}{f'(x)}. -\]
-

Having in mind an iterative procedure, it is natural to start iterating with

-
-\[ -x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. -\]
-
-
-

4.12. Simple geometric interpretation

-

The above is Newton-Raphson’s method. It has a simple geometric -interpretation, namely \(x_{n+1}\) is the point where the tangent from -\((x_n,f(x_n))\) crosses the \(x\)-axis. Close to the solution, -Newton-Raphson converges fast to the desired result. However, if we -are far from a root, where the higher-order terms in the series are -important, the Newton-Raphson formula can give grossly inaccurate -results. For instance, the initial guess for the root might be so far -from the true root as to let the search interval include a local -maximum or minimum of the function. If an iteration places a trial -guess near such a local extremum, so that the first derivative nearly -vanishes, then Newton-Raphson may fail totally

-
-
-

4.13. Extending to more than one variable

-

Newton’s method can be generalized to systems of several non-linear equations -and variables. Consider the case with two equations

-
-\[\begin{split} -\begin{array}{cc} f_1(x_1,x_2) &=0\\ - f_2(x_1,x_2) &=0,\end{array} -\end{split}\]
-

which we Taylor expand to obtain

-
-\[\begin{split} -\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1 - \partial f_1/\partial x_1+h_2 - \partial f_1/\partial x_2+\dots\\ - 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1 - \partial f_2/\partial x_1+h_2 - \partial f_2/\partial x_2+\dots - \end{array}. -\end{split}\]
-

Defining the Jacobian matrix \({\bf \boldsymbol{J}}\) we have

-
-\[\begin{split} -{\bf \boldsymbol{J}}=\left( \begin{array}{cc} - \partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\ - \partial f_2/\partial x_1 &\partial f_2/\partial x_2 - \end{array} \right), -\end{split}\]
-

we can rephrase Newton’s method as

-
-\[\begin{split} -\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)= -\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+ -\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right), -\end{split}\]
-

where we have defined

-
-\[\begin{split} -\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)= - -{\bf \boldsymbol{J}}^{-1} - \left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right). -\end{split}\]
-

We need thus to compute the inverse of the Jacobian matrix and it -is to understand that difficulties may -arise in case \({\bf \boldsymbol{J}}\) is nearly singular.

-

It is rather straightforward to extend the above scheme to systems of -more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function.

-
-
-

4.14. Steepest descent

-

The basic idea of gradient descent is -that a function \(F(\mathbf{x})\), -\(\mathbf{x} \equiv (x_1,\cdots,x_n)\), decreases fastest if one goes from \(\bf {x}\) in the -direction of the negative gradient \(-\nabla F(\mathbf{x})\).

-

It can be shown that if

-
-\[ -\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), -\]
-

with \(\gamma_k > 0\).

-

For \(\gamma_k\) small enough, then \(F(\mathbf{x}_{k+1}) \leq -F(\mathbf{x}_k)\). This means that for a sufficiently small \(\gamma_k\) -we are always moving towards smaller function values, i.e a minimum.

-
-
-

4.15. More on Steepest descent

-

The previous observation is the basis of the method of steepest -descent, which is also referred to as just gradient descent (GD). One -starts with an initial guess \(\mathbf{x}_0\) for a minimum of \(F\) and -computes new approximations according to

-
-\[ -\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0. -\]
-

The parameter \(\gamma_k\) is often referred to as the step length or -the learning rate within the context of Machine Learning.

-
-
-

4.16. The ideal

-

Ideally the sequence \(\{\mathbf{x}_k \}_{k=0}\) converges to a global -minimum of the function \(F\). In general we do not know if we are in a -global or local minimum. In the special case when \(F\) is a convex -function, all local minima are also global minima, so in this case -gradient descent can converge to the global solution. The advantage of -this scheme is that it is conceptually simple and straightforward to -implement. However the method in this form has some severe -limitations:

-

In machine learing we are often faced with non-convex high dimensional -cost functions with many local minima. Since GD is deterministic we -will get stuck in a local minimum, if the method converges, unless we -have a very good intial guess. This also implies that the scheme is -sensitive to the chosen initial condition.

-

Note that the gradient is a function of \(\mathbf{x} = -(x_1,\cdots,x_n)\) which makes it expensive to compute numerically.

-
-
-

4.17. The sensitiveness of the gradient descent

-

The gradient descent method -is sensitive to the choice of learning rate \(\gamma_k\). This is due -to the fact that we are only guaranteed that \(F(\mathbf{x}_{k+1}) \leq -F(\mathbf{x}_k)\) for sufficiently small \(\gamma_k\). The problem is to -determine an optimal learning rate. If the learning rate is chosen too -small the method will take a long time to converge and if it is too -large we can experience erratic behavior.

-

Many of these shortcomings can be alleviated by introducing -randomness. One such method is that of Stochastic Gradient Descent -(SGD), see below.

-
-
-

4.18. Convex functions

-

Ideally we want our cost/loss function to be convex(concave).

-

First we give the definition of a convex set: A set \(C\) in -\(\mathbb{R}^n\) is said to be convex if, for all \(x\) and \(y\) in \(C\) and -all \(t \in (0,1)\) , the point \((1 − t)x + ty\) also belongs to -C. Geometrically this means that every point on the line segment -connecting \(x\) and \(y\) is in \(C\) as discussed below.

-

The convex subsets of \(\mathbb{R}\) are the intervals of -\(\mathbb{R}\). Examples of convex sets of \(\mathbb{R}^2\) are the -regular polygons (triangles, rectangles, pentagons, etc…).

-
-
-

4.19. Convex function

-

Convex function: Let \(X \subset \mathbb{R}^n\) be a convex set. Assume that the function \(f: X \rightarrow \mathbb{R}\) is continuous, then \(f\) is said to be convex if $\(f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) \)\( for all \)x_1, x_2 \in X\( and for all \)t \in [0,1]\(. If \)\leq\( is replaced with a strict inequaltiy in the definition, we demand \)x_1 \neq x_2\( and \)t\in(0,1)\( then \)f\( is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting \)f(x_1)\( and \)f(x_2)\(, the value of the function on the interval \)[x_1,x_2]$ is always below the line as illustrated below.

-
-
-

4.20. Conditions on convex functions

-

In the following we state first and second-order conditions which -ensures convexity of a function \(f\). We write \(D_f\) to denote the -domain of \(f\), i.e the subset of \(R^n\) where \(f\) is defined. For more -details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/, 2004).

-

First order condition.

-

Suppose \(f\) is differentiable (i.e \(\nabla f(x)\) is well defined for -all \(x\) in the domain of \(f\)). Then \(f\) is convex if and only if \(D_f\) -is a convex set and $\(f(y) \geq f(x) + \nabla f(x)^T (y-x) \)\( holds -for all \)x,y \in D_f\(. This condition means that for a convex function -the first order Taylor expansion (right hand side above) at any point -a global under estimator of the function. To convince yourself you can -make a drawing of \)f(x) = x^2+1\( and draw the tangent line to \)f(x)$ and -note that it is always below the graph.

-

Second order condition.

-

Assume that \(f\) is twice -differentiable, i.e the Hessian matrix exists at each point in -\(D_f\). Then \(f\) is convex if and only if \(D_f\) is a convex set and its -Hessian is positive semi-definite for all \(x\in D_f\). For a -single-variable function this reduces to \(f''(x) \geq 0\). Geometrically this means that \(f\) has nonnegative curvature -everywhere.

-

This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.

-
-
-

4.21. More on convex functions

-

The next result is of great importance to us and the reason why we are -going on about convex functions. In machine learning we frequently -have to minimize a loss/cost function in order to find the best -parameters for the model we are considering.

-

Ideally we want the -global minimum (for high-dimensional models it is hard to know -if we have local or global minimum). However, if the cost/loss function -is convex the following result provides invaluable information:

-

Any minimum is global for convex functions.

-

Consider the problem of finding \(x \in \mathbb{R}^n\) such that \(f(x)\) -is minimal, where \(f\) is convex and differentiable. Then, any point -\(x^*\) that satisfies \(\nabla f(x^*) = 0\) is a global minimum.

-

This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.

-
-
-

4.22. Some simple problems

-
    -

  1. Show that \(f(x)=x^2\) is convex for \(x \in \mathbb{R}\) using the definition of convexity. Hint: If you re-write the definition, \(f\) is convex if the following holds for all \(x,y \in D_f\) and any \(\lambda \in [0,1]\) \(\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0\).

    2. -

  2. Using the second order condition show that the following functions are convex on the specified domain.

    3. -
-
    -

  • \(f(x) = e^x\) is convex for \(x \in \mathbb{R}\).

    • -

  • \(g(x) = -\ln(x)\) is convex for \(x \in (0,\infty)\).

    • -
-
    -

  1. Let \(f(x) = x^2\) and \(g(x) = e^x\). Show that \(f(g(x))\) and \(g(f(x))\) is convex for \(x \in \mathbb{R}\). Also show that if \(f(x)\) is any convex function than \(h(x) = e^{f(x)}\) is convex.

    2. -

  2. A norm is any function that satisfy the following properties

    3. -
-
    -

  • \(f(\alpha x) = |\alpha| f(x)\) for all \(\alpha \in \mathbb{R}\).

    • -

  • \(f(x+y) \leq f(x) + f(y)\)

    • -

  • \(f(x) \leq 0\) for all \(x \in \mathbb{R}^n\) with equality if and only if \(x = 0\)

    • -
-

Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).

-
-
-

4.23. Friday September 25

-

Video of Lecture and link to handwritten notes.

-
-
-

4.24. Standard steepest descent

-

Before we proceed, we would like to discuss the approach called the -standard Steepest descent (different from the above steepest descent discussion), which again leads to us having to be able -to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).

-

The success of the CG method -for finding solutions of non-linear problems is based on the theory -of conjugate gradients for linear systems of equations. It belongs to -the class of iterative methods for solving problems from linear -algebra of the type

-
-\[ -\boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}. -\]
-

In the iterative process we end up with a problem like

-
-\[ -\boldsymbol{r}= \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}, -\]
-

where \(\boldsymbol{r}\) is the so-called residual or error in the iterative process.

-

When we have found the exact solution, \(\boldsymbol{r}=0\).

-
-
-

4.25. Gradient method

-

The residual is zero when we reach the minimum of the quadratic equation

-
-\[ -P(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T\boldsymbol{b}, -\]
-

with the constraint that the matrix \(\boldsymbol{A}\) is positive definite and -symmetric. This defines also the Hessian and we want it to be positive definite.

-
-
-

4.26. Steepest descent method

-

We denote the initial guess for \(\boldsymbol{x}\) as \(\boldsymbol{x}_0\). -We can assume without loss of generality that

-
-\[ -\boldsymbol{x}_0=0, -\]
-

or consider the system

-
-\[ -\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0, -\]
-

instead.

-
-
-

4.27. Steepest descent method

-

One can show that the solution \(\boldsymbol{x}\) is also the unique minimizer of the quadratic form

-
-\[ -f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n. -\]
-

This suggests taking the first basis vector \(\boldsymbol{r}_1\) (see below for definition) -to be the gradient of \(f\) at \(\boldsymbol{x}=\boldsymbol{x}_0\), -which equals

-
-\[ -\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b}, -\]
-

and -\(\boldsymbol{x}_0=0\) it is equal \(-\boldsymbol{b}\).

-
-
-

4.28. Final expressions

-

We can compute the residual iteratively as

-
-\[ -\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1}, -\]
-

which equals

-
-\[ -\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{r}_k), -\]
-

or

-
-\[ -(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{r}_k, -\]
-

which gives

-
-\[ -\alpha_k = \frac{\boldsymbol{r}_k^T\boldsymbol{r}_k}{\boldsymbol{r}_k^T\boldsymbol{A}\boldsymbol{r}_k} -\]
-

leading to the iterative scheme

-
-\[ -\boldsymbol{x}_{k+1}=\boldsymbol{x}_k-\alpha_k\boldsymbol{r}_{k}, -\]
-
-
-

4.29. Steepest descent example

-
-
-
import numpy as np
-import numpy.linalg as la
-
-import scipy.optimize as sopt
-
-import matplotlib.pyplot as pt
-from mpl_toolkits.mplot3d import axes3d
-
-def f(x):
-    return 0.5*x[0]**2 + 2.5*x[1]**2
-
-def df(x):
-    return np.array([x[0], 5*x[1]])
-
-fig = pt.figure()
-ax = fig.gca(projection="3d")
-
-xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]
-fmesh = f(np.array([xmesh, ymesh]))
-ax.plot_surface(xmesh, ymesh, fmesh)
-
-
-
-
-

And then as countor plot

-
-
-
pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh)
-guesses = [np.array([2, 2./5])]
-
-
-
-
-

Find guesses

-
-
-
x = guesses[-1]
-s = -df(x)
-
-
-
-
-

Run it!

-
-
-
def f1d(alpha):
-    return f(x + alpha*s)
-
-alpha_opt = sopt.golden(f1d)
-next_guess = x + alpha_opt * s
-guesses.append(next_guess)
-print(next_guess)
-
-
-
-
-

What happened?

-
-
-
pt.axis("equal")
-pt.contour(xmesh, ymesh, fmesh, 50)
-it_array = np.array(guesses)
-pt.plot(it_array.T[0], it_array.T[1], "x-")
-
-
-
-
-
-
-

4.30. Conjugate gradient method

-

In the CG method we define so-called conjugate directions and two vectors -\(\boldsymbol{s}\) and \(\boldsymbol{t}\) -are said to be -conjugate if

-
-\[ -\boldsymbol{s}^T\boldsymbol{A}\boldsymbol{t}= 0. -\]
-

The philosophy of the CG method is to perform searches in various conjugate directions -of our vectors \(\boldsymbol{x}_i\) obeying the above criterion, namely

-
-\[ -\boldsymbol{x}_i^T\boldsymbol{A}\boldsymbol{x}_j= 0. -\]
-

Two vectors are conjugate if they are orthogonal with respect to -this inner product. Being conjugate is a symmetric relation: if \(\boldsymbol{s}\) is conjugate to \(\boldsymbol{t}\), then \(\boldsymbol{t}\) is conjugate to \(\boldsymbol{s}\).

-
-
-

4.31. Conjugate gradient method

-

An example is given by the eigenvectors of the matrix

-
-\[ -\boldsymbol{v}_i^T\boldsymbol{A}\boldsymbol{v}_j= \lambda\boldsymbol{v}_i^T\boldsymbol{v}_j, -\]
-

which is zero unless \(i=j\).

-
-
-

4.32. Conjugate gradient method

-

Assume now that we have a symmetric positive-definite matrix \(\boldsymbol{A}\) of size -\(n\times n\). At each iteration \(i+1\) we obtain the conjugate direction of a vector

-
-\[ -\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+\alpha_i\boldsymbol{p}_{i}. -\]
-

We assume that \(\boldsymbol{p}_{i}\) is a sequence of \(n\) mutually conjugate directions. -Then the \(\boldsymbol{p}_{i}\) form a basis of \(R^n\) and we can expand the solution -\( \boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}\) in this basis, namely

-
-\[ -\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i \boldsymbol{p}_i. -\]
-
-
-

4.33. Conjugate gradient method

-

The coefficients are given by

-
-\[ -\mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. -\]
-

Multiplying with \(\boldsymbol{p}_k^T\) from the left gives

-
-\[ -\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{p}_i= \boldsymbol{p}_k^T \boldsymbol{b}, -\]
-

and we can define the coefficients \(\alpha_k\) as

-
-\[ -\alpha_k = \frac{\boldsymbol{p}_k^T \boldsymbol{b}}{\boldsymbol{p}_k^T \boldsymbol{A} \boldsymbol{p}_k} -\]
-
-
-

4.34. Conjugate gradient method and iterations

-

If we choose the conjugate vectors \(\boldsymbol{p}_k\) carefully, -then we may not need all of them to obtain a good approximation to the solution -\(\boldsymbol{x}\). -We want to regard the conjugate gradient method as an iterative method. -This will us to solve systems where \(n\) is so large that the direct -method would take too much time.

-

We denote the initial guess for \(\boldsymbol{x}\) as \(\boldsymbol{x}_0\). -We can assume without loss of generality that

-
-\[ -\boldsymbol{x}_0=0, -\]
-

or consider the system

-
-\[ -\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0, -\]
-

instead.

-
-
-

4.35. Conjugate gradient method

-

One can show that the solution \(\boldsymbol{x}\) is also the unique minimizer of the quadratic form

-
-\[ -f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n. -\]
-

This suggests taking the first basis vector \(\boldsymbol{p}_1\) -to be the gradient of \(f\) at \(\boldsymbol{x}=\boldsymbol{x}_0\), -which equals

-
-\[ -\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b}, -\]
-

and -\(\boldsymbol{x}_0=0\) it is equal \(-\boldsymbol{b}\). -The other vectors in the basis will be conjugate to the gradient, -hence the name conjugate gradient method.

-
-
-

4.36. Conjugate gradient method

-

Let \(\boldsymbol{r}_k\) be the residual at the \(k\)-th step:

-
-\[ -\boldsymbol{r}_k=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k. -\]
-

Note that \(\boldsymbol{r}_k\) is the negative gradient of \(f\) at -\(\boldsymbol{x}=\boldsymbol{x}_k\), -so the gradient descent method would be to move in the direction \(\boldsymbol{r}_k\). -Here, we insist that the directions \(\boldsymbol{p}_k\) are conjugate to each other, -so we take the direction closest to the gradient \(\boldsymbol{r}_k\)
-under the conjugacy constraint. -This gives the following expression

-
-\[ -\boldsymbol{p}_{k+1}=\boldsymbol{r}_k-\frac{\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{r}_k}{\boldsymbol{p}_k^T\boldsymbol{A}\boldsymbol{p}_k} \boldsymbol{p}_k. -\]
-
-
-

4.37. Conjugate gradient method

-

We can also compute the residual iteratively as

-
-\[ -\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1}, -\]
-

which equals

-
-\[ -\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{p}_k), -\]
-

or

-
-\[ -(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{p}_k, -\]
-

which gives

-
-\[ -\boldsymbol{r}_{k+1}=\boldsymbol{r}_k-\boldsymbol{A}\boldsymbol{p}_{k}, -\]
-
-
-

4.38. Revisiting our first homework

-

We will use linear regression as a case study for the gradient descent -methods. Linear regression is a great test case for the gradient -descent methods discussed in the lectures since it has several -desirable properties such as:

-
    -

  1. An analytical solution (recall homework set 1).

    2. -

  2. The gradient can be computed analytically.

    3. -

  3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates

    4. -
-

We revisit an example similar to what we had in the first homework set. We had a function of the type

-
-
-
x = 2*np.random.rand(m,1)
-y = 4+3*x+np.random.randn(m,1)
-
-
-
-
-

with \(x_i \in [0,1] \) is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution \(\cal {N}(0,1)\). -The linear regression model is given by

-
-\[ -h_\beta(x) = \boldsymbol{y} = \beta_0 + \beta_1 x, -\]
-

such that

-
-\[ -\boldsymbol{y}_i = \beta_0 + \beta_1 x_i. -\]
-
-
-

4.39. Gradient descent example

-

Let \(\mathbf{y} = (y_1,\cdots,y_n)^T\), \(\mathbf{\boldsymbol{y}} = (\boldsymbol{y}_1,\cdots,\boldsymbol{y}_n)^T\) and \(\beta = (\beta_0, \beta_1)^T\)

-

It is convenient to write \(\mathbf{\boldsymbol{y}} = X\beta\) where \(X \in \mathbb{R}^{100 \times 2} \) is the design matrix given by (we keep the intercept here)

-
-\[\begin{split} -X \equiv \begin{bmatrix} -1 & x_1 \\ -\vdots & \vdots \\ -1 & x_{100} & \\ -\end{bmatrix}. -\end{split}\]
-

The cost/loss/risk function is given by (

-
-\[ -C(\beta) = \frac{1}{n}||X\beta-\mathbf{y}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{100}\left[ (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2\right] -\]
-

and we want to find \(\beta\) such that \(C(\beta)\) is minimized.

-
-
-

4.40. The derivative of the cost/loss function

-

Computing \(\partial C(\beta) / \partial \beta_0\) and \(\partial C(\beta) / \partial \beta_1\) we can show that the gradient can be written as

-
-\[\begin{split} -\nabla_{\beta} C(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} = \frac{2}{n}X^T(X\beta - \mathbf{y}), -\end{split}\]
-

where \(X\) is the design matrix defined above.

-
-
-

4.41. The Hessian matrix

-

The Hessian matrix of \(C(\beta)\) is given by

-
-\[\begin{split} -\boldsymbol{H} \equiv \begin{bmatrix} -\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ -\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\ -\end{bmatrix} = \frac{2}{n}X^T X. -\end{split}\]
-

This result implies that \(C(\beta)\) is a convex function since the matrix \(X^T X\) always is positive semi-definite.

-
-
-

4.42. Simple program

-

We can now write a program that minimizes \(C(\beta)\) using the gradient descent method with a constant learning rate \(\gamma\) according to

-
-\[ -\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots -\]
-

We can use the expression we computed for the gradient and let use a -\(\beta_0\) be chosen randomly and let \(\gamma = 0.001\). Stop iterating -when \(||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8}\). Note that the code below does not include the latter stop criterion.

-

And finally we can compare our solution for \(\beta\) with the analytic result given by -\(\beta= (X^TX)^{-1} X^T \mathbf{y}\).

-
-
-

4.43. Gradient Descent Example

-

Here our simple example

-
-
-
# Importing various packages
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.mplot3d import Axes3D
-from matplotlib import cm
-from matplotlib.ticker import LinearLocator, FormatStrFormatter
-import sys
-
-# the number of datapoints
-n = 100
-x = 2*np.random.rand(n,1)
-y = 4+3*x+np.random.randn(n,1)
-
-X = np.c_[np.ones((n,1)), x]
-# Hessian matrix
-H = (2.0/n)* X.T @ X
-# Get the eigenvalues
-EigValues, EigVectors = np.linalg.eig(H)
-print(EigValues)
-
-beta_linreg = np.linalg.inv(X.T @ X) @ X.T @ y
-print(beta_linreg)
-beta = np.random.randn(2,1)
-
-eta = 1.0/np.max(EigValues)
-Niterations = 1000
-
-for iter in range(Niterations):
-    gradient = (2.0/n)*X.T @ (X @ beta-y)
-    beta -= eta*gradient
-
-print(beta)
-xnew = np.array([[0],[2]])
-xbnew = np.c_[np.ones((2,1)), xnew]
-ypredict = xbnew.dot(beta)
-ypredict2 = xbnew.dot(beta_linreg)
-plt.plot(xnew, ypredict, "r-")
-plt.plot(xnew, ypredict2, "b-")
-plt.plot(x, y ,'ro')
-plt.axis([0,2.0,0, 15.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Gradient descent example')
-plt.show()
-
-
-
-
-
-
-

4.44. And a corresponding example using scikit-learn

-
-
-
# Importing various packages
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
-
-n = 100
-x = 2*np.random.rand(n,1)
-y = 4+3*x+np.random.randn(n,1)
-
-X = np.c_[np.ones((n,1)), x]
-beta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
-print(beta_linreg)
-sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
-sgdreg.fit(x,y.ravel())
-print(sgdreg.intercept_, sgdreg.coef_)
-
-
-
-
-
-
-

4.45. Gradient descent and Ridge

-

We have also discussed Ridge regression where the loss function contains a regularized term given by the \(L_2\) norm of \(\beta\),

-
-\[ -C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0. -\]
-

In order to minimize \(C_{\text{ridge}}(\beta)\) using GD we only have adjust the gradient as follows

-
-\[\begin{split} -\nabla_\beta C_{\text{ridge}}(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta). -\end{split}\]
-

We can easily extend our program to minimize \(C_{\text{ridge}}(\beta)\) using gradient descent and compare with the analytical solution given by

-
-\[ -\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}. -\]
-
-
-

4.46. Program example for gradient descent with Ridge Regression

-
-
-
from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from mpl_toolkits.mplot3d import Axes3D
-from matplotlib import cm
-from matplotlib.ticker import LinearLocator, FormatStrFormatter
-import sys
-
-# the number of datapoints
-n = 100
-x = 2*np.random.rand(n,1)
-y = 4+3*x+np.random.randn(n,1)
-
-X = np.c_[np.ones((n,1)), x]
-XT_X = X.T @ X
-
-#Ridge parameter lambda
-lmbda  = 0.001
-Id = lmbda* np.eye(XT_X.shape[0])
-
-beta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y
-print(beta_linreg)
-# Start plain gradient descent
-beta = np.random.randn(2,1)
-
-eta = 0.1
-Niterations = 100
-
-for iter in range(Niterations):
-    gradients = 2.0/n*X.T @ (X @ (beta)-y)+2*lmbda*beta
-    beta -= eta*gradients
-
-print(beta)
-ypredict = X @ beta
-ypredict2 = X @ beta_linreg
-plt.plot(x, ypredict, "r-")
-plt.plot(x, ypredict2, "b-")
-plt.plot(x, y ,'ro')
-plt.axis([0,2.0,0, 15.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Gradient descent example for Ridge')
-plt.show()
-
-
-
-
-
-
-

4.47. Using gradient descent methods, limitations

-
    -

  • Gradient descent (GD) finds local minima of our function. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.

    • -

  • GD is sensitive to initial conditions. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.

    • -

  • Gradients are computationally expensive to calculate for large datasets. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, \(E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2\); for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over all \(n\) data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called “mini batches”. This has the added benefit of introducing stochasticity into our algorithm.

    • -

  • GD is very sensitive to choices of learning rates. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would adaptively choose the learning rates to match the landscape.

    • -

  • GD treats all directions in parameter space uniformly. Another major drawback of GD is that unlike Newton’s method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive.

    • -

  • GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.

    • -
-
-
-

4.48. Stochastic Gradient Descent

-

Stochastic gradient descent (SGD) and variants thereof address some of -the shortcomings of the Gradient descent method discussed above.

-

The underlying idea of SGD comes from the observation that the cost -function, which we want to minimize, can almost always be written as a -sum over \(n\) data points \(\{\mathbf{x}_i\}_{i=1}^n\),

-
-\[ -C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, -\mathbf{\beta}). -\]
-
-
-

4.49. Computation of gradients

-

This in turn means that the gradient can be -computed as a sum over \(i\)-gradients

-
-\[ -\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}). -\]
-

Stochasticity/randomness is introduced by only taking the -gradient on a subset of the data called minibatches. If there are \(n\) -data points and the size of each minibatch is \(M\), there will be \(n/M\) -minibatches. We denote these minibatches by \(B_k\) where -\(k=1,\cdots,n/M\).

-
-
-

4.50. SGD example

-

As an example, suppose we have \(10\) data points \((\mathbf{x}_1,\cdots, \mathbf{x}_{10})\) -and we choose to have \(M=5\) minibathces, -then each minibatch contains two data points. In particular we have -\(B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 = -(\mathbf{x}_9,\mathbf{x}_{10})\). Note that if you choose \(M=1\) you -have only a single batch with all data points and on the other extreme, -you may choose \(M=n\) resulting in a minibatch for each datapoint, i.e -\(B_k = \mathbf{x}_k\).

-

The idea is now to approximate the gradient by replacing the sum over -all data points with a sum over the data points in one the minibatches -picked at random in each gradient descent step

-
-\[ -\nabla_{\beta} -C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta -c_i(\mathbf{x}_i, \mathbf{\beta}). -\]
-
-
-

4.51. The gradient step

-

Thus a gradient descent step now looks like

-
-\[ -\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) -\]
-

where \(k\) is picked at random with equal -probability from \([1,n/M]\). An iteration over the number of -minibathces (n/M) is commonly referred to as an epoch. Thus it is -typical to choose a number of epochs and for each epoch iterate over -the number of minibatches, as exemplified in the code below.

-
-
-

4.52. Simple example code

-
-
-
import numpy as np 
-
-n = 100 #100 datapoints 
-M = 5   #size of each minibatch
-m = int(n/M) #number of minibatches
-n_epochs = 10 #number of epochs
-
-j = 0
-for epoch in range(1,n_epochs+1):
-    for i in range(m):
-        k = np.random.randint(m) #Pick the k-th minibatch at random
-        #Compute the gradient using the data in minibatch Bk
-        #Compute new suggestion for 
-        j += 1
-
-
-
-
-

Taking the gradient only on a subset of the data has two important -benefits. First, it introduces randomness which decreases the chance -that our opmization scheme gets stuck in a local minima. Second, if -the size of the minibatches are small relative to the number of -datapoints (\(M < n\)), the computation of the gradient is much -cheaper since we sum over the datapoints in the \(k-th\) minibatch and not -all \(n\) datapoints.

-
-
-

4.53. When do we stop?

-

A natural question is when do we stop the search for a new minimum? -One possibility is to compute the full gradient after a given number -of epochs and check if the norm of the gradient is smaller than some -threshold and stop if true. However, the condition that the gradient -is zero is valid also for local minima, so this would only tell us -that we are close to a local/global minimum. However, we could also -evaluate the cost function at this point, store the result and -continue the search. If the test kicks in at a later stage we can -compare the values of the cost function and keep the \(\beta\) that -gave the lowest value.

-
-
-

4.54. Slightly different approach

-

Another approach is to let the step length \(\gamma_j\) depend on the -number of epochs in such a way that it becomes very small after a -reasonable time such that we do not move at all.

-

As an example, let \(e = 0,1,2,3,\cdots\) denote the current epoch and let \(t_0, t_1 > 0\) be two fixed numbers. Furthermore, let \(t = e \cdot m + i\) where \(m\) is the number of minibatches and \(i=0,\cdots,m-1\). Then the function $\(\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} \)\( goes to zero as the number of epochs gets large. I.e. we start with a step length \)\gamma_j (0; t_0, t_1) = t_0/t_1\( which decays in *time* \)t$.

-

In this way we can fix the number of epochs, compute \(\beta\) and -evaluate the cost function at the end. Repeating the computation will -give a different result since the scheme is random by design. Then we -pick the final \(\beta\) that gives the lowest value of the cost -function.

-
-
-
import numpy as np 
-
-def step_length(t,t0,t1):
-    return t0/(t+t1)
-
-n = 100 #100 datapoints 
-M = 5   #size of each minibatch
-m = int(n/M) #number of minibatches
-n_epochs = 500 #number of epochs
-t0 = 1.0
-t1 = 10
-
-gamma_j = t0/t1
-j = 0
-for epoch in range(1,n_epochs+1):
-    for i in range(m):
-        k = np.random.randint(m) #Pick the k-th minibatch at random
-        #Compute the gradient using the data in minibatch Bk
-        #Compute new suggestion for beta
-        t = epoch*m+i
-        gamma_j = step_length(t,t0,t1)
-        j += 1
-
-print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j))
-
-
-
-
-
-
-

4.55. Program for stochastic gradient

-
-
-
# Importing various packages
-from math import exp, sqrt
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.linear_model import SGDRegressor
-
-m = 100
-x = 2*np.random.rand(m,1)
-y = 4+3*x+np.random.randn(m,1)
-
-X = np.c_[np.ones((m,1)), x]
-theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)
-print("Own inversion")
-print(theta_linreg)
-sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)
-sgdreg.fit(x,y.ravel())
-print("sgdreg from scikit")
-print(sgdreg.intercept_, sgdreg.coef_)
-
-
-theta = np.random.randn(2,1)
-eta = 0.1
-Niterations = 1000
-
-
-for iter in range(Niterations):
-    gradients = 2.0/m*X.T @ ((X @ theta)-y)
-    theta -= eta*gradients
-print("theta from own gd")
-print(theta)
-
-xnew = np.array([[0],[2]])
-Xnew = np.c_[np.ones((2,1)), xnew]
-ypredict = Xnew.dot(theta)
-ypredict2 = Xnew.dot(theta_linreg)
-
-
-n_epochs = 50
-t0, t1 = 5, 50
-def learning_schedule(t):
-    return t0/(t+t1)
-
-theta = np.random.randn(2,1)
-
-for epoch in range(n_epochs):
-    for i in range(m):
-        random_index = np.random.randint(m)
-        xi = X[random_index:random_index+1]
-        yi = y[random_index:random_index+1]
-        gradients = 2 * xi.T @ ((xi @ theta)-yi)
-        eta = learning_schedule(epoch*m+i)
-        theta = theta - eta*gradients
-print("theta from own sdg")
-print(theta)
-
-plt.plot(xnew, ypredict, "r-")
-plt.plot(xnew, ypredict2, "b-")
-plt.plot(x, y ,'ro')
-plt.axis([0,2.0,0, 15.0])
-plt.xlabel(r'$x$')
-plt.ylabel(r'$y$')
-plt.title(r'Random numbers ')
-plt.show()
-
-
-
-
-

Challenge: try to write a similar code for a Logistic Regression case.

-
-
- - - - -
- - -
- - 3. Ridge and Lasso Regression - 5. Support Vector Machines, overarching aims - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

-
-
-
- - -
-
- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter5.html b/doc/LectureNotes/_build/html/chapter5.html deleted file mode 100644 index 975ec3f3e..000000000 --- a/doc/LectureNotes/_build/html/chapter5.html +++ /dev/null @@ -1,1402 +0,0 @@ - - - - - - - - 5. Support Vector Machines, overarching aims — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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- - Contents -
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-

5. Support Vector Machines, overarching aims

-

A Support Vector Machine (SVM) is a very powerful and versatile -Machine Learning method, capable of performing linear or nonlinear -classification, regression, and even outlier detection. It is one of -the most popular models in Machine Learning, and anyone interested in -Machine Learning should have it in their toolbox. SVMs are -particularly well suited for classification of complex but small-sized or -medium-sized datasets.

-

The case with two well-separated classes only can be understood in an -intuitive way in terms of lines in a two-dimensional space separating -the two classes (see figure below).

-

The basic mathematics behind the SVM is however less familiar to most of us. -It relies on the definition of hyperplanes and the -definition of a margin which separates classes (in case of -classification problems) of variables. It is also used for regression -problems.

-

With SVMs we distinguish between hard margin and soft margins. The -latter introduces a so-called softening parameter to be discussed -below. We distinguish also between linear and non-linear -approaches. The latter are the most frequent ones since it is rather -unlikely that we can separate classes easily by say straight lines.

-
-

5.1. Hyperplanes and all that

-

The theory behind support vector machines (SVM hereafter) is based on -the mathematical description of so-called hyperplanes. Let us start -with a two-dimensional case. This will also allow us to introduce our -first SVM examples. These will be tailored to the case of two specific -classes, as displayed in the figure here based on the usage of the petal data.

-

We assume here that our data set can be well separated into two -domains, where a straight line does the job in the separating the two -classes. Here the two classes are represented by either squares or -circles.

-
-
-
%matplotlib inline
-
-from sklearn import datasets
-from sklearn.svm import SVC, LinearSVC
-from sklearn.linear_model import SGDClassifier
-from sklearn.preprocessing import StandardScaler
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
-
-
-iris = datasets.load_iris()
-X = iris["data"][:, (2, 3)]  # petal length, petal width
-y = iris["target"]
-
-setosa_or_versicolor = (y == 0) | (y == 1)
-X = X[setosa_or_versicolor]
-y = y[setosa_or_versicolor]
-
-
-
-C = 5
-alpha = 1 / (C * len(X))
-
-lin_clf = LinearSVC(loss="hinge", C=C, random_state=42)
-svm_clf = SVC(kernel="linear", C=C)
-sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha,
-                        max_iter=100000, random_state=42)
-
-scaler = StandardScaler()
-X_scaled = scaler.fit_transform(X)
-
-lin_clf.fit(X_scaled, y)
-svm_clf.fit(X_scaled, y)
-sgd_clf.fit(X_scaled, y)
-
-print("LinearSVC:                   ", lin_clf.intercept_, lin_clf.coef_)
-print("SVC:                         ", svm_clf.intercept_, svm_clf.coef_)
-print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)
-
-# Compute the slope and bias of each decision boundary
-w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]
-b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]
-w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]
-b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]
-w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]
-b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]
-
-# Transform the decision boundary lines back to the original scale
-line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])
-line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])
-line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])
-
-# Plot all three decision boundaries
-plt.figure(figsize=(11, 4))
-plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC")
-plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC")
-plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier")
-plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor"
-plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa"
-plt.xlabel("Petal length", fontsize=14)
-plt.ylabel("Petal width", fontsize=14)
-plt.legend(loc="upper center", fontsize=14)
-plt.axis([0, 5.5, 0, 2])
-
-plt.show()
-
-
-
-
-
LinearSVC:                    [0.28475098] [[1.05364854 1.09903804]]
-SVC:                          [0.31896852] [[1.1203284  1.02625193]]
-SGDClassifier(alpha=0.00200): [0.117] [[0.77714169 0.72981762]]
-
-
-_images/chapter5_1_1.png -
-
-

The aim of the SVM algorithm is to find a hyperplane in a -\(p\)-dimensional space, where \(p\) is the number of features that -distinctly classifies the data points.

-

In a \(p\)-dimensional space, a hyperplane is what we call an affine subspace of dimension of \(p-1\). -As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is -a two-dimensional subspace, or stated simply, a plane.

-

In two dimensions, with the variables \(x_1\) and \(x_2\), the hyperplane is defined as

-
-\[ -b+w_1x_1+w_2x_2=0, -\]
-

where \(b\) is the intercept and \(w_1\) and \(w_2\) define the elements of a vector orthogonal to the line -\(b+w_1x_1+w_2x_2=0\). -In two dimensions we define the vectors \(\boldsymbol{x} =[x1,x2]\) and \(\boldsymbol{w}=[w1,w2]\). -We can then rewrite the above equation as

-
-\[ -\boldsymbol{x}^T\boldsymbol{w}+b=0. -\]
-

We limit ourselves to two classes of outputs \(y_i\) and assign these classes the values \(y_i = \pm 1\). -In a \(p\)-dimensional space of say \(p\) features we have a hyperplane defines as

-
-\[ -b+wx_1+w_2x_2+\dots +w_px_p=0. -\]
-

If we define a -matrix \(\boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right]\) -of dimension \(n\times p\), where \(n\) represents the observations for each feature and each vector \(x_i\) is a column vector of the matrix \(\boldsymbol{X}\),

-
-\[\begin{split} -\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}. -\end{split}\]
-

If the above condition is not met for a given vector \(\boldsymbol{x}_i\) we have

-
-\[ -b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0, -\]
-

if our output \(y_i=1\). -In this case we say that \(\boldsymbol{x}_i\) lies on one of the sides of the hyperplane and if

-
-\[ -b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0, -\]
-

for the class of observations \(y_i=-1\), -then \(\boldsymbol{x}_i\) lies on the other side.

-

Equivalently, for the two classes of observations we have

-
-\[ -y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0. -\]
-

When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.

-
-

5.1.1. The two-dimensional case

-

Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional -plane. To separate the two classes of data points, there are many -possible lines (hyperplanes if you prefer a more strict naming)
-that could be chosen. Our objective is to find a -plane that has the maximum margin, i.e the maximum distance between -data points of both classes. Maximizing the margin distance provides -some reinforcement so that future data points can be classified with -more confidence.

-

What a linear classifier attempts to accomplish is to split the -feature space into two half spaces by placing a hyperplane between the -data points. This hyperplane will be our decision boundary. All -points on one side of the plane will belong to class one and all points -on the other side of the plane will belong to the second class two.

-

Unfortunately there are many ways in which we can place a hyperplane -to divide the data. Below is an example of two candidate hyperplanes -for our data sample.

-

Let us define the function

-
-\[ -f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0, -\]
-

as the function that determines the line \(L\) that separates two classes (our two features), see the figure here.

-

Any point defined by \(\boldsymbol{x}_i\) and \(\boldsymbol{x}_2\) on the line \(L\) will satisfy \(\boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0\).

-

The signed distance \(\delta\) from any point defined by a vector \(\boldsymbol{x}\) and a point \(\boldsymbol{x}_0\) on the line \(L\) is then

-
-\[ -\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b). -\]
-

How do we find the parameter \(b\) and the vector \(\boldsymbol{w}\)? What we could -do is to define a cost function which now contains the set of all -misclassified points \(M\) and attempt to minimize this function

-
-\[ -C(\boldsymbol{w},b) = -\sum_{i\in M} y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b). -\]
-

We could now for example define all values \(y_i =1\) as misclassified in case we have \(\boldsymbol{w}^T\boldsymbol{x}_i+b < 0\) and the opposite if we have \(y_i=-1\). Taking the derivatives gives us

-
-\[ -\frac{\partial C}{\partial b} = -\sum_{i\in M} y_i, -\]
-

and

-
-\[ -\frac{\partial C}{\partial \boldsymbol{w}} = -\sum_{i\in M} y_ix_i. -\]
-

We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations

-
-\[ -b \leftarrow b +\eta \frac{\partial C}{\partial b}, -\]
-

and

-
-\[ -\boldsymbol{w} \leftarrow \boldsymbol{w} +\eta \frac{\partial C}{\partial \boldsymbol{w}}, -\]
-

where \(\eta\) is our by now well-known learning rate.

-

The equations we discussed above can be coded rather easily (the -framework is similar to what we developed for logistic -regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way.

-

There are however problems with this approach, although it looks -pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.

-

For small -gaps between the entries, we may also end up needing many iterations -before the solutions converge and if the data cannot be separated -properly into two distinct classes, we may not experience a converge -at all.

-
-
-

5.1.2. A better approach

-

A better approach is rather to try to define a large margin between -the two classes (if they are well separated from the beginning).

-

Thus, we wish to find a margin \(M\) with \(\boldsymbol{w}\) normalized to -\(\vert\vert \boldsymbol{w}\vert\vert =1\) subject to the condition

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. -\]
-

All points are thus at a signed distance from the decision boundary defined by the line \(L\). The parameters \(b\) and \(w_1\) and \(w_2\) define this line.

-

We seek thus the largest value \(M\) defined by

-
-\[ -\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, -\]
-

or just

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i. -\]
-

If we scale the equation so that \(\vert \vert \boldsymbol{w}\vert\vert = 1/M\), we have to find the minimum of -\(\boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert\) (the norm) subject to the condition

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. -\]
-

We have thus defined our margin as the invers of the norm of -\(\boldsymbol{w}\). We want to minimize the norm in order to have a as large as -possible margin \(M\). Before we proceed, we need to remind ourselves -about Lagrangian multipliers.

-
-
-
-

5.2. A quick Reminder on Lagrangian Multipliers

-

Consider a function of three independent variables \(f(x,y,z)\) . For the function \(f\) to be an -extreme we have

-
-\[ -df=0. -\]
-

A necessary and sufficient condition is

-
-\[ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, -\]
-

due to

-
-\[ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. -\]
-

In many problems the variables \(x,y,z\) are often subject to constraints (such as those above for the margin) -so that they are no longer all independent. It is possible at least in principle to use each -constraint to eliminate one variable -and to proceed with a new and smaller set of independent varables.

-

The use of so-called Lagrangian multipliers is an alternative technique when the elimination -of variables is incovenient or undesirable. Assume that we have an equation of constraint on -the variables \(x,y,z\)

-
-\[ -\phi(x,y,z) = 0, -\]
-

resulting in

-
-\[ -d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. -\]
-

Now we cannot set anymore

-
-\[ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, -\]
-

if \(df=0\) is wanted -because there are now only two independent variables! Assume \(x\) and \(y\) are the independent -variables. -Then \(dz\) is no longer arbitrary.

-

However, we can add to

-
-\[ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz, -\]
-

a multiplum of \(d\phi\), viz. \(\lambda d\phi\), resulting in

-
-\[ -df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda -\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+ -(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0. -\]
-

Our multiplier is chosen so that

-
-\[ -\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0. -\]
-

We need to remember that we took \(dx\) and \(dy\) to be arbitrary and thus we must have

-
-\[ -\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0, -\]
-

and

-
-\[ -\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0. -\]
-

When all these equations are satisfied, \(df=0\). We have four unknowns, \(x,y,z\) and -\(\lambda\). Actually we want only \(x,y,z\), \(\lambda\) needs not to be determined, -it is therefore often called -Lagrange’s undetermined multiplier. -If we have a set of constraints \(\phi_k\) we have the equations

-
-\[ -\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0. -\]
-

In order to solve the above problem, we define the following Lagrangian function to be minimized

-
-\[ -\cal{L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], -\]
-

where \(\lambda_i\) is a so-called Lagrange multiplier subject to the condition \(\lambda_i \geq 0\).

-

Taking the derivatives with respect to \(b\) and \(\boldsymbol{w}\) we obtain

-
-\[ -\frac{\partial \cal{L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, -\]
-

and

-
-\[ -\frac{\partial \cal{L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i. -\]
-

Inserting these constraints into the equation for \(\cal{L}\) we obtain

-
-\[ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -\]
-

subject to the constraints \(\lambda_i\geq 0\) and \(\sum_i\lambda_iy_i=0\). -We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition

-
-\[ -\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i. -\]
-
    -

  1. If \(\lambda_i > 0\), then \(y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1\) and we say that \(x_i\) is on the boundary.

    2. -

  2. If \(y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1\), we say \(x_i\) is not on the boundary and we set \(\lambda_i=0\).

    3. -
-

When \(\lambda_i > 0\), the vectors \(\boldsymbol{x}_i\) are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin \(M\).

-

We can rewrite

-
-\[ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -\]
-

and its constraints in terms of a matrix-vector problem where we minimize w.r.t. \(\lambda\) the following problem

-
-\[\begin{split} -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\ -y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, -\end{split}\]
-

subject to \(\boldsymbol{y}^T\boldsymbol{\lambda}=0\). Here we defined the vectors \(\boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n]\) and -\(\boldsymbol{y}=[y_1,y_2,\dots,y_n]\).

-

Solving the above problem, yields the values of \(\lambda_i\). -To find the coefficients of your hyperplane we need simply to compute

-
-\[ -\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i. -\]
-

With our vector \(\boldsymbol{w}\) we can in turn find the value of the intercept \(b\) (here in two dimensions) via

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, -\]
-

resulting in

-
-\[ -b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i, -\]
-

or if we write it out in terms of the support vectors only, with \(N_s\) being their number, we have

-
-\[ -b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right). -\]
-

With our hyperplane coefficients we can use our classifier to assign any observation by simply using

-
-\[ -y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b). -\]
-

Below we discuss how to find the optimal values of \(\lambda_i\). Before we proceed however, we discuss now the so-called soft classifier.

-
-
-

5.3. A soft classifier

-

Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.

-

Suppose now that classes overlap in feature space, as shown in the -figure here. One way to deal with this problem before we define the -so-called kernel approach, is to allow a kind of slack in the sense -that we allow some points to be on the wrong side of the margin.

-

We introduce thus the so-called slack variables \(\boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n]\) and -modify our previous equation

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, -\]
-

to

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i, -\]
-

with the requirement \(\xi_i\geq 0\). The total violation is now \(\sum_i\xi\). -The value \(\xi_i\) in the constraint the last constraint corresponds to the amount by which the prediction -\(y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1\) is on the wrong side of its margin. Hence by bounding the sum \(\sum_i \xi_i\), -we bound the total amount by which predictions fall on the wrong side of their margins.

-

Misclassifications occur when \(\xi_i > 1\). Thus bounding the total sum by some value \(C\) bounds in turn the total number of -misclassifications.

-

This has in turn the consequences that we change our optmization problem to finding the minimum of

-
-\[ -\cal{L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i, -\]
-

subject to

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, -\]
-

with the requirement \(\xi_i\geq 0\).

-

Taking the derivatives with respect to \(b\) and \(\boldsymbol{w}\) we obtain

-
-\[ -\frac{\partial \cal{L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, -\]
-

and

-
-\[ -\frac{\partial \cal{L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i, -\]
-

and

-
-\[ -\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. -\]
-

Inserting these constraints into the equation for \(\cal{L}\) we obtain the same equation as before

-
-\[ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -\]
-

but now subject to the constraints \(\lambda_i\geq 0\), \(\sum_i\lambda_iy_i=0\) and \(0\leq\lambda_i \leq C\). -We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads

-

5 -0

-

< -< -< -! -! -M -A -T -H -_ -B -L -O -C -K

-
-\[ -\gamma_i\xi_i = 0, -\]
-

and

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. -\]
-
-
-

5.4. Kernels and non-linearity

-

The cases we have studied till now, were all characterized by two classes -with a close to linear separability. The classifiers we have described -so far find linear boundaries in our input feature space. It is -possible to make our procedure more flexible by exploring the feature -space using other basis expansions such as higher-order polynomials, -wavelets, splines etc.

-

If our feature space is not easy to separate, as shown in the figure -here, we can achieve a better separation by introducing more complex -basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to -obtain a separation between the classes which is almost linear.

-

The change of basis, from \(x\rightarrow z=\phi(x)\) leads to the same type of equations to be solved, except that -we need to introduce for example a polynomial transformation to a two-dimensional training set.

-
-
-
import numpy as np
-import os
-
-np.random.seed(42)
-
-# To plot pretty figures
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
-
-
-from sklearn.svm import SVC
-from sklearn import datasets
-
-
-
-X1D = np.linspace(-4, 4, 9).reshape(-1, 1)
-X2D = np.c_[X1D, X1D**2]
-y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs")
-plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^")
-plt.gca().get_yaxis().set_ticks([])
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.axis([-4.5, 4.5, -0.2, 0.2])
-
-plt.subplot(122)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.axvline(x=0, color='k')
-plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs")
-plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^")
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
-plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])
-plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3)
-plt.axis([-4.5, 4.5, -1, 17])
-plt.subplots_adjust(right=1)
-plt.show()
-
-
-
-
-_images/chapter5_109_0.png -
-
-

Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with \(x_i\) and \(y_i\) as variables)

-
-\[ -z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). -\]
-

With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)

-
-\[ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, -\]
-

subject to the constraints \(\lambda_i\geq 0\), \(\sum_i\lambda_iy_i=0\), and for the support vectors

-
-\[ -y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i, -\]
-

from which we also find \(b\). -To compute \(\boldsymbol{z}_i^T\boldsymbol{z}_j\) we define the kernel \(K(\boldsymbol{x}_i,\boldsymbol{x}_j)\) as

-
-\[ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). -\]
-

For the above example, the kernel reads

-
-\[\begin{split} -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2. -\end{split}\]
-

We note that this is nothing but the dot product of the two original -vectors \((\boldsymbol{x}_i^T\boldsymbol{x}_j)^2\). Instead of thus computing the -product in the Lagrangian of \(\boldsymbol{z}_i^T\boldsymbol{z}_j\) we simply compute -the dot product \((\boldsymbol{x}_i^T\boldsymbol{x}_j)^2\).

-

This leads to the so-called -kernel trick and the result leads to the same as if we went through -the trouble of performing the transformation -\(\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j)\) during the SVM calculations.

-

Using our definition of the kernel We can rewrite again the Lagrangian

-
-\[ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, -\]
-

subject to the constraints \(\lambda_i\geq 0\), \(\sum_i\lambda_iy_i=0\) in terms of a convex optimization problem

-
-\[\begin{split} -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, -\end{split}\]
-

subject to \(\boldsymbol{y}^T\boldsymbol{\lambda}=0\). Here we defined the vectors \(\boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n]\) and -\(\boldsymbol{y}=[y_1,y_2,\dots,y_n]\). -If we add the slack constants this leads to the additional constraint \(0\leq \lambda_i \leq C\).

-

We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type

-
-\[\begin{split} -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} -\end{split}\]
-

Below we discuss how to solve these equations. Here we note that the matrix \(\boldsymbol{P}\) has matrix elements \(p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j)\). -Given a kernel \(K\) and the targets \(y_i\) this matrix is easy to set up. The constraint \(\boldsymbol{y}^T\boldsymbol{\lambda}=0\) leads to \(f=0\) and \(\boldsymbol{A}=\boldsymbol{y}\). How to set up the matrix \(\boldsymbol{G}\) is discussed later. Here note that the inequalities \(0\leq \lambda_i \leq C\) can be split up into -\(0\leq \lambda_i\) and \(\lambda_i \leq C\). These two inequalities define then the matrix \(\boldsymbol{G}\) and the vector \(\boldsymbol{h}\).

-
-
-

5.5. Different kernels and Mercer’s theorem

-

There are several popular kernels being used. These are

-
    -

  1. Linear: \(K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y}\),

    2. -

  2. Polynomial: \(K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d\),

    3. -

  3. Gaussian Radial Basis Function: \(K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)}\),

    4. -

  4. Tanh: \(K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)}\),

    5. -
-

and many other ones.

-

An important theorem for us is Mercer’s -theorem. The -theorem states that if a kernel function \(K\) is symmetric, continuous -and leads to a positive semi-definite matrix \(\boldsymbol{P}\) then there -exists a function \(\phi\) that maps \(\boldsymbol{x}_i\) and \(\boldsymbol{x}_j\) into -another space (possibly with much higher dimensions) such that

-
-\[ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). -\]
-

So you can use \(K\) as a kernel since you know \(\phi\) exists, even if -you don’t know what \(\phi\) is.

-

Note that some frequently used kernels (such as the Sigmoid kernel) -don’t respect all of Mercer’s conditions, yet they generally work well -in practice.

-
-
-

5.6. The moons example

-
-
-
from __future__ import division, print_function, unicode_literals
-
-import numpy as np
-np.random.seed(42)
-
-import matplotlib
-import matplotlib.pyplot as plt
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
-
-
-from sklearn.svm import SVC
-from sklearn import datasets
-
-
-
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import StandardScaler
-from sklearn.svm import LinearSVC
-
-
-from sklearn.datasets import make_moons
-X, y = make_moons(n_samples=100, noise=0.15, random_state=42)
-
-def plot_dataset(X, y, axes):
-    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs")
-    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^")
-    plt.axis(axes)
-    plt.grid(True, which='both')
-    plt.xlabel(r"$x_1$", fontsize=20)
-    plt.ylabel(r"$x_2$", fontsize=20, rotation=0)
-
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.show()
-
-from sklearn.datasets import make_moons
-from sklearn.pipeline import Pipeline
-from sklearn.preprocessing import PolynomialFeatures
-
-polynomial_svm_clf = Pipeline([
-        ("poly_features", PolynomialFeatures(degree=3)),
-        ("scaler", StandardScaler()),
-        ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42))
-    ])
-
-polynomial_svm_clf.fit(X, y)
-
-def plot_predictions(clf, axes):
-    x0s = np.linspace(axes[0], axes[1], 100)
-    x1s = np.linspace(axes[2], axes[3], 100)
-    x0, x1 = np.meshgrid(x0s, x1s)
-    X = np.c_[x0.ravel(), x1.ravel()]
-    y_pred = clf.predict(X).reshape(x0.shape)
-    y_decision = clf.decision_function(X).reshape(x0.shape)
-    plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)
-    plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)
-
-plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-
-plt.show()
-
-
-from sklearn.svm import SVC
-
-poly_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5))
-    ])
-poly_kernel_svm_clf.fit(X, y)
-
-poly100_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5))
-    ])
-poly100_kernel_svm_clf.fit(X, y)
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.title(r"$d=3, r=1, C=5$", fontsize=18)
-
-plt.subplot(122)
-plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])
-plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-plt.title(r"$d=10, r=100, C=5$", fontsize=18)
-
-plt.show()
-
-def gaussian_rbf(x, landmark, gamma):
-    return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)
-
-gamma = 0.3
-
-x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)
-x2s = gaussian_rbf(x1s, -2, gamma)
-x3s = gaussian_rbf(x1s, 1, gamma)
-
-XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]
-yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red")
-plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs")
-plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^")
-plt.plot(x1s, x2s, "g--")
-plt.plot(x1s, x3s, "b:")
-plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])
-plt.xlabel(r"$x_1$", fontsize=20)
-plt.ylabel(r"Similarity", fontsize=14)
-plt.annotate(r'$\mathbf{x}$',
-             xy=(X1D[3, 0], 0),
-             xytext=(-0.5, 0.20),
-             ha="center",
-             arrowprops=dict(facecolor='black', shrink=0.1),
-             fontsize=18,
-            )
-plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20)
-plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20)
-plt.axis([-4.5, 4.5, -0.1, 1.1])
-
-plt.subplot(122)
-plt.grid(True, which='both')
-plt.axhline(y=0, color='k')
-plt.axvline(x=0, color='k')
-plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs")
-plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^")
-plt.xlabel(r"$x_2$", fontsize=20)
-plt.ylabel(r"$x_3$  ", fontsize=20, rotation=0)
-plt.annotate(r'$\phi\left(\mathbf{x}\right)$',
-             xy=(XK[3, 0], XK[3, 1]),
-             xytext=(0.65, 0.50),
-             ha="center",
-             arrowprops=dict(facecolor='black', shrink=0.1),
-             fontsize=18,
-            )
-plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3)
-plt.axis([-0.1, 1.1, -0.1, 1.1])
-    
-plt.subplots_adjust(right=1)
-
-plt.show()
-
-
-x1_example = X1D[3, 0]
-for landmark in (-2, 1):
-    k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)
-    print("Phi({}, {}) = {}".format(x1_example, landmark, k))
-
-rbf_kernel_svm_clf = Pipeline([
-        ("scaler", StandardScaler()),
-        ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001))
-    ])
-rbf_kernel_svm_clf.fit(X, y)
-
-
-from sklearn.svm import SVC
-
-gamma1, gamma2 = 0.1, 5
-C1, C2 = 0.001, 1000
-hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
-
-svm_clfs = []
-for gamma, C in hyperparams:
-    rbf_kernel_svm_clf = Pipeline([
-            ("scaler", StandardScaler()),
-            ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C))
-        ])
-    rbf_kernel_svm_clf.fit(X, y)
-    svm_clfs.append(rbf_kernel_svm_clf)
-
-plt.figure(figsize=(11, 7))
-
-for i, svm_clf in enumerate(svm_clfs):
-    plt.subplot(221 + i)
-    plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])
-    plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])
-    gamma, C = hyperparams[i]
-    plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16)
-
-plt.show()
-
-
-
-
-_images/chapter5_129_0.png -
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/svm/_base.py:976: ConvergenceWarning: Liblinear failed to converge, increase the number of iterations.
-  warnings.warn("Liblinear failed to converge, increase "
-
-
-_images/chapter5_129_2.png -_images/chapter5_129_3.png -_images/chapter5_129_4.png -
Phi(-1.0, -2) = [0.74081822]
-Phi(-1.0, 1) = [0.30119421]
-
-
-_images/chapter5_129_6.png -
-
-
-
-

5.7. Mathematical optimization of convex functions

-

A mathematical (quadratic) optimization problem, or just optimization problem, has the form

-
-\[\begin{split} -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} -\end{split}\]
-

subject to some constraints for say a selected set \(i=1,2,\dots, n\). -In our case we are optimizing with respect to the Lagrangian multipliers \(\lambda_i\), and the -vector \(\boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n]\) is the optimization variable we are dealing with.

-

In our case we are particularly interested in a class of optimization problems called convex optmization problems. -In our discussion on gradient descent methods we discussed at length the definition of a convex function.

-

Convex optimization problems play a central role in applied mathematics and we recommend strongly Boyd and Vandenberghe’s text on the topics.

-

If we use Python as programming language and wish to venture beyond -scikit-learn, tensorflow and similar software which makes our -lives so much easier, we need to dive into the wonderful world of -quadratic programming. We can, if we wish, solve the minimization -problem using say standard gradient methods or conjugate gradient -methods. However, these methods tend to exhibit a rather slow -converge. So, welcome to the promised land of quadratic programming.

-

The functions we need are contained in the quadratic programming package CVXOPT and we need to import it together with numpy as

-
-
-
import numpy
-import cvxopt
-
-
-
-
-

This will make our life much easier. You don’t need t write your own optimizer.

-

We remind ourselves about the general problem we want to solve

-
-\[\begin{split} -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber - &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. -\end{align*} -\end{split}\]
-

Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem

-
-\[\begin{split} -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber - &\mathrm{subject to} \\ \nonumber - &x, y \geq 0 \\ \nonumber - &x+3y \geq 15 \\ \nonumber - &2x+5y \leq 100 \\ \nonumber - &3x+4y \leq 80. \\ \nonumber -\end{align*} -\end{split}\]
-

The minimization problem can be rewritten in terms of vectors and matrices as (with \(x\) and \(y\) being the unknowns)

-
-\[\begin{split} -\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. -\end{split}\]
-

Similarly, we can now set up the inequalities (we need to change \(\geq\) to \(\leq\) by multiplying with \(-1\) on bot sides) as the following matrix-vector equation

-
-\[\begin{split} -\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -\end{split}\]
-

We have collapsed all the inequalities into a single matrix \(\boldsymbol{G}\). We see also that our matrix

-
-\[\begin{split} -\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} -\end{split}\]
-

is clearly positive semi-definite (all eigenvalues larger or equal zero). -Finally, the vector \(\boldsymbol{h}\) is defined as

-
-\[\begin{split} -\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -\end{split}\]
-

Since we don’t have any equalities the matrix \(\boldsymbol{A}\) is set to zero -The following code solves the equations for us

-
-
-
# Import the necessary packages
-import numpy
-from cvxopt import matrix
-from cvxopt import solvers
-P = matrix(numpy.diag([1,0]), tc=’d’)
-q = matrix(numpy.array([3,4]), tc=’d’)
-G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
-h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
-# Construct the QP, invoke solver
-sol = solvers.qp(P,q,G,h)
-# Extract optimal value and solution
-sol[’x’] 
-sol[’primal objective’]
-
-
-
-
-
  File "<ipython-input-5-c46dd114b2af>", line 5
-    P = matrix(numpy.diag([1,0]), tc=’d’)
-                                       ^
-SyntaxError: invalid character in identifier
-
-
-
-
-

We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the slack parameter \(C\) we have

-
-\[\begin{split} -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, -\end{split}\]
-

subject to \(\boldsymbol{y}^T\boldsymbol{\lambda}=0\). Here we defined the vectors \(\boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n]\) and -\(\boldsymbol{y}=[y_1,y_2,\dots,y_n]\). -With the slack constants this leads to the additional constraint \(0\leq \lambda_i \leq C\).

-

code will be added

-
-
- - - - -
- - -
- - 4. Logistic Regression - 1. Decision trees, overarching aims - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

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- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter6.html b/doc/LectureNotes/_build/html/chapter6.html deleted file mode 100644 index a455b2bb8..000000000 --- a/doc/LectureNotes/_build/html/chapter6.html +++ /dev/null @@ -1,1452 +0,0 @@ - - - - - - - - 1. Decision trees, overarching aims — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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- - Contents -
- - -
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-

1. Decision trees, overarching aims

-

We start here with the most basic algorithm, the so-called decision -tree. With this basic algorithm we can in turn build more complex -networks, spanning from homogeneous and heterogenous forests (bagging, -random forests and more) to one of the most popular supervised -algorithms nowadays, the extreme gradient boosting, or just -XGBoost. But let us start with the simplest possible ingredient.

-

Decision trees are supervised learning algorithms used for both, -classification and regression tasks.

-

The main idea of decision trees -is to find those descriptive features which contain the most -information regarding the target feature and then split the dataset -along the values of these features such that the target feature values -for the resulting underlying datasets are as pure as possible.

-

The descriptive features which reproduce best the target/output features are normally said -to be the most informative ones. The process of finding the most -informative feature is done until we accomplish a stopping criteria -where we then finally end up in so called leaf nodes.

-
-

1.1. Basics of a tree

-

A decision tree is typically divided into a root node, the interior nodes, -and the final leaf nodes or just leaves. These entities are then connected by so-called branches.

-

The leaf nodes -contain the predictions we will make for new query instances presented -to our trained model. This is possible since the model has -learned the underlying structure of the training data and hence can, -given some assumptions, make predictions about the target feature value -(class) of unseen query instances.

-
-
-

1.2. General Features

-

The overarching approach to decision trees is a top-down approach.

-
    -

  • A leaf provides the classification of a given instance.

    • -

  • A node specifies a test of some attribute of the instance.

    • -

  • A branch corresponds to a possible values of an attribute.

    • -

  • An instance is classified by starting at the root node of the tree, testing the attribute specified by this node, then moving down the tree branch corresponding to the value of the attribute in the given example.

    • -
-

This process is then repeated for the subtree rooted at the new -node.

-

In simplified terms, the process of training a decision tree and -predicting the target features of query instances is as follows:

-
    -

  1. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature

    2. -

  2. Train the decision tree model by continuously splitting the target feature along the values of the descriptive features using a measure of information gain during the training process

    3. -

  3. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the predictions we want to make for new query instances

    4. -

  4. Show query instances to the tree and run down the tree until we arrive at leaf nodes

    5. -
-

Then we are essentially done!

-
-
-
%matplotlib inline
-
-import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.linear_model import LinearRegression
-
-steps=250
-
-distance=0
-x=0
-distance_list=[]
-steps_list=[]
-while x<steps:
-    distance+=np.random.randint(-1,2)
-    distance_list.append(distance)
-    x+=1
-    steps_list.append(x)
-plt.plot(steps_list,distance_list, color='green', label="Random Walk Data")
-
-steps_list=np.asarray(steps_list)
-distance_list=np.asarray(distance_list)
-
-X=steps_list[:,np.newaxis]
-
-#Polynomial fits
-
-#Degree 2
-poly_features=PolynomialFeatures(degree=2, include_bias=False)
-X_poly=poly_features.fit_transform(X)
-
-lin_reg=LinearRegression()
-poly_fit=lin_reg.fit(X_poly,distance_list)
-b=lin_reg.coef_
-c=lin_reg.intercept_
-print ("2nd degree coefficients:")
-print ("zero power: ",c)
-print ("first power: ", b[0])
-print ("second power: ",b[1])
-
-z = np.arange(0, steps, .01)
-z_mod=b[1]*z**2+b[0]*z+c
-
-fit_mod=b[1]*X**2+b[0]*X+c
-plt.plot(z, z_mod, color='r', label="2nd Degree Fit")
-plt.title("Polynomial Regression")
-
-plt.xlabel("Steps")
-plt.ylabel("Distance")
-
-#Degree 10
-poly_features10=PolynomialFeatures(degree=10, include_bias=False)
-X_poly10=poly_features10.fit_transform(X)
-
-poly_fit10=lin_reg.fit(X_poly10,distance_list)
-
-y_plot=poly_fit10.predict(X_poly10)
-plt.plot(X, y_plot, color='black', label="10th Degree Fit")
-
-plt.legend()
-plt.show()
-
-
-#Decision Tree Regression
-from sklearn.tree import DecisionTreeRegressor
-regr_1=DecisionTreeRegressor(max_depth=2)
-regr_2=DecisionTreeRegressor(max_depth=5)
-regr_3=DecisionTreeRegressor(max_depth=7)
-regr_1.fit(X, distance_list)
-regr_2.fit(X, distance_list)
-regr_3.fit(X, distance_list)
-
-X_test = np.arange(0.0, steps, 0.01)[:, np.newaxis]
-y_1 = regr_1.predict(X_test)
-y_2 = regr_2.predict(X_test)
-y_3=regr_3.predict(X_test)
-
-# Plot the results
-plt.figure()
-plt.scatter(X, distance_list, s=2.5, c="black", label="data")
-plt.plot(X_test, y_1, color="red",
-         label="max_depth=2", linewidth=2)
-plt.plot(X_test, y_2, color="green", label="max_depth=5", linewidth=2)
-plt.plot(X_test, y_3, color="m", label="max_depth=7", linewidth=2)
-
-plt.xlabel("Data")
-plt.ylabel("Darget")
-plt.title("Decision Tree Regression")
-plt.legend()
-plt.show()
-
-
-
-
-
2nd degree coefficients:
-zero power:  1.3961181500194275
-first power:  -0.0007183308296987448
-second power:  0.0002882855342275337
-
-
-_images/chapter6_1_1.png -_images/chapter6_1_2.png -
-
-
-
-

1.3. Building a tree, regression

-

There are mainly two steps

-
    -

  1. We split the predictor space (the set of possible values \(x_1,x_2,\dots, x_p\)) into \(J\) distinct and non-non-overlapping regions, \(R_1,R_2,\dots,R_J\).

    2. -

  2. For every observation that falls into the region \(R_j\) , we make the same prediction, which is simply the mean of the response values for the training observations in \(R_j\).

    3. -
-

How do we construct the regions \(R_1,\dots,R_J\)? In theory, the -regions could have any shape. However, we choose to divide the -predictor space into high-dimensional rectangles, or boxes, for -simplicity and for ease of interpretation of the resulting predictive -model. The goal is to find boxes \(R_1,\dots,R_J\) that minimize the -MSE, given by

-
-\[ -\sum_{j=1}^J\sum_{i\in R_j}(y_i-\overline{y}_{R_j})^2, -\]
-

where \(\overline{y}_{R_j}\) is the mean response for the training observations -within box \(j\).

-

Unfortunately, it is computationally infeasible to consider every -possible partition of the feature space into \(J\) boxes. The common -strategy is to take a top-down approach

-

The approach is top-down because it begins at the top of the tree (all -observations belong to a single region) and then successively splits -the predictor space; each split is indicated via two new branches -further down on the tree. It is greedy because at each step of the -tree-building process, the best split is made at that particular step, -rather than looking ahead and picking a split that will lead to a -better tree in some future step.

-
-

1.3.1. Making a tree

-

In order to implement the recursive binary splitting we start by selecting -the predictor \(x_j\) and a cutpoint \(s\) that splits the predictor space into two regions \(R_1\) and \(R_2\)

-
-\[ -\left\{X\vert x_j < s\right\}, -\]
-

and

-
-\[ -\left\{X\vert x_j \geq s\right\}, -\]
-

so that we obtain the lowest MSE, that is

-
-\[ -\sum_{i:x_i\in R_j}(y_i-\overline{y}_{R_1})^2+\sum_{i:x_i\in R_2}(y_i-\overline{y}_{R_2})^2, -\]
-

which we want to minimize by considering all predictors -\(x_1,x_2,\dots,x_p\). We consider also all possible values of \(s\) for -each predictor. These values could be determined by randomly assigned -numbers or by starting at the midpoint and then proceed till we find -an optimal value.

-

For any \(j\) and \(s\), we define the pair of half-planes where -\(\overline{y}_{R_1}\) is the mean response for the training -observations in \(R_1(j,s)\), and \(\overline{y}_{R_2}\) is the mean -response for the training observations in \(R_2(j,s)\).

-

Finding the values of \(j\) and \(s\) that minimize the above equation can be -done quite quickly, especially when the number of features \(p\) is not -too large.

-

Next, we repeat the process, looking -for the best predictor and best cutpoint in order to split the data -further so as to minimize the MSE within each of the resulting -regions. However, this time, instead of splitting the entire predictor -space, we split one of the two previously identified regions. We now -have three regions. Again, we look to split one of these three regions -further, so as to minimize the MSE. The process continues until a -stopping criterion is reached; for instance, we may continue until no -region contains more than five observations.

-

The above procedure is rather straightforward, but leads often to -overfitting and unnecessarily large and complicated trees. The basic -idea is to grow a large tree \(T_0\) and then prune it back in order to -obtain a subtree. A smaller tree with fewer splits (fewer regions) can -lead to smaller variance and better interpretation at the cost of a -little more bias.

-

The so-called Cost complexity pruning algorithm gives us a -way to do just this. Rather than considering every possible subtree, -we consider a sequence of trees indexed by a nonnegative tuning -parameter \(\alpha\).

-

Read more at the following Scikit-Learn link on pruning.

-

For each value of \(\alpha\) there corresponds a subtree \(T \in T_0\) such that

-
-\[ -\sum_{m=1}^{\overline{T}}\sum_{i:x_i\in R_m}(y_i-\overline{y}_{R_m})^2+\alpha\overline{T}, -\]
-

is as small as possible. Here \(\overline{T}\) is -the number of terminal nodes of the tree \(T\) , \(R_m\) is the -rectangle (i.e. the subset of predictor space) corresponding to the \(m\)-th terminal node.

-

The tuning parameter \(\alpha\) controls a trade-off between the subtree’s -complexity and its fit to the training data. When \(\alpha = 0\), then the -subtree \(T\) will simply equal \(T_0\), -because then the above equation just measures the -training error. -However, as \(\alpha\) increases, there is a price to pay for -having a tree with many terminal nodes. The above equation will -tend to be minimized for a smaller subtree.

-

It turns out that as we increase \(\alpha\) from zero -branches get pruned from the tree in a nested and predictable fashion, -so obtaining the whole sequence of subtrees as a function of \(\alpha\) is -easy. We can select a value of \(\alpha\) using a validation set or using -cross-validation. We then return to the full data set and obtain the -subtree corresponding to \(\alpha\).

-
-
-

1.3.2. Schematic Regression Procedure

-

Building a Regression Tree

-
    -

  1. Use recursive binary splitting to grow a large tree on the training data, stopping only when each terminal node has fewer than some minimum number of observations.

    2. -

  2. Apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of \(\alpha\).

    3. -

  3. Use for example \(K\)-fold cross-validation to choose \(\alpha\). Divide the training observations into \(K\) folds. For each \(k=1,2,\dots,K\) we:

    4. -
-
    -

  • repeat steps 1 and 2 on all but the \(k\)-th fold of the training data.

    • -

  • Then we valuate the mean squared prediction error on the data in the left-out \(k\)-th fold, as a function of \(\alpha\).

    • -

  • Finally we average the results for each value of \(\alpha\), and pick \(\alpha\) to minimize the average error.

    • -
-
    -

  1. Return the subtree from Step 2 that corresponds to the chosen value of \(\alpha\).

    2. -
-

!eblock

-
-
-
-

1.4. A Classification Tree

-

A classification tree is very similar to a regression tree, except -that it is used to predict a qualitative response rather than a -quantitative one. Recall that for a regression tree, the predicted -response for an observation is given by the mean response of the -training observations that belong to the same terminal node. In -contrast, for a classification tree, we predict that each observation -belongs to the most commonly occurring class of training observations -in the region to which it belongs. In interpreting the results of a -classification tree, we are often interested not only in the class -prediction corresponding to a particular terminal node region, but -also in the class proportions among the training observations that -fall into that region.

-

The task of growing a -classification tree is quite similar to the task of growing a -regression tree. Just as in the regression setting, we use recursive -binary splitting to grow a classification tree. However, in the -classification setting, the MSE cannot be used as a criterion for making -the binary splits. A natural alternative to MSE is the classification -error rate. Since we plan to assign an observation in a given region -to the most commonly occurring error rate class of training -observations in that region, the classification error rate is simply -the fraction of the training observations in that region that do not -belong to the most common class.

-

When building a classification tree, either the Gini index or the -entropy are typically used to evaluate the quality of a particular -split, since these two approaches are more sensitive to node purity -than is the classification error rate.

-

If our targets are the outcome of a classification process that takes -for example \(k=1,2,\dots,K\) values, the only thing we need to think of -is to set up the splitting criteria for each node.

-

We define a PDF \(p_{mk}\) that represents the number of observations of -a class \(k\) in a region \(R_m\) with \(N_m\) observations. We represent -this likelihood function in terms of the proportion \(I(y_i=k)\) of -observations of this class in the region \(R_m\) as

-
-\[ -p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i=k). -\]
-

We let \(p_{mk}\) represent the majority class of observations in region -\(m\). The three most common ways of splitting a node are given by

-
    -

  • Misclassification error

    • -
-
-\[ -p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i\ne k) = 1-p_{mk}. -\]
-
    -

  • Gini index \(g\)

    • -
-
-\[ -g = \sum_{k=1}^K p_{mk}(1-p_{mk}). -\]
-
    -

  • Information entropy or just entropy \(s\)

    • -
-
-\[ -s = -\sum_{k=1}^K p_{mk}\log{p_{mk}}. -\]
-
-

1.4.1. Visualizing the Tree, Classification

-
-
-
import os
-from sklearn.datasets import load_breast_cancer
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.model_selection import train_test_split
-from sklearn.metrics import confusion_matrix
-from sklearn.tree import export_graphviz
-
-from IPython.display import Image 
-from pydot import graph_from_dot_data
-import pandas as pd
-import numpy as np
-
-
-cancer = load_breast_cancer()
-X = pd.DataFrame(cancer.data, columns=cancer.feature_names)
-print(X)
-y = pd.Categorical.from_codes(cancer.target, cancer.target_names)
-y = pd.get_dummies(y)
-print(y)
-X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)
-tree_clf = DecisionTreeClassifier(max_depth=5)
-tree_clf.fit(X_train, y_train)
-
-export_graphviz(
-    tree_clf,
-    out_file="DataFiles/cancer.dot",
-    feature_names=cancer.feature_names,
-    class_names=cancer.target_names,
-    rounded=True,
-    filled=True
-)
-cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
-os.system(cmd)
-
-
-
-
-
---------------------------------------------------------------------------
-ModuleNotFoundError                       Traceback (most recent call last)
-<ipython-input-2-7c394b1e8b71> in <module>
-      7 
-      8 from IPython.display import Image
-----> 9 from pydot import graph_from_dot_data
-     10 import pandas as pd
-     11 import numpy as np
-
-ModuleNotFoundError: No module named 'pydot'
-
-
-
-
-
-
-
# Common imports
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.datasets import make_moons
-from sklearn.tree import export_graphviz
-from pydot import graph_from_dot_data
-import pandas as pd
-import os
-
-np.random.seed(42)
-X, y = make_moons(n_samples=100, noise=0.25, random_state=53)
-X_train, X_test, y_train, y_test = train_test_split(X,y,random_state=0)
-tree_clf = DecisionTreeClassifier(max_depth=5)
-tree_clf.fit(X_train, y_train)
-
-export_graphviz(
-    tree_clf,
-    out_file="DataFiles/moons.dot",
-    rounded=True,
-    filled=True
-)
-cmd = 'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'
-os.system(cmd)
-
-
-
-
-
-
-

1.4.2. Other ways of visualizing the trees

-

Scikit-Learn has also another way to visualize the trees which is very useful, here with the Iris data.

-
-
-
from sklearn.datasets import load_iris
-from sklearn import tree
-X, y = load_iris(return_X_y=True)
-tree_clf = tree.DecisionTreeClassifier()
-tree_clf = tree_clf.fit(X, y)
-# and then plot the tree
-tree.plot_tree(tree_clf)
-
-
-
-
-

Alternatively, the tree can also be exported in textual format with the function exporttext. -This method doesn’t require the installation of external libraries and is more compact:

-
-
-
from sklearn.datasets import load_iris
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.tree import export_text
-iris = load_iris()
-decision_tree = DecisionTreeClassifier(random_state=0, max_depth=2)
-decision_tree = decision_tree.fit(iris.data, iris.target)
-r = export_text(decision_tree, feature_names=iris['feature_names'])
-print(r)
-
-
-
-
-
-
-
-

1.5. Algorithms for Setting up Decision Trees

-

Two algorithms stand out in the set up of decision trees:

-
    -

  1. The CART (Classification And Regression Tree) algorithm for both classification and regression

    2. -

  2. The ID3 algorithm based on the computation of the information gain for classification

    3. -
-

We discuss both algorithms with applications here. The popular library -Scikit-Learn uses the CART algorithm. For classification problems -you can use either the gini index or the entropy to split a tree -in two branches.

-
-

1.5.1. The CART algorithm for Classification

-

For classification, the CART algorithm splits the data set in two subsets using a single feature \(k\) and a threshold \(t_k\). -This could be for example a threshold set by a number below a certain circumference of a malign tumor.

-

How do we find these two quantities? -We search for the pair \((k,t_k)\) that produces the purest subset using for example the gini factor \(G\). -The cost function it tries to minimize is then

-
-\[ -C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}}, -\]
-

where \(G_{\mathrm{left/right}}\) measures the impurity of the left/right subset and \(m_{\mathrm{left/right}}\) -is the number of instances in the left/right subset

-

Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets -and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the -\(max\_depth\) hyperparameter), or if it cannot find a split that will reduce impurity. A few other -hyperparameters control additional stopping conditions such as the \(min\_samples\_split\), -\(min\_samples\_leaf\), \(min\_weight\_fraction\_leaf\), and \(max\_leaf\_nodes\).

-
-
-

1.5.2. The CART algorithm for Regression

-

The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the -training set in a way that minimizes say the gini or entropy impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now

-
-\[ -C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}. -\]
-

Here the MSE for a specific node is defined as

-
-\[ -\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2, -\]
-

with

-
-\[ -\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i, -\]
-

the mean value of all observations in a specific node.

-

Without any regularization, the regression task for decision trees, -just like for classification tasks, is prone to overfitting.

-
-
-

1.5.3. Computing the Gini index

-

The example we will look at is a classical one in many Machine -Learning applications. Based on various meteorological features, we -have several so-called attributes which decide whether we at the end -will do some outdoor activity like skiing, going for a bike ride etc -etc. The table here contains the feautures outlook, temperature, -humidity and wind. The target or output is whether we ride -(True=1) or whether we do something else that day (False=0). The -attributes for each feature are then sunny, overcast and rain for the -outlook, hot, cold and mild for temperature, high and normal for -humidity and weak and strong for wind.

-

The table here summarizes the various attributes and

- - - - - - - - - - - - - - - - - - - - -
Day Outlook Temperature Humidity Wind Ride
1 Sunny Hot High Weak 0
2 Sunny Hot High Strong 1
3 Overcast Hot High Weak 1
4 Rain Mild High Weak 1
5 Rain Cool Normal Weak 1
6 Rain Cool Normal Strong 0
7 Overcast Cool Normal Strong 1
8 Sunny Mild High Weak 0
9 Sunny Cool Normal Weak 1
10 Rain Mild Normal Weak 1
11 Sunny Mild Normal Strong 1
12 Overcast Mild High Strong 1
13 Overcast Hot Normal Weak 1
14 Rain Mild High Strong 0
-
-
-

1.5.4. Simple Python Code to read in Data and perform Classification

-
-
-
# Common imports
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.model_selection import train_test_split
-from sklearn.tree import export_graphviz
-from sklearn.preprocessing import StandardScaler, OneHotEncoder
-from sklearn.compose import ColumnTransformer
-from IPython.display import Image 
-from pydot import graph_from_dot_data
-import os
-
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
-
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
-
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
-
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("rideclass.csv"),'r')
-
-# Read the experimental data with Pandas
-from IPython.display import display
-ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))
-ridedata = pd.DataFrame(ridedata)
-
-# Features and targets
-X = ridedata.loc[:, ridedata.columns != 'Ride'].values
-y = ridedata.loc[:, ridedata.columns == 'Ride'].values
-
-# Create the encoder.
-encoder = OneHotEncoder(handle_unknown="ignore")
-# Assume for simplicity all features are categorical.
-encoder.fit(X)    
-# Apply the encoder.
-X = encoder.transform(X)
-print(X)
-# Then do a Classification tree
-tree_clf = DecisionTreeClassifier(max_depth=2)
-tree_clf.fit(X, y)
-print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
-#transfer to a decision tree graph
-export_graphviz(
-    tree_clf,
-    out_file="DataFiles/ride.dot",
-    rounded=True,
-    filled=True
-)
-cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
-os.system(cmd)
-
-
-
-
-

The above functions (gini, entropy and misclassification error) are -important components of the so-called CART algorithm. We will discuss -this algorithm below after we have discussed the information gain -algorithm ID3.

-

In the example here we have converted all our attributes into numerical values \(0,1,2\) etc.

-
-
-
# Split a dataset based on an attribute and an attribute value
-def test_split(index, value, dataset):
-	left, right = list(), list()
-	for row in dataset:
-		if row[index] < value:
-			left.append(row)
-		else:
-			right.append(row)
-	return left, right
- 
-# Calculate the Gini index for a split dataset
-def gini_index(groups, classes):
-	# count all samples at split point
-	n_instances = float(sum([len(group) for group in groups]))
-	# sum weighted Gini index for each group
-	gini = 0.0
-	for group in groups:
-		size = float(len(group))
-		# avoid divide by zero
-		if size == 0:
-			continue
-		score = 0.0
-		# score the group based on the score for each class
-		for class_val in classes:
-			p = [row[-1] for row in group].count(class_val) / size
-			score += p * p
-		# weight the group score by its relative size
-		gini += (1.0 - score) * (size / n_instances)
-	return gini
-
-# Select the best split point for a dataset
-def get_split(dataset):
-	class_values = list(set(row[-1] for row in dataset))
-	b_index, b_value, b_score, b_groups = 999, 999, 999, None
-	for index in range(len(dataset[0])-1):
-		for row in dataset:
-			groups = test_split(index, row[index], dataset)
-			gini = gini_index(groups, class_values)
-			print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
-			if gini < b_score:
-				b_index, b_value, b_score, b_groups = index, row[index], gini, groups
-	return {'index':b_index, 'value':b_value, 'groups':b_groups}
- 
-dataset = [[0,0,0,0,0],
-            [0,0,0,1,1],
-            [1,0,0,0,1],
-            [2,1,0,0,1],
-            [2,2,1,0,1],
-            [2,2,1,1,0],
-            [1,2,1,1,1],
-            [0,1,0,0,0],
-            [0,2,1,0,1],
-            [2,1,1,0,1],
-            [0,1,1,1,1],
-            [1,1,0,1,1],
-            [1,0,1,0,1],
-            [2,1,0,1,0]]
-
-split = get_split(dataset)
-print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))
-
-
-
-
-
-
-
-

1.6. Entropy and the ID3 algorithm

-

The ID3 algorithm learns decision trees by constructing -them in a top down way, beginning with the question which attribute should be tested at the root of the tree?

-
    -

  1. Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.

    2. -

  2. The best attribute is selected and used as the test at the root node of the tree.

    3. -

  3. A descendant of the root node is then created for each possible value of this attribute.

    4. -

  4. Training examples are sorted to the appropriate descendant node.

    5. -

  5. The entire process is then repeated using the training examples associated with each descendant node to select the best attribute to test at that point in the tree.

    6. -

  6. This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.

    7. -
-

The ID3 algorithm selects which attribute to test at each node in the -tree.

-

We would like to select the attribute that is most useful for classifying -examples.

-

What is a good quantitative measure of the worth of an attribute?

-

Information gain measures how well a given attribute separates the -training examples according to their target classification.

-

The ID3 algorithm uses this information gain measure to select among the candidate -attributes at each step while growing the tree.

-
-

1.6.1. Cancer Data again now with Decision Trees and other Methods

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.svm import SVC
-from sklearn.linear_model import LogisticRegression
-from sklearn.tree import DecisionTreeClassifier
-
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-# Logistic Regression
-logreg = LogisticRegression(solver='lbfgs')
-logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
-# Support vector machine
-svm = SVC(gamma='auto', C=100)
-svm.fit(X_train, y_train)
-print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
-# Decision Trees
-deep_tree_clf = DecisionTreeClassifier(max_depth=None)
-deep_tree_clf.fit(X_train, y_train)
-print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Logistic Regression
-logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-# Support Vector Machine
-svm.fit(X_train_scaled, y_train)
-print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-# Decision Trees
-deep_tree_clf.fit(X_train_scaled, y_train)
-print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
-
-
-
-
-
-
-

1.6.2. Another example, the moons again

-
-
-
from __future__ import division, print_function, unicode_literals
-
-# Common imports
-import numpy as np
-import os
-
-# to make this notebook's output stable across runs
-np.random.seed(42)
-
-# To plot pretty figures
-import matplotlib
-import matplotlib.pyplot as plt
-from matplotlib.colors import ListedColormap
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
-
-
-from sklearn.svm import SVC
-from sklearn import datasets
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.datasets import make_moons
-from sklearn.tree import export_graphviz
-
-Xm, ym = make_moons(n_samples=100, noise=0.25, random_state=53)
-
-deep_tree_clf1 = DecisionTreeClassifier(random_state=42)
-deep_tree_clf2 = DecisionTreeClassifier(min_samples_leaf=4, random_state=42)
-deep_tree_clf1.fit(Xm, ym)
-deep_tree_clf2.fit(Xm, ym)
-
-
-def plot_decision_boundary(clf, X, y, axes=[0, 7.5, 0, 3], iris=True, legend=False, plot_training=True):
-    x1s = np.linspace(axes[0], axes[1], 100)
-    x2s = np.linspace(axes[2], axes[3], 100)
-    x1, x2 = np.meshgrid(x1s, x2s)
-    X_new = np.c_[x1.ravel(), x2.ravel()]
-    y_pred = clf.predict(X_new).reshape(x1.shape)
-    custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
-    plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
-    if not iris:
-        custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
-        plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
-    if plot_training:
-        plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", label="Iris-Setosa")
-        plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", label="Iris-Versicolor")
-        plt.plot(X[:, 0][y==2], X[:, 1][y==2], "g^", label="Iris-Virginica")
-        plt.axis(axes)
-    if iris:
-        plt.xlabel("Petal length", fontsize=14)
-        plt.ylabel("Petal width", fontsize=14)
-    else:
-        plt.xlabel(r"$x_1$", fontsize=18)
-        plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
-    if legend:
-        plt.legend(loc="lower right", fontsize=14)
-plt.figure(figsize=(11, 4))
-plt.subplot(121)
-plot_decision_boundary(deep_tree_clf1, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
-plt.title("No restrictions", fontsize=16)
-plt.subplot(122)
-plot_decision_boundary(deep_tree_clf2, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
-plt.title("min_samples_leaf = {}".format(deep_tree_clf2.min_samples_leaf), fontsize=14)
-plt.show()
-
-
-
-
-
-
-
np.random.seed(6)
-Xs = np.random.rand(100, 2) - 0.5
-ys = (Xs[:, 0] > 0).astype(np.float32) * 2
-
-angle = np.pi/4
-rotation_matrix = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])
-Xsr = Xs.dot(rotation_matrix)
-
-tree_clf_s = DecisionTreeClassifier(random_state=42)
-tree_clf_s.fit(Xs, ys)
-tree_clf_sr = DecisionTreeClassifier(random_state=42)
-tree_clf_sr.fit(Xsr, ys)
-
-plt.figure(figsize=(11, 4))
-plt.subplot(121)
-plot_decision_boundary(tree_clf_s, Xs, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
-plt.subplot(122)
-plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
-
-plt.show()
-
-
-
-
-
-
-
# Quadratic training set + noise
-np.random.seed(42)
-m = 200
-X = np.random.rand(m, 1)
-y = 4 * (X - 0.5) ** 2
-y = y + np.random.randn(m, 1) / 10
-
-
-
-
-
-
-
from sklearn.tree import DecisionTreeRegressor
-
-tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)
-tree_reg.fit(X, y)
-
-
-
-
-
-
-
from sklearn.tree import DecisionTreeRegressor
-
-tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)
-tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)
-tree_reg1.fit(X, y)
-tree_reg2.fit(X, y)
-
-def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel="$y$"):
-    x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)
-    y_pred = tree_reg.predict(x1)
-    plt.axis(axes)
-    plt.xlabel("$x_1$", fontsize=18)
-    if ylabel:
-        plt.ylabel(ylabel, fontsize=18, rotation=0)
-    plt.plot(X, y, "b.")
-    plt.plot(x1, y_pred, "r.-", linewidth=2, label=r"$\hat{y}$")
-
-plt.figure(figsize=(11, 4))
-plt.subplot(121)
-plot_regression_predictions(tree_reg1, X, y)
-for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
-    plt.plot([split, split], [-0.2, 1], style, linewidth=2)
-plt.text(0.21, 0.65, "Depth=0", fontsize=15)
-plt.text(0.01, 0.2, "Depth=1", fontsize=13)
-plt.text(0.65, 0.8, "Depth=1", fontsize=13)
-plt.legend(loc="upper center", fontsize=18)
-plt.title("max_depth=2", fontsize=14)
-
-plt.subplot(122)
-plot_regression_predictions(tree_reg2, X, y, ylabel=None)
-for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
-    plt.plot([split, split], [-0.2, 1], style, linewidth=2)
-for split in (0.0458, 0.1298, 0.2873, 0.9040):
-    plt.plot([split, split], [-0.2, 1], "k:", linewidth=1)
-plt.text(0.3, 0.5, "Depth=2", fontsize=13)
-plt.title("max_depth=3", fontsize=14)
-
-plt.show()
-
-
-
-
-
-
-
tree_reg1 = DecisionTreeRegressor(random_state=42)
-tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)
-tree_reg1.fit(X, y)
-tree_reg2.fit(X, y)
-
-x1 = np.linspace(0, 1, 500).reshape(-1, 1)
-y_pred1 = tree_reg1.predict(x1)
-y_pred2 = tree_reg2.predict(x1)
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.plot(X, y, "b.")
-plt.plot(x1, y_pred1, "r.-", linewidth=2, label=r"$\hat{y}$")
-plt.axis([0, 1, -0.2, 1.1])
-plt.xlabel("$x_1$", fontsize=18)
-plt.ylabel("$y$", fontsize=18, rotation=0)
-plt.legend(loc="upper center", fontsize=18)
-plt.title("No restrictions", fontsize=14)
-
-plt.subplot(122)
-plt.plot(X, y, "b.")
-plt.plot(x1, y_pred2, "r.-", linewidth=2, label=r"$\hat{y}$")
-plt.axis([0, 1, -0.2, 1.1])
-plt.xlabel("$x_1$", fontsize=18)
-plt.title("min_samples_leaf={}".format(tree_reg2.min_samples_leaf), fontsize=14)
-
-plt.show()
-
-
-
-
-
-
-
-

1.7. Pros and cons of trees, pros

-
    -

  • White box, easy to interpret model. Some people believe that decision trees more closely mirror human decision-making than do the regression and classification approaches discussed earlier (think of support vector machines)

    • -

  • Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!

    • -

  • No feature normalization needed

    • -

  • Tree models can handle both continuous and categorical data (Classification and Regression Trees)

    • -

  • Can model nonlinear relationships

    • -

  • Can model interactions between the different descriptive features

    • -

  • Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)

    • -
-
-

1.7.1. Disadvantages

-
    -

  • Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches

    • -

  • If continuous features are used the tree may become quite large and hence less interpretable

    • -

  • Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented

    • -

  • Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests

    • -

  • Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones.

    • -

  • If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data

    • -

  • Features with many levels may be preferred over features with less levels since for them it is more easy to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain

    • -
-

However, by aggregating many decision trees, using methods like -bagging, random forests, and boosting, the predictive performance of -trees can be substantially improved.

-
-
-
- - - - -
- - -
- - 5. Support Vector Machines, overarching aims - 2. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods - -
- -
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-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

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- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter7.html b/doc/LectureNotes/_build/html/chapter7.html deleted file mode 100644 index 7047e89a5..000000000 --- a/doc/LectureNotes/_build/html/chapter7.html +++ /dev/null @@ -1,1462 +0,0 @@ - - - - - - - - 2. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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- - -
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- -
- -
-

2. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods

-

As stated previously and seen in many of the examples discussed in the previous chapter about -a single decision tree, we often end up overfitting our training -data. This normally means that we have a high variance. Can we reduce -the variance of a statistical learning method?

-

This leads us to a set of different methods that can combine different -machine learning algorithms or just use one of them to construct -forests and jungles of trees, homogeneous ones or heterogenous -ones. These methods are recognized by different names which we will -try to explain here. These are

-
    -

  1. Voting classifiers

    2. -

  2. Bagging and Pasting

    3. -

  3. Random forests

    4. -

  4. Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)

    5. -
-

We discuss these methods here.

-
-

2.1. An Overview of Ensemble Methods

- -
-
-

2.2. Bagging

-

The plain decision trees suffer from high -variance. This means that if we split the training data into two parts -at random, and fit a decision tree to both halves, the results that we -get could be quite different. In contrast, a procedure with low -variance will yield similar results if applied repeatedly to distinct -data sets; linear regression tends to have low variance, if the ratio -of \(n\) to \(p\) is moderately large.

-

Bootstrap aggregation, or just bagging, is a -general-purpose procedure for reducing the variance of a statistical -learning method.

-

Bagging typically results in improved accuracy -over prediction using a single tree. Unfortunately, however, it can be -difficult to interpret the resulting model. Recall that one of the -advantages of decision trees is the attractive and easily interpreted -diagram that results.

-

However, when we bag a large number of trees, it is no longer -possible to represent the resulting statistical learning procedure -using a single tree, and it is no longer clear which variables are -most important to the procedure. Thus, bagging improves prediction -accuracy at the expense of interpretability. Although the collection -of bagged trees is much more difficult to interpret than a single -tree, one can obtain an overall summary of the importance of each -predictor using the MSE (for bagging regression trees) or the Gini -index (for bagging classification trees). In the case of bagging -regression trees, we can record the total amount that the MSE is -decreased due to splits over a given predictor, averaged over all \(B\) possible -trees. A large value indicates an important predictor. Similarly, in -the context of bagging classification trees, we can add up the total -amount that the Gini index is decreased by splits over a given -predictor, averaged over all \(B\) trees.

-
-
-
heads_proba = 0.51
-coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
-cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)
-plt.figure(figsize=(8,3.5))
-plt.plot(cumulative_heads_ratio)
-plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%")
-plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%")
-plt.xlabel("Number of coin tosses")
-plt.ylabel("Heads ratio")
-plt.legend(loc="lower right")
-plt.axis([0, 10000, 0.42, 0.58])
-save_fig("votingsimple")
-plt.show()
-
-
-
-
-
---------------------------------------------------------------------------
-NameError                                 Traceback (most recent call last)
-<ipython-input-1-eface79dac2c> in <module>
-      1 heads_proba = 0.51
-----> 2 coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
-      3 cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)
-      4 plt.figure(figsize=(8,3.5))
-      5 plt.plot(cumulative_heads_ratio)
-
-NameError: name 'np' is not defined
-
-
-
-
-
-
-
from sklearn.model_selection import train_test_split
-from sklearn.datasets import make_moons
-
-X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
-X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
-
-from sklearn.ensemble import RandomForestClassifier
-from sklearn.ensemble import VotingClassifier
-from sklearn.linear_model import LogisticRegression
-from sklearn.svm import SVC
-
-log_clf = LogisticRegression(solver="liblinear", random_state=42)
-rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
-svm_clf = SVC(gamma="auto", random_state=42)
-
-voting_clf = VotingClassifier(
-    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
-    voting='hard')
-
-voting_clf.fit(X_train, y_train)
-
-from sklearn.metrics import accuracy_score
-
-for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
-    clf.fit(X_train, y_train)
-    y_pred = clf.predict(X_test)
-    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
-
-log_clf = LogisticRegression(solver="liblinear", random_state=42)
-rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
-svm_clf = SVC(gamma="auto", probability=True, random_state=42)
-
-voting_clf = VotingClassifier(
-    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
-    voting='soft')
-voting_clf.fit(X_train, y_train)
-
-from sklearn.metrics import accuracy_score
-
-for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
-    clf.fit(X_train, y_train)
-    y_pred = clf.predict(X_test)
-    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
-
-
-
-
-
-
-
from sklearn.model_selection import train_test_split
-from sklearn.datasets import make_moons
-
-X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
-X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
-from sklearn.ensemble import RandomForestClassifier
-from sklearn.ensemble import VotingClassifier
-from sklearn.linear_model import LogisticRegression
-from sklearn.svm import SVC
-
-log_clf = LogisticRegression(random_state=42)
-rnd_clf = RandomForestClassifier(random_state=42)
-svm_clf = SVC(random_state=42)
-
-voting_clf = VotingClassifier(
-    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
-    voting='hard')
-voting_clf.fit(X_train, y_train)
-
-
-
-
-
-
-
from sklearn.metrics import accuracy_score
-
-for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
-    clf.fit(X_train, y_train)
-    y_pred = clf.predict(X_test)
-    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
-
-
-
-
-
-
-
log_clf = LogisticRegression(random_state=42)
-rnd_clf = RandomForestClassifier(random_state=42)
-svm_clf = SVC(probability=True, random_state=42)
-
-voting_clf = VotingClassifier(
-    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
-    voting='soft')
-voting_clf.fit(X_train, y_train)
-
-
-
-
-
-
-
from sklearn.metrics import accuracy_score
-
-for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
-    clf.fit(X_train, y_train)
-    y_pred = clf.predict(X_test)
-    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
-
-
-
-
-
-
-

2.3. Bagging Examples

-
-
-
from sklearn.ensemble import BaggingClassifier
-from sklearn.tree import DecisionTreeClassifier
-
-bag_clf = BaggingClassifier(
-    DecisionTreeClassifier(random_state=42), n_estimators=500,
-    max_samples=100, bootstrap=True, n_jobs=-1, random_state=42)
-bag_clf.fit(X_train, y_train)
-y_pred = bag_clf.predict(X_test)
-
-
-
-
-
-
-
from sklearn.metrics import accuracy_score
-print(accuracy_score(y_test, y_pred))
-
-
-
-
-
-
-
tree_clf = DecisionTreeClassifier(random_state=42)
-tree_clf.fit(X_train, y_train)
-y_pred_tree = tree_clf.predict(X_test)
-print(accuracy_score(y_test, y_pred_tree))
-
-
-
-
-
-
-
%matplotlib inline
-
-from matplotlib.colors import ListedColormap
-
-def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True):
-    x1s = np.linspace(axes[0], axes[1], 100)
-    x2s = np.linspace(axes[2], axes[3], 100)
-    x1, x2 = np.meshgrid(x1s, x2s)
-    X_new = np.c_[x1.ravel(), x2.ravel()]
-    y_pred = clf.predict(X_new).reshape(x1.shape)
-    custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
-    plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
-    if contour:
-        custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
-        plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
-    plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", alpha=alpha)
-    plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", alpha=alpha)
-    plt.axis(axes)
-    plt.xlabel(r"$x_1$", fontsize=18)
-    plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
-plt.figure(figsize=(11,4))
-plt.subplot(121)
-plot_decision_boundary(tree_clf, X, y)
-plt.title("Decision Tree", fontsize=14)
-plt.subplot(122)
-plot_decision_boundary(bag_clf, X, y)
-plt.title("Decision Trees with Bagging", fontsize=14)
-save_fig("baggingtree")
-plt.show()
-
-
-
-
-
-

2.3.1. Making your own Bootstrap: Changing the Level of the Decision Tree

-

Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with -a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points \(n\)).

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.pipeline import make_pipeline
-from sklearn.utils import resample
-from sklearn.tree import DecisionTreeRegressor
-
-n = 100
-n_boostraps = 100
-maxdepth = 8
-
-# Make data set.
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-error = np.zeros(maxdepth)
-bias = np.zeros(maxdepth)
-variance = np.zeros(maxdepth)
-polydegree = np.zeros(maxdepth)
-X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
-
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-# we produce a simple tree first as benchmark
-simpletree = DecisionTreeRegressor(max_depth=3) 
-simpletree.fit(X_train_scaled, y_train)
-simpleprediction = simpletree.predict(X_test_scaled)
-for degree in range(1,maxdepth):
-    model = DecisionTreeRegressor(max_depth=degree) 
-    y_pred = np.empty((y_test.shape[0], n_boostraps))
-    for i in range(n_boostraps):
-        x_, y_ = resample(X_train_scaled, y_train)
-        model.fit(x_, y_)
-        y_pred[:, i] = model.predict(X_test_scaled)#.ravel()
-
-    polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-    print('Polynomial degree:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
- 
-mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2)
-print(mse_simpletree)
-plt.xlim(1,maxdepth)
-plt.plot(polydegree, error, label='MSE')
-plt.plot(polydegree, bias, label='bias')
-plt.plot(polydegree, variance, label='Variance')
-plt.legend()
-save_fig("baggingboot")
-plt.show()
-
-
-
-
-
-
-
-

2.4. Random forests

-

Random forests provide an improvement over bagged trees by way of a -small tweak that decorrelates the trees.

-

As in bagging, we build a -number of decision trees on bootstrapped training samples. But when -building these decision trees, each time a split in a tree is -considered, a random sample of \(m\) predictors is chosen as split -candidates from the full set of \(p\) predictors. The split is allowed to -use only one of those \(m\) predictors.

-

A fresh sample of \(m\) predictors is -taken at each split, and typically we choose

-
-\[ -m\approx \sqrt{p}. -\]
-

In building a random forest, at -each split in the tree, the algorithm is not even allowed to consider -a majority of the available predictors.

-

The reason for this is rather clever. Suppose that there is one very -strong predictor in the data set, along with a number of other -moderately strong predictors. Then in the collection of bagged -variable importance random forest trees, most or all of the trees will -use this strong predictor in the top split. Consequently, all of the -bagged trees will look quite similar to each other. Hence the -predictions from the bagged trees will be highly correlated. -Unfortunately, averaging many highly correlated quantities does not -lead to as large of a reduction in variance as averaging many -uncorrelated quantities. In particular, this means that bagging will -not lead to a substantial reduction in variance over a single tree in -this setting.

-

The algorithm described here can be applied to both classification and regression problems.

-

We will grow of forest of say \(B\) trees.

-
    -

  1. For \(b=1:B\)

    2. -
-
    -

  • Draw a bootstrap sample from the training data organized in our \(\boldsymbol{X}\) matrix.

    • -

  • We grow then a random forest tree \(T_b\) based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached

    • -
-
    -

  1. we select \(m \le p\) variables at random from the \(p\) predictors/features

    2. -

  2. pick the best split point among the \(m\) features using for example the CART algorithm and create a new node

    3. -

  3. split the node into daughter nodes

    4. -

  4. Output then the ensemble of trees \(\{T_b\}_1^{B}\) and make predictions for either a regression type of problem or a classification type of problem.

    5. -
-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.svm import SVC
-from sklearn.linear_model import LogisticRegression
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.ensemble import BaggingClassifier
-
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-# Logistic Regression
-logreg = LogisticRegression(solver='lbfgs')
-logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
-# Support vector machine
-svm = SVC(gamma='auto', C=100)
-svm.fit(X_train, y_train)
-print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
-# Decision Trees
-deep_tree_clf = DecisionTreeClassifier(max_depth=None)
-deep_tree_clf.fit(X_train, y_train)
-print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Logistic Regression
-logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-# Support Vector Machine
-svm.fit(X_train_scaled, y_train)
-print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-# Decision Trees
-deep_tree_clf.fit(X_train_scaled, y_train)
-print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
-
-
-from sklearn.ensemble import RandomForestClassifier
-from sklearn.preprocessing import LabelEncoder
-from sklearn.model_selection import cross_validate
-# Data set not specificied
-#Instantiate the model with 500 trees and entropy as splitting criteria
-Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy")
-Random_Forest_model.fit(X_train_scaled, y_train)
-#Cross validation
-accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
-
-
-import scikitplot as skplt
-y_pred = Random_Forest_model.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-plt.show()
-y_probas = Random_Forest_model.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-plt.show()
-
-
-
-
-

Recall that the cumulative gains curve shows the percentage of the -overall number of cases in a given category gained by targeting a -percentage of the total number of cases.

-

Similarly, the receiver operating characteristic curve, or ROC curve, -displays the diagnostic ability of a binary classifier system as its -discrimination threshold is varied. It plots the true positive rate against the false positive rate.

-
-

2.4.1. Compare Bagging on Trees with Random Forests

-
-
-
bag_clf = BaggingClassifier(
-    DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
-    n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
-
-
-
-
-
-
-
bag_clf.fit(X_train, y_train)
-y_pred = bag_clf.predict(X_test)
-from sklearn.ensemble import RandomForestClassifier
-rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
-rnd_clf.fit(X_train, y_train)
-y_pred_rf = rnd_clf.predict(X_test)
-np.sum(y_pred == y_pred_rf) / len(y_pred)
-
-
-
-
-
-
-
-

2.5. Boosting, a Bird’s Eye View

-

The basic idea is to combine weak classifiers in order to create a good -classifier. With a weak classifier we often intend a classifier which -produces results which are only slightly better than we would get by -random guesses.

-

This is done by applying in an iterative way a weak (or a standard -classifier like decision trees) to modify the data. In each iteration -we emphasize those observations which are misclassified by weighting -them with a factor.

-

Boosting is a way of fitting an additive expansion in a set of -elementary basis functions like for example some simple polynomials. -Assume for example that we have a function

-
-\[ -f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), -\]
-

where \(\beta_m\) are the expansion parameters to be determined in a -minimization process and \(b(x;\gamma_m)\) are some simple functions of -the multivariable parameter \(x\) which is characterized by the -parameters \(\gamma_m\).

-

As an example, consider the Sigmoid function we used in logistic -regression. In that case, we can translate the function -\(b(x;\gamma_m)\) into the Sigmoid function

-
-\[ -\sigma(t) = \frac{1}{1+\exp{(-t)}}, -\]
-

where \(t=\gamma_0+\gamma_1 x\) and the parameters \(\gamma_0\) and -\(\gamma_1\) were determined by the Logistic Regression fitting -algorithm.

-

As another example, consider the cost function we defined for linear regression

-
-\[ -C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. -\]
-

In this case the function \(f(x)\) was replaced by the design matrix -\(\boldsymbol{X}\) and the unknown linear regression parameters \(\boldsymbol{\beta}\), -that is \(\boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta}\). In linear regression we can -simply invert a matrix and obtain the parameters \(\beta\) by

-
-\[ -\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. -\]
-

In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \(\beta_m\) and \(\gamma_m\).

-
-

2.5.1. Iterative Fitting, Regression and Squared-error Cost Function

-

The way we proceed is as follows (here we specialize to the squared-error cost function)

-
    -

  1. Establish a cost function, here \(\cal{C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2\) with \(f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)\).

    2. -

  2. Initialize with a guess \(f_0(x)\). It could be one or even zero or some random numbers.

    3. -

  3. For \(m=1:M\)

    4. -
-

a. minimize \(\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2\) wrt \(\gamma\) and \(\beta\)

-

b. This gives the optimal values \(\beta_m\) and \(\gamma_m\)

-

c. Determine then the new values \(f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)\)

-

We could use any of the algorithms we have discussed till now. If we -use trees, \(\gamma\) parameterizes the split variables and split points -at the internal nodes, and the predictions at the terminal nodes.

-

To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.

-

For simplicity we assume also that our functions \(b(x;\gamma)=1+\gamma x\).

-

This means that for every iteration \(m\), we need to optimize

-
-\[ -(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2. -\]
-

We start our iteration by simply setting \(f_0(x)=0\). -Taking the derivatives with respect to \(\beta\) and \(\gamma\) we obtain

-
-\[ -\frac{\partial \cal{C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0, -\]
-

and

-
-\[ -\frac{\partial \cal{C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0. -\]
-

We can then rewrite these equations as (defining \(\boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x})\) with \(\boldsymbol{e}\) being the unit vector)

-
-\[ -\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, -\]
-

which gives us \(\beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w})\). Similarly we have

-
-\[ -\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0, -\]
-

which leads to \(\gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x})\). Inserting -for \(\beta\) gives us an equation for \(\gamma\). This is a non-linear equation in the unknown \(\gamma\) and has to be solved numerically.

-

The solution to these two equations gives us in turn \(\beta_1\) and \(\gamma_1\) leading to the new expression for \(f_1(x)\) as -\(f_1(x) = \beta_1(1+\gamma_1x)\). Doing this \(M\) times results in our final estimate for the function \(f\).

-
-
-

2.5.2. Iterative Fitting, Classification and AdaBoost

-

Let us consider a binary classification problem with two outcomes \(y_i \in \{-1,1\}\) and \(i=0,1,2,\dots,n-1\) as our set of -observations. We define a classification function \(G(x)\) which produces a prediction taking one or the other of the two values -\(\{-1,1\}\).

-

The error rate of the training sample is then

-
-\[ -\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)). -\]
-

The iterative procedure starts with defining a weak classifier whose -error rate is barely better than random guessing. The iterative -procedure in boosting is to sequentially apply a weak -classification algorithm to repeatedly modified versions of the data -producing a sequence of weak classifiers \(G_m(x)\).

-

Here we will express our function \(f(x)\) in terms of \(G(x)\). That is

-
-\[ -f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), -\]
-

will be a function of

-
-\[ -G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x). -\]
-

In our iterative procedure we define thus

-
-\[ -f_m(x) = f_{m-1}(x)+\beta_mG_m(x). -\]
-

The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the -exponential cost/loss function defined as

-
-\[ -C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}. -\]
-

We optimize \(\beta\) and \(G\) for each value of \(m=1:M\) as we did in the regression case. -This is normally done in two steps. Let us however first rewrite the cost function as

-
-\[ -C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))}, -\]
-

where we have defined \(w_i^m= \exp{(-y_if_{m-1}(x_i))}\).

-

First, for any \(\beta > 0\), we optimize \(G\) by setting

-
-\[ -G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)), -\]
-

which is the classifier that minimizes the weighted error rate in predicting \(y\).

-

We can do this by rewriting

-
-\[ -\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m, -\]
-

which can be rewritten as

-
-\[ -(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0, -\]
-

which leads to

-
-\[ -\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}}, -\]
-

where we have redefined the error as

-
-\[ -\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m}, -\]
-

which leads to an update of

-
-\[ -f_m(x) = f_{m-1}(x) +\beta_m G_m(x). -\]
-

This leads to the new weights

-
-\[ -w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))} -\]
-
-
-

2.5.3. Adaptive boosting: AdaBoost, Basic Algorithm

-

The algorithm here is rather straightforward. Assume that our weak -classifier is a decision tree and we consider a binary set of outputs -with \(y_i \in \{-1,1\}\) and \(i=0,1,2,\dots,n-1\) as our set of -observations. Our design matrix is given in terms of the -feature/predictor vectors -\(\boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}]\). Finally, we define also a -classifier determined by our data via a function \(G(x)\). This function tells us how well we are able to classify our outputs/targets \(\boldsymbol{y}\).

-

We have already defined the misclassification error \(\mathrm{err}\) as

-
-\[ -\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)), -\]
-

where the function \(I()\) is one if we misclassify and zero if we classify correctly.

-

With the above definitions we are now ready to set up the algorithm for AdaBoost. -The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.

-
    -

  1. We start by initializing all weights to \(w_i = 1/n\), with \(i=0,1,2,\dots n-1\). It is easy to see that we must have \(\sum_{i=0}^{n-1}w_i = 1\).

    2. -

  2. We rewrite the misclassification error as

    3. -
-
-\[ -\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i}, -\]
-
    -

  1. Then we start looping over all attempts at classifying, namely we start an iterative process for \(m=1:M\), where \(M\) is the final number of classifications. Our given classifier could for example be a plain decision tree.

    2. -
-

a. Fit then a given classifier to the training set using the weights \(w_i\).

-

b. Compute then \(\mathrm{err}\) and figure out which events are classified properly and which are classified wrongly.

-

c. Define a quantity \(\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}\)

-

d. Set the new weights to \(w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}\).

-
    -

  1. Compute the new classifier \(G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)\).

    2. -
-

For the iterations with \(m \le 2\) the weights are modified -individually at each steps. The observations which were misclassified -at iteration \(m-1\) have a weight which is larger than those which were -classified properly. As this proceeds, the observations which were -difficult to classifiy correctly are given a larger influence. Each -new classification step \(m\) is then forced to concentrate on those -observations that are missed in the previous iterations.

-

Using Scikit-Learn it is easy to apply the adaptive boosting algorithm, as done here.

-
-
-
from sklearn.ensemble import AdaBoostClassifier
-
-ada_clf = AdaBoostClassifier(
-    DecisionTreeClassifier(max_depth=1), n_estimators=200,
-    algorithm="SAMME.R", learning_rate=0.5, random_state=42)
-ada_clf.fit(X_train, y_train)
-
-from sklearn.ensemble import AdaBoostClassifier
-
-ada_clf = AdaBoostClassifier(
-    DecisionTreeClassifier(max_depth=1), n_estimators=200,
-    algorithm="SAMME.R", learning_rate=0.5, random_state=42)
-ada_clf.fit(X_train_scaled, y_train)
-y_pred = ada_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-plt.show()
-y_probas = ada_clf.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-plt.show()
-
-
-
-
-
-
-
-

2.6. Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent

-

Gradient boosting is again a similar technique to Adaptive boosting, -it combines so-called weak classifiers or regressors into a strong -method via a series of iterations.

-

In order to understand the method, let us illustrate its basics by -bringing back the essential steps in linear regression, where our cost -function was the least squares function.

-

We start again with our cost function \(\cal{C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}\cal{L}(y_i, f(x_i))\) where we want to minimize -This means that for every iteration, we need to optimize

-
-\[ -(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. -\]
-

We define a real function \(h_m(x)\) that defines our final function \(f_M(x)\) as

-
-\[ -f_M(x) = \sum_{m=0}^M h_m(x). -\]
-

In the steepest decent approach we approximate \(h_m(x) = -\rho_m g_m(x)\), where \(\rho_m\) is a scalar and \(g_m(x)\) the gradient defined as

-
-\[ -g_m(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. -\]
-

With the new gradient we can update \(f_m(x) = f_{m-1}(x) -\rho_m g_m(x)\). Using the above squared-error function we see that -the gradient is \(g_m(x_i) = -2(y_i-f(x_i))\).

-

Choosing \(f_0(x)=0\) we obtain \(g_m(x) = -2y_i\) and inserting this into the minimization problem for the cost function we have

-
-\[ -(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. -\]
-

Optimizing with respect to \(\rho\) we obtain (taking the derivative) that \(\rho_1 = -1/2\). We have then that

-
-\[ -f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i. -\]
-

We can then proceed and compute

-
-\[ -g_2(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i, -\]
-

and find a new value for \(\rho_2=-1/2\) and continue till we have reached \(m=M\). We can modify the steepest descent method, or steepest boosting, by introducing what is called gradient boosting.

-

Steepest descent is however not much used, since it only optimizes \(f\) at a fixed set of \(n\) points, -so we do not learn a function that can generalize. However, we can modify the algorithm by -fitting a weak learner to approximate the negative gradient signal.

-

Suppose we have a cost function \(C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i))\) where \(y_i\) is our target and \(f(x_i)\) the function which is meant to model \(y_i\). The above cost function could be our standard squared-error function

-
-\[ -C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. -\]
-

The way we proceed in an iterative fashion is to

-
    -

  1. Initialize our estimate \(f_0(x)\).

    2. -

  2. For \(m=1:M\), we

    3. -
-

a. compute the negative gradient vector \(\boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x)\) at \(f(x) = f_{m-1}(x)\);

-

b. fit the so-called base-learner to the negative gradient \(h_m(u_m,x)\);

-

c. update the estimate \(f_m(x) = f_{m-1}(x)+h_m(u_m,x)\);

-
    -

  1. The final estimate is then \(f_M(x) = \sum_{m=1}^M h_m(u_m,x)\).

    2. -
-
-
-

2.7. Gradient Boosting, Examples of Regression

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.ensemble import GradientBoostingRegressor
-from sklearn.preprocessing import StandardScaler
-import scikitplot as skplt
-from sklearn.metrics import mean_squared_error
-
-n = 100
-maxdegree = 6
-
-# Make data set.
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-
-error = np.zeros(maxdegree)
-bias = np.zeros(maxdegree)
-variance = np.zeros(maxdegree)
-polydegree = np.zeros(maxdegree)
-X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-for degree in range(1,maxdegree):
-    model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0)  
-    model.fit(X_train_scaled,y_train)
-    y_pred = model.predict(X_test_scaled)
-    polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
-    variance[degree] = np.mean( np.var(y_pred) )
-    print('Max depth:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
-
-plt.xlim(1,maxdegree-1)
-plt.plot(polydegree, error, label='Error')
-plt.plot(polydegree, bias, label='bias')
-plt.plot(polydegree, variance, label='Variance')
-plt.legend()
-save_fig("gdregression")
-plt.show()
-
-
-
-
-
-
-

2.8. Gradient Boosting, Classification Example

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-import scikitplot as skplt
-from sklearn.ensemble import GradientBoostingClassifier
-from sklearn.model_selection import cross_validate
-
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0)  
-gd_clf.fit(X_train_scaled, y_train)
-#Cross validation
-accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']
-print(accuracy)
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test)))
-
-import scikitplot as skplt
-y_pred = gd_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-save_fig("gdclassiffierconfusion")
-plt.show()
-y_probas = gd_clf.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-save_fig("gdclassiffierroc")
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-save_fig("gdclassiffiercgain")
-plt.show()
-
-
-
-
-
-
-

2.9. XGBoost: Extreme Gradient Boosting

-

XGBoost or Extreme Gradient -Boosting, is an optimized distributed gradient boosting library -designed to be highly efficient, flexible and portable. It implements -machine learning algorithms under the Gradient Boosting -framework. XGBoost provides a parallel tree boosting that solve many -data science problems in a fast and accurate way. See the article by Chen and Guestrin.

-

The authors design and build a highly scalable end-to-end tree -boosting system. It has a theoretically justified weighted quantile -sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.

-

It is now the algorithm which wins essentially all ML competitions!!!

-
-
-

2.10. Regression Case

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-import xgboost as xgb
-from sklearn.preprocessing import StandardScaler
-import scikitplot as skplt
-from sklearn.metrics import mean_squared_error
-
-n = 100
-maxdegree = 6
-
-# Make data set.
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-
-error = np.zeros(maxdegree)
-bias = np.zeros(maxdegree)
-variance = np.zeros(maxdegree)
-polydegree = np.zeros(maxdegree)
-X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-for degree in range(maxdegree):
-    model =  xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200)
-
-    model.fit(X_train_scaled,y_train)
-    y_pred = model.predict(X_test_scaled)
-    polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )
-    variance[degree] = np.mean( np.var(y_pred) )
-    print('Max depth:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
-
-plt.xlim(1,maxdegree-1)
-plt.plot(polydegree, error, label='Error')
-plt.plot(polydegree, bias, label='bias')
-plt.plot(polydegree, variance, label='Variance')
-plt.legend()
-plt.show()
-
-
-
-
-

As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.preprocessing import LabelEncoder
-from sklearn.model_selection import cross_validate
-import scikitplot as skplt
-import xgboost as xgb
-# Load the data
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-
-xg_clf = xgb.XGBClassifier()
-xg_clf.fit(X_train_scaled,y_train)
-
-y_test = xg_clf.predict(X_test_scaled)
-
-print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test)))
-
-import scikitplot as skplt
-y_pred = xg_clf.predict(X_test_scaled)
-skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
-save_fig("xdclassiffierconfusion")
-plt.show()
-y_probas = xg_clf.predict_proba(X_test_scaled)
-skplt.metrics.plot_roc(y_test, y_probas)
-save_fig("xdclassiffierroc")
-plt.show()
-skplt.metrics.plot_cumulative_gain(y_test, y_probas)
-save_fig("gdclassiffiercgain")
-plt.show()
-
-
-xgb.plot_tree(xg_clf,num_trees=0)
-plt.rcParams['figure.figsize'] = [50, 10]
-save_fig("xgtree")
-plt.show()
-
-xgb.plot_importance(xg_clf)
-plt.rcParams['figure.figsize'] = [5, 5]
-save_fig("xgparams")
-plt.show()
-
-
-
-
-
-
- - - - -
- - -
- - 1. Decision trees, overarching aims - 1. Basic ideas of the Principal Component Analysis (PCA) - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

-
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-
- - -
-
- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter8.html b/doc/LectureNotes/_build/html/chapter8.html deleted file mode 100644 index e74b1342b..000000000 --- a/doc/LectureNotes/_build/html/chapter8.html +++ /dev/null @@ -1,1492 +0,0 @@ - - - - - - - - 1. Basic ideas of the Principal Component Analysis (PCA) — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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1. Basic ideas of the Principal Component Analysis (PCA)

-

The principal component analysis deals with the problem of fitting a -low-dimensional affine subspace \(S\) of dimension \(d\) much smaller than -the total dimension \(D\) of the problem at hand (our data -set). Mathematically it can be formulated as a statistical problem or -a geometric problem. In our discussion of the theorem for the -classical PCA, we will stay with a statistical approach. -Historically, the PCA was first formulated in a statistical setting in order to estimate the principal component of a multivariate random variable.

-

We have a data set defined by a design/feature matrix \(\boldsymbol{X}\) (see below for its definition)

-
    -

  • Each data point is determined by \(p\) extrinsic (measurement) variables

    • -

  • We may want to ask the following question: Are there fewer intrinsic variables (say \(d << p\)) that still approximately describe the data?

    • -

  • If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do.

    • -
-

A good read is for example Vidal, Ma and Sastry.

-
-

1.1. Introducing the Covariance and Correlation functions

-

Before we discuss the PCA theorem, we need to remind ourselves about -the definition of the covariance and the correlation function. These are quantities

-

Suppose we have defined two vectors -\(\hat{x}\) and \(\hat{y}\) with \(n\) elements each. The covariance matrix \(\boldsymbol{C}\) is defined as

-
-\[\begin{split} -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ - \end{bmatrix}, -\end{split}\]
-

where for example

-
-\[ -\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -\]
-

With this definition and recalling that the variance is defined as

-
-\[ -\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, -\]
-

we can rewrite the covariance matrix as

-
-\[\begin{split} -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ - \end{bmatrix}. -\end{split}\]
-

The covariance takes values between zero and infinity and may thus -lead to problems with loss of numerical precision for particularly -large values. It is common to scale the covariance matrix by -introducing instead the correlation matrix defined via the so-called -correlation function

-
-\[ -\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. -\]
-

The correlation function is then given by values \(\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] -\in [-1,1]\). This avoids eventual problems with too large values. We -can then define the correlation matrix for the two vectors \(\boldsymbol{x}\) -and \(\boldsymbol{y}\) as

-
-\[\begin{split} -\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\ - \end{bmatrix}, -\end{split}\]
-

In the above example this is the function we constructed using pandas.

-

In our derivation of the various regression algorithms like Ordinary Least Squares or Ridge regression -we defined the design/feature matrix \(\boldsymbol{X}\) as

-
-\[\begin{split} -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -\end{split}\]
-

with \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\), with the predictors/features \(p\) refering to the column numbers and the -entries \(n\) being the row elements. -We can rewrite the design/feature matrix in terms of its column vectors as

-
-\[ -\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, -\]
-

with a given vector

-
-\[ -\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. -\]
-

With these definitions, we can now rewrite our \(2\times 2\) -correaltion/covariance matrix in terms of a moe general design/feature -matrix \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\). This leads to a \(p\times p\) -covariance matrix for the vectors \(\boldsymbol{x}_i\) with \(i=0,1,\dots,p-1\)

-
-\[\begin{split} -\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} -\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ -\end{bmatrix}, -\end{split}\]
-

and the correlation matrix

-
-\[\begin{split} -\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} -1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ -\end{bmatrix}, -\end{split}\]
-

The Numpy function np.cov calculates the covariance elements using -the factor \(1/(n-1)\) instead of \(1/n\) since it assumes we do not have -the exact mean values. The following simple function uses the -np.vstack function which takes each vector of dimension \(1\times n\) -and produces a \(2\times n\) matrix \(\boldsymbol{W}\)

-
-\[\begin{split} -\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ - x_1 & y_1 \\ - x_2 & y_2\\ - \dots & \dots \\ - x_{n-2} & y_{n-2}\\ - x_{n-1} & y_{n-1} & - \end{bmatrix}, -\end{split}\]
-

which in turn is converted into into the \(2\times 2\) covariance matrix -\(\boldsymbol{C}\) via the Numpy function np.cov(). We note that we can also calculate -the mean value of each set of samples \(\boldsymbol{x}\) etc using the Numpy -function np.mean(x). We can also extract the eigenvalues of the -covariance matrix through the np.linalg.eig() function.

-
-
-
# Importing various packages
-import numpy as np
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-W = np.vstack((x, y))
-C = np.cov(W)
-print(C)
-
-
-
-
-
-0.2310524795427768
-3.1831903689627863
-[[0.86117632 2.59603122]
- [2.59603122 9.06044826]]
-
-
-
-
-
-
-

1.2. Correlation Matrix

-

The previous example can be converted into the correlation matrix by -simply scaling the matrix elements with the variances. We should also -subtract the mean values for each column. This leads to the following -code which sets up the correlations matrix for the previous example in -a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the \(2\times 2\) correlation matrix (since we have only two vectors).

-
-
-
import numpy as np
-n = 100
-# define two vectors                                                                                           
-x = np.random.random(size=n)
-y = 4+3*x+np.random.normal(size=n)
-#scaling the x and y vectors                                                                                   
-x = x - np.mean(x)
-y = y - np.mean(y)
-variance_x = np.sum(x@x)/n
-variance_y = np.sum(y@y)/n
-print(variance_x)
-print(variance_y)
-cov_xy = np.sum(x@y)/n
-cov_xx = np.sum(x@x)/n
-cov_yy = np.sum(y@y)/n
-C = np.zeros((2,2))
-C[0,0]= cov_xx/variance_x
-C[1,1]= cov_yy/variance_y
-C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
-C[1,0]= C[0,1]
-print(C)
-
-
-
-
-
0.08745913868064381
-2.1128123336365077
-[[1.        0.6798478]
- [0.6798478 1.       ]]
-
-
-
-
-

We see that the matrix elements along the diagonal are one as they -should be and that the matrix is symmetric. Furthermore, diagonalizing -this matrix we easily see that it is a positive definite matrix.

-

The above procedure with numpy can be made more compact if we use pandas.

-

We whow here how we can set up the correlation matrix using pandas, as done in this simple code

-
-
-
import numpy as np
-import pandas as pd
-n = 10
-x = np.random.normal(size=n)
-x = x - np.mean(x)
-y = 4+3*x+np.random.normal(size=n)
-y = y - np.mean(y)
-X = (np.vstack((x, y))).T
-print(X)
-Xpd = pd.DataFrame(X)
-print(Xpd)
-correlation_matrix = Xpd.corr()
-print(correlation_matrix)
-
-
-
-
-
[[ 1.21030078  4.50152769]
- [ 0.85231126  2.49259646]
- [-0.27754082 -0.99161035]
- [-0.05499028 -0.95681341]
- [ 0.65405197  0.57844946]
- [ 0.12802926  0.88441436]
- [ 0.43424077  1.60051748]
- [-1.57205214 -3.89834457]
- [-0.77949144 -1.89899948]
- [-0.59485937 -2.31173763]]
-          0         1
-0  1.210301  4.501528
-1  0.852311  2.492596
-2 -0.277541 -0.991610
-3 -0.054990 -0.956813
-4  0.654052  0.578449
-5  0.128029  0.884414
-6  0.434241  1.600517
-7 -1.572052 -3.898345
-8 -0.779491 -1.898999
-9 -0.594859 -2.311738
-          0         1
-0  1.000000  0.957565
-1  0.957565  1.000000
-
-
-
-
-

We expand this model to the Franke function discussed above.

-
-
-
# Common imports
-import numpy as np
-import pandas as pd
-
-
-def FrankeFunction(x,y):
-	term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
-	term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
-	term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
-	term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
-	return term1 + term2 + term3 + term4
-
-
-def create_X(x, y, n ):
-	if len(x.shape) > 1:
-		x = np.ravel(x)
-		y = np.ravel(y)
-
-	N = len(x)
-	l = int((n+1)*(n+2)/2)		# Number of elements in beta
-	X = np.ones((N,l))
-
-	for i in range(1,n+1):
-		q = int((i)*(i+1)/2)
-		for k in range(i+1):
-			X[:,q+k] = (x**(i-k))*(y**k)
-
-	return X
-
-
-# Making meshgrid of datapoints and compute Franke's function
-n = 4
-N = 100
-x = np.sort(np.random.uniform(0, 1, N))
-y = np.sort(np.random.uniform(0, 1, N))
-z = FrankeFunction(x, y)
-X = create_X(x, y, n=n)    
-
-Xpd = pd.DataFrame(X)
-# subtract the mean values and set up the covariance matrix
-Xpd = Xpd - Xpd.mean()
-covariance_matrix = Xpd.cov()
-print(covariance_matrix)
-
-
-
-
-
     0         1         2         3         4         5         6         7   \
-0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   
-1   0.0  0.091583  0.077830  0.092209  0.086333  0.080182  0.084446  0.080360   
-2   0.0  0.077830  0.067325  0.077735  0.073465  0.068967  0.071391  0.068389   
-3   0.0  0.092209  0.077735  0.099350  0.092714  0.085784  0.094735  0.090145   
-4   0.0  0.086333  0.073465  0.092714  0.086992  0.080974  0.088533  0.084586   
-5   0.0  0.080182  0.068967  0.085784  0.080974  0.075877  0.082067  0.078752   
-6   0.0  0.084446  0.071391  0.094735  0.088533  0.082067  0.092739  0.088427   
-7   0.0  0.080360  0.068389  0.090145  0.084586  0.078752  0.088427  0.084585   
-8   0.0  0.076446  0.065511  0.085713  0.080765  0.075532  0.084250  0.080847   
-9   0.0  0.072627  0.062704  0.081361  0.076998  0.072349  0.080136  0.077151   
-10  0.0  0.076483  0.065094  0.088105  0.082612  0.076881  0.087850  0.084014   
-11  0.0  0.073208  0.062649  0.084453  0.079459  0.074213  0.084413  0.080946   
-12  0.0  0.070145  0.060358  0.081016  0.076483  0.071690  0.081166  0.078040   
-13  0.0  0.067263  0.058198  0.077760  0.073657  0.069286  0.078077  0.075268   
-14  0.0  0.064527  0.056149  0.074647  0.070949  0.066978  0.075114  0.072600   
-
-          8         9         10        11        12        13        14  
-0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  
-1   0.076446  0.072627  0.076483  0.073208  0.070145  0.067263  0.064527  
-2   0.065511  0.062704  0.065094  0.062649  0.060358  0.058198  0.056149  
-3   0.085713  0.081361  0.088105  0.084453  0.081016  0.077760  0.074647  
-4   0.080765  0.076998  0.082612  0.079459  0.076483  0.073657  0.070949  
-5   0.075532  0.072349  0.076881  0.074213  0.071690  0.069286  0.066978  
-6   0.084250  0.080136  0.087850  0.084413  0.081166  0.078077  0.075114  
-7   0.080847  0.077151  0.084014  0.080946  0.078040  0.075268  0.072600  
-8   0.077525  0.074225  0.080285  0.077563  0.074977  0.072501  0.070112  
-9   0.074225  0.071304  0.076603  0.074208  0.071924  0.069731  0.067608  
-10  0.080285  0.076603  0.084360  0.081287  0.078374  0.075595  0.072921  
-11  0.077563  0.074208  0.081287  0.078509  0.075868  0.073341  0.070901  
-12  0.074977  0.071924  0.078374  0.075868  0.073479  0.071184  0.068960  
-13  0.072501  0.069731  0.075595  0.073341  0.071184  0.069105  0.067084  
-14  0.070112  0.067608  0.072921  0.070901  0.068960  0.067084  0.065252  
-
-
-
-
-

We note here that the covariance is zero for the first rows and -columns since all matrix elements in the design matrix were set to one -(we are fitting the function in terms of a polynomial of degree \(n\)). We would however not include the intercept -and wee can simply -drop these elements and construct a correlation -matrix without them.

-

We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix \(\boldsymbol{X}\) as

-
-\[ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}]. -\]
-

To see this let us simply look at a design matrix \(\boldsymbol{X}\in {\mathbb{R}}^{2\times 2}\)

-
-\[\begin{split} -\boldsymbol{X}=\begin{bmatrix} -x_{00} & x_{01}\\ -x_{10} & x_{11}\\ -\end{bmatrix}=\begin{bmatrix} -\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\ -\end{bmatrix}. -\end{split}\]
-

If we then compute the expectation value

-
-\[\begin{split} -\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix} -x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ -x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ -\end{bmatrix}, -\end{split}\]
-

which is just

-
-\[\begin{split} -\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\ - \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\ - \end{bmatrix}, -\end{split}\]
-

where we wrote $\(\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]\)\( to indicate that this the covariance of the vectors \)\boldsymbol{x}\( of the design/feature matrix \)\boldsymbol{X}$.

-

It is easy to generalize this to a matrix \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\).

-
-
-

1.3. Towards the PCA theorem

-

We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as

-
-\[ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}]. -\]
-

Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices \(\boldsymbol{S}\). -These matrices are defined as \(\boldsymbol{S}\in {\mathbb{R}}^{p\times p}\) and obey the orthogonality requirements \(\boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I}\). The matrix can be written out in terms of the column vectors \(\boldsymbol{s}_i\) as \(\boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}]\) and \(\boldsymbol{s}_i \in {\mathbb{R}}^{p}\).

-

Assume also that there is a transformation \(\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}=\boldsymbol{C}[\boldsymbol{y}]\) such that the new matrix \(\boldsymbol{C}[\boldsymbol{y}]\) is diagonal with elements \([\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}]\).

-

That is we have

-
-\[ -\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}^T\boldsymbol{X}^T\boldsymbol{X}T\boldsymbol{S}]=\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}, -\]
-

since the matrix \(\boldsymbol{S}\) is not a data dependent matrix. Multiplying with \(\boldsymbol{S}\) from the left we have

-
-\[ -\boldsymbol{S}\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}, -\]
-

and since \(\boldsymbol{C}[\boldsymbol{y}]\) is diagonal we have for a given eigenvalue \(i\) of the covariance matrix that

-
-\[ -\boldsymbol{S}_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}_i. -\]
-

In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is -\(\lambda_0 > \lambda_1 > \dots > \lambda_{p-1}\).

-

The eigenvalues tell us then how much we need to stretch the -corresponding eigenvectors. Dimensions with large eigenvalues have -thus large variations (large variance) and define therefore useful -dimensions. The data points are more spread out in the direction of -these eigenvectors. Smaller eigenvalues mean on the other hand that -the corresponding eigenvectors are shrunk accordingly and the data -points are tightly bunched together and there is not much variation in -these specific directions. Hopefully then we could leave it out -dimensions where the eigenvalues are very small. If \(p\) is very large, -we could then aim at reducing \(p\) to \(l << p\) and handle only \(l\) -features/predictors.

-
-

1.3.1. The Algorithm before theorem

-

Here’s how we would proceed in setting up the algorithm for the PCA, see also discussion below here.

-
    -

  • Set up the datapoints for the design/feature matrix \(\boldsymbol{X}\) with \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\), with the predictors/features \(p\) referring to the column numbers and the entries \(n\) being the row elements.

    • -
-
-\[\begin{split} -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -\end{split}\]
-
    -

  • Center the data by subtracting the mean value for each column. This leads to a new matrix \(\boldsymbol{X}\rightarrow \overline{\boldsymbol{X}}\).

    • -

  • Compute then the covariance/correlation matrix \(\mathbb{E}[\overline{\boldsymbol{X}}^T\overline{\boldsymbol{X}}]\).

    • -

  • Find the eigenpairs of \(\boldsymbol{C}\) with eigenvalues \([\lambda_0,\lambda_1,\dots,\lambda_{p-1}]\) and eigenvectors \([\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}]\).

    • -

  • Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.

    • -

  • Keep only those \(l\) eigenvalues larger than a selected threshold value, discarding thus \(p-l\) features since we expect small variations in the data here.

    • -
-
-
-

1.3.2. Writing our own PCA code

-

We will use a simple example first with two-dimensional data -drawn from a multivariate normal distribution with the following mean and covariance matrix (we have fixed these quantities but will play around with them below):

-
-\[\begin{split} -\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\ -2 & 2 -\end{bmatrix} -\end{split}\]
-

Note that the mean refers to each column of data. -We will generate \(n = 10000\) points \(X = \{ x_1, \ldots, x_N \}\) from -this distribution, and store them in the \(1000 \times 2\) matrix \(\boldsymbol{X}\). This is our design matrix where we have forced the covariance and mean values to take specific values.

-

The following Python code aids in setting up the data and writing out the design matrix. -Note that the function multivariate returns also the covariance discussed above and that it is defined by dividing by \(n-1\) instead of \(n\).

-
-
-
%matplotlib inline
-
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from IPython.display import display
-n = 10000
-mean = (-1, 2)
-cov = [[4, 2], [2, 2]]
-X = np.random.multivariate_normal(mean, cov, n)
-
-
-
-
-

Now we are going to implement the PCA algorithm. We will break it down into various substeps.

-

The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is

-
-\[ -\mu_n = \frac{1}{n} \sum_{i=1}^n x_i -\]
-

and the mean-centered data \(\bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \}\) takes the form

-
-\[ -\bar{x}_i = x_i - \mu_n. -\]
-

When you are done with these steps, print out \(\mu_n\) to verify it is -close to \(\mu\) and plot your mean centered data to verify it is -centered at the origin! -The following code elements perform these operations using pandas or using our own functionality for doing so. The latter, using numpy is rather simple through the mean() function.

-
-
-
df = pd.DataFrame(X)
-# Pandas does the centering for us
-df = df -df.mean()
-# we center it ourselves
-X_centered = X - X.mean(axis=0)
-
-
-
-
-

Alternatively, we could use the functions we discussed -earlier for scaling the data set. That is, we could have used the -StandardScaler function in Scikit-Learn, a function which ensures -that for each feature/predictor we study the mean value is zero and -the variance is one (every column in the design/feature matrix). You -would then not get the same results, since we divide by the -variance. The diagonal covariance matrix elements will then be one, -while the non-diagonal ones need to be divided by \(2\sqrt{2}\) for our -specific case.

-

Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation

-
-\[ -\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n) -\]
-

where the data points \(x_i \in \mathbb{R}^p\) (here in this example \(p = 2\)) are column vectors and \(x^T\) is the transpose of \(x\). -We can write our own code or simply use either the functionaly of numpy or that of pandas, as follows

-
-
-
print(df.cov())
-print(np.cov(X_centered.T))
-
-
-
-
-
          0         1
-0  3.923640  1.961854
-1  1.961854  1.947452
-[[3.92363958 1.96185372]
- [1.96185372 1.94745179]]
-
-
-
-
-

Note that the way we define the covariance matrix here has a factor \(n-1\) instead of \(n\). This is included in the cov() function by numpy and pandas. -Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific \(2\times 2\) covariance matrix.

-
-
-
# extract the relevant columns from the centered design matrix of dim n x 2
-x = X_centered[:,0]
-y = X_centered[:,1]
-Cov = np.zeros((2,2))
-Cov[0,1] = np.sum(x.T@y)/(n-1.0)
-Cov[0,0] = np.sum(x.T@x)/(n-1.0)
-Cov[1,1] = np.sum(y.T@y)/(n-1.0)
-Cov[1,0]= Cov[0,1]
-print("Centered covariance using own code")
-print(Cov)
-plt.plot(x, y, 'x')
-plt.axis('equal')
-plt.show()
-
-
-
-
-
Centered covariance using own code
-[[3.92363958 1.96185372]
- [1.96185372 1.94745179]]
-
-
-_images/chapter8_65_1.png -
-
-

Depending on the number of points \(n\), we will get results that are close to the covariance values defined above. -The plot shows how the data are clustered around a line with slope close to one. Is this expected? Try to change the covariance and the mean values. For example, try to make the variance of the first element much larger than that of the second diagonal element. Try also to shrink the covariance (the non-diagonal elements) and see how the data points are distributed.

-
-
-

1.3.3. Diagonalize the sample covariance matrix to obtain the principal components

-

Now we are ready to solve for the principal components! To do so we -diagonalize the sample covariance matrix \(\Sigma\). We can use the -function np.linalg.eig to do so. It will return the eigenvalues and -eigenvectors of \(\Sigma\). Once we have these we can perform the -following tasks:

-
    -

  • We compute the percentage of the total variance captured by the first principal component

    • -

  • We plot the mean centered data and lines along the first and second principal components

    • -

  • Then we project the mean centered data onto the first and second principal components, and plot the projected data.

    • -

  • Finally, we approximate the data as

    • -
-
-\[ -x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0 -\]
-

where \(v_0\) is the first principal component.

-

Collecting all these steps we can write our own PCA function and -compare this with the functionality included in Scikit-Learn.

-

The code here outlines some of the elements we could include in the -analysis. Feel free to extend upon this in order to address the above -questions.

-
-
-
# diagonalize and obtain eigenvalues, not necessarily sorted
-EigValues, EigVectors = np.linalg.eig(Cov)
-# sort eigenvectors and eigenvalues
-#permute = EigValues.argsort()
-#EigValues = EigValues[permute]
-#EigVectors = EigVectors[:,permute]
-print("Eigenvalues of Covariance matrix")
-for i in range(2):
-    print(EigValues[i])
-FirstEigvector = EigVectors[:,0]
-SecondEigvector = EigVectors[:,1]
-print("First eigenvector")
-print(FirstEigvector)
-print("Second eigenvector")
-print(SecondEigvector)
-#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2Dsl = pca.fit_transform(X)
-print("Eigenvector of largest eigenvalue")
-print(pca.components_.T[:, 0])
-
-
-
-
-
Eigenvalues of Covariance matrix
-5.132179379442221
-0.7389119925478163
-First eigenvector
-[0.85141702 0.52448933]
-Second eigenvector
-[-0.52448933  0.85141702]
-
-
-
Eigenvector of largest eigenvalue
-[0.85141702 0.52448933]
-
-
-
-
-

This code does not contain all the above elements, but it shows how we can use Scikit-Learn to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then?

-
-
-
-

1.4. Classical PCA Theorem

-

We assume now that we have a design matrix \(\boldsymbol{X}\) which has been -centered as discussed above. For the sake of simplicity we skip the -overline symbol. The matrix is defined in terms of the various column -vectors \([\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}]\) each with dimension -\(\boldsymbol{x}\in {\mathbb{R}}^{n}\).

-

The PCA theorem states that minimizing the above reconstruction error -corresponds to setting \(\boldsymbol{W}=\boldsymbol{S}\), the orthogonal matrix which -diagonalizes the empirical covariance(correlation) matrix. The optimal -low-dimensional encoding of the data is then given by a set of vectors -\(\boldsymbol{z}_i\) with at most \(l\) vectors, with \(l << p\), defined by the -orthogonal projection of the data onto the columns spanned by the -eigenvectors of the covariance(correlations matrix).

-

To show the PCA theorem let us start with the assumption that there is one vector \(\boldsymbol{s}_0\) which corresponds to a solution which minimized the reconstruction error \(J\). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \(\boldsymbol{w}_0\) and \(\boldsymbol{z}_0\) as

-

We are almost there, we have obtained a relation between minimizing -the reconstruction error and the variance and the covariance -matrix. Minimizing the error is equivalent to maximizing the variance -of the projected data.

-

We could trivially maximize the variance of the projection (and -thereby minimize the error in the reconstruction function) by letting -the norm-2 of \(\boldsymbol{w}_0\) go to infinity. However, this norm since we -want the matrix \(\boldsymbol{W}\) to be an orthogonal matrix, is constrained by -\(\vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1\). Imposing this condition via a -Lagrange multiplier we can then in turn maximize

-
-\[ -J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0). -\]
-

Taking the derivative with respect to \(\boldsymbol{w}_0\) we obtain

-
-\[ -\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0, -\]
-

meaning that

-
-\[ -\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0. -\]
-

The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \(\boldsymbol{w}_0^T\) we have the variance of the projected data is

-
-\[ -\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. -\]
-

If we want to maximize the variance (minimize the construction error) -we simply pick the eigenvector of the covariance matrix with the -largest eigenvalue. This establishes the link between the minimization -of the reconstruction function \(J\) in terms of an orthogonal matrix -and the maximization of the variance and thereby the covariance of our -observations encoded in the design/feature matrix \(\boldsymbol{X}\).

-

The proof -for the other eigenvectors \(\boldsymbol{w}_1,\boldsymbol{w}_2,\dots\) can be -established by applying the above arguments and using the fact that -our basis of eigenvectors is orthogonal, see Murphy chapter -12.2. The -discussion in chapter 12.2 of Murphy’s text has also a nice link with -the Singular Value Decomposition theorem. For categorical data, see -chapter 12.4 and discussion therein.

-

For more details, see for example Vidal, Ma and Sastry, chapter 2.

-
- -
-

1.6. PCA and scikit-learn

-

Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The -following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note -that it automatically takes care of centering the data):

-
-
-
#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D = pca.fit_transform(X)
-print(X2D)
-
-
-
-
-
[[ 1.5378811  -0.94639099]
- [-0.86145244  0.89288636]
- [ 0.00445655  0.81633628]
- [-0.07145103 -1.00433417]
- [-2.03707133 -0.48476997]
- [-0.72174172 -1.4557763 ]
- [ 0.55854694  1.60673226]
- [-1.6999536   0.43766686]
- [ 1.10405456  0.31718909]
- [ 2.18673098 -0.17953942]]
-
-
-
-
-

After fitting the PCA transformer to the dataset, you can access the principal components using the -components variable (note that it contains the PCs as horizontal vectors, so, for example, the first -principal component is equal to

-
-
-
pca.components_.T[:, 0]
-
-
-
-
-
array([-0.62373464, -0.5303329 ,  0.317367  ,  0.01873344,  0.47815203])
-
-
-
-
-

Another very useful piece of information is the explained variance ratio of each principal component, -available via the \(explained\_variance\_ratio\) variable. It indicates the proportion of the dataset’s -variance that lies along the axis of each principal component.

-
-
-

1.7. Back to the Cancer Data

-

We can now repeat the above but applied to real data, in this case our breast cancer data. -Here we compute performance scores on the training data using logistic regression.

-
-
-
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.datasets import load_breast_cancer
-from sklearn.linear_model import LogisticRegression
-cancer = load_breast_cancer()
-
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-
-logreg = LogisticRegression()
-logreg.fit(X_train, y_train)
-print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
-# We scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Then perform again a log reg fit
-logreg.fit(X_train_scaled, y_train)
-print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
-#thereafter we do a PCA with Scikit-learn
-from sklearn.decomposition import PCA
-pca = PCA(n_components = 2)
-X2D_train = pca.fit_transform(X_train_scaled)
-# and finally compute the log reg fit and the score on the training data	
-logreg.fit(X2D_train,y_train)
-print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
-
-
-
-
-
Train set accuracy from Logistic Regression: 0.95
-Train set accuracy scaled data: 0.99
-Train set accuracy scaled and PCA data: 0.96
-
-
-
/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/linear_model/_logistic.py:762: ConvergenceWarning: lbfgs failed to converge (status=1):
-STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
-
-Increase the number of iterations (max_iter) or scale the data as shown in:
-    https://scikit-learn.org/stable/modules/preprocessing.html
-Please also refer to the documentation for alternative solver options:
-    https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
-  n_iter_i = _check_optimize_result(
-
-
-
-
-

We see that our training data after the PCA decomposition has a performance similar to the non-scaled data.

-

Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to -choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%). -Unless, of course, you are reducing dimensionality for data visualization — in that case you will -generally want to reduce the dimensionality down to 2 or 3. -The following code computes PCA without reducing dimensionality, then computes the minimum number -of dimensions required to preserve 95% of the training set’s variance:

-
-
-
pca = PCA()
-pca.fit(X)
-cumsum = np.cumsum(pca.explained_variance_ratio_)
-d = np.argmax(cumsum >= 0.95) + 1
-
-
-
-
-

You could then set \(n\_components=d\) and run PCA again. However, there is a much better option: instead -of specifying the number of principal components you want to preserve, you can set \(n\_components\) to be -a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:

-
-
-
pca = PCA(n_components=0.95)
-X_reduced = pca.fit_transform(X)
-
-
-
-
-
-

1.7.1. Incremental PCA

-

One problem with the preceding implementation of PCA is that it requires the whole training set to fit in -memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have -been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch -at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new -instances arrive).

-
-
-

1.7.2. Randomized PCA

-

Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic -algorithm that quickly finds an approximation of the first d principal components. Its computational -complexity is \(O(m \times d^2)+O(d^3)\), instead of \(O(m \times n^2) + O(n^3)\), so it is dramatically faster than the -previous algorithms when \(d\) is much smaller than \(n\).

-
-
-

1.7.3. Kernel PCA

-

The kernel trick is a mathematical technique that implicitly maps instances into a -very high-dimensional space (called the feature space), enabling nonlinear classification and regression -with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature -space corresponds to a complex nonlinear decision boundary in the original space. -It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear -projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at -preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a -twisted manifold. -For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an

-
-
-
from sklearn.decomposition import KernelPCA
-rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
-X_reduced = rbf_pca.fit_transform(X)
-
-
-
-
-
-
-
-

1.8. Other techniques

-

There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.

-

Here are some of the most popular:

-
    -

  • Multidimensional Scaling (MDS) reduces dimensionality while trying to preserve the distances between the instances.

    • -

  • Isomap creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.

    • -

  • t-Distributed Stochastic Neighbor Embedding (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).

    • -

  • Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.

    • -
-
-
- - - - -
- - -
- - 2. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods - 1. Neural networks - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

-
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- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/chapter9.html b/doc/LectureNotes/_build/html/chapter9.html deleted file mode 100644 index 340760d2f..000000000 --- a/doc/LectureNotes/_build/html/chapter9.html +++ /dev/null @@ -1,1157 +0,0 @@ - - - - - - - - 1. Neural networks — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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1. Neural networks

-

Artificial neural networks are computational systems that can learn to -perform tasks by considering examples, generally without being -programmed with any task-specific rules. It is supposed to mimic a -biological system, wherein neurons interact by sending signals in the -form of mathematical functions between layers. All layers can contain -an arbitrary number of neurons, and each connection is represented by -a weight variable.

-

The field of artificial neural networks has a long history of -development, and is closely connected with the advancement of computer -science and computers in general. A model of artificial neurons was -first developed by McCulloch and Pitts in 1943 to study signal -processing in the brain and has later been refined by others. The -general idea is to mimic neural networks in the human brain, which is -composed of billions of neurons that communicate with each other by -sending electrical signals. Each neuron accumulates its incoming -signals, which must exceed an activation threshold to yield an -output. If the threshold is not overcome, the neuron remains inactive, -i.e. has zero output.

-

This behaviour has inspired a simple mathematical model for an artificial neuron.

- -
-
-\[ -\begin{equation} - y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u) -\label{artificialNeuron} \tag{1} -\end{equation} -\]
-

Here, the output \(y\) of the neuron is the value of its activation function, which have as input -a weighted sum of signals \(x_i, \dots ,x_n\) received by \(n\) other neurons.

-

Conceptually, it is helpful to divide neural networks into four -categories:

-
    -

  1. general purpose neural networks for supervised learning,

    2. -

  2. neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),

    3. -

  3. neural networks for sequential data such as Recurrent Neural Networks (RNNs), and

    4. -

  4. neural networks for unsupervised learning such as Deep Boltzmann Machines.

    5. -
-

In natural science, DNNs and CNNs have already found numerous -applications. In statistical physics, they have been applied to detect -phase transitions in 2D Ising and Potts models, lattice gauge -theories, and different phases of polymers, or solving the -Navier-Stokes equation in weather forecasting. Deep learning has also -found interesting applications in quantum physics. Various quantum -phase transitions can be detected and studied using DNNs and CNNs, -topological phases, and even non-equilibrium many-body -localization. Representing quantum states as DNNs quantum state -tomography are among some of the impressive achievements to reveal the -potential of DNNs to facilitate the study of quantum systems.

-

In quantum information theory, it has been shown that one can perform -gate decompositions with the help of neural.

-

The applications are not limited to the natural sciences. There is a -plethora of applications in essentially all disciplines, from the -humanities to life science and medicine.

-

An artificial neural network (ANN), is a computational model that -consists of layers of connected neurons, or nodes or units. We will -refer to these interchangeably as units or nodes, and sometimes as -neurons.

-

It is supposed to mimic a biological nervous system by letting each -neuron interact with other neurons by sending signals in the form of -mathematical functions between layers. A wide variety of different -ANNs have been developed, but most of them consist of an input layer, -an output layer and eventual layers in-between, called hidden -layers. All layers can contain an arbitrary number of nodes, and each -connection between two nodes is associated with a weight variable.

-

Neural networks (also called neural nets) are neural-inspired -nonlinear models for supervised learning. As we will see, neural nets -can be viewed as natural, more powerful extensions of supervised -learning methods such as linear and logistic regression and soft-max -methods we discussed earlier.

-
-

1.1. Feed-forward neural networks

-

The feed-forward neural network (FFNN) was the first and simplest type -of ANNs that were devised. In this network, the information moves in -only one direction: forward through the layers.

-

Nodes are represented by circles, while the arrows display the -connections between the nodes, including the direction of information -flow. Additionally, each arrow corresponds to a weight variable -(figure to come). We observe that each node in a layer is connected -to all nodes in the subsequent layer, making this a so-called -fully-connected FFNN.

-
-
-

1.2. Convolutional Neural Network

-

A different variant of FFNNs are convolutional neural networks -(CNNs), which have a connectivity pattern inspired by the animal -visual cortex. Individual neurons in the visual cortex only respond to -stimuli from small sub-regions of the visual field, called a receptive -field. This makes the neurons well-suited to exploit the strong -spatially local correlation present in natural images. The response of -each neuron can be approximated mathematically as a convolution -operation. (figure to come)

-

Convolutional neural networks emulate the behaviour of neurons in the -visual cortex by enforcing a local connectivity pattern between -nodes of adjacent layers: Each node in a convolutional layer is -connected only to a subset of the nodes in the previous layer, in -contrast to the fully-connected FFNN. Often, CNNs consist of several -convolutional layers that learn local features of the input, with a -fully-connected layer at the end, which gathers all the local data and -produces the outputs. They have wide applications in image and video -recognition.

-
-
-

1.3. Recurrent neural networks

-

So far we have only mentioned ANNs where information flows in one -direction: forward. Recurrent neural networks on the other hand, -have connections between nodes that form directed cycles. This -creates a form of internal memory which are able to capture -information on what has been calculated before; the output is -dependent on the previous computations. Recurrent NNs make use of -sequential information by performing the same task for every element -in a sequence, where each element depends on previous elements. An -example of such information is sentences, making recurrent NNs -especially well-suited for handwriting and speech recognition.

-
-
-

1.4. Other types of networks

-

There are many other kinds of ANNs that have been developed. One type -that is specifically designed for interpolation in multidimensional -space is the radial basis function (RBF) network. RBFs are typically -made up of three layers: an input layer, a hidden layer with -non-linear radial symmetric activation functions and a linear output -layer (‘’linear’’ here means that each node in the output layer has a -linear activation function). The layers are normally fully-connected -and there are no cycles, thus RBFs can be viewed as a type of -fully-connected FFNN. They are however usually treated as a separate -type of NN due the unusual activation functions.

-
-
-

1.5. Multilayer perceptrons

-

One uses often so-called fully-connected feed-forward neural networks -with three or more layers (an input layer, one or more hidden layers -and an output layer) consisting of neurons that have non-linear -activation functions.

-

Such networks are often called multilayer perceptrons (MLPs).

-

According to the Universal approximation theorem, a feed-forward -neural network with just a single hidden layer containing a finite -number of neurons can approximate a continuous multidimensional -function to arbitrary accuracy, assuming the activation function for -the hidden layer is a non-constant, bounded and -monotonically-increasing continuous function.

-

Note that the requirements on the activation function only applies to -the hidden layer, the output nodes are always assumed to be linear, so -as to not restrict the range of output values.

-

The output \(y\) is produced via the activation function \(f\)

-
-\[ -y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z), -\]
-

This function receives \(x_i\) as inputs. -Here the activation \(z=(\sum_{i=1}^n w_ix_i+b_i)\). -In an FFNN of such neurons, the inputs \(x_i\) are the outputs of -the neurons in the preceding layer. Furthermore, an MLP is -fully-connected, which means that each neuron receives a weighted sum -of the outputs of all neurons in the previous layer.

-

First, for each node \(i\) in the first hidden layer, we calculate a weighted sum \(z_i^1\) of the input coordinates \(x_j\),

- -
-
-\[ -\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1 -\label{_auto1} \tag{2} -\end{equation} -\]
-

Here \(b_i\) is the so-called bias which is normally needed in -case of zero activation weights or inputs. How to fix the biases and -the weights will be discussed below. The value of \(z_i^1\) is the -argument to the activation function \(f_i\) of each node \(i\), The -variable \(M\) stands for all possible inputs to a given node \(i\) in the -first layer. We define the output \(y_i^1\) of all neurons in layer 1 as

- -
-
-\[ -\begin{equation} - y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right) -\label{outputLayer1} \tag{3} -\end{equation} -\]
-

where we assume that all nodes in the same layer have identical -activation functions, hence the notation \(f\). In general, we could assume in the more general case that different layers have different activation functions. -In this case we would identify these functions with a superscript \(l\) for the \(l\)-th layer,

- -
-
-\[ -\begin{equation} - y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right) -\label{generalLayer} \tag{4} -\end{equation} -\]
-

where \(N_l\) is the number of nodes in layer \(l\). When the output of -all the nodes in the first hidden layer are computed, the values of -the subsequent layer can be calculated and so forth until the output -is obtained.

-

The output of neuron \(i\) in layer 2 is thus,

- -
-
-\[ -\begin{equation} - y_i^2 = f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) -\label{_auto2} \tag{5} -\end{equation} -\]
- -
-
-\[ -\begin{equation} - = f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right] -\label{outputLayer2} \tag{6} -\end{equation} -\]
-

where we have substituted \(y_k^1\) with the inputs \(x_k\). Finally, the ANN output reads

- -
-
-\[ -\begin{equation} - y_i^3 = f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) -\label{_auto3} \tag{7} -\end{equation} -\]
- -
-
-\[ -\begin{equation} - = f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right) - + b_1^3\right] -\label{_auto4} \tag{8} -\end{equation} -\]
-

We can generalize this expression to an MLP with \(l\) hidden -layers. The complete functional form is,

- -
-
-\[ -\begin{equation} -y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] -\label{completeNN} \tag{9} -\end{equation} -\]
-

which illustrates a basic property of MLPs: The only independent -variables are the input values \(x_n\).

-

This confirms that an MLP, despite its quite convoluted mathematical -form, is nothing more than an analytic function, specifically a -mapping of real-valued vectors \(\hat{x} \in \mathbb{R}^n \rightarrow -\hat{y} \in \mathbb{R}^m\).

-

Furthermore, the flexibility and universality of an MLP can be -illustrated by realizing that the expression is essentially a nested -sum of scaled activation functions of the form

- -
-
-\[ -\begin{equation} - f(x) = c_1 f(c_2 x + c_3) + c_4 -\label{_auto5} \tag{10} -\end{equation} -\]
-

where the parameters \(c_i\) are weights and biases. By adjusting these -parameters, the activation functions can be shifted up and down or -left and right, change slope or be rescaled which is the key to the -flexibility of a neural network.

-

We can introduce a more convenient notation for the activations in an A NN.

-

Additionally, we can represent the biases and activations -as layer-wise column vectors \(\hat{b}_l\) and \(\hat{y}_l\), so that the \(i\)-th element of each vector -is the bias \(b_i^l\) and activation \(y_i^l\) of node \(i\) in layer \(l\) respectively.

-

We have that \(\mathrm{W}_l\) is an \(N_{l-1} \times N_l\) matrix, while \(\hat{b}_l\) and \(\hat{y}_l\) are \(N_l \times 1\) column vectors. -With this notation, the sum becomes a matrix-vector multiplication, and we can write -the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as

- -
-
-\[\begin{split} -\begin{equation} - \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = - f_2\left(\left[\begin{array}{ccc} - w^2_{11} &w^2_{12} &w^2_{13} \\ - w^2_{21} &w^2_{22} &w^2_{23} \\ - w^2_{31} &w^2_{32} &w^2_{33} \\ - \end{array} \right] \cdot - \left[\begin{array}{c} - y^1_1 \\ - y^1_2 \\ - y^1_3 \\ - \end{array}\right] + - \left[\begin{array}{c} - b^2_1 \\ - b^2_2 \\ - b^2_3 \\ - \end{array}\right]\right). -\label{_auto6} \tag{11} -\end{equation} -\end{split}\]
-
-

1.5.1. Matrix-vector notation and activation

-

The activation of node \(i\) in layer 2 is

- -
-
-\[ -\begin{equation} - y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = - f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). -\label{_auto7} \tag{12} -\end{equation} -\]
-

This is not just a convenient and compact notation, but also a useful -and intuitive way to think about MLPs: The output is calculated by a -series of matrix-vector multiplications and vector additions that are -used as input to the activation functions. For each operation -\(\mathrm{W}_l \hat{y}_{l-1}\) we move forward one layer.

-
-
-

1.5.2. Activation functions

-

A property that characterizes a neural network, other than its -connectivity, is the choice of activation function(s). As described -in, the following restrictions are imposed on an activation function -for a FFNN to fulfill the universal approximation theorem

-
    -

  • Non-constant

    • -

  • Bounded

    • -

  • Monotonically-increasing

    • -

  • Continuous

    • -
-

The second requirement excludes all linear functions. Furthermore, in -a MLP with only linear activation functions, each layer simply -performs a linear transformation of its inputs.

-

Regardless of the number of layers, the output of the NN will be -nothing but a linear function of the inputs. Thus we need to introduce -some kind of non-linearity to the NN to be able to fit non-linear -functions Typical examples are the logistic Sigmoid

-
-\[ -f(x) = \frac{1}{1 + e^{-x}}, -\]
-

and the hyperbolic tangent function

-
-\[ -f(x) = \tanh(x) -\]
-

The sigmoid function are more biologically plausible because the -output of inactive neurons are zero. Such activation function are -called one-sided. However, it has been shown that the hyperbolic -tangent performs better than the sigmoid for training MLPs. has -become the most popular for deep neural networks

-
-
-
%matplotlib inline
-
-"""The sigmoid function (or the logistic curve) is a 
-function that takes any real number, z, and outputs a number (0,1).
-It is useful in neural networks for assigning weights on a relative scale.
-The value z is the weighted sum of parameters involved in the learning algorithm."""
-
-import numpy
-import matplotlib.pyplot as plt
-import math as mt
-
-z = numpy.arange(-5, 5, .1)
-sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
-sigma = sigma_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, sigma)
-ax.set_ylim([-0.1, 1.1])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sigmoid function')
-
-plt.show()
-
-"""Step Function"""
-z = numpy.arange(-5, 5, .02)
-step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
-step = step_fn(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, step)
-ax.set_ylim([-0.5, 1.5])
-ax.set_xlim([-5,5])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('step function')
-
-plt.show()
-
-"""Sine Function"""
-z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
-t = numpy.sin(z)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, t)
-ax.set_ylim([-1.0, 1.0])
-ax.set_xlim([-2*mt.pi,2*mt.pi])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('sine function')
-
-plt.show()
-
-"""Plots a graph of the squashing function used by a rectified linear
-unit"""
-z = numpy.arange(-2, 2, .1)
-zero = numpy.zeros(len(z))
-y = numpy.max([zero, z], axis=0)
-
-fig = plt.figure()
-ax = fig.add_subplot(111)
-ax.plot(z, y)
-ax.set_ylim([-2.0, 2.0])
-ax.set_xlim([-2.0, 2.0])
-ax.grid(True)
-ax.set_xlabel('z')
-ax.set_title('Rectified linear unit')
-
-plt.show()
-
-
-
-
-_images/chapter9_29_0.png -_images/chapter9_29_1.png -_images/chapter9_29_2.png -_images/chapter9_29_3.png -
-
-
-
-
-

1.6. The multilayer perceptron (MLP)

-

The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of

-
    -

  1. A neural network with one or more layers of nodes between the input and the output nodes.

    2. -

  2. The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.

    3. -

  3. The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.

    4. -
-

As a convention it is normal to call a network with one layer of input units, one layer of hidden -units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.

-

For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units. -Hereafter we will call the various entities of a layer for nodes. -There are also no connections within a single layer.

-

The number of input nodes does not need to equal the number of output -nodes. This applies also to the hidden layers. Each layer may have its -own number of nodes and activation functions.

-

The hidden layers have their name from the fact that they are not -linked to observables and as we will see below when we define the -so-called activation \(\hat{z}\), we can think of this as a basis -expansion of the original inputs \(\hat{x}\). The difference however -between neural networks and say linear regression is that now these -basis functions (which will correspond to the weights in the network) -are learned from data. This results in an important difference between -neural networks and deep learning approaches on one side and methods -like logistic regression or linear regression and their modifications on the other side.

-
-

1.6.1. From one to many layers, the universal approximation theorem

-

A neural network with only one layer, what we called the simple -perceptron, is best suited if we have a standard binary model with -clear (linear) boundaries between the outcomes. As such it could -equally well be replaced by standard linear regression or logistic -regression. Networks with one or more hidden layers approximate -systems with more complex boundaries.

-

As stated earlier, -an important theorem in studies of neural networks, restated without -proof here, is the universal approximation -theorem.

-

It states that a feed-forward network with a single hidden layer -containing a finite number of neurons can approximate continuous -functions on compact subsets of real functions. The theorem thus -states that simple neural networks can represent a wide variety of -interesting functions when given appropriate parameters. It is the -multilayer feedforward architecture itself which gives neural networks -the potential of being universal approximators.

-
-
-
-

1.7. Deriving the back propagation code for a multilayer perceptron model

-

As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications. -The unknowwn quantities are our weights \(w_{ij}\) and we need to find an algorithm for changing them so that our errors are as small as possible. -This leads us to the famous back propagation algorithm.

-

The questions we want to ask are how do changes in the biases and the -weights in our network change the cost function and how can we use the -final output to modify the weights?

-

To derive these equations let us start with a plain regression problem -and define our cost function as

-
-\[ -{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, -\]
-

where the \(t_i\)s are our \(n\) targets (the values we want to -reproduce), while the outputs of the network after having propagated -all inputs \(\hat{x}\) are given by \(y_i\). Below we will demonstrate -how the basic equations arising from the back propagation algorithm -can be modified in order to study classification problems with \(K\) -classes.

-

With our definition of the targets \(\hat{t}\), the outputs of the -network \(\hat{y}\) and the inputs \(\hat{x}\) we -define now the activation \(z_j^l\) of node/neuron/unit \(j\) of the -\(l\)-th layer as a function of the bias, the weights which add up from -the previous layer \(l-1\) and the forward passes/outputs -\(\hat{a}^{l-1}\) from the previous layer as

-
-\[ -z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l, -\]
-

where \(b_k^l\) are the biases from layer \(l\). Here \(M_{l-1}\) -represents the total number of nodes/neurons/units of layer \(l-1\). The -figure here illustrates this equation. We can rewrite this in a more -compact form as the matrix-vector products we discussed earlier,

-
-\[ -\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l. -\]
-

With the activation values \(\hat{z}^l\) we can in turn define the -output of layer \(l\) as \(\hat{a}^l = f(\hat{z}^l)\) where \(f\) is our -activation function. In the examples here we will use the sigmoid -function discussed in our logistic regression lectures. We will also use the same activation function \(f\) for all layers -and their nodes. It means we have

-
-\[ -a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}. -\]
-
-

1.7.1. Derivatives and the chain rule

-

From the definition of the activation \(z_j^l\) we have

-
-\[ -\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1}, -\]
-

and

-
-\[ -\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. -\]
-

With our definition of the activation function we have that (note that this function depends only on \(z_j^l\))

-
-\[ -\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). -\]
-

With these definitions we can now compute the derivative of the cost function in terms of the weights.

-

Let us specialize to the output layer \(l=L\). Our cost function is

-
-\[ -{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, -\]
-

The derivative of this function with respect to the weights is

-
-\[ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, -\]
-

The last partial derivative can easily be computed and reads (by applying the chain rule)

-
-\[ -\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}, -\]
-
-
-

1.7.2. Bringing it together, first back propagation equation

-

We have thus

-
-\[ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, -\]
-

Defining

-
-\[ -\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, -\]
-

and using the Hadamard product of two vectors we can write this as

-
-\[ -\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}. -\]
-

This is an important expression. The second term on the right handside -measures how fast the cost function is changing as a function of the \(j\)th -output activation. If, for example, the cost function doesn’t depend -much on a particular output node \(j\), then \(\delta_j^L\) will be small, -which is what we would expect. The first term on the right, measures -how fast the activation function \(f\) is changing at a given activation -value \(z_j^L\).

-

Notice that everything in the above equations is easily computed. In -particular, we compute \(z_j^L\) while computing the behaviour of the -network, and it is only a small additional overhead to compute -\(f'(z^L_j)\). The exact form of the derivative with respect to the -output depends on the form of the cost function. -However, provided the cost function is known there should be little -trouble in calculating

-
-\[ -\frac{\partial {\cal C}}{\partial (a_j^L)} -\]
-

With the definition of \(\delta_j^L\) we have a more compact definition of the derivative of the cost function in terms of the weights, namely

-
-\[ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. -\]
-

It is also easy to see that our previous equation can be written as

-
-\[ -\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L}, -\]
-

which can also be interpreted as the partial derivative of the cost function with respect to the biases \(b_j^L\), namely

-
-\[ -\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L}, -\]
-

That is, the error \(\delta_j^L\) is exactly equal to the rate of change of the cost function as a function of the bias.

-

We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are

-

The starting equations.

- -
-
-\[ -\begin{equation} -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}, -\label{_auto8} \tag{13} -\end{equation} -\]
-

and

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-\[ -\begin{equation} -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, -\label{_auto9} \tag{14} -\end{equation} -\]
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and

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-\[ -\begin{equation} -\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}, -\label{_auto10} \tag{15} -\end{equation} -\]
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An interesting consequence of the above equations is that when the -activation \(a_k^{L-1}\) is small, the gradient term, that is the -derivative of the cost function with respect to the weights, will also -tend to be small. We say then that the weight learns slowly, meaning -that it changes slowly when we minimize the weights via say gradient -descent. In this case we say the system learns slowly.

-

Another interesting feature is that is when the activation function, -represented by the sigmoid function here, is rather flat when we move towards -its end values \(0\) and \(1\) (see the above Python codes). In these -cases, the derivatives of the activation function will also be close -to zero, meaning again that the gradients will be small and the -network learns slowly again.

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We need a fourth equation and we are set. We are going to propagate -backwards in order to the determine the weights and biases. In order -to do so we need to represent the error in the layer before the final -one \(L-1\) in terms of the errors in the final output layer.

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1.7.3. Final back propagating equation

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We have that (replacing \(L\) with a general layer \(l\))

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-\[ -\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}. -\]
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We want to express this in terms of the equations for layer \(l+1\). Using the chain rule and summing over all \(k\) entries we have

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-\[ -\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, -\]
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and recalling that

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-\[ -z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1}, -\]
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with \(M_l\) being the number of nodes in layer \(l\), we obtain

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-\[ -\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), -\]
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This is our final equation.

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We are now ready to set up the algorithm for back propagation and learning the weights and biases.

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1.7.4. Setting up the Back propagation algorithm

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The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.

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First, we set up the input data \(\hat{x}\) and the activations -\(\hat{z}_1\) of the input layer and compute the activation function and -the pertinent outputs \(\hat{a}^1\).

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Secondly, we perform then the feed forward till we reach the output -layer and compute all \(\hat{z}_l\) of the input layer and compute the -activation function and the pertinent outputs \(\hat{a}^l\) for -\(l=2,3,\dots,L\).

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Thereafter we compute the ouput error \(\hat{\delta}^L\) by computing all

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-\[ -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. -\]
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Then we compute the back propagate error for each \(l=L-1,L-2,\dots,2\) as

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-\[ -\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). -\]
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Finally, we update the weights and the biases using gradient descent for each \(l=L-1,L-2,\dots,2\) and update the weights and biases according to the rules

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-\[ -w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, -\]
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-\[ -b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l, -\]
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The parameter \(\eta\) is the learning parameter discussed in connection with the gradient descent methods. -Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.

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- - 1. Basic ideas of the Principal Component Analysis (PCA) - 2. Building a Feed Forward Neural Network - -
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Applied Data Analysis and Machine Learning

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Introduction

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Probability theory and statistical methods play a central role in science. Nowadays we are -surrounded by huge amounts of data. For example, there are about one trillion web pages; more than one -hour of video is uploaded to YouTube every second, amounting to years of content every -day; the genomes of 1000s of people, each of which has a length of more than a billion base pairs, have -been sequenced by various labs and so on. This deluge of data calls for automated methods of data analysis, -which is exactly what machine learning aims at providing.

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Learning outcomes

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This course aims at giving you insights and knowledge about many of the central algorithms used in Data Analysis and Machine Learning. The course is project based and through various numerical projects, normally three, you will be exposed to fundamental research problems in these fields, with the aim to reproduce state of the art scientific results. Both supervised and unsupervised methods will be covered. The emphasis is on a frequentist approach, although we will try to link it with a Bayesian approach as well. You will learn to develop and structure large codes for studying different cases where Machine Learning is applied to, get acquainted with computing facilities and learn to handle large scientific projects. A good scientific and ethical conduct is emphasized throughout the course. More specifically, after this course you will

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  • Learn about basic data analysis, statistical analysis, Bayesian statistics, Monte Carlo sampling, data optimization and machine learning;

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  • Be capable of extending the acquired knowledge to other systems and cases;

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  • Have an understanding of central algorithms used in data analysis and machine learning;

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  • Understand linear methods for regression and classification, from ordinary least squares, via Lasso and Ridge to Logistic regression;

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  • Learn about neural networks and deep learning methods for supervised and unsupervised learning. Emphasis on feed forward neural networks, convolutional and recurrent neural networks;

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  • Learn about about decision trees, random forests, bagging and boosting methods;

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  • Learn about support vector machines and kernel transformations;

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  • Reduction of data sets, from PCA to clustering;

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  • Autoencoders and Reinforcement Learning;

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  • Work on numerical projects to illustrate the theory. The projects play a central role and you are expected to know modern programming languages like Python or C++ and/or Fortran (Fortran2003 or later).

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Prerequisites

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Basic knowledge in programming and mathematics, with an emphasis on -linear algebra. Knowledge of Python or/and C++ as programming -languages is strongly recommended and experience with Jupiter notebook -is recommended. Required courses are the equivalents to the University -of Oslo mathematics courses MAT1100, MAT1110, MAT1120 and at least one -of the corresponding computing and programming courses INF1000/INF1110 -or MAT-INF1100/MAT-INF1100L/BIOS1100/KJM-INF1100. Most universities -offer nowadays a basic programming course (often compulsory) where -Python is the recurring programming language.

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The course has two central parts

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  1. Statistical analysis and optimization of data

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  2. Machine learning

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These topics will be scattered thorughout the course and may not necessarily be taught separately. Rather, we will often take an approach (during the lectures and project/exercise sessions) where say elements from statistical data analysis are mixed with specific Machine Learning algorithms.

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Statistical analysis and optimization of data

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The following topics will be covered

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  • Basic concepts, expectation values, variance, covariance, correlation functions and errors;

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  • Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;

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  • Central elements of Bayesian statistics and modeling;

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  • Gradient methods for data optimization,

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  • Monte Carlo methods, Markov chains, Gibbs sampling and Metropolis-Hastings sampling;

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  • Estimation of errors and resampling techniques such as the cross-validation, blocking, bootstrapping and jackknife methods;

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  • Principal Component Analysis (PCA) and its mathematical foundation

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Machine learning

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The following topics will be covered:

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  • Linear Regression and Logistic Regression;

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  • Neural networks and deep learning, including convolutional and recurrent neural networks

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  • Decisions trees, Random Forests, Bagging and Boosting

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  • Support vector machines

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  • Bayesian linear and logistic regression

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  • Boltzmann Machines

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  • Unsupervised learning Dimensionality reduction, from PCA to cluster models

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Hands-on demonstrations, exercises and projects aim at deepening your understanding of these topics.

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Computational aspects play a central role and you are -expected to work on numerical examples and projects which illustrate -the theory and varous algorithms discussed during the lectures. We recommend strongly to form small project groups of 2-3 participants, if possible.

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Required Technologies

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Course participants are expected to have their own laptops/PCs. We use Git as version control software and the usage of providers like GitHub, GitLab or similar are strongly recommended.

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We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -Jupyter notebooks invaluable in your work. You can run R -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be mainly -on Python.

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If you have Python installed and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via pip as

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  • pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow

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For OSX users we recommend, after having installed Xcode, to -install brew. Brew allows for a seamless installation of additional -software via for example

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  • brew install python3

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For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use pip as well and simply install Python as

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  • sudo apt-get install python3

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Python installers

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If you don’t want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely

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which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system conda.

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is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license.

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Furthermore, Google’s Colab:https://colab.research.google.com/notebooks/welcome.ipynb is a free Jupyter notebook environment that requires -no setup and runs entirely in the cloud. Try it out!

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Useful Python libraries

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Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)

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  • NumPy:https://www.numpy.org/ is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays

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  • The pandas:https://pandas.pydata.org/ library provides high-performance, easy-to-use data structures and data analysis tools

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  • Xarray:http://xarray.pydata.org/en/stable/ is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!

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  • Scipy:https://www.scipy.org/ (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering.

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  • Matplotlib:https://matplotlib.org/ is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.

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  • Autograd:https://github.com/HIPS/autograd can automatically differentiate native Python and Numpy code. It can handle a large subset of Python’s features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives

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  • SymPy:https://www.sympy.org/en/index.html is a Python library for symbolic mathematics.

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  • scikit-learn:https://scikit-learn.org/stable/ has simple and efficient tools for machine learning, data mining and data analysis

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  • TensorFlow:https://www.tensorflow.org/ is a Python library for fast numerical computing created and released by Google

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  • Keras:https://keras.io/ is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano

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  • And many more such as pytorch:https://pytorch.org/, Theano:https://pypi.org/project/Theano/ etc

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2. Linear Algebra, Handling of Arrays and more Python Features

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2.1. Introduction

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The aim of this set of lectures is to review some central linear algebra algorithms that we will need in our -data analysis part and in the construction of Machine Learning algorithms (ML). -This will allow us to introduce some central programming features of high-level languages like Python and -compiled languages like C++ and/or Fortran.

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As discussed in the introductory notes, these series of lectures focuses both on using -central Python packages like tensorflow and scikit-learn as well -as writing your own codes for some central ML algorithms. The -latter can be written in a language of your choice, be it Python, Julia, R, -Rust, C++, Fortran etc. In order to avoid confusion however, in these lectures we will limit our -attention to Python, C++ and Fortran.

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2.2. Important Matrix and vector handling packages

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There are several central software packages for linear algebra and eigenvalue problems. Several of the more -popular ones have been wrapped into ofter software packages like those from the widely used text Numerical Recipes. The original source codes in many of the available packages are often taken from the widely used -software package LAPACK, which follows two other popular packages -developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.

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  • LINPACK: package for linear equations and least square problems.

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  • LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK’s website http://www.netlib.org it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.

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  • BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from http://www.netlib.org.

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When dealing with matrices and vectors a central issue is memory -handling and allocation. If our code is written in Python the way we -declare these objects and the way they are handled, interpreted and -used by say a linear algebra library, requires codes that interface -our Python program with such libraries. For Python programmers, -Numpy is by now the standard Python package for numerical arrays in -Python as well as the source of functions which act on these -arrays. These functions span from eigenvalue solvers to functions that -compute the mean value, variance or the covariance matrix. If you are -not familiar with how arrays are handled in say Python or compiled -languages like C++ and Fortran, the sections in this chapter may be -useful. For C++ programmer, Armadillo is widely used library for -linear algebra and eigenvalue problems. In addition it offers a -convenient way to handle and organize arrays. We discuss this library -as well. Before we proceed we believe it may be convenient to repeat some basic features of -matrices and vectors.

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2.3. Basic Matrix Features

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Matrix properties reminder

-
-\[\begin{split} -\mathbf{A} = - \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix}\qquad -\mathbf{I} = - \begin{bmatrix} 1 & 0 & 0 & 0 \\ - 0 & 1 & 0 & 0 \\ - 0 & 0 & 1 & 0 \\ - 0 & 0 & 0 & 1 - \end{bmatrix} -\end{split}\]
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The inverse of a matrix is defined by

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-\[ -\mathbf{A}^{-1} \cdot \mathbf{A} = I -\]
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Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \left (A^{T} \right )^{-1}$ real orthogonal $\sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \left (A^{\dagger} \right )^{-1}$ unitary $\sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij}$
-### Some famous Matrices -
    -

  • Diagonal if \(a_{ij}=0\) for \(i\ne j\)

    • -

  • Upper triangular if \(a_{ij}=0\) for \(i > j\)

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  • Lower triangular if \(a_{ij}=0\) for \(i < j\)

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  • Upper Hessenberg if \(a_{ij}=0\) for \(i > j+1\)

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  • Lower Hessenberg if \(a_{ij}=0\) for \(i < j+1\)

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  • Upper banded with bandwidth \(p\): \(a_{ij}=0\) for \(i < j+p\)

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  • Banded, block upper triangular, block lower triangular….

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Some Equivalent Statements. For an \(N\times N\) matrix \(\mathbf{A}\) the following properties are all equivalent

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  • If the inverse of \(\mathbf{A}\) exists, \(\mathbf{A}\) is nonsingular.

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  • The equation \(\mathbf{Ax}=0\) implies \(\mathbf{x}=0\).

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  • The columns of \(\mathbf{A}\) form a basis of \(R^N\).

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  • \(\mathbf{A}\) is a product of elementary matrices.

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  • \(0\) is not eigenvalue of \(\mathbf{A}\).

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2.4. Numpy and arrays

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Numpy provides an easy way to handle arrays in Python. The standard way to import this library is as

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import numpy as np
-n = 10
-x = np.random.normal(size=n)
-print(x)
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[ 0.44960994 -0.31524949 -0.60668732 -1.03920139 -0.23088568 -0.05148059
- -1.4727093   0.29019465  0.82846181 -0.09720925]
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Here we have defined a vector \(x\) with \(n=10\) elements with its values given by the Normal distribution \(N(0,1)\). -Another alternative is to declare a vector as follows

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import numpy as np
-x = np.array([1, 2, 3])
-print(x)
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[1 2 3]
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Here we have defined a vector with three elements, with \(x_0=1\), \(x_1=2\) and \(x_2=3\). Note that both Python and C++ -start numbering array elements from \(0\) and on. This means that a vector with \(n\) elements has a sequence of entities \(x_0, x_1, x_2, \dots, x_{n-1}\). We could also let (recommended) Numpy to compute the logarithms of a specific array as

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import numpy as np
-x = np.log(np.array([4, 7, 8]))
-print(x)
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[1.38629436 1.94591015 2.07944154]
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Here we have used Numpy’s unary function \(np.log\). This function is -highly tuned to compute array elements since the code is vectorized -and does not require looping. We normaly recommend that you use the -Numpy intrinsic functions instead of the corresponding log function -from Python’s math module. The looping is done explicitely by the -np.log function. The alternative, and slower way to compute the -logarithms of a vector would be to write

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import numpy as np
-from math import log
-x = np.array([4, 7, 8])
-for i in range(0, len(x)):
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[1 1 2]
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We note that our code is much longer already and we need to import the log function from the math module. -The attentive reader will also notice that the output is \([1, 1, 2]\). Python interprets automacally our numbers as integers (like the automatic keyword in C++). To change this we could define our array elements to be double precision numbers as

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import numpy as np
-x = np.log(np.array([4, 7, 8], dtype = np.float64))
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[1.38629436 1.94591015 2.07944154]
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or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is

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import numpy as np
-x = np.log(np.array([4.0, 7.0, 8.0])
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  File "<ipython-input-6-f6d7a289d493>", line 3
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-SyntaxError: invalid syntax
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import numpy as np
-x = np.log(np.array([4.0, 7.0, 8.0])
-print(x.itemsize)
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Having defined vectors, we are now ready to try out matrices. We can define a \(3 \times 3 \) real matrix \(\hat{A}\) -as (recall that we user lowercase letters for vectors and uppercase letters for matrices)

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-
import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-print(A)
-
-
-
-
-

If we use the shape function we would get \((3, 3)\) as output, that is verifying that our matrix is a \(3\times 3\) matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as

-
-
-
import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-# print the first column, row-major order and elements start with 0
-print(A[:,0])
-
-
-
-
-

We can continue this was by printing out other columns or rows. The example here prints out the second column

-
-
-
import numpy as np
-A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))
-# print the first column, row-major order and elements start with 0
-print(A[1,:])
-
-
-
-
-

Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the Numpy website for more details. Useful functions when defining a matrix are the np.zeros function which declares a matrix of a given dimension and sets all elements to zero

-
-
-
import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to zero
-A = np.zeros( (n, n) )
-print(A)
-
-
-
-
-

or initializing all elements to

-
-
-
import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to one
-A = np.ones( (n, n) )
-print(A)
-
-
-
-
-

or as unitarily distributed random numbers (see the material on random number generators in the statistics part)

-
-
-
import numpy as np
-n = 10
-# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1]
-A = np.random.rand(n, n)
-print(A)
-
-
-
-
-

As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. -As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors -\(\hat{x}, \hat{y}, \hat{z}\) with \(n\) elements each. The covariance matrix is defined as

-
-\[\begin{split} -\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ - \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ - \sigma_{zx} & \sigma_{zy} & \sigma_{zz} - \end{bmatrix}, -\end{split}\]
-

where for example

-
-\[ -\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -\]
-

The Numpy function np.cov calculates the covariance elements using the factor \(1/(n-1)\) instead of \(1/n\) since it assumes we do not have the exact mean values. For a more in-depth discussion of the covariance and covariance matrix and its meaning, we refer you to the lectures on statistics. -The following simple function uses the np.vstack function which takes each vector of dimension \(1\times n\) and produces a \( 3\times n\) matrix \(\hat{W}\)

-
-\[\begin{split} -\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ - x_1 & y_1 & z_1 \\ - x_2 & y_2 & z_2 \\ - \dots & \dots & \dots \\ - x_{n-2} & y_{n-2} & z_{n-2} \\ - x_{n-1} & y_{n-1} & z_{n-1} - \end{bmatrix}, -\end{split}\]
-

which in turn is converted into into the \(3 times 3\) covariance matrix -\(\hat{\Sigma}\) via the Numpy function np.cov(). In our review of -statistical functions and quantities we will discuss more about the -meaning of the covariance matrix. Here we note that we can calculate -the mean value of each set of samples \(\hat{x}\) etc using the Numpy -function np.mean(x). We can also extract the eigenvalues of the -covariance matrix through the np.linalg.eig() function.

-
-
-
# Importing various packages
-import numpy as np
-
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-z = x**3+np.random.normal(size=n)
-print(np.mean(z))
-W = np.vstack((x, y, z))
-Sigma = np.cov(W)
-print(Sigma)
-Eigvals, Eigvecs = np.linalg.eig(Sigma)
-print(Eigvals)
-
-
-
-
-
-
-
%matplotlib inline
-
-import numpy as np
-import matplotlib.pyplot as plt
-from scipy import sparse
-eye = np.eye(4)
-print(eye)
-sparse_mtx = sparse.csr_matrix(eye)
-print(sparse_mtx)
-x = np.linspace(-10,10,100)
-y = np.sin(x)
-plt.plot(x,y,marker='x')
-plt.show()
-
-
-
-
-
-
-

2.5. Gaussian Elimination

-

We start with the linear set of equations

-
-\[ -\mathbf{A}\mathbf{x} = \mathbf{w}. -\]
-

We assume also that the matrix \(\mathbf{A}\) is non-singular and that the -matrix elements along the diagonal satisfy \(a_{ii} \ne 0\). Simple \(4\times 4 \) example

-
-\[\begin{split} -\begin{bmatrix} - a_{11}& a_{12} &a_{13}& a_{14}\\ - a_{21}& a_{22} &a_{23}& a_{24}\\ - a_{31}& a_{32} &a_{33}& a_{34}\\ - a_{41}& a_{42} &a_{43}& a_{44}\\ - \end{bmatrix} \begin{bmatrix} - x_1\\ - x_2\\ - x_3 \\ - x_4 \\ - \end{bmatrix} - =\begin{bmatrix} - w_1\\ - w_2\\ - w_3 \\ - w_4\\ - \end{bmatrix}. -\end{split}\]
-

or

-
-\[ -a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \nonumber -\]
-
-\[ -a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \nonumber -\]
-
-\[ -a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \nonumber -\]
-
-\[ -a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \nonumber -\]
-

The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown \(x_1\) -from the remaining \(n-1\) equations. Then we use the new second equation to eliminate the second unknown -\(x_2\) from the remaining \(n-2\) equations. With \(n-1\) such eliminations -we obtain a so-called upper triangular set of equations of the form

-
-\[ -b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=y_1 \nonumber -\]
-
-\[ -b_{22}x_2 + b_{23}x_3 + b_{24}x_4=y_2 \nonumber -\]
-
-\[ -b_{33}x_3 + b_{34}x_4=y_3 \nonumber -\]
- -
-
-\[ -b_{44}x_4=y_4. \nonumber -\label{eq:gaussbacksub} \tag{1} -\]
-

We can solve this system of equations recursively starting from \(x_n\) (in our case \(x_4\)) and proceed with -what is called a backward substitution.

-

This process can be expressed mathematically as

- -
-
-\[ -\begin{equation} - x_m = \frac{1}{b_{mm}}\left(y_m-\sum_{k=m+1}^nb_{mk}x_k\right)\quad m=n-1,n-2,\dots,1. -\label{_auto1} \tag{2} -\end{equation} -\]
-

To arrive at such an upper triangular system of equations, we start by eliminating -the unknown \(x_1\) for \(j=2,n\). We achieve this by multiplying the first equation by \(a_{j1}/a_{11}\) and then subtract -the result from the \(j\)th equation. We assume obviously that \(a_{11}\ne 0\) and that -\(\mathbf{A}\) is not singular.

-

Our actual \(4\times 4\) example reads after the first operation

-
-\[\begin{split} -\begin{bmatrix} - a_{11}& a_{12} &a_{13}& a_{14}\\ - 0& (a_{22}-\frac{a_{21}a_{12}}{a_{11}}) &(a_{23}-\frac{a_{21}a_{13}}{a_{11}}) & (a_{24}-\frac{a_{21}a_{14}}{a_{11}})\\ -0& (a_{32}-\frac{a_{31}a_{12}}{a_{11}})& (a_{33}-\frac{a_{31}a_{13}}{a_{11}})& (a_{34}-\frac{a_{31}a_{14}}{a_{11}})\\ -0&(a_{42}-\frac{a_{41}a_{12}}{a_{11}}) &(a_{43}-\frac{a_{41}a_{13}}{a_{11}}) & (a_{44}-\frac{a_{41}a_{14}}{a_{11}}) \\ - \end{bmatrix} \begin{bmatrix} - x_1\\ - x_2\\ - x_3 \\ - x_4 \\ - \end{bmatrix} - =\begin{bmatrix} - y_1\\ - w_2^{(2)}\\ - w_3^{(2)} \\ - w_4^{(2)}\\ - \end{bmatrix}, -\end{split}\]
-

or

-
-\[ -b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=y_1 \nonumber -\]
-
-\[ -a^{(2)}_{22}x_2 + a^{(2)}_{23}x_3 + a^{(2)}_{24}x_4=w^{(2)}_2 \nonumber -\]
-
-\[ -a^{(2)}_{32}x_2 + a^{(2)}_{33}x_3 + a^{(2)}_{34}x_4=w^{(2)}_3 \nonumber -\]
-
-\[ -a^{(2)}_{42}x_2 + a^{(2)}_{43}x_3 + a^{(2)}_{44}x_4=w^{(2)}_4, \nonumber -\]
- -
-
-\[ -\begin{equation} -\label{_auto2} \tag{3} -\end{equation} -\]
-

The new coefficients are

- -
-
-\[ -\begin{equation} - b_{1k} = a_{1k}^{(1)} \quad k=1,\dots,n, -\label{_auto3} \tag{4} -\end{equation} -\]
-

where each \(a_{1k}^{(1)}\) is equal to the original \(a_{1k}\) element. The other coefficients are

- -
-
-\[ -\begin{equation} -a_{jk}^{(2)} = a_{jk}^{(1)}-\frac{a_{j1}^{(1)}a_{1k}^{(1)}}{a_{11}^{(1)}} \quad j,k=2,\dots,n, -\label{_auto4} \tag{5} -\end{equation} -\]
-

with a new right-hand side given by

- -
-
-\[ -\begin{equation} -y_{1}=w_1^{(1)}, \quad w_j^{(2)} =w_j^{(1)}-\frac{a_{j1}^{(1)}w_1^{(1)}}{a_{11}^{(1)}} \quad j=2,\dots,n. -\label{_auto5} \tag{6} -\end{equation} -\]
-

We have also set \(w_1^{(1)}=w_1\), the original vector element. -We see that the system of unknowns \(x_1,\dots,x_n\) is transformed into an \((n-1)\times (n-1)\) problem.

-

This step is called forward substitution. -Proceeding with these substitutions, we obtain the -general expressions for the new coefficients

- -
-
-\[ -\begin{equation} - a_{jk}^{(m+1)} = a_{jk}^{(m)}-\frac{a_{jm}^{(m)}a_{mk}^{(m)}}{a_{mm}^{(m)}} \quad j,k=m+1,\dots,n, -\label{_auto6} \tag{7} -\end{equation} -\]
-

with \(m=1,\dots,n-1\) and a -right-hand side given by

- -
-
-\[ -\begin{equation} - w_j^{(m+1)} =w_j^{(m)}-\frac{a_{jm}^{(m)}w_m^{(m)}}{a_{mm}^{(m)}}\quad j=m+1,\dots,n. -\label{_auto7} \tag{8} -\end{equation} -\]
-

This set of \(n-1\) elimations leads us to an equations which is solved by back substitution. -If the arithmetics is exact and the matrix \(\mathbf{A}\) is not singular, then the computed answer will be exact.

-

Even though the matrix elements along the diagonal are not zero, -numerically small numbers may appear and subsequent divisions may lead to large numbers, which, if added -to a small number may yield losses of precision. Suppose for example that our first division in \((a_{22}-a_{21}a_{12}/a_{11})\) -results in \(-10^{-7}\) and that \(a_{22}\) is one. -one. We are then -adding \(10^7+1\). With single precision this results in \(10^7\).

-
    -

  • Gaussian elimination, \(O(2/3n^3)\) flops, general matrix

    • -

  • LU decomposition, upper triangular and lower tridiagonal matrices, \(O(2/3n^3)\) flops, general matrix. Get easily the inverse, determinant and can solve linear equations with back-substitution only, \(O(n^2)\) flops

    • -

  • Cholesky decomposition. Real symmetric or hermitian positive definite matrix, \(O(1/3n^3)\) flops.

    • -

  • Tridiagonal linear systems, important for differential equations. Normally positive definite and non-singular. \(O(8n)\) flops for symmetric. Special case of banded matrices.

    • -

  • Singular value decomposition

    • -

  • the QR method will be discussed in chapter 7 in connection with eigenvalue systems. \(O(4/3n^3)\) flops.

    • -
-

The LU decomposition method means that we can rewrite -this matrix as the product of two matrices \(\mathbf{L}\) and \(\mathbf{U}\) -where

-
-\[\begin{split} -\begin{bmatrix} - a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix} - = \begin{bmatrix} - 1 & 0 & 0 & 0 \\ - l_{21} & 1 & 0 & 0 \\ - l_{31} & l_{32} & 1 & 0 \\ - l_{41} & l_{42} & l_{43} & 1 - \end{bmatrix} - \begin{bmatrix} - u_{11} & u_{12} & u_{13} & u_{14} \\ - 0 & u_{22} & u_{23} & u_{24} \\ - 0 & 0 & u_{33} & u_{34} \\ - 0 & 0 & 0 & u_{44} - \end{bmatrix}. -\end{split}\]
-

LU decomposition forms the backbone of other algorithms in linear algebra, such as the -solution of linear equations given by

-
-\[ -a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \nonumber -\]
-
-\[ -a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \nonumber -\]
-
-\[ -a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \nonumber -\]
-
-\[ -a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \nonumber -\]
-

The above set of equations is conveniently solved by using LU decomposition as an intermediate step.

-

The matrix \(\mathbf{A}\in \mathbb{R}^{n\times n}\) has an LU factorization if the determinant -is different from zero. If the LU factorization exists and \(\mathbf{A}\) is non-singular, then the LU factorization -is unique and the determinant is given by

-
-\[ -det\{\mathbf{A}\}=det\{\mathbf{LU}\}= det\{\mathbf{L}\}det\{\mathbf{U}\}=u_{11}u_{22}\dots u_{nn}. -\]
-

There are at least three main advantages with LU decomposition compared with standard Gaussian elimination:

-
    -

  • It is straightforward to compute the determinant of a matrix

    • -

  • If we have to solve sets of linear equations with the same matrix but with different vectors \(\mathbf{y}\), the number of FLOPS is of the order \(n^3\).

    • -

  • The inverse is such an operation

    • -
-

With the LU decomposition it is rather -simple to solve a system of linear equations

-
-\[ -a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \nonumber -\]
-
-\[ -a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \nonumber -\]
-
-\[ -a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \nonumber -\]
-
-\[ -a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \nonumber -\]
-

This can be written in matrix form as

-
-\[ -\mathbf{Ax}=\mathbf{w}. -\]
-

where \(\mathbf{A}\) and \(\mathbf{w}\) are known and we have to solve for -\(\mathbf{x}\). Using the LU dcomposition we write

-
-\[ -\mathbf{A} \mathbf{x} \equiv \mathbf{L} \mathbf{U} \mathbf{x} =\mathbf{w}. -\]
-

The previous equation can be calculated in two steps

-
-\[ -\mathbf{L} \mathbf{y} = \mathbf{w};\qquad \mathbf{Ux}=\mathbf{y}. -\]
-

To show that this is correct we use to the LU decomposition -to rewrite our system of linear equations as

-
-\[ -\mathbf{LUx}=\mathbf{w}, -\]
-

and since the determinant of \(\mathbf{L}\) is equal to 1 (by construction -since the diagonals of \(\mathbf{L}\) equal 1) we can use the inverse of -\(\mathbf{L}\) to obtain

-
-\[ -\mathbf{Ux}=\mathbf{L^{-1}w}=\mathbf{y}, -\]
-

which yields the intermediate step

-
-\[ -\mathbf{L^{-1}w}=\mathbf{y} -\]
-

and as soon as we have \(\mathbf{y}\) we can obtain \(\mathbf{x}\) -through \(\mathbf{Ux}=\mathbf{y}\).

-

For our four-dimentional example this takes the form

-
-\[ -y_1=w_1 \nonumber -\]
-
-\[ -l_{21}y_1 + y_2=w_2\nonumber -\]
-
-\[ -l_{31}y_1 + l_{32}y_2 + y_3 =w_3\nonumber -\]
-
-\[ -l_{41}y_1 + l_{42}y_2 + l_{43}y_3 + y_4=w_4. \nonumber -\]
-

and

-
-\[ -u_{11}x_1 +u_{12}x_2 +u_{13}x_3 + u_{14}x_4=y_1 \nonumber -\]
-
-\[ -u_{22}x_2 + u_{23}x_3 + u_{24}x_4=y_2\nonumber -\]
-
-\[ -u_{33}x_3 + u_{34}x_4=y_3\nonumber -\]
-
-\[ -u_{44}x_4=y_4 \nonumber -\]
-

This example shows the basis for the algorithm -needed to solve the set of \(n\) linear equations.

-

The algorithm goes as follows

-
    -

  • Set up the matrix \(\bf A\) and the vector \(\bf w\) with their correct dimensions. This determines the dimensionality of the unknown vector \(\bf x\).

    • -

  • Then LU decompose the matrix \(\bf A\) through a call to the function ludcmp(double a, int n, int indx, double &d). This functions returns the LU decomposed matrix \(\bf A\), its determinant and the vector indx which keeps track of the number of interchanges of rows. If the determinant is zero, the solution is malconditioned.

    • -

  • Thereafter you call the function lubksb(double a, int n, int indx, double w) which uses the LU decomposed matrix \(\bf A\) and the vector \(\bf w\) and returns \(\bf x\) in the same place as \(\bf w\). Upon exit the original content in \(\bf w\) is destroyed. If you wish to keep this information, you should make a backup of it in your calling function.

    • -
-
-

2.5.1. LU Decomposition, the inverse of a matrix

-

If the inverse exists then

-
-\[ -\mathbf{A}^{-1}\mathbf{A}=\mathbf{I}, -\]
-

the identity matrix. With an LU decomposed matrix we can rewrite the last equation as

-
-\[ -\mathbf{LU}\mathbf{A}^{-1}=\mathbf{I}. -\]
-

If we assume that the first column (that is column 1) of the inverse matrix -can be written as a vector with unknown entries

-
-\[\begin{split} -\mathbf{A}_1^{-1}= \begin{bmatrix} - a_{11}^{-1} \\ - a_{21}^{-1} \\ - \dots \\ - a_{n1}^{-1} \\ - \end{bmatrix}, -\end{split}\]
-

then we have a linear set of equations

-
-\[\begin{split} -\mathbf{LU}\begin{bmatrix} - a_{11}^{-1} \\ - a_{21}^{-1} \\ - \dots \\ - a_{n1}^{-1} \\ - \end{bmatrix} =\begin{bmatrix} - 1 \\ - 0 \\ - \dots \\ - 0 \\ - \end{bmatrix}. -\end{split}\]
-

In a similar way we can compute the unknow entries of the second column,

-
-\[\begin{split} -\mathbf{LU}\begin{bmatrix} - a_{12}^{-1} \\ - a_{22}^{-1} \\ - \dots \\ - a_{n2}^{-1} \\ - \end{bmatrix}=\begin{bmatrix} - 0 \\ - 1 \\ - \dots \\ - 0 \\ - \end{bmatrix}, -\end{split}\]
-

and continue till we have solved all \(n\) sets of linear equations.

-
-
-
- - - - -
- - -
- - 1. Elements of Probability Theory and Statistical Data Analysis - 1. Linear Regression, basic Elements - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

-
-
-
- - -
-
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File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete - return future.result() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute - await self.async_execute_cell( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell - self._check_raise_for_error(cell, exec_reply) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error - raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content']) -nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell: ------------------- -import numpy as np -x = np.log(np.array([4.0, 7.0, 8.0]) -print(x) ------------------- - - File "", line 3 - print(x) - ^ -SyntaxError: invalid syntax - -SyntaxError: invalid syntax (, line 3) - diff --git a/doc/LectureNotes/_build/html/reports/chapter10.log b/doc/LectureNotes/_build/html/reports/chapter10.log deleted file mode 100644 index 5c0d14ab9..000000000 --- a/doc/LectureNotes/_build/html/reports/chapter10.log +++ /dev/null @@ -1,29 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete - return future.result() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute - await self.async_execute_cell( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell - self._check_raise_for_error(cell, exec_reply) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error - raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content']) -nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell: ------------------- -pip3 install tensorflow ------------------- - - File "", line 1 - pip3 install tensorflow - ^ -SyntaxError: invalid syntax - -SyntaxError: invalid syntax (, line 1) - diff --git a/doc/LectureNotes/_build/html/reports/chapter11.log b/doc/LectureNotes/_build/html/reports/chapter11.log deleted file mode 100644 index 42ec51108..000000000 --- a/doc/LectureNotes/_build/html/reports/chapter11.log +++ /dev/null @@ -1,39 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 609, in _async_poll_for_reply - msg = await ensure_async(self.kc.shell_channel.get_msg(timeout=new_timeout)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 85, in ensure_async - result = await obj - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_client/asynchronous/channels.py", line 48, in get_msg - raise Empty -_queue.Empty - -During handling of the above exception, another exception occurred: - -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete - return future.result() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute - await self.async_execute_cell( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 816, in async_execute_cell - exec_reply = await self.task_poll_for_reply - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 633, in _async_poll_for_reply - await self._async_handle_timeout(timeout, cell) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 683, in _async_handle_timeout - raise CellTimeoutError.error_from_timeout_and_cell( -nbclient.exceptions.CellTimeoutError: A cell timed out while it was being executed, after 30 seconds. -The message was: Cell execution timed out. -Here is a preview of the cell contents: -------------------- -['import autograd.numpy as np', 'from autograd import jacobian,hessian,grad', 'import autograd.numpy.random as npr', 'from matplotlib import cm', 'from matplotlib import pyplot as plt'] -... -[' plt.plot(x, res3)', ' plt.plot(x,res_analytical3)', " plt.legend(['dnn','analytical'])", '', ' plt.show()'] -------------------- - diff --git a/doc/LectureNotes/_build/html/reports/chapter2.log b/doc/LectureNotes/_build/html/reports/chapter2.log deleted file mode 100644 index 825b97e1f..000000000 --- a/doc/LectureNotes/_build/html/reports/chapter2.log +++ /dev/null @@ -1,104 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete - return future.result() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute - await self.async_execute_cell( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell - self._check_raise_for_error(cell, exec_reply) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error - raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content']) -nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell: ------------------- -%matplotlib inline - -from numpy import * -from numpy.random import randint, randn -from time import time -import matplotlib.mlab as mlab -import matplotlib.pyplot as plt - -# Returns mean of bootstrap samples -def stat(data): - return mean(data) - -# Bootstrap algorithm -def bootstrap(data, statistic, R): - t = zeros(R); n = len(data); inds = arange(n); t0 = time() - # non-parametric bootstrap - for i in range(R): - t[i] = statistic(data[randint(0,n,n)]) - - # analysis - print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :") - print("original bias std. error") - print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t))) - return t - - -mu, sigma = 100, 15 -datapoints = 10000 -x = mu + sigma*random.randn(datapoints) -# bootstrap returns the data sample -t = bootstrap(x, stat, datapoints) -# the histogram of the bootstrapped data -n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75) - -# add a 'best fit' line -y = mlab.normpdf( binsboot, mean(t), std(t)) -lt = plt.plot(binsboot, y, 'r--', linewidth=1) -plt.xlabel('Smarts') -plt.ylabel('Probability') -plt.axis([99.5, 100.6, 0, 3.0]) -plt.grid(True) - -plt.show() ------------------- - ---------------------------------------------------------------------------- -AttributeError Traceback (most recent call last) - in  - 31 t = bootstrap(x, stat, datapoints) - 32 # the histogram of the bootstrapped data ----> 33 n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75) - 34  - 35 # add a 'best fit' line - -~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/pyplot.py in hist(x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, data, **kwargs) - 2683 orientation='vertical', rwidth=None, log=False, color=None, - 2684 label=None, stacked=False, *, data=None, **kwargs): --> 2685 return gca().hist( - 2686 x, bins=bins, range=range, density=density, weights=weights, - 2687 cumulative=cumulative, bottom=bottom, histtype=histtype, - -~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/__init__.py in inner(ax, data, *args, **kwargs) - 1445 def inner(ax, *args, data=None, **kwargs): - 1446 if data is None: --> 1447 return func(ax, *map(sanitize_sequence, args), **kwargs) - 1448  - 1449 bound = new_sig.bind(ax, *args, **kwargs) - -~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/axes/_axes.py in hist(self, x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, **kwargs) - 6813 if patch: - 6814 p = patch[0] --> 6815 p.update(kwargs) - 6816 if lbl is not None: - 6817 p.set_label(lbl) - -~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/artist.py in update(self, props) - 994 func = getattr(self, f"set_{k}", None) - 995 if not callable(func): ---> 996 raise AttributeError(f"{type(self).__name__!r} object " - 997 f"has no property {k!r}") - 998 ret.append(func(v)) - -AttributeError: 'Rectangle' object has no property 'normed' -AttributeError: 'Rectangle' object has no property 'normed' - diff --git a/doc/LectureNotes/_build/html/reports/chapter4.log b/doc/LectureNotes/_build/html/reports/chapter4.log deleted file mode 100644 index f2d559b52..000000000 --- a/doc/LectureNotes/_build/html/reports/chapter4.log +++ /dev/null @@ -1,89 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete - return future.result() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute - await self.async_execute_cell( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell - self._check_raise_for_error(cell, exec_reply) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error - raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content']) -nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell: ------------------- -%matplotlib inline - -# Common imports -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression, Ridge, Lasso -from sklearn.model_selection import train_test_split -from sklearn.utils import resample -from sklearn.metrics import mean_squared_error -from IPython.display import display -from pylab import plt, mpl -plt.style.use('seaborn') -mpl.rcParams['font.family'] = 'serif' - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("chddata.csv"),'r') - -# Read the chd data as csv file and organize the data into arrays with age group, age, and chd -chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD')) -chd.columns = ['ID', 'Age', 'Agegroup', 'CHD'] -output = chd['CHD'] -age = chd['Age'] -agegroup = chd['Agegroup'] -numberID = chd['ID'] -display(chd) - -plt.scatter(age, output, marker='o') -plt.axis([18,70.0,-0.1, 1.2]) -plt.xlabel(r'Age') -plt.ylabel(r'CHD') -plt.title(r'Age distribution and Coronary heart disease') -plt.show() ------------------- - ---------------------------------------------------------------------------- -FileNotFoundError Traceback (most recent call last) - in  - 38 plt.savefig(image_path(fig_id) + ".png", format='png') - 39  ----> 40 infile = open(data_path("chddata.csv"),'r') - 41  - 42 # Read the chd data as csv file and organize the data into arrays with age group, age, and chd - -FileNotFoundError: [Errno 2] No such file or directory: 'DataFiles/chddata.csv' -FileNotFoundError: [Errno 2] No such file or directory: 'DataFiles/chddata.csv' - diff --git a/doc/LectureNotes/_build/html/reports/chapter5.log b/doc/LectureNotes/_build/html/reports/chapter5.log deleted file mode 100644 index 828ec304d..000000000 --- a/doc/LectureNotes/_build/html/reports/chapter5.log +++ /dev/null @@ -1,41 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete - return future.result() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute - await self.async_execute_cell( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell - self._check_raise_for_error(cell, exec_reply) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error - raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content']) -nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell: ------------------- -# Import the necessary packages -import numpy -from cvxopt import matrix -from cvxopt import solvers -P = matrix(numpy.diag([1,0]), tc=’d’) -q = matrix(numpy.array([3,4]), tc=’d’) -G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’) -h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’) -# Construct the QP, invoke solver -sol = solvers.qp(P,q,G,h) -# Extract optimal value and solution -sol[’x’] -sol[’primal objective’] ------------------- - - File "", line 5 - P = matrix(numpy.diag([1,0]), tc=’d’) - ^ -SyntaxError: invalid character in identifier - -SyntaxError: invalid character in identifier (, line 5) - diff --git a/doc/LectureNotes/_build/html/reports/chapter6.log b/doc/LectureNotes/_build/html/reports/chapter6.log deleted file mode 100644 index b9dbedf5f..000000000 --- a/doc/LectureNotes/_build/html/reports/chapter6.log +++ /dev/null @@ -1,66 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete - return future.result() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute - await self.async_execute_cell( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell - self._check_raise_for_error(cell, exec_reply) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error - raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content']) -nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell: ------------------- -import os -from sklearn.datasets import load_breast_cancer -from sklearn.tree import DecisionTreeClassifier -from sklearn.model_selection import train_test_split -from sklearn.metrics import confusion_matrix -from sklearn.tree import export_graphviz - -from IPython.display import Image -from pydot import graph_from_dot_data -import pandas as pd -import numpy as np - - -cancer = load_breast_cancer() -X = pd.DataFrame(cancer.data, columns=cancer.feature_names) -print(X) -y = pd.Categorical.from_codes(cancer.target, cancer.target_names) -y = pd.get_dummies(y) -print(y) -X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1) -tree_clf = DecisionTreeClassifier(max_depth=5) -tree_clf.fit(X_train, y_train) - -export_graphviz( - tree_clf, - out_file="DataFiles/cancer.dot", - feature_names=cancer.feature_names, - class_names=cancer.target_names, - rounded=True, - filled=True -) -cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png' -os.system(cmd) ------------------- - ---------------------------------------------------------------------------- -ModuleNotFoundError Traceback (most recent call last) - in  - 7  - 8 from IPython.display import Image -----> 9 from pydot import graph_from_dot_data - 10 import pandas as pd - 11 import numpy as np - -ModuleNotFoundError: No module named 'pydot' -ModuleNotFoundError: No module named 'pydot' - diff --git a/doc/LectureNotes/_build/html/reports/chapter7.log b/doc/LectureNotes/_build/html/reports/chapter7.log deleted file mode 100644 index 727b82aa1..000000000 --- a/doc/LectureNotes/_build/html/reports/chapter7.log +++ /dev/null @@ -1,46 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/asyncio/base_events.py", line 616, in run_until_complete - return future.result() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 540, in async_execute - await self.async_execute_cell( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 832, in async_execute_cell - self._check_raise_for_error(cell, exec_reply) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 740, in _check_raise_for_error - raise CellExecutionError.from_cell_and_msg(cell, exec_reply['content']) -nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell: ------------------- -heads_proba = 0.51 -coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32) -cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1) -plt.figure(figsize=(8,3.5)) -plt.plot(cumulative_heads_ratio) -plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%") -plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%") -plt.xlabel("Number of coin tosses") -plt.ylabel("Heads ratio") -plt.legend(loc="lower right") -plt.axis([0, 10000, 0.42, 0.58]) -save_fig("votingsimple") -plt.show() ------------------- - ---------------------------------------------------------------------------- -NameError Traceback (most recent call last) - in  - 1 heads_proba = 0.51 -----> 2 coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32) - 3 cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1) - 4 plt.figure(figsize=(8,3.5)) - 5 plt.plot(cumulative_heads_ratio) - -NameError: name 'np' is not defined -NameError: name 'np' is not defined - diff --git a/doc/LectureNotes/_build/html/reports/linalg.log b/doc/LectureNotes/_build/html/reports/linalg.log deleted file mode 100644 index 4cb123d39..000000000 --- a/doc/LectureNotes/_build/html/reports/linalg.log +++ /dev/null @@ -1,31 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - 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- File "", line 3 - print(x) - ^ -SyntaxError: invalid syntax - -SyntaxError: invalid syntax (, line 3) - diff --git a/doc/LectureNotes/_build/html/reports/statistics.log b/doc/LectureNotes/_build/html/reports/statistics.log deleted file mode 100644 index 279cbb99d..000000000 --- a/doc/LectureNotes/_build/html/reports/statistics.log +++ /dev/null @@ -1,58 +0,0 @@ -Traceback (most recent call last): - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/jupyter_cache/executors/utils.py", line 51, in single_nb_execution - executenb( - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/client.py", line 1087, in execute - return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute() - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 74, in wrapped - return just_run(coro(*args, **kwargs)) - File "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/nbclient/util.py", line 53, in just_run - return loop.run_until_complete(coro) - 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Elements of Probability Theory and Statistical Data Analysis — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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1. Elements of Probability Theory and Statistical Data Analysis

-
-

1.1. Domains and probabilities

-

Consider the following simple example, namely the tossing of two dice, resulting in the following possible values

-
-\[ -\{2,3,4,5,6,7,8,9,10,11,12\}. -\]
-

These values are called the domain. -To this domain we have the corresponding probabilities

-
-\[ -\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\}. -\]
-
-
-

1.2. Tossing the dice

-

The numbers in the domain are the outcomes of the physical process of tossing say two dice. -We cannot tell beforehand whether the outcome is 3 or 5 or any other number in this domain. -This defines the randomness of the outcome, or unexpectedness or any other synonimous word which -encompasses the uncertitude of the final outcome.

-

The only thing we can tell beforehand -is that say the outcome 2 has a certain probability.
-If our favorite hobby is to spend an hour every evening throwing dice and -registering the sequence of outcomes, we will note that the numbers in the above domain

-
-\[ -\{2,3,4,5,6,7,8,9,10,11,12\}, -\]
-

appear in a random order. After 11 throws the results may look like

-
-\[ -\{10,8,6,3,6,9,11,8,12,4,5\}. -\]
-
-
-

1.3. Stochastic variables

-

Random variables are characterized by a domain which contains all possible values that the random value may take. This domain has a corresponding probability distribution function(PDF).

-
-
-

1.4. Stochastic variables and the main concepts, the discrete case

-

There are two main concepts associated with a stochastic variable. The -domain is the set \(\mathbb D = \{x\}\) of all accessible values -the variable can assume, so that \(X \in \mathbb D\). An example of a -discrete domain is the set of six different numbers that we may get by -throwing of a dice, \(x\in\{1,\,2,\,3,\,4,\,5,\,6\}\).

-

The probability distribution function (PDF) is a function -\(p(x)\) on the domain which, in the discrete case, gives us the -probability or relative frequency with which these values of \(X\) -occur

-
-\[ -p(x) = \mathrm{Prob}(X=x). -\]
-
-
-

1.5. Stochastic variables and the main concepts, the continuous case

-

In the continuous case, the PDF does not directly depict the -actual probability. Instead we define the probability for the -stochastic variable to assume any value on an infinitesimal interval -around \(x\) to be \(p(x)dx\). The continuous function \(p(x)\) then gives us -the density of the probability rather than the probability -itself. The probability for a stochastic variable to assume any value -on a non-infinitesimal interval \([a,\,b]\) is then just the integral

-
-\[ -\mathrm{Prob}(a\leq X\leq b) = \int_a^b p(x)dx. -\]
-

Qualitatively speaking, a stochastic variable represents the values of -numbers chosen as if by chance from some specified PDF so that the -selection of a large set of these numbers reproduces this PDF.

-
-
-

1.6. The cumulative probability

-

Of interest to us is the cumulative probability -distribution function (CDF), \(P(x)\), which is just the probability -for a stochastic variable \(X\) to assume any value less than \(x\)

-
-\[ -P(x)=\mathrm{Prob(}X\leq x\mathrm{)} = -\int_{-\infty}^x p(x^{\prime})dx^{\prime}. -\]
-

The relation between a CDF and its corresponding PDF is then

-
-\[ -p(x) = \frac{d}{dx}P(x). -\]
-
-
-

1.7. Properties of PDFs

-

There are two properties that all PDFs must satisfy. The first one is -positivity (assuming that the PDF is normalized)

-
-\[ -0 \leq p(x) \leq 1. -\]
-

Naturally, it would be nonsensical for any of the values of the domain -to occur with a probability greater than \(1\) or less than \(0\). Also, -the PDF must be normalized. That is, all the probabilities must add up -to unity. The probability of “anything” to happen is always unity. For -both discrete and continuous PDFs, this condition is

-
-\[\begin{split} -\begin{align*} -\sum_{x_i\in\mathbb D} p(x_i) & = 1,\\ -\int_{x\in\mathbb D} p(x)\,dx & = 1. -\end{align*} -\end{split}\]
-
-
-

1.8. Important distributions, the uniform distribution

-

The first one -is the most basic PDF; namely the uniform distribution

- -
-
-\[ -\begin{equation} -p(x) = \frac{1}{b-a}\theta(x-a)\theta(b-x). -\label{eq:unifromPDF} \tag{1} -\end{equation} -\]
-

For \(a=0\) and \(b=1\) we have

-
-\[ -\begin{array}{ll} -p(x)dx = dx & \in [0,1]. -\end{array} -\]
-

The latter distribution is used to generate random numbers. For other PDFs, one needs normally a mapping from this distribution to say for example the exponential distribution.

-
-
-

1.9. Gaussian distribution

-

The second one is the Gaussian Distribution

-
-\[ -p(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{(x-\mu)^2}{2\sigma^2})}, -\]
-

with mean value \(\mu\) and standard deviation \(\sigma\). If \(\mu=0\) and \(\sigma=1\), it is normally called the standard normal distribution

-
-\[ -p(x) = \frac{1}{\sqrt{2\pi}} \exp{(-\frac{x^2}{2})}, -\]
-

The following simple Python code plots the above distribution for different values of \(\mu\) and \(\sigma\).

-
-
-
%matplotlib inline
-
-import numpy as np
-from math import acos, exp, sqrt
-from  matplotlib import pyplot as plt
-from matplotlib import rc, rcParams
-import matplotlib.units as units
-import matplotlib.ticker as ticker
-rc('text',usetex=True)
-rc('font',**{'family':'serif','serif':['Gaussian distribution']})
-font = {'family' : 'serif',
-        'color'  : 'darkred',
-        'weight' : 'normal',
-        'size'   : 16,
-        }
-pi = acos(-1.0)
-mu0 = 0.0
-sigma0 = 1.0
-mu1= 1.0
-sigma1 = 2.0
-mu2 = 2.0
-sigma2 = 4.0
-
-x = np.linspace(-20.0, 20.0)
-v0 = np.exp(-(x*x-2*x*mu0+mu0*mu0)/(2*sigma0*sigma0))/sqrt(2*pi*sigma0*sigma0)
-v1 = np.exp(-(x*x-2*x*mu1+mu1*mu1)/(2*sigma1*sigma1))/sqrt(2*pi*sigma1*sigma1)
-v2 = np.exp(-(x*x-2*x*mu2+mu2*mu2)/(2*sigma2*sigma2))/sqrt(2*pi*sigma2*sigma2)
-plt.plot(x, v0, 'b-', x, v1, 'r-', x, v2, 'g-')
-plt.title(r'{\bf Gaussian distributions}', fontsize=20)
-plt.text(-19, 0.3, r'Parameters: $\mu = 0$, $\sigma = 1$', fontdict=font)
-plt.text(-19, 0.18, r'Parameters: $\mu = 1$, $\sigma = 2$', fontdict=font)
-plt.text(-19, 0.08, r'Parameters: $\mu = 2$, $\sigma = 4$', fontdict=font)
-plt.xlabel(r'$x$',fontsize=20)
-plt.ylabel(r'$p(x)$ [MeV]',fontsize=20)
-
-# Tweak spacing to prevent clipping of ylabel                                                                       
-plt.subplots_adjust(left=0.15)
-plt.savefig('gaussian.pdf', format='pdf')
-plt.show()
-
-
-
-
-_images/statistics_29_0.png -
-
-
-
-

1.10. Exponential distribution

-

Another important distribution in science is the exponential distribution

-
-\[ -p(x) = \alpha\exp{-(\alpha x)}. -\]
-
-
-

1.11. Expectation values

-

Let \(h(x)\) be an arbitrary continuous function on the domain of the stochastic -variable \(X\) whose PDF is \(p(x)\). We define the expectation value -of \(h\) with respect to \(p\) as follows

- -
-
-\[ -\begin{equation} -\langle h \rangle_X \equiv \int\! h(x)p(x)\,dx -\label{eq:expectation_value_of_h_wrt_p} \tag{2} -\end{equation} -\]
-

Whenever the PDF is known implicitly, like in this case, we will drop -the index \(X\) for clarity.
-A particularly useful class of special expectation values are the -moments. The \(n\)-th moment of the PDF \(p\) is defined as -follows

-
-\[ -\langle x^n \rangle \equiv \int\! x^n p(x)\,dx -\]
-
-
-

1.12. Stochastic variables and the main concepts, mean values

-

The zero-th moment \(\langle 1\rangle\) is just the normalization condition of -\(p\). The first moment, \(\langle x\rangle\), is called the mean of \(p\) -and often denoted by the letter \(\mu\)

-
-\[ -\langle x\rangle = \mu \equiv \int x p(x)dx, -\]
-

for a continuous distribution and

-
-\[ -\langle x\rangle = \mu \equiv \sum_{i=1}^N x_i p(x_i), -\]
-

for a discrete distribution. -Qualitatively it represents the centroid or the average value of the -PDF and is therefore simply called the expectation value of \(p(x)\).

-
-
-

1.13. Stochastic variables and the main concepts, central moments, the variance

-

A special version of the moments is the set of central moments, the n-th central moment defined as

-
-\[ -\langle (x-\langle x\rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx -\]
-

The zero-th and first central moments are both trivial, equal \(1\) and -\(0\), respectively. But the second central moment, known as the -variance of \(p\), is of particular interest. For the stochastic -variable \(X\), the variance is denoted as \(\sigma^2_X\) or \(\mathrm{Var}(X)\)

-
-\[\begin{split} -\begin{align*} -\sigma^2_X &=\mathrm{Var}(X) = \langle (x-\langle x\rangle)^2\rangle = -\int (x-\langle x\rangle)^2 p(x)dx\\ -& = \int\left(x^2 - 2 x \langle x\rangle^{2} +\langle x\rangle^2\right)p(x)dx\\ -& = \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2\\ -& = \langle x^2 \rangle - \langle x\rangle^2 -\end{align*} -\end{split}\]
-

The square root of the variance, \(\sigma =\sqrt{\langle (x-\langle x\rangle)^2\rangle}\) is called the -standard deviation of \(p\). It is the RMS (root-mean-square) -value of the deviation of the PDF from its mean value, interpreted -qualitatively as the “spread” of \(p\) around its mean.

-
-
-

1.14. Probability Distribution Functions

-

The following table collects properties of probability distribution functions. -In our notation we reserve the label \(p(x)\) for the probability of a certain event, -while \(P(x)\) is the cumulative probability.

- - - - - - - - - - - - - -
Discrete PDF Continuous PDF
Domain $\left\{x_1, x_2, x_3, \dots, x_N\right\}$ $[a,b]$
Probability $p(x_i)$ $p(x)dx$
Cumulative $P_i=\sum_{l=1}^ip(x_l)$ $P(x)=\int_a^xp(t)dt$
Positivity $0 \le p(x_i) \le 1$ $p(x) \ge 0$
Positivity $0 \le P_i \le 1$ $0 \le P(x) \le 1$
Monotonic $P_i \ge P_j$ if $x_i \ge x_j$ $P(x_i) \ge P(x_j)$ if $x_i \ge x_j$
Normalization $P_N=1$ $P(b)=1$
-
-
-

1.15. Probability Distribution Functions

-

With a PDF we can compute expectation values of selected quantities such as

-
-\[ -\langle x^k\rangle=\sum_{i=1}^{N}x_i^kp(x_i), -\]
-

if we have a discrete PDF or

-
-\[ -\langle x^k\rangle=\int_a^b x^kp(x)dx, -\]
-

in the case of a continuous PDF. We have already defined the mean value \(\mu\) -and the variance \(\sigma^2\).

-
-
-

1.16. The three famous Probability Distribution Functions

-

There are at least three PDFs which one may encounter. These are the

-

Uniform distribution

-
-\[ -p(x)=\frac{1}{b-a}\Theta(x-a)\Theta(b-x), -\]
-

yielding probabilities different from zero in the interval \([a,b]\).

-

The exponential distribution

-
-\[ -p(x)=\alpha \exp{(-\alpha x)}, -\]
-

yielding probabilities different from zero in the interval \([0,\infty)\) and with mean value

-
-\[ -\mu = \int_0^{\infty}xp(x)dx=\int_0^{\infty}x\alpha \exp{(-\alpha x)}dx=\frac{1}{\alpha}, -\]
-

with variance

-
-\[ -\sigma^2=\int_0^{\infty}x^2p(x)dx-\mu^2 = \frac{1}{\alpha^2}. -\]
-
-
-

1.17. Probability Distribution Functions, the normal distribution

-

Finally, we have the so-called univariate normal distribution, or just the normal distribution

-
-\[ -p(x)=\frac{1}{b\sqrt{2\pi}}\exp{\left(-\frac{(x-a)^2}{2b^2}\right)} -\]
-

with probabilities different from zero in the interval \((-\infty,\infty)\). -The integral \(\int_{-\infty}^{\infty}\exp{\left(-(x^2\right)}dx\) appears in many calculations, its value -is \(\sqrt{\pi}\), a result we will need when we compute the mean value and the variance. -The mean value is

-
-\[ -\mu = \int_0^{\infty}xp(x)dx=\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}x \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, -\]
-

which becomes with a suitable change of variables

-
-\[ -\mu =\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(a+b\sqrt{2}y)\exp{-y^2}dy=a. -\]
-
-
-

1.18. Probability Distribution Functions, the normal distribution

-

Similarly, the variance becomes

-
-\[ -\sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}(x-\mu)^2 \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, -\]
-

and inserting the mean value and performing a variable change we obtain

-
-\[ -\sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(b\sqrt{2}y)^2\exp{\left(-y^2\right)}dy= -\frac{2b^2}{\sqrt{\pi}}\int_{-\infty}^{\infty}y^2\exp{\left(-y^2\right)}dy, -\]
-

and performing a final integration by parts we obtain the well-known result \(\sigma^2=b^2\). -It is useful to introduce the standard normal distribution as well, defined by \(\mu=a=0\), viz. a distribution -centered around zero and with a variance \(\sigma^2=1\), leading to

- -
-
-\[ -\begin{equation} - p(x)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right)}. -\label{_auto1} \tag{3} -\end{equation} -\]
-
-
-

1.19. Probability Distribution Functions, the cumulative distribution

-

The exponential and uniform distributions have simple cumulative functions, -whereas the normal distribution does not, being proportional to the so-called -error function \(erf(x)\), given by

-
-\[ -P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x\exp{\left(-\frac{t^2}{2}\right)}dt, -\]
-

which is difficult to evaluate in a quick way.

-
-
-

1.20. Probability Distribution Functions, other important distribution

-

Some other PDFs which one encounters often in the natural sciences are the binomial distribution

-
-\[\begin{split} -p(x) = \left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} \hspace{0.5cm}x=0,1,\dots,n, -\end{split}\]
-

where \(y\) is the probability for a specific event, such as the tossing of a coin or moving left or right -in case of a random walker. Note that \(x\) is a discrete stochastic variable.

-

The sequence of binomial trials is characterized by the following definitions

-
    -

  • Every experiment is thought to consist of \(N\) independent trials.

    • -

  • In every independent trial one registers if a specific situation happens or not, such as the jump to the left or right of a random walker.

    • -

  • The probability for every outcome in a single trial has the same value, for example the outcome of tossing (either heads or tails) a coin is always \(1/2\).

    • -
-
-
-

1.21. Probability Distribution Functions, the binomial distribution

-

In order to compute the mean and variance we need to recall Newton’s binomial -formula

-
-\[\begin{split} -(a+b)^m=\sum_{n=0}^m \left(\begin{array}{c} m \\ n\end{array}\right)a^nb^{m-n}, -\end{split}\]
-

which can be used to show that

-
-\[\begin{split} -\sum_{x=0}^n\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1, -\end{split}\]
-

the PDF is normalized to one. -The mean value is

-
-\[\begin{split} -\mu = \sum_{x=0}^n x\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = -\sum_{x=0}^n x\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x}, -\end{split}\]
-

resulting in

-
-\[ -\mu = -\sum_{x=0}^n x\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)}, -\]
-

which we rewrite as

-
-\[\begin{split} -\mu=ny\sum_{\nu=0}^n\left(\begin{array}{c} n-1 \\ \nu\end{array}\right)y^{\nu}(1-y)^{n-1-\nu} =ny(y+1-y)^{n-1}=ny. -\end{split}\]
-

The variance is slightly trickier to get. It reads \(\sigma^2=ny(1-y)\).

-
-
-

1.22. Probability Distribution Functions, Poisson’s distribution

-

Another important distribution with discrete stochastic variables \(x\) is
-the Poisson model, which resembles the exponential distribution and reads

-
-\[ -p(x) = \frac{\lambda^x}{x!} e^{-\lambda} \hspace{0.5cm}x=0,1,\dots,;\lambda > 0. -\]
-

In this case both the mean value and the variance are easier to calculate,

-
-\[ -\mu = \sum_{x=0}^{\infty} x \frac{\lambda^x}{x!} e^{-\lambda} = \lambda e^{-\lambda}\sum_{x=1}^{\infty} -\frac{\lambda^{x-1}}{(x-1)!}=\lambda, -\]
-

and the variance is \(\sigma^2=\lambda\).

-
-
-

1.23. Probability Distribution Functions, Poisson’s distribution

-

An example of applications of the Poisson distribution could be the counting -of the number of \(\alpha\)-particles emitted from a radioactive source in a given time interval. -In the limit of \(n\rightarrow \infty\) and for small probabilities \(y\), the binomial distribution -approaches the Poisson distribution. Setting \(\lambda = ny\), with \(y\) the probability for an event in -the binomial distribution we can show that

-
-\[\begin{split} -\lim_{n\rightarrow \infty}\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} e^{-\lambda}=\sum_{x=1}^{\infty}\frac{\lambda^x}{x!} e^{-\lambda}. -\end{split}\]
-
-
-

1.24. Meet the covariance!

-

An important quantity in a statistical analysis is the so-called covariance.

-

Consider the set \(\{X_i\}\) of \(n\) -stochastic variables (not necessarily uncorrelated) with the -multivariate PDF \(P(x_1,\dots,x_n)\). The covariance of two -of the stochastic variables, \(X_i\) and \(X_j\), is defined as follows

- -
-
-\[ -\begin{equation} -\mathrm{Cov}(X_i,\,X_j) = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle -\label{_auto2} \tag{4} -\end{equation} -\]
- -
-
-\[ -\begin{equation} -=\int\cdots\int (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)P(x_1,\dots,x_n)\,dx_1\dots dx_n, -\label{eq:def_covariance} \tag{5} -\end{equation} -\]
-

with

-
-\[ -\langle x_i\rangle = -\int\cdots\int x_i P(x_1,\dots,x_n)\,dx_1\dots dx_n. -\]
-
-
-

1.25. Meet the covariance in matrix disguise

-

If we consider the above covariance as a matrix

-
-\[ -C_{ij} =\mathrm{Cov}(X_i,\,X_j), -\]
-

then the diagonal elements are just the familiar -variances, \(C_{ii} = \mathrm{Cov}(X_i,\,X_i) = \mathrm{Var}(X_i)\). It turns out that -all the off-diagonal elements are zero if the stochastic variables are -uncorrelated.

-
-
-

1.26. Covariance

-
-
-
# Importing various packages
-from math import exp, sqrt
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-
-def covariance(x, y, n):
-    sum = 0.0
-    mean_x = np.mean(x)
-    mean_y = np.mean(y)
-    for i in range(0, n):
-        sum += (x[(i)]-mean_x)*(y[i]-mean_y)
-    return  sum/n
-
-n = 10
-
-x=np.random.normal(size=n)
-y = 4+3*x+np.random.normal(size=n)
-covxy = covariance(x,y,n)
-print(covxy)
-z = np.vstack((x, y))
-c = np.cov(z.T)
-
-print(c)
-
-
-
-
-
4.752993882123058
-[[ 4.70339379  7.80064137 18.20965525  1.99515112 14.01360746 10.49675434
-   5.81431951  6.88529462  8.48275939  8.04252289]
- [ 7.80064137 12.93746782 30.20095624  3.30898476 23.24175499 17.40900715
-   9.64312649 11.41935301 14.06876966 13.33863154]
- [18.20965525 30.20095624 70.50048516  7.72442532 54.25507027 40.63922482
-  22.51071426 26.65710056 32.84184377 31.13742451]
- [ 1.99515112  3.30898476  7.72442532  0.84633101  5.9444873   4.45265953
-   2.46639907  2.92070022  3.59833509  3.41158944]
- [14.01360746 23.24175499 54.25507027  5.9444873  41.75308359 31.27473511
-  17.32357417 20.51450938 25.27410326 23.96243304]
- [10.49675434 17.40900715 40.63922482  4.45265953 31.27473511 23.42603161
-  12.97605223 15.36619075 18.93131756 17.94882393]
- [ 5.81431951  9.64312649 22.51071426  2.46639907 17.32357417 12.97605223
-   7.18764212  8.51157793 10.48635849  9.94213961]
- [ 6.88529462 11.41935301 26.65710056  2.92070022 20.51450938 15.36619075
-   8.51157793 10.07937759 12.41790507 11.77344318]
- [ 8.48275939 14.06876966 32.84184377  3.59833509 25.27410326 18.93131756
-  10.48635849 12.41790507 15.29899688 14.50501268]
- [ 8.04252289 13.33863154 31.13742451  3.41158944 23.96243304 17.94882393
-   9.94213961 11.77344318 14.50501268 13.75223451]]
-
-
-
-
-
-
-

1.27. Meet the covariance, uncorrelated events

-

Consider the stochastic variables \(X_i\) and \(X_j\), (\(i\neq j\)). We have

-
-\[\begin{split} -\begin{align*} -Cov(X_i,\,X_j) &= \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle\\ -&=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle\\ -&=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j \rangle + -\langle \langle x_i\rangle\langle x_j\rangle\rangle \\ -&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle + -\langle x_i\rangle\langle x_j\rangle \\ -&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle -\end{align*} -\end{split}\]
-

If \(X_i\) and \(X_j\) are independent (assuming \(i \neq j\)), we have that

-
-\[ -\langle x_i x_j\rangle = \langle x_i\rangle\langle x_j\rangle, -\]
-

leading to

-
-\[ -Cov(X_i, X_j) = 0 \hspace{0.1cm} (i\neq j). -\]
-
-
-

1.28. Numerical experiments and the covariance

-

Now that we have constructed an idealized mathematical framework, let -us try to apply it to empirical observations. Examples of relevant -physical phenomena may be spontaneous decays of nuclei, or a purely -mathematical set of numbers produced by some deterministic -mechanism. It is the latter we will deal with, using so-called pseudo-random -number generators. In general our observations will contain only a limited set of -observables. We remind the reader that -a stochastic process is a process that produces sequentially a -chain of values

-
-\[ -\{x_1, x_2,\dots\,x_k,\dots\}. -\]
-
-
-

1.29. Numerical experiments and the covariance

-

We will call these -values our measurements and the entire set as our measured -sample. The action of measuring all the elements of a sample -we will call a stochastic experiment (since, operationally, -they are often associated with results of empirical observation of -some physical or mathematical phenomena; precisely an experiment). We -assume that these values are distributed according to some -PDF \(p_X^{\phantom X}(x)\), where \(X\) is just the formal symbol for the -stochastic variable whose PDF is \(p_X^{\phantom X}(x)\). Instead of -trying to determine the full distribution \(p\) we are often only -interested in finding the few lowest moments, like the mean -\(\mu_X^{\phantom X}\) and the variance \(\sigma_X^{\phantom X}\).

-
-
-

1.30. Numerical experiments and the covariance, actual situations

-

In practical situations however, a sample is always of finite size. Let that -size be \(n\). The expectation value of a sample \(\alpha\), the sample mean, is then defined as follows

-
-\[ -\langle x_{\alpha} \rangle \equiv \frac{1}{n}\sum_{k=1}^n x_{\alpha,k}. -\]
-

The sample variance is:

-
-\[ -\mathrm{Var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_{\alpha,k} - \langle x_{\alpha} \rangle)^2, -\]
-

with its square root being the standard deviation of the sample.

-
-
-

1.31. Numerical experiments and the covariance, our observables

-

You can think of the above observables as a set of quantities which define -a given experiment. This experiment is then repeated several times, say \(m\) times. -The total average is then

- -
-
-\[ -\begin{equation} -\langle X_m \rangle= \frac{1}{m}\sum_{\alpha=1}^mx_{\alpha}=\frac{1}{mn}\sum_{\alpha, k} x_{\alpha,k}, -\label{eq:exptmean} \tag{6} -\end{equation} -\]
-

where the last sums end at \(m\) and \(n\). -The total variance is

-
-\[ -\sigma^2_m= \frac{1}{mn^2}\sum_{\alpha=1}^m(\langle x_{\alpha} \rangle-\langle X_m \rangle)^2, -\]
-

which we rewrite as

- -
-
-\[ -\begin{equation} -\sigma^2_m=\frac{1}{m}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle). -\label{eq:exptvariance} \tag{7} -\end{equation} -\]
-
-
-

1.32. Numerical experiments and the covariance, the sample variance

-

We define also the sample variance \(\sigma^2\) of all \(mn\) individual experiments as

- -
-
-\[ -\begin{equation} -\sigma^2=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n (x_{\alpha,k}-\langle X_m \rangle)^2. -\label{eq:sampleexptvariance} \tag{8} -\end{equation} -\]
-

These quantities, being known experimental values or the results from our calculations, -may differ, in some cases -significantly, from the similarly named -exact values for the mean value \(\mu_X\), the variance \(\mathrm{Var}(X)\) -and the covariance \(\mathrm{Cov}(X,Y)\).

-
-
-

1.33. Numerical experiments and the covariance, central limit theorem

-

The central limit theorem states that the PDF \(\tilde{p}(z)\) of -the average of \(m\) random values corresponding to a PDF \(p(x)\) -is a normal distribution whose mean is the -mean value of the PDF \(p(x)\) and whose variance is the variance -of the PDF \(p(x)\) divided by \(m\), the number of values used to compute \(z\).

-

The central limit theorem leads then to the well-known expression for the -standard deviation, given by

-
-\[ -\sigma_m= -\frac{\sigma}{\sqrt{m}}. -\]
-

In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic. We need therefore a more precise defintion of the error and the variance in our results.

-
-
-

1.34. Definition of Correlation Functions and Standard Deviation

-

Our estimate of the true average \(\mu_{X}\) is the sample mean \(\langle X_m \rangle\)

-
-\[ -\mu_{X}^{\phantom X} \approx X_m=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n x_{\alpha,k}. -\]
-

We can then use Eq. (7)

-
-\[ -\sigma^2_m=\frac{1}{mn^2}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), -\]
-

and rewrite it as

-
-\[ -\sigma^2_m=\frac{\sigma^2}{n}+\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k<l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), -\]
-

where the first term is the sample variance of all \(mn\) experiments divided by \(n\) -and the last term is nothing but the covariance which arises when \(k\ne l\).

-
-
-

1.35. Definition of Correlation Functions and Standard Deviation

-

Our estimate of the true average \(\mu_{X}\) is the sample mean \(\langle X_m \rangle\)

-

If the -observables are uncorrelated, then the covariance is zero and we obtain a total variance -which agrees with the central limit theorem. Correlations may often be present in our data set, resulting in a non-zero covariance. The first term is normally called the uncorrelated -contribution. -Computationally the uncorrelated first term is much easier to treat -efficiently than the second. -We just accumulate separately the values \(x^2\) and \(x\) for every -measurement \(x\) we receive. The correlation term, though, has to be -calculated at the end of the experiment since we need all the -measurements to calculate the cross terms. Therefore, all measurements -have to be stored throughout the experiment.

-
-
-

1.36. Definition of Correlation Functions and Standard Deviation

-

Let us analyze the problem by splitting up the correlation term into -partial sums of the form

-
-\[ -f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), -\]
-

The correlation term of the total variance can now be rewritten in terms of -\(f_d\)

-
-\[ -\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k<l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle)= -\frac{2}{n}\sum_{d=1}^{n-1} f_d -\]
-
-
-

1.37. Definition of Correlation Functions and Standard Deviation

-

The value of \(f_d\) reflects the correlation between measurements -separated by the distance \(d\) in the samples. Notice that for -\(d=0\), \(f\) is just the sample variance, \(\sigma^2\). If we divide \(f_d\) -by \(\sigma^2\), we arrive at the so called autocorrelation function

- -
-
-\[ -\begin{equation} -\kappa_d = \frac{f_d}{\sigma^2} -\label{eq:autocorrelformal} \tag{9} -\end{equation} -\]
-

which gives us a useful measure of the correlation pair correlation -starting always at \(1\) for \(d=0\).

-
-
-

1.38. Definition of Correlation Functions and Standard Deviation, sample variance

-

The sample variance of the \(mn\) experiments can now be -written in terms of the autocorrelation function

- -
-
-\[ -\begin{equation} -\sigma_m^2=\frac{\sigma^2}{n}+\frac{2}{n}\cdot\sigma^2\sum_{d=1}^{n-1} -\frac{f_d}{\sigma^2}=\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\sigma^2=\frac{\tau}{n}\cdot\sigma^2 -\label{eq:error_estimate_corr_time} \tag{10} -\end{equation} -\]
-

and we see that \(\sigma_m\) can be expressed in terms of the -uncorrelated sample variance times a correction factor \(\tau\) which -accounts for the correlation between measurements. We call this -correction factor the autocorrelation time

- -
-
-\[ -\begin{equation} -\tau = 1+2\sum_{d=1}^{n-1}\kappa_d -\label{eq:autocorrelation_time} \tag{11} -\end{equation} -\]
- - -

For a correlation free experiment, \(\tau\) -equals 1.

-
-
-

1.39. Definition of Correlation Functions and Standard Deviation

-

From the point of view of -Eq. (10) we can interpret a sequential -correlation as an effective reduction of the number of measurements by -a factor \(\tau\). The effective number of measurements becomes

-
-\[ -n_\mathrm{eff} = \frac{n}{\tau} -\]
-

To neglect the autocorrelation time \(\tau\) will always cause our -simple uncorrelated estimate of \(\sigma_m^2\approx \sigma^2/n\) to -be less than the true sample error. The estimate of the error will be -too “good”. On the other hand, the calculation of the full -autocorrelation time poses an efficiency problem if the set of -measurements is very large. The solution to this problem is given by -more practically oriented methods like the blocking technique.

- -
-
-

1.40. Code to compute the Covariance matrix and the Covariance

-
-
-
# Importing various packages
-from math import exp, sqrt
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-
-# Sample covariance, note the factor 1/(n-1)
-def covariance(x, y, n):
-    sum = 0.0
-    mean_x = np.mean(x)
-    mean_y = np.mean(y)
-    for i in range(0, n):
-        sum += (x[(i)]-mean_x)*(y[i]-mean_y)
-    return  sum/(n-1.)
-
-n = 100
-x = np.random.normal(size=n)
-print(np.mean(x))
-y = 4+3*x+np.random.normal(size=n)
-print(np.mean(y))
-z = x**3+np.random.normal(size=n)
-print(np.mean(z))
-covxx = covariance(x,x,n)
-covyy = covariance(y,y,n)
-covzz = covariance(z,z,n)
-covxy = covariance(x,y,n)
-covxz = covariance(x,z,n)
-covyz = covariance(y,z,n)
-print(covxx,covyy, covzz)
-print(covxy,covxz, covyz)
-w = np.vstack((x, y, z))
-#print(w)
-c = np.cov(w)
-print(c)
-#eigen = np.zeros(n)
-Eigvals, Eigvecs = np.linalg.eig(c)
-print(Eigvals)
-
-
-
-
-
-0.027477551848353866
-3.7802249527556957
--0.4029931539585397
-0.8404538734963756 8.457307018163291 11.102672153764402
-2.530192962097468 2.272095736442208 6.697993128260892
-[[ 0.84045387  2.53019296  2.27209574]
- [ 2.53019296  8.45730702  6.69799313]
- [ 2.27209574  6.69799313 11.10267215]]
-[17.29496013  0.06631095  3.03916197]
-
-
-
-
-
-
-

1.41. Random Numbers

-

Uniform deviates are just random numbers that lie within a specified range -(typically 0 to 1), with any one number in the range just as likely as any other. They -are, in other words, what you probably think random numbers are. However, -we want to distinguish uniform deviates from other sorts of random numbers, for -example numbers drawn from a normal (Gaussian) distribution of specified mean -and standard deviation. These other sorts of deviates are almost always generated by -performing appropriate operations on one or more uniform deviates, as we will see -in subsequent sections. So, a reliable source of random uniform deviates, the subject -of this section, is an essential building block for any sort of stochastic modeling -or Monte Carlo computer work.

-
-
-

1.42. Random Numbers, better name: pseudo random numbers

-

A disclaimer is however appropriate. It should be fairly obvious that -something as deterministic as a computer cannot generate purely random numbers.

-

Numbers generated by any of the standard algorithms are in reality pseudo random -numbers, hopefully abiding to the following criteria:

-
    -

  • they produce a uniform distribution in the interval [0,1].

    • -

  • correlations between random numbers are negligible

    • -

  • the period before the same sequence of random numbers is repeated is as large as possible and finally

    • -

  • the algorithm should be fast.

    • -
-
-
-

1.43. Random number generator RNG

-

The most common random number generators are based on so-called -Linear congruential relations of the type

-
-\[ -N_i=(aN_{i-1}+c) \mathrm{MOD} (M), -\]
-

which yield a number in the interval [0,1] through

-
-\[ -x_i=N_i/M -\]
-

The number -\(M\) is called the period and it should be as large as possible -and -\(N_0\) is the starting value, or seed. The function \(\mathrm{MOD}\) means the remainder, -that is if we were to evaluate \((13)\mathrm{MOD}(9)\), the outcome is the remainder -of the division \(13/9\), namely \(4\).

-
-
-

1.44. Random number generator RNG and periodic outputs

-

The problem with such generators is that their outputs are periodic; -they -will start to repeat themselves with a period that is at most \(M\). If however -the parameters \(a\) and \(c\) are badly chosen, the period may be even shorter.

-

Consider the following example

-
-\[ -N_i=(6N_{i-1}+7) \mathrm{MOD} (5), -\]
-

with a seed \(N_0=2\). This generator produces the sequence -\(4,1,3,0,2,4,1,3,0,2,...\dots\), i.e., a sequence with period \(5\). -However, increasing \(M\) may not guarantee a larger period as the following -example shows

-
-\[ -N_i=(27N_{i-1}+11) \mathrm{MOD} (54), -\]
-

which still, with \(N_0=2\), results in \(11,38,11,38,11,38,\dots\), a period of -just \(2\).

-
-
-

1.45. Random number generator RNG and its period

-

Typical periods for the random generators provided in the program library -are of the order of \(\sim 10^9\) or larger. Other random number generators which have -become increasingly popular are so-called shift-register generators. -In these generators each successive number depends on many preceding -values (rather than the last values as in the linear congruential -generator). -For example, you could make a shift register generator whose \(l\)th -number is the sum of the \(l-i\)th and \(l-j\)th values with modulo \(M\),

-
-\[ -N_l=(aN_{l-i}+cN_{l-j})\mathrm{MOD}(M). -\]
-
-
-

1.46. Random number generator RNG, other examples

-

Such a generator again produces a sequence of pseudorandom numbers -but this time with a period much larger than \(M\). -It is also possible to construct more elaborate algorithms by including -more than two past terms in the sum of each iteration. -One example is the generator of Marsaglia and Zaman -which consists of two congruential relations

- -
-
-\[ -\begin{equation} - N_l=(N_{l-3}-N_{l-1})\mathrm{MOD}(2^{31}-69), -\label{eq:mz1} \tag{12} -\end{equation} -\]
-

followed by

- -
-
-\[ -\begin{equation} - N_l=(69069N_{l-1}+1013904243)\mathrm{MOD}(2^{32}), -\label{eq:mz2} \tag{13} -\end{equation} -\]
-

which according to the authors has a period larger than \(2^{94}\).

-
-
-

1.47. Random number generator RNG, other examples

-

Instead of using modular addition, we could use the bitwise -exclusive-OR (\(\oplus\)) operation so that

-
-\[ -N_l=(N_{l-i})\oplus (N_{l-j}) -\]
-

where the bitwise action of \(\oplus\) means that if \(N_{l-i}=N_{l-j}\) the result is -\(0\) whereas if \(N_{l-i}\ne N_{l-j}\) the result is -\(1\). As an example, consider the case where \(N_{l-i}=6\) and \(N_{l-j}=11\). The first -one has a bit representation (using 4 bits only) which reads \(0110\) whereas the -second number is \(1011\). Employing the \(\oplus\) operator yields -\(1101\), or \(2^3+2^2+2^0=13\).

-

In Fortran90, the bitwise \(\oplus\) operation is coded through the intrinsic -function \(\mathrm{IEOR}(m,n)\) where \(m\) and \(n\) are the input numbers, while in \(C\) -it is given by \(m\wedge n\).

-
-
-

1.48. Random number generator RNG, RAN0

-

We show here how the linear congruential algorithm can be implemented, namely

-
-\[ -N_i=(aN_{i-1}) \mathrm{MOD} (M). -\]
-

However, since \(a\) and \(N_{i-1}\) are integers and their multiplication -could become greater than the standard 32 bit integer, there is a trick via -Schrage’s algorithm which approximates the multiplication -of large integers through the factorization

-
-\[ -M=aq+r, -\]
-

where we have defined

-
-\[ -q=[M/a], -\]
-

and

-
-\[ -r = M\hspace{0.1cm}\mathrm{MOD} \hspace{0.1cm}a. -\]
-

where the brackets denote integer division. In the code below the numbers -\(q\) and \(r\) are chosen so that \(r < q\).

-
-
-

1.49. Random number generator RNG, RAN0

-

To see how this works we note first that

- -
-
-\[ -\begin{equation} -(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\mathrm{MOD} (M), -\label{eq:rntrick1} \tag{14} -\end{equation} -\]
-

since we can add or subtract any integer multiple of \(M\) from \(aN_{i-1}\). -The last term \([N_{i-1}/q]M\mathrm{MOD}(M)\) is zero since the integer division -\([N_{i-1}/q]\) just yields a constant which is multiplied with \(M\).

-
-
-

1.50. Random number generator RNG, RAN0

-

We can now rewrite Eq. (14) as

- -
-
-\[ -\begin{equation} -(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\mathrm{MOD} (M), -\label{eq:rntrick2} \tag{15} -\end{equation} -\]
-

which results -in

- -
-
-\[ -\begin{equation} -(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\right)\mathrm{MOD} (M), -\label{eq:rntrick3} \tag{16} -\end{equation} -\]
-

yielding

- -
-
-\[ -\begin{equation} -(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}\mathrm{MOD} (q)) -[N_{i-1}/q]r)\right)\mathrm{MOD} (M). -\label{eq:rntrick4} \tag{17} -\end{equation} -\]
-
-
-

1.51. Random number generator RNG, RAN0

-

The term \([N_{i-1}/q]r\) is always smaller or equal \(N_{i-1}(r/q)\) and with \(r < q\) we obtain always a -number smaller than \(N_{i-1}\), which is smaller than \(M\). -And since the number \(N_{i-1}\mathrm{MOD} (q)\) is between zero and \(q-1\) then -\(a(N_{i-1}\mathrm{MOD} (q))< aq\). Combined with our definition of \(q=[M/a]\) ensures that -this term is also smaller than \(M\) meaning that both terms fit into a -32-bit signed integer. None of these two terms can be negative, but their difference could. -The algorithm below adds \(M\) if their difference is negative. -Note that the program uses the bitwise \(\oplus\) operator to generate -the starting point for each generation of a random number. The period -of \(ran0\) is \(\sim 2.1\times 10^{9}\). A special feature of this -algorithm is that is should never be called with the initial seed -set to \(0\).

-
-
-

1.52. Random number generator RNG, RAN0 code

-
        /*
-         ** The function
-         **           ran0()
-         ** is an "Minimal" random number generator of Park and Miller
-         ** Set or reset the input value
-         ** idum to any integer value (except the unlikely value MASK)
-         ** to initialize the sequence; idum must not be altered between
-         ** calls for sucessive deviates in a sequence.
-         ** The function returns a uniform deviate between 0.0 and 1.0.
-         */
-    double ran0(long &idum)
-    {
-       const int a = 16807, m = 2147483647, q = 127773;
-       const int r = 2836, MASK = 123459876;
-       const double am = 1./m;
-       long     k;
-       double   ans;
-       idum ^= MASK;
-       k = (*idum)/q;
-       idum = a*(idum - k*q) - r*k;
-       // add m if negative difference
-       if(idum < 0) idum += m;
-       ans=am*(idum);
-       idum ^= MASK;
-       return ans;
-    } // End: function ran0() 
-
-
-
-
-

1.53. Properties of Selected Random Number Generators

-

As mentioned previously, the underlying PDF for the generation of -random numbers is the uniform distribution, meaning that the -probability for finding a number \(x\) in the interval [0,1] is \(p(x)=1\).

-

A random number generator should produce numbers which are uniformly distributed -in this interval. The table shows the distribution of \(N=10000\) random -numbers generated by the functions in the program library. -We note in this table that the number of points in the various -intervals \(0.0-0.1\), \(0.1-0.2\) etc are fairly close to \(1000\), with some minor -deviations.

-

Two additional measures are the standard deviation \(\sigma\) and the mean -\(\mu=\langle x\rangle\).

-
-
-

1.54. Properties of Selected Random Number Generators

-

For the uniform distribution, the mean value \(\mu\) is then

-
-\[ -\mu=\langle x\rangle=\frac{1}{2} -\]
-

while the standard deviation is

-
-\[ -\sigma=\sqrt{\langle x^2\rangle-\mu^2}=\frac{1}{\sqrt{12}}=0.2886. -\]
-
-
-

1.55. Properties of Selected Random Number Generators

-

The various random number generators produce results which agree rather well with -these limiting values.

- - - - - - - - - - - - - - - - - - -
$x$-bin ran0 ran1 ran2 ran3
0.0-0.1 1013 991 938 1047
0.1-0.2 1002 1009 1040 1030
0.2-0.3 989 999 1030 993
0.3-0.4 939 960 1023 937
0.4-0.5 1038 1001 1002 992
0.5-0.6 1037 1047 1009 1009
0.6-0.7 1005 989 1003 989
0.7-0.8 986 962 985 954
0.8-0.9 1000 1027 1009 1023
0.9-1.0 991 1015 961 1026
$\mu$ 0.4997 0.5018 0.4992 0.4990
$\sigma$ 0.2882 0.2892 0.2861 0.2915
-
-
-

1.56. Simple demonstration of RNGs using python

-

The following simple Python code plots the distribution of the produced random numbers using the linear congruential RNG employed by Python. The trend displayed in the previous table is seen rather clearly.

-
-
-
#!/usr/bin/env python
-import numpy as np
-import matplotlib.mlab as mlab
-import matplotlib.pyplot as plt
-import random
-
-# initialize the rng with a seed
-random.seed() 
-counts = 10000
-values = np.zeros(counts)   
-for i in range (1, counts, 1):
-    values[i] = random.random()
-
-# the histogram of the data
-n, bins, patches = plt.hist(values, 10, facecolor='green')
-
-plt.xlabel('$x$')
-plt.ylabel('Number of counts')
-plt.title(r'Test of uniform distribution')
-plt.axis([0, 1, 0, 1100])
-plt.grid(True)
-plt.show()
-
-
-
-
-_images/statistics_178_0.png -
-
-
-
-

1.57. Properties of Selected Random Number Generators

-

Since our random numbers, which are typically generated via a linear congruential algorithm, -are never fully independent, we can then define -an important test which measures the degree of correlation, namely the so-called
-auto-correlation function defined previously, see again Eq. (9). -We rewrite it here as

-
-\[ -C_k=\frac{f_d} - {\sigma^2}, -\]
-

with \(C_0=1\). Recall that -\(\sigma^2=\langle x_i^2\rangle-\langle x_i\rangle^2\) and that

-
-\[ -f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), -\]
-

The non-vanishing of \(C_k\) for \(k\ne 0\) means that the random -numbers are not independent. The independence of the random numbers is crucial -in the evaluation of other expectation values. If they are not independent, our -assumption for approximating \(\sigma_N\) is no longer valid.

-
-
-

1.58. Autocorrelation function

-

This program computes the autocorrelation function as discussed in the equation on the previous slide for random numbers generated with the normal distribution \(N(0,1)\).

-
-
-
# Importing various packages
-from math import exp, sqrt
-from random import random, seed
-import numpy as np
-import matplotlib.pyplot as plt
-
-def autocovariance(x, n, k, mean_x):
-    sum = 0.0
-    for i in range(0, n-k):
-        sum += (x[(i+k)]-mean_x)*(x[i]-mean_x)
-    return  sum/n
-
-n = 1000
-x=np.random.normal(size=n)
-autocor = np.zeros(n)
-figaxis = np.zeros(n)
-mean_x=np.mean(x)
-var_x = np.var(x)
-print(mean_x, var_x)
-for i in range (0, n):
-    figaxis[i] = i
-    autocor[i]=(autocovariance(x, n, i, mean_x))/var_x    
-
-plt.plot(figaxis, autocor, "r-")
-plt.axis([0,n,-0.1, 1.0])
-plt.xlabel(r'$i$')
-plt.ylabel(r'$\gamma_i$')
-plt.title(r'Autocorrelation function')
-plt.show()
-
-
-
-
-
0.07147629989718486 0.9457130616859395
-
-
-_images/statistics_184_1.png -
-
-

As can be seen from the plot, the first point gives back the variance and a value of one. -For the remaining values we notice that there are still non-zero values for the auto-correlation function.

-
-
-

1.59. Correlation function and which random number generators should I use

-

The program here computes the correlation function for one of the standard functions included with the c++ compiler.

-
    //  This function computes the autocorrelation function for 
-    //  the standard c++ random number generator
-    
-    #include <fstream>
-    #include <iomanip>
-    #include <iostream>
-    #include <cmath>
-    using namespace std;
-    // output file as global variable
-    ofstream ofile;  
-    
-    //     Main function begins here     
-    int main(int argc, char* argv[])
-    {
-         int n;
-         char *outfilename;
-    
-         cin >> n;
-         double MCint = 0.;      double MCintsqr2=0.;
-         double invers_period = 1./RAND_MAX; // initialise the random number generator
-         srand(time(NULL));  // This produces the so-called seed in MC jargon
-         // Compute the variance and the mean value of the uniform distribution
-         // Compute also the specific values x for each cycle in order to be able to
-         // the covariance and the correlation function  
-         // Read in output file, abort if there are too few command-line arguments
-         if( argc <= 2 ){
-           cout << "Bad Usage: " << argv[0] << 
-    	 " read also output file and number of cycles on same line" << endl;
-           exit(1);
-         }
-         else{
-           outfilename=argv[1];
-         }
-         ofile.open(outfilename); 
-         // Get  the number of Monte-Carlo samples
-         n = atoi(argv[2]);
-         double *X;  
-         X = new double[n];
-         for (int i = 0;  i < n; i++){
-               double x = double(rand())*invers_period; 
-               X[i] = x;
-               MCint += x;
-               MCintsqr2 += x*x;
-         }
-         double Mean = MCint/((double) n );
-         MCintsqr2 = MCintsqr2/((double) n );
-         double STDev = sqrt(MCintsqr2-Mean*Mean);
-         double Variance = MCintsqr2-Mean*Mean;
-    //   Write mean value and standard deviation 
-         cout << " Standard deviation= " << STDev << " Integral = " << Mean << endl;
-    
-         // Now we compute the autocorrelation function
-         double *autocor;  autocor = new double[n];
-         for (int j = 0; j < n; j++){
-           double sum = 0.0;
-           for (int k = 0; k < (n-j); k++){
-    	 sum  += (X[k]-Mean)*(X[k+j]-Mean); 
-           }
-           autocor[j] = sum/Variance/((double) n );
-           ofile << setiosflags(ios::showpoint | ios::uppercase);
-           ofile << setw(15) << setprecision(8) << j;
-           ofile << setw(15) << setprecision(8) << autocor[j] << endl;
-         }
-         ofile.close();  // close output file
-         return 0;
-    }  // end of main program 
-
-
-
-
-

1.60. Which RNG should I use?

-
    -

  • C++ has a class called random. The random class contains a large selection of RNGs and is highly recommended. Some of these RNGs have very large periods making it thereby very safe to use these RNGs in case one is performing large calculations. In particular, the Mersenne twister random number engine has a period of \(2^{19937}\).

    • -

  • Add RNGs in Python

    • -
-
-
-

1.61. How to use the Mersenne generator

-

The following part of a c++ code (from project 4) sets up the uniform distribution for \(x\in [0,1]\).

-
    /*
-    
-    //  You need this 
-    #include <random>
-    
-    // Initialize the seed and call the Mersienne algo
-    std::random_device rd;
-    std::mt19937_64 gen(rd());
-    // Set up the uniform distribution for x \in [[0, 1]
-    std::uniform_real_distribution<double> RandomNumberGenerator(0.0,1.0);
-    
-    // Now use the RNG
-    int ix = (int) (RandomNumberGenerator(gen)*NSpins);
-
-
-
-
-

1.62. Why blocking?

-

Statistical analysis.

-
* Monte Carlo simulations can be treated as *computer experiments*
-
-* The results can be analysed with the same statistical tools as we would use analysing experimental data.
-
-* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.
-
-
-

A very good article which explains blocking is H. Flyvbjerg and H. G. Petersen, Error estimates on averages of correlated data, Journal of Chemical Physics 91, 461-466 (1989).

-
-
-

1.63. Why blocking?

-

Statistical analysis.

-
* As in other experiments, Monte Carlo experiments have two classes of errors:
-
-  * Statistical errors
-
-  * Systematical errors
-
-
-* Statistical errors can be estimated using standard tools from statistics
-
-* Systematical errors are method specific and must be treated differently from case to case. (In VMC a common source is the step length or time step in importance sampling)
-
-
-
-
-

1.64. Code to demonstrate the calculation of the autocorrelation function

-

The following code computes the autocorrelation function, the covariance and the standard deviation -for standard RNG. -The following file gives the code.

-
    //  This function computes the autocorrelation function for 
-    //  the Mersenne random number generator with a uniform distribution
-    #include <iostream>
-    #include <fstream>
-    #include <iomanip>
-    #include <cstdlib>
-    #include <random>
-    #include <armadillo>
-    #include <string>
-    #include <cmath>
-    using namespace  std;
-    using namespace arma;
-    // output file
-    ofstream ofile;
-    
-    //     Main function begins here     
-    int main(int argc, char* argv[])
-    {
-      int MonteCarloCycles;
-      string filename;
-      if (argc > 1) {
-        filename=argv[1];
-        MonteCarloCycles = atoi(argv[2]);
-        string fileout = filename;
-        string argument = to_string(MonteCarloCycles);
-        fileout.append(argument);
-        ofile.open(fileout);
-      }
-    
-      // Compute the variance and the mean value of the uniform distribution
-      // Compute also the specific values x for each cycle in order to be able to
-      // compute the covariance and the correlation function  
-    
-      vec X  = zeros<vec>(MonteCarloCycles);
-      double MCint = 0.;      double MCintsqr2=0.;
-      std::random_device rd;
-      std::mt19937_64 gen(rd());
-      // Set up the uniform distribution for x \in [[0, 1]
-      std::uniform_real_distribution<double> RandomNumberGenerator(0.0,1.0);
-      for (int i = 0;  i < MonteCarloCycles; i++){
-        double x =   RandomNumberGenerator(gen); 
-        X(i) = x;
-        MCint += x;
-        MCintsqr2 += x*x;
-      }
-      double Mean = MCint/((double) MonteCarloCycles );
-      MCintsqr2 = MCintsqr2/((double) MonteCarloCycles );
-      double STDev = sqrt(MCintsqr2-Mean*Mean);
-      double Variance = MCintsqr2-Mean*Mean;
-      //   Write mean value and variance
-      cout << " Sample variance= " << Variance  << " Mean value = " << Mean << endl;
-      // Now we compute the autocorrelation function
-      vec autocorrelation = zeros<vec>(MonteCarloCycles);
-      for (int j = 0; j < MonteCarloCycles; j++){
-        double sum = 0.0;
-        for (int k = 0; k < (MonteCarloCycles-j); k++){
-          sum  += (X(k)-Mean)*(X(k+j)-Mean); 
-        }
-        autocorrelation(j) = sum/Variance/((double) MonteCarloCycles );
-        ofile << setiosflags(ios::showpoint | ios::uppercase);
-        ofile << setw(15) << setprecision(8) << j;
-        ofile << setw(15) << setprecision(8) << autocorrelation(j) << endl;
-      }
-      // Now compute the exact covariance using the autocorrelation function
-      double Covariance = 0.0;
-      for (int j = 0; j < MonteCarloCycles; j++){
-        Covariance  += autocorrelation(j);
-      }
-      Covariance *=  2.0/((double) MonteCarloCycles);
-      // Compute now the total variance, including the covariance, and obtain the standard deviation
-      double TotalVariance = (Variance/((double) MonteCarloCycles ))+Covariance;
-      cout << "Covariance =" << Covariance << "Totalvariance= " << TotalVariance << "Sample Variance/n= " << (Variance/((double) MonteCarloCycles )) << endl;
-      cout << " STD from sample variance= " << sqrt(Variance/((double) MonteCarloCycles )) << " STD with covariance = " << sqrt(TotalVariance) << endl;
-    
-      ofile.close();  // close output file
-      return 0;
-    }  // end of main program 
-
-
-
-
-

1.65. What is blocking?

-

Blocking.

-
* Say that we have a set of samples from a Monte Carlo experiment
-
-* Assuming (wrongly) that our samples are uncorrelated our best estimate of the standard deviation of the mean $\langle \mathbf{M}\rangle$ is given by
-
-
-
-\[ -\sigma=\sqrt{\frac{1}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} -\]
-
    -

  • If the samples are correlated we can rewrite our results to show that

    • -
-
-\[ -\sigma=\sqrt{\frac{1+2\tau/\Delta t}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} -\]
-

where \(\tau\) is the correlation time (the time between a sample and the next uncorrelated sample) and \(\Delta t\) is time between each sample

-
-
-

1.66. What is blocking?

-

Blocking.

-
* If $\Delta t\gg\tau$ our first estimate of $\sigma$ still holds
-
-* Much more common that $\Delta t<\tau$
-
-* In the method of data blocking we divide the sequence of samples into blocks
-
-* We then take the mean $\langle \mathbf{M}_i\rangle$ of block $i=1\ldots n_{blocks}$ to calculate the total mean and variance
-
-* The size of each block must be so large that sample $j$ of block $i$ is not correlated with sample $j$ of block $i+1$
-
-* The correlation time $\tau$ would be a good choice
-
-
-
-
-

1.67. What is blocking?

-

Blocking.

-
* Problem: We don't know $\tau$ or it is too expensive to compute
-
-* Solution: Make a plot of std. dev. as a function of blocksize
-
-* The estimate of std. dev. of correlated data is too low $\to$ the error will increase with increasing block size until the blocks are uncorrelated, where we reach a plateau
-
-* When the std. dev. stops increasing the blocks are uncorrelated
-
-
-
-
-

1.68. Implementation

-
* Do a Monte Carlo simulation, storing all samples to file
-
-* Do the statistical analysis on this file, independently of your Monte Carlo program
-
-* Read the file into an array
-
-* Loop over various block sizes
-
-* For each block size $n_b$, loop over the array in steps of $n_b$ taking the mean of elements $i n_b,\ldots,(i+1) n_b$
-
-* Take the mean and variance of the resulting array
-
-* Write the results for each block size to file for later
-  analysis
-
-
-
-
-

1.69. Actual implementation with code, main function

-

When the file gets large, it can be useful to write your data in binary mode instead of ascii characters. -The following python file reads data from file with the output from every Monte Carlo cycle.

-
-
-
# Blocking
-    @timeFunction
-    def blocking(self, blockSizeMax = 500):
-        blockSizeMin = 1
-
-        self.blockSizes = []
-        self.meanVec = []
-        self.varVec = []
-
-        for i in range(blockSizeMin, blockSizeMax):
-            if(len(self.data) % i != 0):
-                pass#continue
-            blockSize = i
-            meanTempVec = []
-            varTempVec = []
-            startPoint = 0
-            endPoint = blockSize
-
-            while endPoint <= len(self.data):
-                meanTempVec.append(np.average(self.data[startPoint:endPoint]))
-                startPoint = endPoint
-                endPoint += blockSize
-            mean, var = np.average(meanTempVec), np.var(meanTempVec)/len(meanTempVec)
-            self.meanVec.append(mean)
-            self.varVec.append(var)
-            self.blockSizes.append(blockSize)
-
-        self.blockingAvg = np.average(self.meanVec[-200:])
-        self.blockingVar = (np.average(self.varVec[-200:]))
-        self.blockingStd = np.sqrt(self.blockingVar)
-
-
-
-
-
  File "<ipython-input-6-2ff97f4bf03b>", line 2
-    @timeFunction
-    ^
-IndentationError: unexpected indent
-
-
-
-
-
-
-

1.70. The Bootstrap method

-

The Bootstrap resampling method is also very popular. It is very simple:

-
    -

  1. Start with your sample of measurements and compute the sample variance and the mean values

    2. -

  2. Then start again but pick in a random way the numbers in the sample and recalculate the mean and the sample variance.

    3. -

  3. Repeat this \(K\) times.

    4. -
-

It can be shown, see the article by Efron -that it produces the correct standard deviation.

-

This method is very useful for small ensembles of data points.

-
-
-

1.71. Bootstrapping

-

Given a set of \(N\) data, assume that we are interested in some -observable \(\theta\) which may be estimated from that set. This observable can also be for example the result of a fit based on all \(N\) raw data. -Let us call the value of the observable obtained from the original -data set \(\hat{\theta}\). One recreates from the sample repeatedly -other samples by choosing randomly \(N\) data out of the original set. -This costs essentially nothing, since we just recycle the original data set for the building of new sets.

-
-
-

1.72. Bootstrapping, recipe

-

Let us assume we have done this \(K\) times and thus have \(K\) sets of \(N\) -data values each. -Of course some values will enter more than once in the new sets. For each of these sets one computes the observable \(\theta\) resulting in values \(\theta_k\) with \(k = 1,...,K\). Then one determines

-
-\[ -\tilde{\theta} = \frac{1}{K} \sum_{k=1}^K \theta_k, -\]
-

and

-
-\[ -sigma^2_{\tilde{\theta}} = \frac{1}{K} \sum_{k=1}^K \left(\theta_k-\tilde{\theta}\right)^2. -\]
-

These are estimators for \(\angle\theta\rangle\) and its variance. They are not unbiased and therefore -\(\tilde{\theta}\neq\hat{\theta}\) for finite K.

-

The difference is called bias and gives an idea on how far away the result may be from -the true \(\angle\theta\rangle\). As final result for the observable one quotes \(\angle\theta\rangle = \tilde{\theta} \pm \sigma_{\tilde{\theta}}\) .

-
-
-

1.73. Bootstrapping, code

-
    # Bootstrap
-        @timeFunction
-        def bootstrap(self, nBoots = 1000):
-            bootVec = np.zeros(nBoots)
-            for k in range(0,nBoots):
-                bootVec[k] = np.average(np.random.choice(self.data, len(self.data)))
-            self.bootAvg = np.average(bootVec)
-            self.bootVar = np.var(bootVec)
-            self.bootStd = np.std(bootVec)
-
-
-
-
-

1.74. Jackknife, code

-
    # Jackknife
-        @timeFunction
-        def jackknife(self):
-            jackknVec = np.zeros(len(self.data))
-            for k in range(0,len(self.data)):
-                jackknVec[k] = np.average(np.delete(self.data, k))
-            self.jackknAvg = self.avg - (len(self.data) - 1) * (np.average(jackknVec) - self.avg)
-            self.jackknVar = float(len(self.data) - 1) * np.var(jackknVec)
-            self.jackknStd = np.sqrt(self.jackknVar)
-
-
-
-
- - - - -
- - -
- - Textbooks - 2. Linear Algebra, Handling of Arrays and more Python Features - -
- -
-
-
-
-

- - By Morten Hjorth-Jensen
- - © Copyright 2020.
-

-
-
-
- - -
-
- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/teachers.html b/doc/LectureNotes/_build/html/teachers.html deleted file mode 100644 index d6b2b7373..000000000 --- a/doc/LectureNotes/_build/html/teachers.html +++ /dev/null @@ -1,393 +0,0 @@ - - - - - - - - Teachers and Grading — Applied Data Analysis and Machine Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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Teachers and Grading

-
-

Instructor information

-
    -

  • Name: Morten Hjorth-Jensen

    • -

  • Email: morten.hjorth-jensen@fys.uio.no

    • -

  • Phone: +47-48257387

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  • Office: Department of Physics, University of Oslo, Eastern wing, room FØ470

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  • Office hours: Anytime! In Fall Semester 2020 (FS20), as a rule of thumb office hours are planned via computer or telephone. Individual or group office hours will be performed via zoom. Feel free to send an email for planning. In person meetings may also be possible if allowed by the University of Oslo’s COVID-19 instructions (see below for links).

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Grading

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Grading scale: Grades are awarded on a scale from A to F, where A is the best grade and F is a fail. There are three projects which are graded and each project counts 1/3 of the final grade. The total score is thus the average from all three projects.

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The final number of points is based on the average of all projects (including eventual additional points) and the grade follows the following table:

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Textbooks

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Recommended textbooks:

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The books by Bishop and Hastie et al. can be downloaded for free if you access the university library via an IP number of your home university.

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General learning book on statistical analysis:

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  • Christian Robert and George Casella, Monte Carlo Statistical Methods, Springer

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  • Peter Hoff, A first course in Bayesian statistical models, Springer

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General Machine Learning Books:

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  • Kevin Murphy, Machine Learning: A Probabilistic Perspective, MIT Press

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  • Christopher M. Bishop, Pattern Recognition and Machine Learning, Springer

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  • David J.C. MacKay, Information Theory, Inference, and Learning Algorithms, Cambridge University Press

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  • David Barber, Bayesian Reasoning and Machine Learning, Cambridge University Press

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- - - - - - - - \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/Clustering.ipynb b/doc/LectureNotes/_build/jupyter_execute/Clustering.ipynb deleted file mode 100644 index 2c1c48b1e..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/Clustering.ipynb +++ /dev/null @@ -1,917 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Clustering Analysis\n", - "In this chapter we will concern ourselves with the study of **cluster analysis**.\n", - "In general terms cluster analysis, or clustering, is the task of grouping a\n", - "data-set into different distinct categories based on some measure of equality of\n", - "the data. This measure is often referred to as a **metric** or **similarity\n", - "measure** in the literature (note: sometimes we deal with a **dissimilarity\n", - "measure** instead). Usually, these metrics are formulated as some kind of\n", - "distance function between points in a high-dimensional space.\n", - "\n", - "There exists a lot of such distance measures. The simplest, and also the most\n", - "common is the **Euclidean distance** (i.e. Pythagoras). A good source for those of\n", - "you wanting a thorough overview is the article (DOI:10.5120/ijca2016907841\n", - "Irani, Pise, Phatak). A few other metrics mentioned there are: *cosine\n", - "similarity*, *Manhattan distance*, *Chebychev distance* and the *Minkowski\n", - "distance*. The Minkowski distance is a general formulation which encapsulates a\n", - "range of metrics. All of these, and many more, can be used in clustering. There\n", - "exists different categories of clustering algorithms. A few of the most\n", - "common are: *centroid-*, *distribution-*, *density-* and *hierarchical-\n", - "clustering*. We will concern ourselves primarily with the first one." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Basic Idea of the K-means Clustering Algorithm\n", - "The simplest of all clustering algorithms is the aptly named **k-means algorithm**\n", - ", sometimes also referred to as *Lloyds algorithm*. It is the simplest and also\n", - "the most common. From its simplicity it obtains both strengths and weaknesses.\n", - "These will be discussed in more detail later. The k-means algorithm is a\n", - "**centroid based** clustering algorithm.\n", - "\n", - "Assume, we are given $n$ data points and we wish to split the data into $K < n$\n", - "different categories, or clusters. We label each cluster by an integer $k\\in\\{\n", - "1, \\cdots, K \\}$. In the basic k-means algorithm each point is assigned to only\n", - "one cluster $k$, and these assignments are *non-injective* i.e. many-to-one. We\n", - "can think of these mappings as an encoder $k = C(i)$, which assigns the $i$-th\n", - "data-point $\\bf x_i$ to the $k$-th cluster. Before we jump into the mathematics\n", - "let us describe the k-means algorithm in words:\n", - "1. We start with guesses / random initializations of our $k$ cluster centers / centroids\n", - "\n", - "2. For each centroid the points that are most similar are identified\n", - "\n", - "3. Then we move / replace each centroid with a coordinate average of all the points that were assigned to that centroid.\n", - "\n", - "4. Iterate this points 2, 3) until the centroids no longer move (to some tolerance)\n", - "\n", - "Now we consider the method formally. Again, we assume we have $n$ data-points\n", - "(vectors)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\\label{eq:kmeanspoints} \\tag{1}\n", - " \\boldsymbol{x_i} = \\{x_{i, 1}, \\cdots, x_{i, p}\\}\\in\\mathbb{R}^p.\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which we wish to group into $K < n$ clusters. For our dissimilarity measure we\n", - "will use the *squared Euclidean distance*" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\\label{eq:squaredeuclidean} \\tag{2}\n", - " d(\\boldsymbol{x_i}, \\boldsymbol{x_i'}) = \\sum_{j=1}^p(x_{ij} - x_{i'j})^2\n", - " = ||\\boldsymbol{x_i} - \\boldsymbol{x_{i'}}||^2\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Next we define the so called *within-cluster point scatter* which gives us a\n", - "measure of how close each data point assigned to the same cluster tends to be to\n", - "the all the others." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\\label{eq:withincluster} \\tag{3}\n", - " W(C) = \\frac{1}{2}\\sum_{k=1}^K\\sum_{C(i)=k}\n", - " \\sum_{C(i')=k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}}) =\n", - " \\sum_{k=1}^KN_k\\sum_{C(i)=k}||\\boldsymbol{x_i} - \\boldsymbol{\\overline{x_k}}||^2\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\boldsymbol{\\overline{x_k}}$ is the mean vector associated with the $k$-th\n", - "cluster, and $N_k = \\sum_{i=1}^nI(C(i) = k)$, where the $I()$ notation is\n", - "similar to the Kronecker delta (*Commonly used in statistics, it just means that\n", - "when $i = k$ we have the encoder $C(i)$*). In other words, the within-cluster\n", - "scatter measures the compactness of each cluster with respect to the data points\n", - "assigned to each cluster. This is the quantity that the $k$-means algorithm aims\n", - "to minimize. We refer to this quantity $W(C)$ as the within cluster scatter\n", - "because of its relation to the *total scatter*." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\\label{eq:totalscatter} \\tag{4}\n", - " T = W(C) + B(C) = \\frac{1}{2}\\sum_{i=1}^n\n", - " \\sum_{i'=1}^nd(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\n", - " = \\frac{1}{2}\\sum_{k=1}^K\\sum_{C(i)=k}\n", - " \\Big(\\sum_{C(i') = k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\n", - " + \\sum_{C(i')\\neq k}d(\\boldsymbol{x_i}, \\boldsymbol{x_{i'}})\\Big)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Which is a quantity that is conserved throughout the $k$-means algorithm. It can\n", - "be thought of as the total amount of information in the data, and it is composed\n", - "of the aforementioned within-cluster scatter and the *between-cluster scatter*\n", - "$B(C)$. In methods such as principle component analysis the total scatter is not\n", - "conserved.\n", - "\n", - "Given a cluster mean $\\boldsymbol{m_k}$ we define the **total cluster variance**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\\label{eq:totalclustervariance} \\tag{5}\n", - " \\min_{C, \\{\\boldsymbol{m_k}\\}_1^K}\\sum_{k=1}^KN_k\\sum||\\boldsymbol{x_i} - \\boldsymbol{m_k}||^2\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now we have all the pieces necessary to formally revisit the k-means algorithm.\n", - "If you at this point feel like some of the above definitions came a bit out of\n", - "no-where, don't fret, the method does get a whole lot simpler once we start\n", - "programming." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The K-means Clustering Algorithm\n", - "The k-means clustering algorithm goes as follows (note in my opinion this\n", - "description is a bit complicated and is lifted directly out of ESL HASTIE for\n", - "deeper understanding purposes)\n", - "\n", - "1. For a given cluster assignment $C$, and $k$ cluster means $\\{m_1, \\cdots, m_k\\}$. We minimize the total cluster variance with respect to the cluster means $\\{m_k\\}$ yielding the means of the currently assigned clusters.\n", - "\n", - "2. Given a current set of $k$ means $\\{m_k\\}$ the total cluster variance is minimized by assigning each observation to the closest (current) cluster mean. That is $$C(i) = \\underset{1\\leq k\\leq K}{\\mathrm{argmin}} ||\\boldsymbol{x_i} - \\boldsymbol{m_k}||^2$$\n", - "\n", - "3. Steps 1 and 2 are repeated until the assignments do not change.\n", - "\n", - "As previously stated the above formulation can be a bit difficult to understand,\n", - "*at least the first time*, due to the dense notation used. But all in all the\n", - "concept is fairly simple when explained in words. The math needs to be\n", - "understood but to help you along the way we summarize the algorithm as follows\n", - "(try to look at the terms above to match with the summary).\n", - "\n", - "1. Before we start we specify a number $k$ which is the number of clusters we want to try to separate our data into.\n", - "\n", - "2. We initially choose $k$ random data points in our data as our initial centroids, *or means* (this is where the name comes from).\n", - "\n", - "3. Assign each data point to their closest centroid, based on the squared Euclidean distance.\n", - "\n", - "4. For each of the $k$ cluster we update the centroid by calculating new mean values for all the data points in the cluster.\n", - "\n", - "5. Iteratively minimize the within cluster scatter by performing steps (3, 4) until the new assignments stop changing (can be to some tolerance) or until a maximum number of iterations have passed.\n", - "\n", - "That's it, nothing magical happening." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Writing Our Own Code\n", - "In the following section we will work to develop a deeper understanding of the\n", - "previously discussed mathematics through developing codes to do k-means cluster\n", - "analysis." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Basic Python\n", - "\n", - "Let us now program the most basic version of the algorithm using nothing but\n", - "Python with numpy arrays. This code is kept intentionally simple to gradually\n", - "progress our understanding. There is no vectorization of any kind, and even most\n", - "helper functions are not utilized. Throughout our implementation process it will\n", - "be helpful to keep in mind both the mathematical description of the algorithm\n", - "*and* our summary from above. In addition, try to think of ways to optimize this\n", - "while reading the next section. We will get to it, take it as a challenge to see\n", - "if your optimizations are better.\n", - "\n", - "First of all we need a dataset to do our cluster analysis on, for clarity (and\n", - "lack of googling beforehand) we generate it ourselves using Gaussians. First we\n", - "import" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": {}, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "%matplotlib inline\n", - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "import time\n", - "from IPython.display import display\n", - "\n", - "np.random.seed(2021)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Next we define functions, for ease of use later, to generate Gaussians and to\n", - "set up our toy data set." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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IZ5wWvLY9eigBrUXGfX8sAvljgI86rvwocML1vbDcmv8sKapP9ZDFverGH+da3iu43hmIXtkaMs3KecPvo1BFbw2U7y8Sw9tG7VfbIXGg2B6mjLOSZujRCL0HXix+6flT/MYbuxxMc5YaPheWmxhr+NoTS7Qjn51RyjevHCCATuhRacswKR6wDLJEUZn5e59kJa9uDWlF/n3VDk8y+muHU75yfmkx+HYfLAL5Y4BPsnTxKCcNf1S41zl/3Gt5tyz0zkBUaehEPtvDlGfX/dvO4V74sPIC8KHZqpKCtNTzchu4RTbwHrz1dWG5yZ/+2rn5a57qRozTEiUF47Tgm1cOMNrw3EaXysCN44TzS416UeSepZA7F82rB1MsYl7KuTVYWwTPnWoDzIeKQk8ikXMmziI7vzsWgfwxwSdVujihqQ2Sgmle0QpdMLmzBvuopQM+zvHuVz75KNfyYUfrW6GiqAzTvJr/7GCSczjN+cbl/YcarFJS8GE7jJNRe/h4O7U7r8fJe31te4QAntvo0grfLz0dz3KOZjkvXh+gjWGpHggaJgWXVlscz4r5dTr59yApKLTmS2e6tCP/A8Ha8wSvbg350tn+LaqLglb00XYznzUsAvljjJMv3PYwZZpVtCKPjV78SLehS82A37q8Tzf251/ArUHCn/rq2dvO46PWNx8mA33QjOxRl6Ie5P3duXhs9GJ+sDmkHTplwYNJzmtbQ75wpveBY4ALUi/dOCZQikurLaZ5xfYw5WiWszfK+UOfu30qMvYVWal/KDu1k8C+PUwZzArePZjSjQJW2yGNQHHjeMYwLVhrR+SlYWc0QckJT622ePdgyhdveY9XD6a3cdePpwXvHBzz4vVjtIa1TsjFlSaXVlr8YHPI1YPpfLw/rzQXltvz9/tp3gV+0lgE8scUJ8FlklW8fHNAXllCJajO9R+ohvmgOJ4VfOFMj0FSMMk1rcjn3FKT41kxlzZ9GH71nQ3BUVqydkdD8G4Z6INmZI+6FHXvmjvzxXRnlHJxuUXkS64dzRhnJaEnuDlImGYVXzjTY60T3XaMVzaHaGNpBB4SSVZW/N2XbnI0K2h6Hhu9iFlRzbPVW1UIhbh3aePjLPQnjJhGoMiKirSoeO+oYqnhM0xLsIK3dsZzrnngSb71zgFPr7Xvet+WmgH/9LUdxlmF1hZPCW4cz2iFHk+ttfnSmS6X98ZYBAbLs+vt+6ouLuCwCOSfctzLkssY+MHmEE8Iuu2AWaZ5ZXPIzzy58si2oScNtZMABMypj7c+50Hq6B9oCG4OmeQVy3c0BF/bHvETF5bue7x74VGWou72/m4OEoZJQSPwONdvEHmKN3bGJEXFmV7Mzz25gqfkvPwR+ZLLu+N5iep0N+LK/pQvbHRdIJfwzu6MzeOUwBPEDcWN44TQV7y1M+F77x0RBR7N0GOlGbHc8gk8xTipGGUFhTac68ec6sS0I49RWtLwPd47nHE8K0iLilPdmG9dOeBUJ8JaaEUe7chd91tZKdcOZ6y2Ql7ZdJriaVEwSkvek4InVhrcPErxPUUndg3QcVY6iqqxH7hOJ/fteFaw1AooKstmpjFYluJgvvs4t9SYNzYfVHVxAYdFIP8UY5gUH+Bwbw4Sfv7pVcZpyfEsRxtLpxEgEDRDxSA1HM9yPPVhJMGHx4PQ9R6U0veBhqCxdD+kISiwn6pJ0bu9v2lWsdIK5z+PA8U0L0kLTeSr+eI0zSpevH7ENNMst0LO9GJKbXh1a0hluG3I6HBSMExKSq0ZJRpPWkLfw5OCpKhYUYpJUjJOK24OJM+stRmmJUVp2BrOKCsDFraGFm0cdXR/nNGOfARweXdMHCh2hylPn2pzOM0ZZyVJXnG6F+NJybeuHDJMCqy1LLUClCw5mOREvuKptRZFZZnlFW0pqLStz81gLXjy9s/drfdtnJZUFUgp6tfOEEJw/XBGUrhg/sL5HsCnbi7h045FIP8U45XNIZuDlF4c0IgleWXYHKS8sukoaG/tjulGPmVlCDxFqS3dyOM4KXmqZgB8XFxcafKtKwd1cDH4StJr+PzcLQNBD1qT/mBD0KOoNNP8/QnGtNQ8tdYmKSqmWcXxLOc4KVBS8kvPn3ok7+lhcbf314q8D/hb5qUhUIJSGy7vjjnTb7B5PCPJNXGg0Nry3uGU070GFkEzkPNFYpprCq3JS40QgkBBUhrGWUY39rmw3OLCcpOi0nz/xoDV2p5trR1xlORUxvLu/tT1MvKK59c7fPvqEef6MYGnOJzmSCnAwiit8JTizZ0he+OUM90YIVIEgkq7Jm2/ETBKSqQUPLfeoagsWanJSkNWGaqkIAoUaWGQQvL0qTatyCcpqg/9HHRin1FWIBG0Yo/Ak2wOZgyykk7D52sX+nhKznsHJ/2DBe6PRSD/EeGjsDDe2Z/Qjfy6gw+Rr+han3f2J/zrXz3Ht64ccDTL59lSJ/Y53YlQkgcaj3/Qc7IAAoT7zwfm/B60Jv1hDcFXNge0Iv+2LfRXzvcZpSW/8cYu2sBSI2CpGXL1YEo39n/kmdnd3t+1w9n8/ZwwLRrByb1yUq7/7I09Qk+SFIaLqy2S3DDKCoZJwU9fWmaYlvNF6+rBxN3LQNGJXHM5G2cUlSEtDKttJwlbWcs4LSi1xlOSZqDYPEqoKjduvz/JOJ4VPLHUYJpX7I4zjIXtYULse2xnBUVlSSrN4TgjChS+L7myN+XzG108JXlrd8okryg1pGXJE0tO18dYwVo74NlTLd7enzJOS55abdJtBASe4peeP8XxrOCt3TFv7Y4YpZpvXtnn608s8/mNLlIKZkWFXwmksKSFYanh4yvFzihjoxfPJRMWDJUHxyKQ/wjwUVkdSWE4GM9ICkNauu16M5A06wbQajtkf5yz1AjItGacFqy1Q37p+fUHambdeU7fvHJAN/Y/UCtda0c8cZeBoDsXgy+f6z2w6JGSgrP9Bp3YZ5AUCAFKCn5wc8juKOPicuu22vydg0h3Y7w8ageluy14F1eYv58Tf8tW6IFwFL3tYcLRJOXJ1TZhK2B7mPH0WpsLyw3GWYmnJBu9mKVmwH//8hb745yi0vi+pNKWrKqotEZJQ+j7bA5SAiXZn6RM6gAdKMn+OEcK8KVkrRthjEAJwavbIxCWaVailGSUliSFJskrSm2Y5AVJbmhFHpGv2J/kDN49xJcCgcWTAVlRsj/OAYGwgmdONdnoN0iKiqfX2qSV5p2DGeeWDP/GV8/RjX3e2B7zjcv7VNpwph+jhOJfvLXPOC350pkeNwcz9sc5WanR1nCu16Df9Cm15fLuhGdOtT4g0rXAvbEI5D8CfBTVvJMa5f7EfeB9T5HkJUmgaIQer2wOubjS4vxSszYG0HgKzi815mySk+PcrVl65yTd5iBllJRcXGny+vaIb769j5SSr5zr3XZuJw2sh12gPiyz/bmnV2+bjDTGBcHv3zimGXh88UwPbS3TXNMMJI3Q4yu1ycLJa3tS8PrWiH/2+i6NwOP50x1W2+HHpkE+yOj4yftx/paKF873mWYVv/HGDoezAqQk9BWVsdzYnzDJSp4/3a4zdld2eGVziK8EXzjTnZs+DJOCwPd5fqPH5jCh4SusNVw/nHFjkOJ7gsCTSATbw4R25HO6zmaNNZzuxtwczPjKuSVuDmYcTgtOd2N2RimltWjAR2Bw4mtH04JTnYiirEApKmMYZSWzXNMIFHlpwFoOZzl5qWlHgbuHkcdyI+D3PbnCe0czrh/NeHVzjJKCwPMZzCrWez5Lcci1wxk/+1TM02ttvny2z42jGaHnrs1qJ57vPK8dzfj8RvdBvloL1FgE8h8BHoTVcWcgcV/4LnvjrGYCWJSSRL7i+dNd3tmf8PULywgh5o3CO9kkdwbHt3bHvHh9wC89f+oD57Q9TOlGjlr29t6UyFestEKuH6W8sjXiy2d7H6CCfZQF6l6TkcbAjeMZka9Y78TcHCT8je/e4Km1FqutgKRQBFk1v1aNwEMbO3eDx8IoyblxPCMO1Px8HySLv1uwflCZ2JNewmvbQ947mDHOHHOlGwe8tjXkwnKTJ1ebXD9K+P6NIZ/f6NCvNbpf2Ryx1PBphQGTvMICq+2AUhsakeKFc46XvTlIeH17jK8Eoac43Y3RxpJXhlJbelFAK/SQAqZFSaktG72Iz53u8HvvHSERZKVGJIUb4MkcBVS4ohlZpTEW/Lo2f7g7phMHnOqESCGoKs3OKOMHN4d0Yx9jDYOJRknL1jAhKw2NQDErSqRwzd+TQaFTnZDDSY617zcy98YZ55ZiDqcFW8MEXVmkAiEEf+xLG/f/Yi0wxyKQfwQ8bL37fqyOD8v6Xr4x4GsXljjTb5AVmrTUxL4kCjxW2yHvHc/uesyT83vpxjGVsVTa0m8ErLYixmnBb7yxyzOn2rf9/jTXVFpz7WBKM/ToNXxWWiG9po/AcvVwypfO9G5rYP3g5vCBx/cfZDLyeOZq/ZHv0Qo9to5naAvGWNJCczDJ+emaWnmyEJ0E8cj3qIxFCrfYnTBhbj2fB52cvHN0/Gwv5uZxchtt8MO2/hY4mhYEnqIZukWmMpYLy03yyiCl4OxSg2bo0W0EnOs3SEvN1iAh8lostyKePdXh5nHC5iCh0pZT7YiNbsS1wwSAZqCIlPscdOKAUmu6sc9xUlBaw1NrTbaGGUrC86fbzPKKg0mOAApjEFLSCDwKbWhhiesmqNYGJQWRFAgp5kJpnchDSslKMyDLNeO84GiW8d1rR2SVphV5XFhq8VtvH5CVmvVejLCQV5bAc4yWtDQkuaEZeXTqHsfJwno8LUgKzSSrsFjKCvqNhw9Lj3qy+HHDIpA/JD5Kvft+rI4Py/qWWuGcy1tqQ+R7rsSiRP3FtHzn6hFLTZcNnvCV17ut+flJJMezlKzQdOIAIQSdOOBgmgPcNt7tHNXHeDUrZZpVXNmb0G8FPL3aYpiUH2hk3j6+r2mFaj6+f7fBH0+KednmhfN9vnS2Nz/WW7tjVluuJj4rKlpx4Lbz0xwLSAFv705QUtCOXHnprd0xy82Q1U6ErwTgstXxhzgo3S27vpW3PslKfvfdIzaPE4ZpQex7fM8e8fWLy4Se5N2DKb937YjnTrd5YqU5H0ffHWWstEJOdxt0ItfovLI/Zus44cm1JqOs4sJSAykFvhQUmjk9cb0b873rA9Y7Ed3YY63jFtDPb3SY5Zp/8uo2hbYkuWZSaI5mKWc7DQrtczx19e/lpk9eaP7JaztobVGe5Fy/wTAtMQbakZswDZXgIC1ACOK65FMaiyclSoAUktBT6MDirqYgzSpGQtAIJNrA5nGKxiKEoCyd7njkKUZZSV5pnjnV5ejIKTK2Iw+s4DjN+fqF/m1N+BP1xUa9A8srQ1bquZbLgzY7F8qJC2OJh8ZHMSY4qaXeTRD/wwwLLi67INFvBKRFxSjJScsKX0m3VV9q8bULfRDw4vVj8kpzabXFt9894u29CTePE6SErLA0Q4/DSQZAXjmWwMkW9+ScfCXY6DeIfMW7+1N+sDniMClRCGaF5miS36Yb4ib/Uv7BD7Z4c2eMqptqr20N8aS4zX7sxlEyr7+/vTdFCclKK+TGUTI3Gri40kRJyTh1vYFhUtIMPFbaEc3QY7UVcaoTMUgLrh1M2R66hl879Enykiu7E5SUtEKPcVrQDNQHDC7uZgxxwlsHeGVzxJs7Q0qjWeuEYC2bg4Q3d0a8dzhF13rdk6zi731/k+Opu0fDtOD68QwpXKkDIPIUB9OMy7sTPCl4dr2DrSlArfB92qKSzBe717fHfPe9YyJP8qWzPa4dTDmYFQzSEouhGSqsge1xzvZxgrGWpq+4tNKm2/RJS4MG1jsRlTbsTwpGacGs0Fxaa/NTl5bpNkLyyu1yrIVWoNDGMkhKCu2aoYfTHF+peudh0cbtKAptWO3GdKKAhq/wPME0L9kaJSRZyc4o4zjJWW0HSAGDtKQVSX7xuTV+35MrvLI55P/1rav8v7/1Lq9sDvEkHM4Kvn/jmO1hwplezGo7fCibwM+SZ+zd8EgyciFED/h/Al/A7TD/V9ba330Ux/604aOqAd5r0vDDSi+ekrxw3tWlsyqej1jfOer95bMBSVGRV07XYpiWrDRDCm3Yn+TsjBKEhTCQtCMfKQTnlxof2OICZKXmm28fcHOQgIVISF7ZHHJ21uBzGx2SvKKoDN+6coDFBZ4vnOlymDi62TPrHb5wpscb22OeWGneNvjjS8FvvLFDMwzoNTxWWiFVPZp+Ii97phfzjbf3CZQk8CQXlhtc3p2w3o2orOHmYUpWajwhCTzFl8/2eGd/4oZcfEUjkCgpeGtvwtl+g1PdaM5FfunGgHcOpoSe5NKKY+C8cnPED7YGZKXmyt6En760yutbQ0JP4kvFUjOiqDLO9mPe3pvy1Qt9uqHHuXaD60cJZ3sRg6RgrROx3AyZ1ruAwaxgb5IRKMHnTneYFRXNWnPFk4JRVvLlsz3ATea+ujliZ5yx1HD16Hbozbn17x5OERamWck0r1BC0ot8BmnJ3jTn0orHc+tdEJZASYyxFJW7rsdJweEkp9/wCT1JM/CY5ZrlVsDuOGOalbSjgH4joBEYtgYJx8biS0kn9BECTnXfv09l5QSyIl8xmpXEgUepNYNpBdbV9YWQXD+cISSstSP+zNfPz5k5r24O2RwkdCIfELx4fcD1o4Rn1lpcOr9EXmm2hglKigcyxjjB46jQ+ajxqEorvwr8E2vtnxZCBEDjER33U4cfhsnD3UovtzIjTvCNy/t4UtSj3nruqXgy6r3cDCi0RVvmzaXjWY5MJa1gwk9eWkF+CM+8E/vcHCRcXG1x/TghKSomWUVeWcZ5hScFP9gacnlvwtbQDaGsd2NaoUcgFb6nOJzkPLXa4mCS8bnTnfmxpYCbg4RBUnGm36DScGV/yqXV5m2j7kvNgD/+pQ2uHc7YGqYEnuTSahNjLO/sTQg8xdcu9NkaZrx3OOULZ7p85fwST621ubI35gc3h3z1whJ//Esb81LTKC25ejClEXg8d6rNq1tDvn31iKRwpaPIl1xcbjLKSn7z8j7744wzSw1OdWJiXxH5EmM9AuWoc0KIeZZ6MngzyUpmRcXbe2N8T3G6E+MpN4n5zHqH052Y/WnGSzcHNAJnw+bOSfE77xwwSiqEsOyOUg4mGb//6TUCT/DK5pBxUlJog7VuDL4oLYEnWG5ENCNFI/B4dr3FNNcMZjkCwfEsY3OQYowhN6amG1rWOxF7EzfJmRYVseeRl4bdIiNQigvLTQ6mOXGg8KTAGMt7BwmtyMOT4EvFLHPURSlhlmlsna17SlAZWGn5dGOP/VHG8bTgcOLOZWeU0G+ErLVDIt99d8rK0AoVR7OCfjOsjTE0145mfO2JBx8G+qSNVz4N+NiBXAjRBX4/8L8EsNYWwI/tUvjDMHl4kIGak5rzq1sjJzQUeCgp8ZXgyv6EaVagEEgJ00xzOM04muVMMk079ogDj2nhGmv9Zpcf3BzewYlu8s2399kdpgyTAm0h9BRxKBgnJb/9ziGt0OOJlSbSCrYGKUmuUVLQjHzSvGJrkDBMS7qxzytbQyrt5FyT3FBpQ7vhPCAFUJQVm8eO6TBKS752vo82lp2RY+mc6cUst9wW+/XtEU+vtznTa9AMPIZJSSGZNzTbkY9Skq9eWOLL526/D99+9+i23cGXzvb5xuV9XtseuUZiv0HsKRqBT1aWaB0RexIlnM5HM/DYGSasdUKysprblp1fajDJSkS9qEa+4snVNvuTnFe3RvSbHkIK3tmfcGVvQl5p9iY5Xzvfpxv5XNmf8MbWCOkJpkVFO1R4QjDNK/7hqzv8yRc2eGVzxHI75O39CUfTAiXBV5JKGwgsG70YIQRSCFZbAduDxMnFVpaqMpTWkBUVk7RkmJV1zV8xKzRSCkprUPXCVGmLks7qzRaaaeaGiJqRxxOtmFB5NEJHgd0fZUxLt9BX2mIM7ncrTehLRmnFMKsQacU/e32Pz5/pMs00W4Mhv/DM+yqOpTbEvmKpFVJqzbsHE7DQbz5cAP4kjVc+LXgUGflF4AD4r4UQXwZeBP68tfa2ApUQ4leAXwE4f/78I3jZTwYPOsV4L9zZYV9qBrdpON/acR8mBa9sDnn5xpClZkBZafZHKePA54llN7l39WDK2aWYyhiOJwV7k5z3DqeUlWW5HfDMegdPSA5nGVcOpnztiaX5B/7WptBTp9r81tsHhL6kKC2VgYNZiq4M+xPJ184vYaxAKkFeuBH0cVahpGumrbcjkqLieJqjpOBUO6SoDG/ujsmKksNpySs3hiy3Ak53YqTS+BICJXhrd+Tqu9bOaWhbo5Q/+xPn2Ru7RmJUS7e2Ig9rPY5mOeO04NrRjBffO+aFcz0mWTmnHca++sDuoB35nO7GvL035uJKEylcm8hXgrwSnOpG+FKQlSV5JQDLxZUmnz/TZXuYstQK64EVw2tbQ+JQ0Qo8QCCl5GsXlhiku+yPc84tNbh57KRoLW5R+ObbBzRDj+fW24SB4qAeAtorNZFyC8g0L/kXbx2w0vL5+gV3b44mOVlpafiSyPdoRz6FNlxcaTo1Qiybo5S80kzzEoubmhQCPCmxxrA9SGnHHu3QRwiBMYYokJSVITOGvNIoIUi1Gy4y1pXE3j2YcX65RRx4+J5HaXOSUuNLia8sFou1AmPh+uEMY01d3lFcP5qxN86JfEkz9tkapTyz5tfXXFJUmmag0AaeXG1jrcFYHrkt3o87HkUg94CvAv+BtfY7QohfBf4PwH9665Ostb8G/BrA17/+9Q+6uT5G+LB694PSn+7ssB9Mcn7r8j5fONP7wBALuA/01jBlpRVyPCv4jTf38RR4mabUlpWWy6bzyvKDm0MOpznDrGRnmNFteHzxTBeB4HCac2XfZYa//c4h1roGW14aXrw+4Fy/wY1BggX6cUDqaSZpidaGSVGiK3h585i39id0Qg8pLdvDDGMNIBgkJYfTgksrbhpzqelGtm8cJ7x3OGWUlpxfarDaanFzmPL2/pSfeKLPc6ddgHz55oDjpGSjGzHNKoZJCRJe3x7xwvkeN44TxllFK1S8cK5PWmhuHDv3+KVmwJfP9bAWLu9O5vKnaalZbUcf7D94gk4jJMkrWpG7R6W2WGt5+lSHn35ymW+/e8TBJGO1HfHTTy5zYbl52z0+TnI2B65ZG3iKL5zp8Ic/d5pxVtLwFUfA/rjgaOYGuo6nBRYIPEmca+LAJysMe5OM2JMYC5mxFFrTiQKKskLJgLd2J2hjWe/GHM0KZL1ohp5iZ5Ryrh+DtfzExWWK0vIvxlntvelkbmPPNQADzzV1LTAr3PWYpCVZZQiVohSasrJ4gUBaKK3BGJjlGmst0zSnGyle2x6Rl67UhhR0osA1Tsuylllwr+15kqRw/9dUJIXlYJZhKk2Saxq+wmDxlXQ0xsAHXKP42fXOPU2aH/Q7+VnCowjkm8CmtfY79b//Ni6Qf2bwMPSnOylwg6SgW4+or3Wi2wZOwFHkKg15VfLq1ghrDT6KKFBsHk+pdIOVVsD+pKAyhkmm8YXAk4KsMLy0OaAbB1TasDfK6DV8Lu+M6TUdRbAb++zXutPDacGFlSavb41oBIqk0OSlIcvB81wAyCpLlldoYymNQRtnAhAGHr3Y573jmRsKyTVPr7V5c2dEHKg5jzrTjnmhjWGYFry9N0EK2BqkSOGolVIIpABj4G/93k3+zE+epxv7rLWj+U5CSljvxvjKiUUJ4UoUTV+xNUg4v9wkKSp++sllrh5MAai0mdffl5s+e+MMay2+pxjVmdxJ0L51OvYEJ8Hi1c0hf/P3blBozdl+g0wbLu9N6cbH9JsB1sLZrmuQpqXBk4JZqcFCr+FTGsO7B45GaY0lLW19vSuUEEShRzNwVBZtLIEUNFoBVkCSueeU2rDUCDG4EoU2lrSsaIYBZ/uwPc7wEFgBWekeb/iKaVG5EohyE8IAjVAyLS1CWLLSIIBcg6qPLYB3DmbcHGQUlbv2FldKmeQus1fCKSEWWqMNUBqkrxDWHUNaQRQoJnmFNppZYVltR/yBZ1f5xtv7vHE4ZJxpOg0PJQVPrt59TP+zzhn/MHzsQG6t3RVC3BRCPGutvQz8IeCNj39qjw8eZsLxzg77NK/qptn7H9qTBuDeOEUiOZrl9WsoTnVjdgYpuYXQ95ikBZ6EnVFC5Cs8JfGlZKMbkRSa3aFTtIt8SSN0ComTomR/6sarZ0XKSsOVQLoNn7BUtGOfw3HOcZKTlYbAh8Bz9e2qqpgal4Uv1YuWxTJLS/KywhiYZE77enuYoq0bsFnvhISeZFa4bbexlp1hynY7xVOCtNJI666d8gAhiDyBNpqXbwyREg7bOWudiI1ezHq3xd9/aZOVVkgn8skr7YZYBOyNM5461Z5vr7uxX5enBiy1Qn7uqRWy0vDijWOGSYEoNE+davOLnzv1oQH8Tvzj13YpjcEiqKxjiQRS8vbehOfW2xjAk04UrJEqNgepm0LVhiTXNCN3LadZxVIz5HCWkxQWK5xkbagkS42Apu+x3oupjOVgnOEDvifBOhnZpWbAwbSg6St+sDnk3YMpx9OUXFt0qalwNW8lIQ4DZkWJ0YbQE1TGkFQlq80YrQ0WhfAFw6SgqMB3L8OJvLjWUOmTLF8ghEQqSOtZhFbk0Qx9pmlBrl2JzBoocAuBtbDSCek3fPqNiOVWgCcFm4MUa93j55YaYJ2C49GscPTaW3BnmXG1HXLzOOGbbx/wwvnefCbhzt+5W9D/cVoQHhVr5T8A/puasXIV+Hce0XEfCzwM/enODnsr9JhkJa3o/QbPwSRnZ+SGLKRwmeD14wMuLjfohh77UoA2tBo+aeFcyitjSbKKpVZAqS1R4JFrg1Kyns4TNEOPrUFKmhuyyk1ybg8zLq0acm144XyP/b0JOElrlJRI3DCSAMrKUhmDNRAFzvi3Mm5QRSkw9TDOO/sJ610nrHQwdouBmxSNaFeal4czBklJJwrYGiRYLJO0QgCf2wiZFRXWwix3wy651jyz3MbgMtelZsC33z3icFqQlYaNvmuChp4T1VJC8tKNYyZZOf9ytyOfn7q0Mr/unRh+4Zm1Oaf/QXDyxf/ee0fM0pKVTkQrdFovg1lOMtW04z6XVt2gSzdWTDIYzgqkcAF1kJZU1hJ5glJrGqHPV8/12RmnjFKnQ96O3KDOOC+wY1ip7+n2MCErKrwwoB16nOqE7IwyvnnlgLyWMl5q+kxzwyQra40ed09muUZKyXIzQEjJIM0JpGCWlzRDj/VmAymcMqG2rraeV07P/KQOaoBQCcrKEvgWTwgm2gV9JQWnOxF7wrFRxlmJwCKlROAWBIlBa8PhNEMKGKUF//LyPk+vtfGUQiDwfUmoDcezDzclOSkzpqXmG5f3eXKt7WYSjhO0sbftgu+1U4aPZyf4acMjCeTW2peBrz+KYz2OeBj6050d9n7DBbNzS825lOu1oykXl1toa/nOtSO0gXbkZD6XWwEbvQadSHE4K9C24uZxQlKWlFrQin1W2yFaW45nllOdiKVWgK8ku8MMYyxbw1md8Xr0YqcJPs5Krh3MiH1FJ/IYJgWxp9CVIS0qjAVP1KYBAhq+R64r8tIgRZ29GcFKK8JTgtd3Rnz5XI9O5NGJPN7dm9XZnmGQlHQjn6W2z5X9Gb2mx5MrTd7YHfP9GwOkEJzuhiglWW25TH5vnHI0K0iKite2RgSe4um1Fu8cTLmyO2ajH3PtYMbmIOGnLi5zMC54a2ebH9wc8Sde2PjAYnviPbo3zub35V5f4FuDQjP0mOTVXNNbSYmuo91qy9XUZ0XFe4cJvWZAP/bItSUpKpZin07DfV4qXbHaCjnVjVnphOwMM3bGKZOs4ivnexgN2+OUq/vOGMJXEitglBdUeHz3vQH9RoA2rnyRa0NSaCJP8fSpNnvjjHFWUlaWvNK4nqyiqCragc8kL6ikZakZsFYHx9D3MKZyRhFA7EmkgEK7+6yNRQoIPEVVWQLldlGdMKATeRgTsTNOXPO4NChhWW4EeJ5knDmZ3Td2xhxPC1qRR1ZWfOfaIUs1bbYV+qy23MDaKzeHtKP3VThPyoydSLE/yeZJUL8RMM6q+UzCyS74Xjvlk39/VDvBTxsWI/qPAA9Df7qzw77UCvhTXz3L8ayYezvujjJ2RynDxA1shApOtSPe3JlwcaXJOK24duiC0Fo7xJOwFAZsT3IOJhm+dFtwbZxWRyNwI+P7kwxtDXlp8DxJaTQrLZc5L3uS944TAiWJPFjvRgzTiqysSEuXUSnfIi1IJaisoRMHpEUGwpUSTnfdNCEIQiVZ78QcjDOeXG3SChSX96fklaEXhzy93nRThaHBGMvRrOBcr0EcKo6mOcZYGoELeAfTHGstndpNaJYbvnimg6cUZ3oN3tod8y/e2sOX7jWdV6ZitR0xSvIPaMucmEAI3EJXVOa+2ditQeGp1TZH04KsqDiYuKZzURnOL7UYZQX/v9+7Tit0AT/2PX7x+XV+990jfE8RKjGvV7eWYrSBvXGKNpZu7HO6EzLMKma54er+mFlpSPOK7VFC5PtYK8gLTV7mnCg/SikwVuArp2Y4LUryY0c7VJ4gUJLKQFq62YLKup0gForScO0w4WhasN6N3LCOtaSVwTP1mxfUw0YGbSAKJb4QaGFYbgYgBKUxrqErcEbdoYenJAf1hGgnkkyEYDzL6TUDNgcpkS/JKkOaa9qBz6lexDipuHaUcH6pcdu9SQrNuX6DVuh2uu8dzdxWYZrTjny6sf+BXfD9dso/TkNEi0D+gLhXPe1h6U8f1mHv1mJXkae4vDdmd+S2n73Y0bqeXG25L2mmGSQ5u6MUC+TacJRUtEKfp1ZbDJKSUVbRDBTPn+7Qa/i1MXCGtk5TY7UdkZSupqytYb0bUWhDqASd2CPyFHmZsdoKGCeObTEtNMZA6DtVQWsNKy2nijfJCwLluQzOV0S+IvQE33r7gE7sow34Hqw0A5qhj8GVZ2alwZdwYzCjrCwbXZfNZ6Vma5QRyALfAyHcxGKgXL15qeHz6uaIceZYFyvNgMhzk5Obw4RJ7tNvhPQaPlI4fRB4X1tma5DMGRxn+o3bsrETM+UPqiMeI5G0Io9zSzGBt8KrmyOuHkzpNwPOr9Za6AcJo3qB7jZ83t6bkJUuW+43nABVqARR4BEox8xQ0qfUhsNJTmkM/aZrTi+3IpLjGYfTnLy0lKGjSap62KnQpeOCIxHCLbaBklgrGSQ52oJfQWYqCu1KI7V6AFXiroXvuSx7f1K4Zmngoa1FWEBAUbmaehwqjJHYvCIKPHwl8JSPMSeaK5p3jqZ4QrDWCWjHAbHv8aWzPaZZRb8Z8sbOkFle4SsP3xNUlaXUBqRhP8nRAmZ5hbUGaNx2b/bHudvlRj7fvz7AWoGrAQre3Z/wB5499dAWhD9OQ0SLQP4AeBBWysehPw2Tgn/62i7DtGSalSw1QraHGYHnPCA7UcDlvQlfPtvjvaMZq52Q0lhGs4JJWqGUMwLoN31n2hs4v0gpJcO0ZJJpksIJbnXiwNViK7d1nqQljUBSVtAK3fj64bTg/HKTceasvGZ1SQAJTd8Da5nmhqK0PH/aDRdlVYXCGRpUxoBVTHOnzNdvemwep0gpCH3B0cyp3RXaMWBmhabfCLBY3jtK6DU8nllr8/LmiNGo4HQ3IvAkV/bH7I9zVtqhyyhxgQbjJFjdFKajsB1MM64eTVlrhZzpRx+QTz3ViTjTb9zGOb/VTPlWsw0BBMr1K0ptmeYu4P3ic2s8farF6W7M7107phEq3tlzE7dX9iesdSKsdQ1bISwGWG0GCOlUEiNf0o19Ql85Kz1jSDI3zr8/yYk8ibbC6bOIutmIojTOj1MJ0NoyzUuMZS4sFipX35aAUM6R3tzxmTtpresKfGf8RJobyqpgqR0ReoppViKEIPQEae5ecKMXs9oJGeUVk6R0x64Ms8JNj/pKMM40S03Xm7m40uS9wxnPnGry3mDKRi+k4fuEvmRrkIIFgWS5GaIQZEVV90GcG9Ll3fF8crYRKkZpyaXVJlvDhGsHCRdXm5zpRexPMnxP3LYLvt9O+cdpiGgRyB8AD8JKuV8H/F7GBd+6csDr2yMCJZzmRjNkvRMyySpGWUU39pkVFZuDhPcOE55db7PcDLG19+Jyw9VgdeWysllW0fA9jO+y2EbgpF0neYW1JWlZkWlDUWpGScneJKMZeKy2I64eTNnoxexPcsZZxXPrHd47nFJUhihUCCO4MXIj/OODMVcPJ87KrNSkhaZVeWwsNTia5YSBYm+SM6oDQjtyAcrigtL+OKfQhn4cstQMmBUaYw1podkeZzRChScdM8PUxPduFLAzTDndjVlph273UFWcW3JKf2mp68cjFxhKzXFSclHcvtgeT4vakGM6V22800y5ETgneixcWm3VE5yCXu38XlnoN0MOprnznzxOGSYVpa4YpBVHScmZXkiSOQ9OH81rWyM8JelEiiR3O4LDaU479mkHip1xxtEkJw499kdlzcO2WOuCc6m1m/CUUBjXWKTOxi0gLMxuaVJmpalt+j5o0XeC0rrHfQUnFJNu5GHqZrm10G8o2nHAaidke5iyN8koS42vFKUxSCEIlCvzlJXhaOrkAv7Ht/Yx2k2SrrZCwk7M1cMZe5OMSerkB3ylOL/UwFqLto7xMkiK+cRs6Cl8zzVNnZia4vMbPX7fpVXGWck0qyi0/kBp7H475R+nIaJFIH8A3K/Wdr+M/W6PX1pt8c/f3OWtnQmltSzFAXHgcTDOWO86RkQ3FmSVphm8L7r05s6E0BPMco3Whrym83kerDcjitJw/TjhtBLEzZB+M+Ld/RmjNOdAQ6Dc97XSzL04tXEGAFcPLQfTjG7k0/AdQ+SNbactEnoBWVFRltqZFFQWXSsHRj70mwGRp9geJoSeR1E46mJRSVqhYJKVBJ5yVEEraAYeLSEotXMn0tq5zCslmeaaWaYZzgoKrWlEHnmpSYxjQkS+JC00vdjH9zwqbZ1fKZJxVlEaV+ePfEnsvy/yOUwKdkcZ//T1HVaaAavtiM3jKYezgjP9Br2GX2uPOx2bE8XEduTz7HrHqS5mJUoJ/vDnTnH9aMbf/L0bvLkzpqgcJTEpKxRut7AzzGkGisCXHKfO3m256ZOXGoRA4sol5/tNKmOxOArgJCuZlZoG4EuBFk4/R1t33wLpmn66HrM3xjFN4rruXJ6k4PXv3G8Cz/J+Fn88dYyXRuBxph9z/XBG4LtyyPE0d0827rg+YKzFGIuSjtHSCBQHk5yDac5Gp8EXzrQZpQVCSCrr1B7BWRXujJx2+hvbIza6zuEoqyreO5pyYakBCPJK8+x6GyXdZ/5WyYXTxCRFReDJe7hSPZzJyeOIRSC/Ax+WOd+v1na/jP3DHnd2YLvcOEpYaQXkGjaPE1bbAYkQ7IwyvnS2x/44Iy01n9/oMssrPC9z/5eStbbP9siwPU5ZaQRcWm2yNy7oNVzWejjJ2Z8WRMplzNq4IRuhBKVxNWJfUgsfuXr5zijjYCr42SdXWW2HHE5dbdJXrsm2M0qRysmZGtz2XlgoSpctCVGiK4OOLQ3f53w3ptSOBz3JCqLA43Q3ZnU5YpQWHM8Kd6zKYD1VB1DwA49RVjItSgIlwLrFpqycwUYceLTDgKSsKHTF0TSfqz7+5BNLNAJFVhnaoc8Xz/So6oXqpRsDbg5m9Bo+148Tvn9jwHPrHb6w0eXmccpvvLnH59Y79BsBeWXYGWacW44B6mDusz/OuH485e+/tEkUqPlrZWWF1u877SgBU+149vnMmSo/1++w3A7ZGqbkpcvQQ9+N5rdCn/VehKfc8FXkKZJaB6W6IxJboBEo8kqgfKiModTM2SYnGfjDOF/W8dkxYEonKZAVhrQ0zMoCUcsia2sIfOXq3Aaqwr2eq5MIplmJtoJ2pFhq+Sy3Yuw0Ayx7owprBef6MZ04YLkVMMwq0kIzzTWfO92hHXm8fHPAKzdHrHVDVpoR7+y7CVenJOka2Sclkf1JRjf2+cbl/U+ED/5p4KMvAvktuFfmfDId+GH1tPtl7Lc+PslKtocpb+6MnOAQrpnXjRQsNZjWdem9cUpSaJSU/KHPrbDRa/D9G8ecyxsYM2MwK5C13OeFpZjn1rtc2Z8AdQZnYW+So40bmbYWpBA0AqfXPMtLtHZ6KkJYWoHCGkc1bEWeK7ukJVuDlPV2xKx0DavLe2OwbvJPwrz2WuEyUCVBKkmWa7qRM6jIK402hlYQYIHBNGcwK4gDRVHWI+CF408XpWFWQDu2xJ4klRKlHCOj1wiYZiUNvy55AO3QQ+P0y2dFxdl+g9Bz9+hUO+KF833nABTKuZ3c9cOZ45aHPkmhefdwhq8kN4cpRmteN5av1r+31PTJS0NSVLV2S+40VgJnhbc5yFBSEnmSvHC9Cilc1mxqWmbDdw3FNNe8czhlkBS0I4+yFskCj6R0AmRL1k2wXt2fMEhyRqnBw31RT9TgT/TLI09RVrW6oXE/U8I1NB13qC6ZyPd/di8I3O+D6wXM8opCGwLP/X5RWSeXKyWFdp+pSjttFAMYDc3Q7bCMMfjKpzKgsSy3nOjYpTWf/UlGWTsSfelsnze2R6im4Ey/wRMrTQYzZw2HEKy1Ql7ePCbXlvO9Bqf7Ub170M62ri4bhZ76UP2gHzY+LaYWi0B+C+6WWR/PinvW0+6WsQtxuw72Witia+gmMLV2GdUk1xzNclaaEe1QkeSSylQ8tdpmpRWwO8p4Y2dMO3JBxxjDcFayNcpZavicaoeMkopvvXuIEK62+d3rUyZZUXO7DUlh5xla7LtzK0pnwIsFoSHwBNoqpHAc76NZzv4kY5xrF4grww09Q0lBWlgUuAbcLZliYdxWu+FJ8rJib1wwTSuUJ3l+vUvgC97emzJISyJfUmpDszaC8KWkGwcopclqZT03Pu94zHmlOdWJajcjx2tfbvl0ooBprvlXvtgn9j12RylJURH5imaoUFLcZk13PMvpNAJGWcnbBxMkoI1zCRJC8PnTHY6mBW/ujHlmvc1PXlxmmJZzA47Dac4XzvS4OUjQ1ikwlsZQWRDSce21FVSyVgUUuEzdQBS593xjkLDeiTHWzq9Dkrkm3+4oZ5g4it4o00Sq4GToV+GCuJTQCF3PoxE6k4uToRtbB9V6wn+OoC5n5Ob9Movi9ow9VG6wx+nOQKYNSalphz6Bet/ZaJy5ec1+7Ne1cHcsT0FauGvhS7fQRL7TJ5fCcjxzA0/aWOLA0R/HmXOeOpoW3BymDJKcjZ5jrGwPM17edDMDgRLsjXNWOyGNWjTsK+f7vHRj4ByNau/WaS25rKTg99+itPjDwkfxrf1hYBHIb8FHFaj/sO74/iSbZwonOtiXdydcWIoBD6UEK+2IlTZcO5yiMUxnmlHiHHJk0w2IlNawc5ghheDm8Ywr+1MqbVlqOJrYW3tjlpohrVCxOcy4eZQCTsa0qNkNJ1ttJZ2XIrjx7fn228LuKCP0Be3IJw48hrkGY6m0+6pr3G5CG7cASN4f374Vxrix7civ9VQSl7ofTFN+5qlVZpnmPa2xxh0kKzVx6HGmG6OUQCYFp9o+OyMXcOPQOEf31A3gKAHPn+mipOTzpzu0I5+Dac6ptjPaUFLwzKn2fNgn8ORt1nRv7Y7pxwEvXR+gBCghyUrXdLu00nKTpkKQTQxFZejFAeeXG/OhlFc3h0jhFsPdkZMX8JQkkBIv8tHWMh1XBEpQOUoGUgpagaSsXKZqrOV46hqaWEHkO2bQs80O7x1MaYSK1XbM0bRkkhbzYHvibWQtxL5HN3KGEUmhGafVXAgtcIO/VEDDd01Ix3V//4YJbg/ivnTWc5WxyPpzUVYGa90wUNNzNXJjndJhoNzfBYLAc7TUrDT4nsUzbo7hOCk4oxuMkoxR3diMfXcdBon7LDlFSydTcarWKn/xxoAvn+1ytt9gMCsQOLZTtxHSb4Qcz3K8euuwPUw5nGS8vTehGwec7sVIAS/fGHzoyP6jxqfF1GJh9XYLhIBXtoa8eH0wpz3dmlmfWJedDCkM65t10h3PK813rx/z+vaQg0lG7LuVuhMHfOlsHwHsjnJ8Jfipi8vI2jex3wgwdaYzzkuSeghmnJUoIYh9ycubA64fzkjSal43HSSF42NnmpvHKZNMM84qhqlmWlpy7TLmk1rpiXaGG5eGyHNfeomjrTUDD1XrXbd8ySwrmZVOS3xcM0IagYcv775NlwJCT5JXmqS0FHVzbm+c891rxygFnzvdqXU3Ajcxihv3NxZGSclbu1Py0mlVr7cjrKHO0ktOdyJ8JVlrh5xfavLseofVVkheORnWVuiytaVmyGo7YpyWXDuc3WYntzfO6EQ+7dgnKQxSSk61Q2Z5wWs74/q6avYnKb/51h5pUc3v/6lOxCyvuLo/ZXvoJid3hjmhLwl9p2WjhERKiaeEC6yewgLN0CMMFYEn54urkK6mLqVknJWYmoGi6nRaCclJYq2p75+BaVqyP87q1xMo6XZVnnTHVKrOyrHMCkOpLaW9PRs/Oa6oP/uVcaU+W+usgJMYFlDLCDsjCiEsjdBzzdXAvc+s0HQij1bg6JS+lCzHIdePp063vd4telK6ngCOVrk7SikNXFppoaTg+tGMstK18bXlqbUWZ3oxK62YXhwQepLjpJybjF89mPG994aMkor9Sc7bexPS0sw9b3/YONmN34pPgo++COQ1honTA59mJYFy9d4fbA7Zr70uG4E33769uTNma5DyyubwtmNoY/nCRpevX1im1HD9yH2IwTXKvnimy2on4tn1Dqd7MWd6MZd3x7yyOQJh+bmnV8kKJ/MJlu1RyrevHnF5d8KNwwTfk6x0QgLltuNVzVYZpjmTrELWjUfLXVgK9v2GVhy4r7HvSzzlvvxSShqeorKw0g7ptwICKZnmFcbWqniVqbm/7+Pk7x4uACSFcep5wgUTW1/frUHC9cMZu8MUgNWO20kczuqmZ6VZbkcY63wxlXTOQsPU+Xe2AsVyO0TVqoBv7IxJiorT3Yhh6rwpT3cj9scZr20NWWmFc7Pnv/Y713hlc8hPXVwiKZz36alWxNcv9Flthfie4L2jlF7kYw1sjTIOpyXWwt/+/ibG1HXuUvP9m0NeuTnkcJI69x4sceDYJtY6lxxrcYM11HVk4/S3m7WwWVZW7I4zDqcZw6TEGMPuMEUJQVEa3juakmtDVhm895URqKwL0kiJJwX7k5TjWUlaubp2oR37BOHs/ZyImit7nUDWB5Pi/X/LeuHQ2tW6VT16H/qK0JNM0pLrhzMOJzlFqZlmFYNZSVo66mGpHdXR8yRLrZD1fsxGP8KTkudOtXlqrY2UYq6cOC40gZIIIcC6HddhvUt54VyPaabZHqZ4UnA4y7l6NGEwy3l9Z0RRaS6uNHllc0hlDGlZEQUKIeBgnHPtYMrF5eZD+X5+VFxcac69Ya21H/CJ/VFhUVqpce1wxmo7YqkZ1vziinbo0Y39eVPn7b0Jka/oRD5ZWd22fbuzVrbcdI25ExcbgKVmyChzmdTmYMbr22PGWcnXn+iz0op4fXtM6Ev2xjlXj5wxhJAWaQXtOOA4KejFAWf7DY6mOdpakkJTVgZRj2nfk2Ym3PCHNhAqxbSqUNLVoU+aZpEnGCUVZzoRa+0IbMZglmMMlMbQjDysNdiqDipA6AmEgjy380VEApGjxGCFY1IMkoJjZwdKO1RMs4qDaQ7W0gjiejEStAKPQeKoirrePfjSBaWdcY4UEE9yOg2ftXZIK/L4/IYzj6iMndex40Bxedfds0bg8dtXjmiGinP9BmXHsj/J6MQ+P/f0Ct9+9xBtDKudBgJBPtXEgZuYPJxkvLU7rrf4zgwh8J2S47kln17f53hWECrJ3qQgq1zj1lM+aeF8Tq21VEZjrWCcVpT1jdIGQl+TFa6o7QalHJc6LXSdKbvrLOtsXQD9yGdSlHOxMV+4EtktU/VkudNH8ZT7optb/thbsnNZSyxUmvnz45ruCoJcG8qspLLO2FlXUJqSyHNUR2Mg9CAKJEU9MRxIwe44w5OCOPI4XcslH6UlnnTj/uPMfXZLT3IwdbsLYzOEgF4zZLUV8t7RDKVc09Nal2Cttd2w0Dv7E061I7I1zdGsoNLQDN0wnKckjfCHn6c+7FT3DwuLQF7jpNYlhJgHXlsPJnRin9e3RkS+mvsNCiHn27evnA+4sjfhxnHCMCnpN3wurbiR+rmLzeGM41nBRi/mjZ0Rm4OEUluywnBlb0roK5LcUbCOp86iLfAEHtJJplrLOKncsIjvpvtmuRvswUKv1nq+VyA/Kako4cbVZT0ZKAREvqId+IzSEm1hZ5zRiQOOZgW9elGyBrJaZa/QGh+3nQeotHP1qepdZh3D547yEjd4IqFmQVh2Ril5ZTjVjZASxmlFVrop1LTQLDV9Yk8wMIbJtORMv0noCQ6mOTuZ5qtP9HmizohurYd+4/I+/UYwX3i1teyOnYXchZZzVSp1xec3umyPUq4ezDicORGnw2nhAmUjIA4UO5Occ/0Gx0lBqBSFFnhKcbbfYHPghLx8JdzYv4Snas/TqDZyOBSCUBlOdUKOZgX7k+y2e2SArAQltFu0JHQCj1RbBqUbRvLUSRPVad1YC9PSDVlp466nkyIwVI6eTlG54CylYxhV3I5bzyH0XA2dumEb+hIpoSgtwoOlOMRYy+FE0w7d9XSlLOOs54ShHbtxfcdGgXp95nQ/ZpiULDcDksrp2+fCoK2ptXac8cUsr+YSCqttyzOnWzQCyele7My3a9vAjV48N504aeGfW3K+rqGnMNZRFO82pfnDoAp+Gvjoi9JKjXvVui6uNDmeFbVusiUrHfXpZPt2/cg51UzT0nHCS8t33zsm8hS+Erx4/RgEfO1Cn6x0IkvaugEaTwl2Jxm/+eY+V/bHHExyCm0JPOcUL4TA8xSl1swKN7ExSEsmqctYV1shnnJZm68UzVuyEHXLe5FQ61JLug0P33OWYS54WDwJx2nGrND0Gj6+EuyMEipjKSpNVdPFepE/n7JDuOap84h0EqftyIlulabmNNs626uzRSXB85y1V2nc6+alxhhBUbkdxiyvWOu4uvcgKTHWEvqKceLq3dNaCvZolnPzOOHtvamTOKh7Fp3Y52CS89bumKsHU17dHNaNS5/IVwSe4omVFlf2JrQjj8+dbtOLA061I5qBq/068+mSotD87JPLpHlJUpTM8gphLYeznDO9GE8I8sowzSuywtCOfNY7kTtGaehHPl+90GepHTLOq3nGe3JP4P0JzKTQZIXG96RzEvJdeUlaVzYBF5iFqCWF60WyrNzPZF0fNzVrpOFLd5PuaGjc+qV3dXq3GAe+a9R5StHwfJqBhzGCWeWYJkZAYdznQJ1Mk9ZDas3Q0SaVgrSWO750qsW5pQZSwLsHs7rh7HatCqeUeaobsdwMyLWhqr9bn9/ocqb/vqzul870+NqFPs+uu+Z27LtBrafXWnUfCS4sN9HW7cYuLDc+lP53QhW8W6/rccYiI69xL12GXiP4gN3YheX2nJ/87XePuLDsOuyVtjQjR5F7e3/MLz53ak6P2h6mvLU7Zpa7YDVJS4rKMElLZllJaU62uRKL+8Kc1DHTUuNLRSAkfujcZBqhIvIk/dhNzikpOZhkrlaNG7nWJ9S1euqvHfpEvsQvTE1dE4zSClsPDHlKUtWTlo3AY6MTsj8tUMrVSkvtsuXAcxngSWYXex6n2gFSCopKM0oqSq3J6qbryfMKA2Wm8RV40g2UHM9KhrMBUjrRJykFfq1tkuJ6AZ6yTCtDUrrG3mo7ZGuQ8cb2qHYYSmiGip97ehVPCn7jjR32JjnWGIapJvQEP/vUKnnt3F5UmvVexM88ucrl3TFPrTXZGWa0QjhMYJJWVAZ+8bk1nlprczDNOJoUzjDBwlIjJA4UvaZPp6ZQjtKKtHSiV1k9udkOJSutWmbVV8h6EjaQ1OPsjgV00nD0lWJzmICQc/5/ecILr6+jL0AIA8ISei77NsbSCH3Hy8exSASghcGXruF5kqYYwBPv18qNdtz3vIQEjZdr8lDXAlyuTJIWZu5mlKHn5+zV1nOqrmVbI4iUdCqTaUXsSSprOdtrcLYfo40l9CTfuXZMpQ3rnZi80jRCj1lW1nTUkuNZjhBOaO1uw3hOCbR0OjXacKYX8/mNDj/39OqHZtmfFqrgDwOLQF7jfrWuL53tOfnRwPtAoP/tKwec7rpJtcOJm7zsxL4bjb6jvr7cDDmcjNgbZ0ghnSuPpyg8zWhWOEqZLx1N0Fq6jYCjaY6SkmaoagMBOeeiy0BythfjeYLIc+JL3YYL9Gmh8XBB3PcE3UaAr4TjBsduTNwCgZRoo1HCqTDm2vHYlBRMC4PyJKu+ZFpqWn5AVuUESjDJnJTtNC/QWrM/y3l2pY1o+JRVxuGswlMgjQvgJ/Vzi9vu5/r9DX5UMyVybWkFHsMkB1wJCwmz3BAGjh7pBJ0MobK8eGPg9EaA946mvHxzgNbQiQKGSclBUqCNq/m+tTvGU4IXzvV5dWtE6Eku7455+eZgbp82zjQrrYBe7Xjzk3VzdKMbc7ob0/A9/vmbOyjpBJwEAg18+Zzbbe2OMyaZphEodFZy/SilqCxfvbDEk2struxNmeR63jw9Ca4Ct4gLCVlmkUJTam6jHooT3r5wu59pbvGUoBm5BcHgmsS9RjDve+SVriduXSNcCbfAx6FCV86TUws7V0c8oaROcoMvDL4S7I4q6p7kh4hvWaSQSGCSVa5RHTptlEGSkxUVaW2wHAc+o8SVF7+40eXq0ZTD2tEoKZxFYT+OqKzLnl8418f3xLzk8mEJ1s89vfrApZJPC1Xwh4FFIL8F99dl+PBAv9qOmGQl3UZIc7kFwCjJiQPvA/X11U7Elb0pRWVZaXu1foSTBm2VFq0t653ISZdWbqLQWEHsSXqxz6wwjpnigdZOZ+Pa8YyVZsjTp1rkZcUs16y2nMZLqS2z0gVpYyyHSUEYeDxzqk3kK7577YijievuKwmDWU5l3La81IbSGBq+otCGQhumWeqybAGBL+eWZ8JaPCsYZiVSwKluiMWQVoaicE5CdxsXP8lGg3rnUlmLMM6mTGAIpMQoS8PzWG4GaOBwWlB4gmlZESuFkIKbRynDtOKZtRZFZTi71ODiapvNQcLBOCP0oBmoWhNEkxSurtoMPK4fzbAWzvQies2QJC+5sNygMpZGKPm5p1cBl71t9BtM0orTvZi1dshGL3aTvwL+4HOneOXmiG9fPWS1FfHcepsbxxmvbg5RgtrQ+n32ELhBnLDWoHG1cFcqu7WOrQF1S4N0mps6uFry0pXivnSmx/mlmK1Bys1Byt44o9IGIYRr3MItjCCJ8qQL5NYtIidTndhbRLhwg00nzdmTQA8uo499hRKQ1l6fgZJ1n8PtGCbWKSReO0zoxD5nerHbKVlD01cMZiWdyKffCJhkmlxXnK6nOqV0CdTJdb/1eweOEnwSwL987v6c8YcxgHncsKiRPwROgvkvPLt2Ww3up59cZpSWjJIcYwyjJGeUlpxbinnpxjH/6NVtfu/aMfsTRy97cq1JHDhN6W7k0W8GiDqItCKP0JP0ms7TUAjJ+aWY5053kEJirGE/ccpx06wizata4sLyzsGMC8stVtohvlIsN6PaasviqzoYVwZrDFd2p87MGQh8gV9P9RnjgsiscLQu50BjGcxKkqwiUAq/rslLnMlzqCSeUoS+Ypo7loY2lk4c4EvpZFfvcV0t1INL1k0HZhXT3FmONWqDhkaoyLVhb5zT8pUrtWRuSz0uKsZZyTir2B9mNR3OMM3cWP1Tqy0urjZ5eq3F/iTnvcMZWeWMEJLC7UqwzuG+E/tzka04UDyx0mSSlfyd72/yd79/k51RyqXVFs1Q0Yk9p7IoBb2GjycFVw+nvLE9ohlKltshz5zu8cRyq3Z+SljrRvSbLnCcTET6dYMx8ATTQtMI5FzY7FbcWhqZKxb6AimkmwGQjt1kwKkXWkNWWseVx5WyWpHTaM9LzfE0w1Mu0/bq+x94Tv8+kK5sYgCp3u+3yPqPhzNtrowl14ZQSdqx201W9VBYVTr6JEBZl9te3XTTtbPMFWdOtQNHkVSS8/0GzVDx8s0hvdjn0mqLa4czfnBzCMCXz/Vus2l72Fr3p4Uq+MPAIpA/AlxYbvKnvnqWOPDYGaXEgcdPXFziu9eOkQi+sNGhqCp+990jRmnB73tyhV945hSr7YhppvHrLGZWVFxYarLcCgmV4BefW+PzG22eOV1T66yhMk5fRCnBUivECjdFt9aJOb/UwFj4qYvLtCKP4zRH1Y0obazz4JSucRb5kqDWxdbaIiX0Y/elOmlkhcrDGMtxWpAUhqJiPkGolMvuikq7L7YSLLdck7QRKFqRRyNwzd67mKHPIe741/tDMZJCW/LS1JmzotsI3E6h7p46GQJb/zFzxovvSYZZicWxexqBsz/rN0OeWGmy3Ax4YrnB9jBlkBRcWHGONDeOExDwxTNdxqnmb3znOi9eHxB7EmvgN9/a462dCc+ud0iKiv/hlW2+994RUrrBmaNpztu7I64dJry5PeL1zRH709SVtAon+HUixhgoZ2RcaPc+PCUJlKDhu0XxftooutY/EdKSaSfA9Ztv7RH7HtNCs9Ft0Gl4+Mrd86VmWEsCaDwlWavNkJVyfRkXxN3dMDhuuakXbFmf80lt3OAmP8vSLfjGGlqB78p9uN6Otk4RsRn4FJUgLSqmWUVWe7weJXltVO0Git7aG/Hq5oj3jmacW3K7nA8L1rfWuoUQ87+fDACdNDW/cXn/Qwf3TuQWTvxaH1fp2luxKK3cAw9KVRombqBlvRvxzHqbiytN/ulru3Rjn24jxPMUFqfTPM0rlBRcWmvSiT2+deWAXqyAiLzUZFpjK+HG5FOnKbLSDJjlJcezgmbg1Xz0iuVmSFG5BeB0N6YyhveOZiRFxXrHie0PZyXW2tqEwCBw2erhtEApRxsUgLCCcV5igMgXtX9oydH+7SPilQVTc9l0ZRASerHPM+ttvnJuic1BQl5W3Bgkjq5XWXzPMSvuFpgsUFgoclPLGtQ6HZ6iwjF8qsKVEry8xFrPTUcaW4+luwEhay2BL7l6OOPJ1RaTpODm8czp3HQivn31kIurTbSxtCPnyvPUWpurBxOUlBxNM5aaAWd6DZQQzPKE3XGGNTUvu9T045BKW/bGGX497NIMPY6nBVf2J9w4SlCeG04ZZxUvbw4422sQKklWVOwM3dUMPUfVzIXzvexGnitLSTdyX1TmNqEsuL2sIerHjHbMkVJrCmHIS81wlmOsm7hUQhCHkqqEtNZ7D5Wjt1QGtHFywnktgmWMJajvlxUWIetavrn9fp3solzT2k2unuqEzuTCWIrKeTCFyj2GNGhtnV5+rvGU233EgWKWVhwnuVtMlMIYy9/63iZ/8Nk1VlpOeuGkHPLK5pAr+1MkbshuoxfPmSyDpLiviNWngSr4w8AikN8FD6pqdrfn3TiecWnF1cubgceFlRb7o5StYUrgSX7+6VVe2RyyO8rJtebcknQ2bbOCg2leb4sd3/ZoWjgPTguV0WwNde0E47KrUsPmIOHagQvipbG8sz8h9BXWuixV1/oYzdCjrAyj1KkPhr5wTJvQJ6knOKvqJGA4euDJl3juKnPLdfLq8fnLO2P2xintwDV5A08R+7VK3v1Sy1vgmnGSMHBfaF051xkt3Lj7qV4MuIGPg4mrAVe1IbA2IIxld5hSak235sGf7oQst0J8JZEI/vlbezQDj1lR0Yt8BonL3PfGKf1mwO+8c8CZXoNBUmG0xQrYHKS8sz+lEznq5ubA48vnevQbAXuTjO1BynevHxFIxbl+zFt7TokyRrqMXDophlHh3Jl0XXcW1i2ck7wizTWd2Ck7mrqpKW8Z8rm1Nn3yb1VPaGoLfr0DOk5KQuVq4EWhySpbl8NMvfBZmrX87rSoMFqQz6qawipZagRuUa957dNUz6eGT+69J6AZeVxYihmljlZ5MM3dbk66spRzJrIMZi77hZoTrwWTzM0SXNuf1YtAhBDOsHmtE5AWzmXp0mpr/tmotOHlG0OWWgGyTkgu7455dr2DkoJO7D80M+XTIEH7KLAordwFH7Z9Mwb+6Wu7t23Z7rbNs4i5RdX1oyk3jhJmpebzZzrzxcBa+P3PrHBhqcGF5SahJxmkJdujlKSwzoCgME6FMC1p+ophVlEZSxRIlJKM6kGKb76zzzSvjRtKzdE0p9JOHlbU3OVZUTFISgqtKbRrpGrjMt5pWpCUBl2ngLLmI5f3CMInJZGkcIMgk6Rkd5IhpWW9EzIr9Dybu1csF0DsuaafV9eG08Jt4qO6DtEKPM72Yy4st/ClYKMTstx0+h+eEISedHRNKQh8yXIrchOA1lIZwaWVFhLJ928csz1wmXbkKX776iFbw5Rppuk3A6eLYgSvbA4ZpwVHScnhNKc6EbuaOW/LtKjYGaVsDxMOJzmzwulsC2GZ5E5LvKgso7wEnBdqaUwtAVxbteG+gLPcMkpKKmMZpyWekMShnE9q3lp6OpnwPIG2UNTR1fG5PSRuArIoNCcEUWMhKTSdho+QkihwzjutQFFoS+x7rHVi/sjn1/nCuS4rLWenJ2t+euhL/PoeBd7JJKhmf5LjK8nTp1rMssI5QYVOx2a5EbqSl3WUQwtkdePbl4J27NFrBCRF5ZyMsLQjhdZ27oh123fyaMZSM+DSSqseNHP3/erBdF7rPjECuRUnvPM78ePEK19k5HfBnVSlSVZy/XhGqQ2fO92ZZ94n7t63IvYVF1cavLs/ZZxV9GJnVjxOC55ZazFMirkaX1EZzvQb/N61I16+OUAIyXPrbfIKtkcZUrovaFlp+s2QaVERewKso2V1QvdFUFpirBPe0tYSBR6H0xyw5JWdC2YJYFbWDSsBYSixhcUIg3TfJRC15sZ9rpGlHkQR4HmSWanpeR7TTFO2rNNbN8YJOVk+YI5wUjoIFYSBIlKuWVppg5SCZhBQaIuxuqY7at7cGTnzYj/g6VMt3twe0Yt9As+JfJXWccyLyjDK3JTsrKj4x6/usD/OqbSh3woYTguOJzlKCFaaIac6EanW5EVFkrsBqMATZKXzFr12MKMsNdN64rQRelSV4fLehBfO9d2uQDpWyihzmXwn9ubOOePUyavGnkIKVy8X1s77B5VxkgbaGrQ2UGvVKMltrBFTX/iTpuPJPYprXr4xrryUFgYPgdauVHLCQpmmlXO8nxWcarvMt9SG88uu1Of6EdJJ3BqD1qIuvVl85cpZ7r268ps2lnFeMskqQt8n8lw5RwvB7jBFSjeR2Qo8jC2JwtAxXZTAaNdgXm1FdBseSroBpl7T7Q4nWVXXyTXHddL0B59dqw0+2k5KI7MYzLwx+u7BhEApLq225n6sd2Om/DjxyheB/C64k6q0PUyRwHIznGfe8L67952UpqfWXO30BzeGHEwzsNBvhFw9nPHr377Ozz61wlIz4NXNIZuDBE8JLi632BymxIFPK4Srh1Oy0rkBrXcifE8RKBcwW1Ky0vB5fqPHt97Zpyt9R/HTJ/6OTsfiZOpTSadKiHWa4UVhyCqL8py5bmEgip3HpTCQ3uf63Dkd6MoyMMoL0lISjTwq4wKVMa45Ku37Lu7iluN4dbCSnqAT+SSlU31qhh6BsRyOK7aHKb4vHfc+9hASzvYbXFptM8lLxmlJhUAJy9E0J/IVceAWBq0NhzOn4RH6iqK0ZJVmqRHSjH2S3Fm3pUnF1tAxOfw6g7y40mJrMGN/VtIJFRuNgIbvUVSGw1nB0TRnlBROWz1QTNLKqUNay/44JfQUvYZHWhhakc84q2hKUZsm67rZd3JNLIGSGGHIKld/llKihcWrFc9OAv9JWeVWzZSicuyUZhAQKo2QAlVnuko4HZ7KGgQWbQyz3JVdzi01+Or5Xq1XYslKw9GsZJyV86GhylpiKTFS0PQUuhb0EjilxmuHKZEvGaUV4yxHSWdDFypFv+HjK8WS57RrrHVBMwoUS42AwcwF6Y1uxHI7JC8Nndjn+Y0OL9085nS3wVIjQKwIrh1OaUf+3K0pKZxxydWDKY3A49lTHV7ZGvHK5oAvnunhKXnXcf0fJ175orRyF9xJVTqaOQPgjV48f06lDQfTnO9cPeIHNweM04L9ccarWyO2hym7o5Rn19t0IscG2R1lJLlme5BwPC14ZXPI9aMZ28OMy3sTSu2+ZLujhLd2xsS1LKnBsjVMmWXOWSb0vLrZ5IKWYzzANC/JKkPoy3nJpNCudmw084ZgN/bxPPc7xji97zQ3JLkzyQ0jdfcLU8NSB+C6blrUxs9J5lgMe9OMrUFC6AkasayplI6pIRW0I0UjlLRCJ/laac1xkjNJS3wl6MY+03pMvnTUCSdXIB0z5L1DZyDxwrkuT6222ejGBEowTp3OdewpdoYpoa9oxQHjpKLXCOYj7hdXWzyx2qDhK5QQXN6dcDR1gVlry3LTTW4OkgIpBJ1Qsd5p8NULS6x1narf86c7nOnHHCUlx9Occ/0mT59q0Y6cGXYzCDjbj7E4nZQnVlq0Isc4ktLpeMeey3ixIJXzBvU8xzTRtbqVkxxWICWBB+3YNRBL63ZVCkhKN5CTaJfpa2PpxgFPLLnrYq3T1JG4xcEXAk8Jnlx1bJ2qHqI6u9TgxtGMvKrcc+v8xDVa3fLrS0Gr4dNtBCgpscb1XNwWzaKN0zUP6uZl7PtklZtlCJSk3/A51Qm5tNIk8hV/8itn+eoTTsYgrwznlmL+yOdP04l8TncbfPFMd66VvjXMeG17eBt9EJhn1p044Mtne7Qin7f2JvdkpnxaJGgfBR5ZRi6EUMD3gC1r7R97VMf9pHDnAFAv9llphfPt2iQreXVrSK/hc3G5ybWjGd9695Cmr/jc6W49Qp7wD36wzdnaIVwpwSApWGuHbA0TZoVzkn96reUCiLE8s97mW+8eMU0LTndi4tCj1JajacbeOKPfCunFHrPSMEwK4lDx+Y1u3bEvEVCXJ1yDKyvd9ltbMKXb/ud1JuWMyRw1Tdbqd54EjMAD7qwqenU25Su3QHi1jIepWR1zCV0LRVG5EX7rtuWelBjrasMCJ7blS8cN13W9wAUAQ5pDK3Db7uNpwSw3xKGbNC0qV6roxR7HSYE9Ejyx1GCcuQWgsk7FcVRTOtc6EU+uthhnBaKuoTcCRS/2Gc4Kt1MJPN7ZHuJLRcNXtEO/NlgOCJRkat34e6ldqWZr6CiKR7OAn31ylWFaApam7zLOSVpxabXJ2V7saJTSaYZMspKNbkwn8nj3cOr0xI1lllVEgcSXgspaQk+gpCIrnINSUWmyely/GfkYa2+TLD6xeFMSAuWonzpzC56q/U6lxJlYhB6R59x+Rnnl7r0QlMaw3grRymn5BJ6iEdTerNKZOysBpzpOtKrQlkBapnmFsBJPGpKywhhLO3Zm2HFt2Fwa1ytphh6B55GVFUopAqU41YnpxB5/8oUzhJ66bWd77WhKVE/fRr5ivRMRKHj3YMqZXmMuDfCdq0cI4XZwqy03oPWlMz0GSTHnnX8Y7iXL8bjhUZZW/jzwJtB5hMf8RHErVemkMXLi3fja9pCtYcbprjNKPmGoYGGt4yhTkf++GpsxjmtdaIunJDeOEwJP8tLNAb9z9YjYd8MorcojVhIvDhBScHG5yf4kZ5Y5ZcNLy008JdgepmSloRMG/MQTSwyTks3jhElWEkqFDJxV10kedesXfpqXtQqh6/wb6+Q/T5zdLU7JMBbvb9t96dgtFY6R0BTOwDkKHD/YuKl+Yglx4FNpNz6ujcGTgmnuIrhzeld04xAENAy0Yo9JLYR1YmIxLSoaVlFWThbVaCiFq9OGnmCYlOj9hIurTsAp8hWnuw2y0mWHuqYlVtrQDBWXVvuEnqy/rK4BPMrcVOHBOKUfBWTaEIcelbUU2rAzSDi35LLotXbIweT9IH66E7PRa7AzTslLw/XDGdO84otnemz0Go5pYw2elDQCZ3k2SUtWmj6VhSR3PQ9fCEZBOTeASAvtpmqNoN8ICJRgZJ3rTq/hY3G9kdCra+d1X+OE7dMOPed12Q5qLe+SYVHg1dOjvhIEniLJ3WvFvsdS0wdjeWV7wkY3QgKNwCc3BmkdO6qqnClz5At2Rjmt0KdV9wkO84KGr9yiLZ0FnGPICFQ9VyCF4HQv5oVzfZK84p3DGXvjlLVOyKXVFt3Y/0BQVVKSV06I7ER1tBkGfP5Mj3bk/AGcI5XGGEtWa5xPsorzSw2WWveuc39aJGgfBR5JIBdCnAX+NeAvA/+7R3HMTxtuvek3BwnXDqY8udpiqRkxSHLe3Bkxytyk5kmjxVjBM6dabA1zfE849/ClJllpGE5zrh1NiQMPY5yy4s4gnUfOE/fwZ051yKsho8jVnEPfI/Il3Vhzqi25tNqiMpb1bsTPPrXK23sTRmnJ7ihF17Zdop7Ss0I4kwLphI00FiEkvnKlFBU615bQc2JJZeVmySvjZFQDX7IUerQiD2Fhf5LTjX1akarLI071Liud2YKQzv1HWyd05SlXIjBAWrqmZqAUDePq2XnpTOTi0OnIFBp8Tzm1KOHqzlprKiHR1qIxbA8TZkWF1o7dvNwMqLQh8iVGMFcnrIxhoxWzP8m5sNxkoxfzzv6EV246E+xWrDgTNRyTxxqwTrSs2/CZpAWr7cgZIUjXR/CU4GCSsT/JyUrNpZUWB9MMXWfUCMvRuOTiWpMnV1q1N2XJUiuiNKZukBrSXM+bk5UpWG0HRL7HzjBxxtlC0Yp8wobk0mqTysDrO6N6AXaSsXV/ei56llWaae6cjza6EbEn2R6nzPKcng2YFSWzTCOFW4TTUHGmFyOVAms51Q3ZHCROf11JrLVOPbIZ8Ln1Luf6Fa9tjzmY5gRK0ot8kE7+IC3cDivyPTqRGxBrRT5VZfn8Rpf3DmfsTVL6jYA/8eWzZKXhN97Y5XQ3diWp2lS5E/v80vOn+PsvbdEKHVc+rwxZqXnmVIvLe2M+v9Hj5nHCmX6D3VGK4KTu7XPtaMrXnjj3gN/rxy9w34lHlZH/FeA/Atp3e4IQ4leAXwE4f/78I3rZHy7u5JguNd0N3xundKIAIdzwxu44Rwnp2AaIObe1FSp6zRCNYKMbszNKyfIKqQR7kwxPSc4tNTmaOi3xrDAcTnOeO93hxvEMbdyXNa+cqt5Sw0NJp543SgtEw2el7WiMs6LieFYwnOUcpaVrbNXlv5avCHxJUTnPRGOpx9BPsiWn3eF8QN37toDveUS+IC2cdkYkJO3I52w/ohV6dBoewkrS2mLNGR+7ycpW6CMFBIHkaFYQ+AKJJA48ZrnTpU6risizdK0PdcC1OCVFLxC1o4xlnDr2S6k1QgkqI+jGAetdZ+VWlJpZoZ3XqXTCWoESzKxFCzjfbyKky9z+9NfOzjOul24c87nTHRqBx+bQ7ZBGmQuuSy1nRdeNPbJK8+buhCQvkULw3OkOw6RgZ5Q5S7/aSeer55c4TgoGSU7k+/z819bcKHxSEHiK//UvPMl3rh3jCWrzDNdUe6bdZlYYOpHicJrz7uGMUjujhmaoiH2PcVZwc5ByuhPy9GqTK/szpllZO+4ItDa0axXGonK7oE7kjJmzyrn4KCEYpoVjt9QlH4vAaMfwWW357E1y2nHAhnWMnaNpwTTXgGOzTAvXa2hFHiuepBN5HE4KdicpcTPkbDcgN26b4EnBMK04nBY8tdp0HPWm78pAyum8bA1TPCFI8oqVVkhSVLfVtO+mOmpxNonTXNNvBIS+4mCccTTLOduPaYTepy6z/mFy1j92IBdC/DFg31r7ohDiD9ztedbaXwN+DeDrX/+6vdvzPi24c9DnYJLzW5f3+cKZHhLJajvg3Xro5qSup6RguRUgcLKqS01XJ//y2R55qZnkPqNZwXPrHS7vTVhrhEicc1CSVyjlygH9ps/WAA4mOYGagRV87nSb9W7smkrAmVqw39b0tpc3h7yxM3bCSwgqIWgELtMrK0ffC6SoZUSdhsj2IGWcVu7LGfustCJsPVq90YsxCKZ5yc4gJfBgKfaojOHlm0M6cVBvY10gO5xmdBshjVpnRQpLI/AZ5wVe7cCupKQROueZtNC0Q49IOVkDY53qoSclnieJal54WS8+UniklSH2nFnxc+ttokBRGYMnnOfnrDSc7kVY6+QGzgZO93ylHfDMqbbrddzyxXnfmKBBUjra4TQrUUpypt9kZ5QyLWZgXDaYlpovne1xpteg1wjmwmZRYLmw0qQZeCy3QkZpyFonYqMXM05LnjrVnn9pbxwnJHlFUhqMsSzVmjr7kynnl7pkpcZo52+ZasPeKCMOnFOSJwWnejGXdyc0fYlSPlldjrGuGUGpDdoKytIgpUanhTP0EMKZJ5eOkeOmLK3T0xGuTJFXltWWo3yGSnDj2EkddIS7fxbL1iBhf5TTChRZZYh8n0unAp461cJXko1+zCAp2TxKOJjkSOWGgrbGGfl7xzQCxVIz4HQ35tWtIRu9BqHn/Eo/jP53N9XRp9ecfk2rVgRtBh6qG3FuqcG5pcZ8AOnTggcdMPyoeBQZ+c8Cf0II8UeBCOgIIX7dWvvLj+DYnxju5JgOkoJu7DsWg4TjiatZX9mbcHGlRSv2+OLZHk+vve/g/tSpNn/qq2c5nhUf+EK/d5QwmOVY69xZ4sAjLzWFsQgheH6jy944Z1ZWPLPeptfwXWaW+oyykoNJxsXVFlIK/uXbB2yNUsZ5iS+gHQWUSY5Ugrw2dXAsCUllodvwafgeZ/sN9mcFQS5pRx5FXc/OS4MxFVnpsrlGqOh5AWmusaWhrNkyrUBhhWCSFnhK0goUgQgJPMn+NKcyBRZHb5zlFctNpzVtMXWt3Q2vrHcjJrlG4jjKvVCRlIayqAg9yXK7STv0wVpWOiHvHsxqJ3nFc+sdSm1pzzLeO0j44kaPWV6SV5ZSW872Iqa5/lA2wuluxO++c4hSkoavMNpirHAOPUWFL91iMi6cbvw011zeHdNrBPQbARvdmKI0dOOAZv05ySunnbPRi7m40pxnYNcOZ1xcgY1eTFG539keJs4T1rhy1v4kczs9BNOiZJprhDwxlbCM0gpPCJ5cafKOsZSzAl9ZwthDSsE4c9LFjcCdtzWWWems4Dzl4UmFF4CQjqKqtSXXmnFS0o4CkkLw1FqLzeOErWFG6EnO9RsczkqSvCT2PZRSbr5BuHPeG6dUxvL86TZfPt/nl55f56UbA660p7x7MGGUlBibE9aTuL6UXFpp0W8EvLU7RgrBKCtphx6TrKQVerfR/+5WxwYnnNVvBFw/npGXTkJ3bbn1qWxY/rA56x87kFtr/2PgPwaoM/K/8LgHcfggx3SaV7Qjt/WkFmLqNwJGaYExhnbg8fRam3bkc365yVOn2vOO+YXlD6qr/atfWOf//pvvOAU7T8yHWX7m0gpCQDMKWGqGHE4yKmNYboa8czBFG+hGvlP+qzQHk4JRWnKqGTCaFVTGzJkgSklWW5JRWpKVhshTrHcCXjjb5+rBjHbk80TodEtmeeUWoWFKVmq0cbzoYVJihCXQBl86pyJPuOnEwjj5VFUbCZzuRuyMUvzKBYCictliVjnVxMo4Jo0UzvoiKwyr7YCn1tqMU83+OHELGbU8a2VptRTnlpp85VyP9U7M1jAlVIqjpKAdBhhruLgSM84iOmHguNie4nCagnD3EZGw3PL5+VqKFlyGZIxlqRWSl5pRWrI3zvjCmQ5r7YhXbg7xlKDfDJnkFWd7DTaHCXujjJdvDLi40mK9G6GNIakMaVEBlnFWcrbfYKkZfGgGdmm1xaubQ75z9cg1fZdb9WfL4+YgwRMlnnK6Lr4SeJ7jnPdjnyBQTGqLu2muOZhltdiVpN8IKE1C6Ek6kV9LDhRY4+iGgSfwZO3NmmmakU8QSI6TjGFaslZpnlrrMZiVTu/ek3ieIpQCTzjWUVoajqYzskqjtWuWtyOPtVbI1jBFW7fRzivD928MyEtNpxHQb7ifb3RjhqkrPeWVM1ceznLiQLHS7HB5d/KhTcq71bFPAnxWOTPoVuSx1Ao+lWP2P2zO+mIg6C64cyCoVWcMaVGx0WvgKcmbu2PSymKSkrWOpRV6c27rs+v9e9bEzi01+MLZLr979ZDR0IkldUI3jamNpRU5q61G6DNOC/75W3tcWGry5GoTr3a2zyvN1sBJ484qg7ACq6FShkA5dkNibW3Y7JpZS01HX1tq+ZzpNfCkpBN7bI9SlpoB//LtAzoNn8E0J/Q8orouOi01F5Ydc2OW69ruS9S6IU6NcHuYuYk/YKkZUFYa2wo5mmRIIKk0wjqKXSOQJLmbmtyb5JR1fVwbw1Gi6UU+S62QRm2cAbA9SlHSDWW9vT9hkBTEnqLUhuc3Ovzhz53iO9eOmGaO+lhUxvHsgeuHM55Yfj9D3h1lrLRCfuri8ty56eJqiwtLDZ5aa3N5d0KhNdujlI1OXCsoStL/P3t/9mvZled3Yp+1533me8+db8xkcCaTOagyS6rKVlsqlFqCBNmQXzy89rMBv/lfMGCgAT8J6Ac3bNgNuBsCZKlVlapSVWVNmVk5cQ4GY77zcMY9D2v54bfPYUQwgoxMMlPJFBfAJPMO55x7zt6/9Vvf33fwbAnWzku+dnHAaltSkY6mKQY5Sb1xYfDUDmwUS1dpWUIAdW3F9c2ueHoXFffOE1qeTVpJcLJvxB7WshS7/YCkLGl7DrsrLfKy4iwuyCqBzzqewyQtSPKata7HWgdmWQnGLKl5t06jJg5O0/Zd2r5cy5O04KPjiPWuj0Hx6k6fW6cx50nJLJNNZW8UYzebgrJkMx/HJWUNG12PuoZ/+/MDvnt9ncC2pBuPc3qhy1lU0A2MuD/aNu8dzhi2fVByfU7SEstSzzykhI8L/KdRDH9T1q/aC/0LLeTGmD8D/uyLfMz/XOtxjulKy2N/nGBbiqqu2R8n9DyHb1xcoShrbp5GPBgn7AzCR45+T8PE3tqbYCn4p6/t4Ds2t04j/sM7h5zMc66uiR3t8TylH3hs9QIRWNiKDw6nXFzpsN71+cHtc7kBlORDdlsuswTGDX+47cuR228CiB2E3vfW3oStbsB5lHN9s4Ol4LWdPj+5PybLKwYtB78fSsG2LXTjjdHxHWZJwbzhho+TAkcpbEdRVBXvHZYM2z7rXdms1nshnm0ROoo4q9kObO6PU/KipqUksSgua8K8ktQiz1m6Lw7aYlc7jkve3htzdb3L67sDxknOnVHM9iBkux+Q5ZpaGQxys7yw2eXPPjil0kK3+/ZzfVZaHnfPYv67//ghr+322R2EHM4y4qLipa0eL271pCv2HeZ5TTdw+eaVFW6fzNmfZpRaWDOWsnhpu08/dDmapZxFOX/42tYTu79ZOvnUDuy1nX4jyxfGUJyXjJKCe6MEZQwbXZ+slDAPjGGjJxa0vtN4mSuYF1K8LwxcbpzMqbSYYdnKYpJWtD0by4JB4NEKbKpasGbHkqH1Vj/kwkrI6TxlmslkvBM65FGxFGMZY/AtJS6QBrq+hVaKeVrhucLV7wQejmUxL0pCxxb5fz/kNBa7iLioGuxdqISWpXh9t8+l1TZJWXM2l1NBktdcHv7mddNfxPpVc9a/6sifsh7H5lY7Hv/yGxf421vnvHswpeM77Ky0aHsOmaV4bafPziB8xPj+0zCxmycRvYYfe/885i9vnlJrzSyv6Mxt+qEU7rwSHvRZBJO0bDxfTtjoBmhtmKUVri0Ebde2cF05jnu2zWY/5NKwha0sjucpeVmz5QeEzYD23ijGKHh5SzIpV1oeV9baSxhpEGrunidcXGlRahgnwtpwbeGUg3RlupLsSFtBnMsR99Iw4MIg5HCW0g0cQk8Gahsdn7SoOItKkrLEsRVbHa9h/Igicdj1sJRFkheM4hLX8tgfpby+O+A8KsmKmt3VkBc3+4DAELOs4HvvHfP6bp+LqyEns5ykrPEdm6SsuTuSBJ+0qMWT4yyS4GpL8fVLq8sTV6cRfF3f6DKKJYfzPJL5yGrbW3bPF1ZCNnrBU4vOwx3YPCs5mKScxyIs2x4EeI7NB3sTai2WtmdxQVpIWHNa1ijLwrGlUFuN940xivWuz3mUsTcWJodvW7R9h0HgcjTLKLRh4CuGoYsGTG7Y6HpcXmtzFouVwVSXdEOba+tt5lmFZdm8eaFDbWC9EzCJS/YmKbsrIeO44KQ5UbUCB8t2UIifva0UrmOz2fPxXZuTecYr25Ife229w04S8OA8IUoLLl1q02+5XN/ooBqWymJI2R52uGQMp1H2iHL609aXzbXwV81Z/6qQf8p6EjbXD12OZylrHZ/AtWUgWNa8sNl9xGHtszAx1bB/749i/vi9I+ZZScu1cSxFWmr6oeDyvjb83b0Rw7Yv9L68ZhTnIi23Ff3QZpqUrLY9uoFLVtSstX22+iG7K60m7V6BNry4Jcf+aVKyN4kJXJv9UYIykNeaFze6fOPKCj+5NxZetpIjeOh5/Ks3dtmfSGRZWs6ZVzW2JfJ8S4Ft2Qy7Lj3fx3fkRhvHBbdOxfp1ZxByFolzYNYIclZCD2MMh9O84X7bWNAwRMTXuuVZDDs+53Ephaqu2ez5uOpjVkKlNe88mBH6EodnWYq00nR8h9N5jlKGstJ0PId7o4Tn17uyyUxT3tk3PLfeWZ64Lq62RYVrKXb6AZ5j8cPbI4pK8+JWC1vJxnFptf2JY/HDxUUpmKYlLVc2TAtJAlrr+BxOMu6eRQybkOYPj2cUteH5jS5FVWM7irwUD3nhhNdsdOF/9fJFwPB398ZkZc3fuzSg1vD24RRlYKsfSv5rXTNJBev+3efWmCQFqvGo+ccvb/GT+yOySvPWgwm2rbi61mGjF+LainvnCa6tKMqaKaBRbPVDTFd/TE01IrVPyppuU9TLusZzZKZwcVUamKtrHVquMIsuDltc3+jwxoUBb+1N2B8n3DtP6Icu24OQutYUlWGelfz5jZPP9P//6f0xWsMozvngaM6P7434g1e2lvOo38RC/6vkrH9VyH/BNWh5vHlphfvnCbNmyn552Ma2FC3/4+LyWZjY8xtdfnxvzA9vj8REyHckLNlSdALxpK6NqEBf2OoxS0tc22aSZdRGMitbgc3RVAQpo7TkyrDNes+jatRww7bPvSZYoRM4DEJXoImziHkiwRTzVMyRoqzk79KS37myyqXVDrdPI9KyZrsXsNoNyCvNB4cTsWtFrE1RTbCALdL0sjJYIcyzmrvjlI2Oj20MWVnzwcEM15Gbv9aGtmuz0vYYxQVGC92x7TsU2vDgPKbf8uj5No7joQ28utNjo+uz0fV5/3CG7YjvR1xUfHQcUdQVV7o9yloT5xVVXVOUhlJZFFoUoYsO0HMsXMdjklTM84r/4a/usLva5pWdLq4jNgpKQdt3uLLW4fn1Dn/50Rkfncx5ZafHpdU2lsUjEWFPopcpSu43jpnDtr/sNm+fRkzTEscWXv2F1TZbvUAKf5M+b9lgSknu6fpKYKSyZpwU/MHLW4ybPNbAtUlqoSyud33unkUMWgHbfckT/RdvXuBklvFXt86oasM4zRm2ha9tW0LbrGoJaX55e4XTeSEeNY5I4gdtnxuHM+K8ZBIV2JZFrTVZIe5n/UBUvOtdn+vtHmWl2d1ssd71ubja4nQe0gtdgWUCVyIR05LayPt3HuW8vTdhvetzabWN79hL6OFp9Lw7ZzFaw/1RQuDarHd8ZqmcyP7VNy8Anw5r/jaurwr5L7Gexm19GO/6LEzsjQsDfv5gQlFVeI10v9aGru/gKJgXmq2ex6DtEToWYy1FPW1sc7Oi5u5JTF5KcUrrmigt2ex1OJpm3B8l7A5CXt3ucTLPsC3FStsjcEXE1A09Sl2T1ZpV38MKFQ/GCX9965QLK202uh6TVNEOXLqezbv7E/nDmo7UApFtA6rWhA5MU+gEFaM4B2VxNpf8zCwt0bVBlY1NQanp+Danc+Fh90MHx1ZkVc319S5RWmJbsrlt90PysmZnEBJllXCEbYVnW6RFyf4opapr1joB/ZaoIldawpu+fRYxX9jHujbnUcrOIMAg9MnzOGezFxA6Ps9vtDlrXBMXyewL74+W5/DfvLbN7bOIvNJPZEY8abi53pWUpt+9toZS4k9/42hOVgqHfmfQarp7GToXtSH0LLTWhK5DaSuSUhN6Nps9yQZ9MEroBQ7XN4XlAeAqxe1xRJwL9XGjF2Ap+bxB7HEncc6rO31uHM/Ja5l9bHR9pmlJnNfcP4+XBXHQFhjgZJ7jOzbDjsvts4iu71DUFVqL4dabFwbsDkIur7XJiprnNzsoFHdHMfdGkXTy8Ehx/t57R1wddvjahQEHkxTHttldbVFUNVfW2s9Ez5ulJaM4bwLNZcbQCz1Oo2wZ9/bbYk/7rOurQv5LrGfBux7/GaVEMPTzB5PlUW+zF9BreYyiglIbNrseKBksYRQXVweNig2ySnM0TfFcm2lWcR5nKIQ9kJQlg5aP7Vh8dBzxzSurRLlAEWlV81+9sMErOz1un0Z8eBzhKDEyGiciZ7aU4mSek2QVcVayP8l5cavDpWGbvKiYZKWYJDkO4zihQV1wbLFVNUBSQcupmSSFBAg7hsqAUdByHRItgpt+6OA4FpOkAsvQ81wc2yIuNKEjmaJvXhowigpevzhgveNjW4oPDudEmaQa/dM3dhgnBTdPIvF43+qy3W8tqZOLJPdhy+P5jQ5VbTicZqx3fXSD9Y+Sgt1+SNt3afkWoeeilHi4LI7kjiUq3SgX4cnVYZtKmyeyJBZQ2gIPj3IZNiZFvTyZHUxSAtem5TugPh50lnXd+L44XB12cZTi9nlMnGu2+h7XN3u4tkXgOqy2PW6fxVzf7GFb8N7hhBvHc9CGqCiYJjVn85zfubZGx7f56f0Rf3PrnLMoY5ZV2Eqx2vHISs3BJGWrF/DGbpuDacooKZjnFdt9CQEPPZv9adowtUJarohveoHLoO2glIQv9wOPP3x1uLSS3R2EpGXN2/tTrg4fLc51A4e81Ovz4pacTo0x/MWHJ08MhHgSPa8XunxwNGe94y+/lle1JBs18OZviz3ts66vCvkvuZ4F71r8zMPH7kVn8v2bp8Jb3unz4dGckygnq2qyQoKGL622GEUFea1Z72nOZxmHk5SWa1MgxlKdQPwsZmlB4IiZ1KwWT+/XdvvUGl7e7uI5FpeHbfqhy8ksJ/RtZmlN6AqT4YOTKdOkEPWeEh+UvbHMAS6vtfmLm6f0A5fQawaqjZ92aQT3RX/slBi6DlmpyWtDN5SAAKVEjGTEgYWWaxMVBR3XZZZXeI5NP5CMyff2Z3z3xQ02eyHfuLRCVWtxmWy7fPeFNRzb4izK+fqlFb77wsYy4aXliQfMwSTlLC4oasO3rgyXBmbzrOSdgwl3TmNe3emz0fU4mUmk3lpXgkF8x2aaFkuM+639KYPQoxc45JXmrf0pr+482ROuF7qcznNuHM+YZ+Ijow3LIeBGNyDKqiX32iAReVWt+egkoh04+I7CoIUGOezyZzeOMSjOo5zVtjBvpknJ4TThB3fO6TVZoUlRNfBGwMZGwL3zhL+7d84buwNRkRYiCrp7FuM1m2XHd5mkJRdXW7R8h5e3+7y41aPdUGgFiqp4bq2D79hUteTNroQeqjFWGyclr2532egFjOLiE11wrTWjhicum1stFN6y5qXt/vK9S8ua9W7wzPS8q2ttfnxvxCwt6IXe0p/l4bnFr5Lq95u4frN0rL+l60lxcIdToa+N4pxhx6MTSFqQ61q8sjvg6noHxxHDor+7M0ZjGLQ8fN+irBVd38GzbYYdj52VFr5ncxqVDFoul9faDYc6472DKT9p4qsGLY8/fG2L33t+neubHWzgw+M5UVERuDbKFndAz1aMk5IPjufcPY/pBy6nUc7JLMMYQ9u38TxhS/iOTSuw6fkWw27AZjegF7hUDW3OArJK7FI922K14/Ha7oB+4GGQPFPHUeKRbYth9geHU8ZxznsHU376YEwncPnahQG90Fu+j4sj9MO+8R3f4eJqixc2O1xdazGKc358b8SNoxkA37m6xu9dX+fV3T6+a+O7FtuD8BOqzMUNr5YxwyCC/qc7S1xda/Pe4ZT9cdI4S8qQz7VYinY0UtzfvLjC1y+uUNaat/enhK7DmxcGeI7NeVTy8wcTbp7OWOt4FGXN3bOYW6cR41iyW7cGYhL1/tGcSVKy2fPZ7IWcJwXawKXVFnVt2BvLCS70HCzbZmcQYtkWd04ToqJkrS3D5qyBrgCuDtuM45KsrJffs5WweBasJIMhzgT+8hybo2nGX9085f55zDz7eOC/2vLYn6TcOJpT1oZe4NByxSd+cS0tPrvvPDd8xP9/8d8PzyEWa9Dy+INXtqgMnEYZjqUemVs8niXwaY/127K+6sh/DeuJsXGnkgr00nafw0lKoQ2OEj+NC4MAhcVq22eWlCKaaLm8fmGFSVywP0vRtcN5JEk6F1dCjuc5tqX4nSur5GXNrZM5z21IN6Uxjwx7fu/6Or3QZX+SYrRBGYlqsy2YpYCj8CxRhL6zP+PiagujoRfaaKMZJxK2u9lxmTYd90ZHsOeoEJViVrkYDa5rkZcaZYHBYrcfcnE1ZKvn8fbBjNC1Gpy2YpxouoHDoO3z/EZXwnszw3rHbzq6OR3fWQ7+FsyEpPF17zTsmK1+h1unAiEtOrYbRzMurbaXFNGra22+f/OUvXH6CVXm1bU2P38w4fXdAR+dRtw6jQChHM6z6omf8aAlnWovcCm1mJJd3+hhKcPBNOW/eX17OTexG8zeshTX1ttcWxNFbeDaeI4MIEdJgUKRVJqi1FRRQa+V88p2T2TtifirV1pzNi/ohzZJIRuGgibaT+PaiguN58s8r8BoOqGN1mJHPElKNvsCs+wMICs1LU+CU5X8D+tdH9uCNy6s8PMHYx6MY7JCPqt/+/M9nt/oNW6QOVFe8+KWKJxX2z7vHkoUn+9Y5FVN6Nn8/efWOItyHFstIce7ZzG2pR5xP/w0et7lYZt/9c0LT2Wm/LbY0z7r+qqQ/xrWgsFSa7NUEc6ykrWeT8d3ub7pcnG15oPDKadRRlkZWj68sNXlcJpyZa3dDPE8AtdmnpeczGv+3tUV8rJmltdsdH2+frFPWtb8+Y0TlBJcfXfQ4nefG2JbajnsGbQ83rgw4Gf3x3Q8hx/fHzOJZfDY8iTswbNr2rbcwFVZc2G1xQdHM1ZbPsrALBe1Zz90RWrecvFtxTgVJ8SVtkelDaN5TqsrRSstpZtbFIabZ+LgV2vDZs8jboy0+qFDbQwfHs84mKT88bvH9FvyPL3QZb3j8eal1SVcdXGltRwmL/xNrg473B8l5JVAVXlZc+c85ptXBN8etDx+//o6b+1N+Ohk/ogqc9DkqY4ikfE/ty4b4iwtOJymy9PN46vlWay22oQPH+mLatnJPz43ySvN67sDDqfZcnAX5xXTtGCzJ8Pdy8M2984ifFtRVZqdXsjts4iPTiI6nsOgLZBO3DCeHowShi2PrX4oA9/TmIurYugldgWiUq5qTVkLnDVNCzxLcdoEdr+y3V9CUsAyTq0buORVzfsHM9Y6NnujhPVuQFVrOr7L3jjhwkqL/XHCpaF0yK9s98Ve9iGG18JP5cpam++9d0ytNastsaSwLJ6ZXfJp8OZviz3ts66vCvmvYV1da/OXN0/ZGycN7GBoeTazRKbvKy0PY8Sc6uVun67vNJ2k+Gq7lk1aau6dx9RaIrsGLZc/fHWb9a6/LGJrHZ9//9YBoStBBrU2HEwToqzPVj94ZNhz5yxmtePT9l0ejBNmaYHVmER1bBvHslBK3AELbTBFKZ2V69BrsO95UYnjX9fDMkJDfG13QNt32OwHfHA4W4qLLq226PouP3kwZm+c8Ppunzd2+9w4nmGj6IUuniPUxvujlPcPpxS1ZE2eRimztKTl2/RTl71JiucIu+RJzIRZWrLe9ZfY7CyraPvOJ6xNBy2P776wwXdf2HjiZ/bje2MJqHbsxmJXgj6exn54fqPLuwczlFJNB6qZZuUjuPrDc5Nbp3P++N0jjmYZV4YtLg47Mij2HIE2UKyEHnHHxxjD7iDknYMpKy2PKK84nmWYU8F/y1rT9W1Wuj6Djsdax0cBN47m3DyesdYk51iWwmk64EFHM2x53DqNuXMa88pOH9tSBK71yJB3uy+Okg97Bz0YJWRFzUrbo6zE7Oy5DXGYXBjGvbglm9ZihrFYSVGhFHzvvSMcpVjtBORVzf1RzKXVp7+/X62nr68K+a9hLTq8TupS1NANBMsFmCQldnNz/e5za1wetvnee0ecRjmrLZffvbbGX948xbFlwJSVgl9eXm3z/tGUeyOb5zfEoOuP3jlisxfgN0PMGjgYJ/z/3jrgd66ustoRI6dZWnLrdM7uoMXBJGOr6zOOcg6mKUlhuLDSIsqEix26FudxziguWO+Igq8fulwattjuhfzw7ojNbkBS1LQ8m5bv8s/f3OHysM2f3zhhpYEcFmuj5/OzB2Pmeck0LXh9e8AoyblzHjNNKozRGKVoezbrnYCf701wbUfCJrRE1vVdm3f2JvzBK1sAHzNFsgqN5vmNLmlZLwN6QYrHL2JtOmh5bPcDkrx6xAv7cXe+h9cbFwZLnvQsrXEcxWpLnn9xSpLXK+nw8yb1KXBs7p4nkjdaG7b7IbdOInotl6LWeI7QTrNSjKaiom6eo6Tr2QSezVbP58E447lNj5e3e1zfkGiAlmfz798+lPzOtTa9wOWtBxNW24K/W817HTg27+1PmKYV33v3GN9VDNs+2/2Q+6OEbzUnmQVMGDUMpLLSuI54i19alVPY71xdXRb9q2t8goZ7Mhcf95vHEVs9H8exl3OKUSyQyy+6PksA9JsoEPoi11eF/BdYn+diMAbe2B08wif2HQurDS9v90iKanms/1ffvPjI83iO4uZxTFyIl0ng2QxbLm3f4dKwvQygPZ1nbPdDHMfmw6MZo7jAdxTTtOJknrM3TvBtm/Wuj2fb3D6LeW6tw/4kkQFk6PFgnDCKhGlQ1YYM8TGxLcFAPcdhsxvw+u6AH909Z5pWtOOClZYnjoe15k/eP+K59S5H00yyGz2bj07m3DyZczDJqGvN339+nWHb5/2jGVFeNukvFXFR0/ZcQkeCCYpK4JbVtk9RaTqBQyd0OJxkS7jqZ/cnRHlFUpT4TVFo+Q4b3eBz+VosLGcf7iZPZhlnUf5U9WE/dDmdZxgUm+2ASusl6+Ot/SkKQ+gKNl1pw8WVFi0v56OTmKjQbPYEYpimJdoYyrrmwkqLXiCnmdNZxkbPp6wFrvIshesId7zbnPYWRbEbuHzj8iq+a8nwtYbzKGPY9bg/islqwziRnM20FOpmVtaAoqiEYlpUNZ3A4eJqyE/vj7l1OsezbSxl6IUeR9OUoqoJPQm1sB8TSg1aHtfWO/ztrXN5zrLGGMH2+4FDWhjuncVcXmvTcm1OI+nmf5H1WV7fv2ov8N+E9VUhf8b1eS+GBT1tnBREeYVtiQTftoWa9/Aw5nF8b5aWvLDZQynhNZe1eaIZ/3o3YJ6V9FsCKwSlTZRWrLRc1joeWovR1UYv4Np6h7f2xtw9j1EYHkwSpg0dLW660GlWstH1KWvxDp8lBVFeMElsbhzN+Nvb52x0fTAQ5zU/uT+mH7pMEpcXN3tMs4I///AE1xbzpiQrmSQFYPizG8dERY2n4CQqJRBBi6zetiQAdJaVDDsSulFUejkYy8qaS2viO33zZM79UUxa1WSNM+XBNOPFLaFdfp5h1+OirtN5zjv7E17bHXziGoCP1YTfujz8BI/6xtGMQegBhlunEVaDVcdFxYtbfXZXWtw7ExWu59j8szd2loIqDZI96jp4Ky02uwGTdCa0yFI3KseUnX7AKM6JspIPjkouN0yOf/Ty1pLj/f6ByOCnqbB8HAXjsuRkmtENXJSy2OoHpEXjRe9JmtFP7o25tNrmxc0eP7w74rCJEgxdG2MMrUCySP/gla1PdMK3TyPWOj5xUYndbS15nLOsxHVsQseR3NSWh21ZvzC75LO8vn/VXuC/CeurQv6M6/NeDKttj7+4cUI/dOkG7tIS919+48IT/cofXg/L/aO8bnjNMlSEj8UO33luyL/5yR4Ada1ZCT08y+IfvrjBwTSjGzjCXEC6tatrHf7o3SNqbUiLirWuxzgpqeuaohbFZVGLj3le1by2MyCramZ5xcGDMRtdj41uIEET04ykrDifF7yw0+XD4zld32Wz63PzJCavaoaN5WvbdcSGt6gY5bUEWihJ/qkBzxFrXQvF9iDg3llCWlS0fBvPsfAsxT96aYOvX1rhP753xNk8Y9D22VnvYivFOC64fRrzT1/f+WU/buCTw8mzKOe13cFyEPjwNbD4/4uv1dpw/zzmaJLy0naP0yhns/vxANG1LcCQFDIIdSyLK+ttXt3pLwe2tqWWPtvHs5QXNrvcPYtJy5rAUSR5zTwXj57tgagoO427YF7WS3dGkA3w3YMJD8bSeW/1A3zbpqw0cV5jWarxRRGaaKvtk1UyzJ6ksqEuDMB8WyyXs1IvHTa/cWmwPFE+vBb3zYNRIsEUyiIIREiktd1g8oqjWU43dD+xETzL+ixfo1+1F/hvwvqqkD/j+rwXwygueG13wDgpmOc1ncDl4mqbUVx8ZiF/uDNsexZHM3HSW+v4cDRjpeWx2vG4PGwvHRrjsqYXSBEHeHd/zFlUMOxKBNl2P+R0nrPe8bmy1uaDozl5WZOUNeOyxndtAkcxTUsC28axLXzPYncYMksrbhzOGbbFR7rWMpSsak1ZauaJyL4nScXhLKfUNSuhR6VFFYgR10RjFIXWBLZNJ3BoBw6nswwbsFHsrITMsxJno41jKVqe+MdcW2/zu88NhVfvOFxda9MNZeh7FhVMkkzyS5/CLvlF1sOnowXm//B6+BpYfG8BnbU9h6SsKWvD2VyCilueiL2iXHDy2hiKBwIRvbrbW0I1j/ts//T+mHcPptTG8NbehLzUnEUZLd/FsiwCV+Cb13f7tDyH59Y7fHA8568+OuNwmnF12OZbl4dcXs35t2/t88Jml1unEbNCHC3X2z61kdjBk1mGUeDZQpPM65pr63KNHkxSfNem7TtkVcn1jS4rLTFse9J7/TCm3gscWp5FWRuMMTy/2eXeeUw7cPjGSuuplsCftT7L1+hX7QX+m7C+KuTPuD7vxbBgUjxM6zKNp/dnrYc7QwPcO0+4PGyx0fGZZyW3Tua8fnHA7IZg6n/42hZ/+NpWM9is+NvbZziW2M+6Fvyn94/43efXGcViz+o7Nistl6p2GLQ9fnR7RK/lEGclVWXIa83FFSngiyT5jZ54UJ/Mc4FObEOpYaPj4do2P7hzTtuzOZllxEVNkmksC57f7HI8k8Dd0laohpznOBZlZWgHDrZlUdaaf/TyJm9cGAA8dTax2vGk8GQFZ5Fwry1L0Q6cz42DPj4TUerTFYOPS/HXuj4H0xQwbPcD7o5idvsyXziLCj46nuO5wt/fWQmXJ6wnratrbW4czTieZ1wZthnFOeO0IC+qJtNU8fXLKww7PsfzjLMooxMIu8hRcH8UE3o2G72AV7b77I0TuoFL23W4vKo4meXMsoJZVpBWNXkzHHZti1d3elxckUJ+GuWcRzkWFsO2R1kb7o0kpedJ1gVKwd/ePuOd/Rm10ay3Q2Z5AUZODYFrszsI+b3r68/0OT1pTvVZvka/ai/w34T1VSF/xvV5L4bPuxE83BkO2/6ys19kcY7ign7g8u7+lO9/eMqblwZcW+/wP/7oAZalWO8EXN3oEmc151HKB0dzvnNtlfvnCXlVs9YNuHcWoVBs930sJayFK2sdTuc5xkA/tHluo8N5VLDZDfjoZE7o2dgKlKXo+zbDjs/Nkxm1gWHLxbYsQg+K2jBNSubpx3a9VSQBCrU2+LbVuAR6tHyXf/G1nUdogU+Dr9640OfH9zQPRokENXsWvcDjle3eUgH6y+CgT5qJTNMSRbnkTt85jxlFOW9eWuHysM3tRjh0MsuZZwWzrOTiaoey1tiWYrMb8MpOn0qLYvKfvbH7Cb72017voOWxMwi5OMt5/3DGySzn4qC1TIzPS7kWsrJmEovx1bW1Dh8eR/RCj3GS8ze3zhh2AnxHEeUVb+z2lzTXcVxw82TOndNY/HfaLt3QY7Xl8vJ2T2YcRUWcVVSVxnUV692AwLUZxznvHszo+M4jG+0kKTiYpHx4EtEPXUZxzu3zGfOkYqXtMk4Ul4ZyOvllP5PFZv1pAqBn8Ub6sq+vCvkzrs97MXxRXcHjnf2NoxlOTzHNKj48nksn2PG5P0qEcw584+IKliXUu5YnFL/9SQoIVnswldCJKK8YJyXDlsvWoMUr2z3WuxLvduc8YhyXrHd8nl/vsj9JCN0BP7k/4iwu2O74XLzQQmvD6SznLM6IcsWV9Q5Gw1mU0fbkpn9uo8PL2z1c2+ZHd0dkZUXgCI3OcSxe3e7x959be6b3Y0H5G8cFw7Z0kB3f4fmN7ufCQZ80E9noCt85r2p+dl8ofN+8vIpjW9w+jbi23uHeecyDsfzuS9t9HMta+oCsdj6GSxYwzeMmWy3feSS0eUFZNAbe3p9gWxIa3fFs5nnF3bOYC6ttXt7ucjLLafs2vdDhjd0+3cCl48t7sPB8v7om7JK80uxPEj5qCvfruwMOpjFrPTlRxVmFY1mEns3RLOMfv7K1PBFaFmz1fFqezf444ed7Y9q+y8lMWEqTpFjeK5U2vLzVY56VYkqWlmS1RqO4tNqi7bkcTDLe2ps8kc//WZ/J4utSzJ9+L/62C4S+KuS/wPo8F8MX1RU83tlHucAdSV7h2S5H05y0qNDAdk8GYAsmS1xU3DuLKCvN9kBCCMBQ1pq0rJuu0aflOzy30V56c692PL555SJv7U24f55wME2xLEU3dHjj4gqTJGenL9DAhycxu4OQfuiABeO4IHRtbMvi4iCgH7q8sNlbyun/wfU17p3H3DyJUEi4wpOGZp/2vv7+9XWSvGaSFkvf727gkhTVIyeeZ6WPTpKCn9wfYyFD4cXjha69lI9/+9rwkdMVyBykG7j845e3uD+KcaxGGFRWn8ii7IUud89jfr5ICbIhLjRxUXL7NOKV7T6Bay0pi6/vDpinFWdRzjgVGmoncDmb5/z43phxkrPdD/nffecSo7igqMQvfGcQ8v7hjLrW+K7QQPfGiYSUFBVv7A7ohS5vH0z56YMpeV7TDmw6gcs8LcmKkmljS7DYhEZRIcKfecaN4zlrHZ+NrsQRPizqmaUCza00qs3LQzibZ+RlTceXeUdZG6ZpwdvPUMj/Sxha/rLrq0L+a1wPq/runMWPWNr+sp29YymmWQkoDqc5gWPj2uI0eG8Us9H1mCZiZDSKC8pKU9SGN3YHtDyHspZQh2trHW4czZYy8Z8/mND2bN5sfElAEm/meUU/cFl4kzy/3ub1C7v8yftHFJWh69ustbvcPJlT1pokr7GVotZa3BWNpOQsDI4GLRnSftZN/Ph6vCh/5zmxUdUa9scJo0RCEP7glc3lzz8LfXTxc75jYaEoa82NoxkvbvWwLVGgPqmgVLXmnYMpaV6x2QvYHbSYZRLa8SRV6Wrb4//zw/uEjoXv2dw7iylriWWbJBJIYVtqSVk8nGb0W55ASHVNUVUczQrSUrPW9eh6wh55e2/C6xcGS5in4zt0A4fTeUaU1PR8YSKttX1Oo4xxXDBKCvZGCVVZYzsWSokAqeXbpKXhkv8xRHV1rc0kKcR/xxj6gU3gOnR8h+OZZG+eRwUv7/TYGYQ4jmoUymJRO81KPMem4zuNK6bCqyzO4s8uxv8lDC1/2fVVIf81r8/LR3+8s780bIm51d506elR1JrLa22qqiZ0Hf7rb2zyt7fO2d+fsj0IeWN3wHbjdlfWGoXiYJKiDdw7j7lzluDY8PWLkoRUa4k+2+gGDNv+0pK0E7j0QpfLwzZfv7RKUWlqbbhxNOf6Rpe//OiUShvisuLKWptOE/g7bgrBF4lfL7jKP7hzTq1ZenfcPhV89lnpo4ufk41NoCrfEehkdyVcys4fLijzrOTt/QmdwMVRwhlPS82LW11e2OxiN3OMh9coLlhpuYzighvHEYFnsdtvEeUlnUC8ym+dRo0cPub+KMGzbeJCTmDH85yy0Ky0G0MqrbkybDNJSkZx8cg1st0PaXkO3UBiCg+mGaWpcR3xG7eUYOubg5Akr1EYSm1I85pey2WzGy59vh++/u6ex2SVJi0LYUB1fQahy2kknjSv7fYpqpof3j2T1KpQ7IBD16YT2Bgkgq+oatYfmhUsPuOHN+rVtsBQP7s/ZrXjc3XYxrGt37qh5S+7virkv+b1tILy1t6EbuB+4tj/JDjg4dUNpJDeO485GCc4jsXloWRLFsbQCcSo6PKwzQtb3U8oFV3bAiWxWXfPYkZJgaUUA8fjaJYt+cPvHkz41uUhSqlHAgEWx9qFm+DhNOX+Wcp5kqON4aWtDnkFW72A9V5Ay7WZZdUXjl8DvHcw4/XmpLFYiwHisx7LFz8nf2e3kf4bNHq52T4uO799GmEQB8iPTsWyteVaHE6EsXNhJeT3r68/8jwHk5Si0gxCj6IndLz7o5h5JkPLS0NDXmrePZhyMsvQBrSGw6koOy+vttkfp2SV5J8Grphi+Y71SHZslFfERcmP7o7o+DbncckoynFtyfCcxAVZWWFZFkPXF7jDaIpKvMStXPHjeyPOYsHxdwbhkimy0vZp+w4fHM7QBs6inKIyBI7F1WGHdw+mPDhPMNowiXMOJilVbVgZShB3lEnc3aDtL9lJ8MmN+nSe8xc3Tnhtd8A3L69y5yzmx/fGvHlp8Fulzvw866tC/mteTzuW/+z+mG9fW3ukS7+23lkq8hZf//7NUxSw3g0e6UbfuDDg5a0+46TgNMoF0rAV7axa8qlX294n3OYGLZcol0GprQwYqI2m0pqirElyCaAwqKemwk+aYpjkFQ9GKbYFl1dbdAOHYSdgte3hNQk3WSlGTJ/nSPy0onw6z3h5u/eJry9mEs9yLH/45xZeLQuflqexIIq65o3dPofTjJWWL7Fjs4yDqRTgWhv64QSQoWWvCfBu+w73zxOOZxlJKeHFli0d/p++f4Tv2ERFRVLUXBy08ByLrZ7PJK0InIx+6OAVipYvjpD3zmJmhThMzrMSbeBnD8bMsorzeUZZu5zPcxwbJllJ1LBQaq2xFGz1QtJSLIonSSEDSpRs/r7N3bOIwLGZJAW2JQZi90cxbd/BGLg7Snh7f8paN2CaFkRZxbDtcXGlTejaTQJTTl6LydZ615e0oZb7SCF/fKPenyTM8oq/uX3GS1s9rq13uL6pHvlM/ktfXxXyX/N6UkFZOBE+3mH+7a3zT+QYTtMSDFwedh752byqsSwRpczSgrDtohE8+uFN4WrDQR4lJdOs5A9e2eLeeUy/5TKNC1xH0Q08bKWYpBXbjUXs9Q2RxEdZ9YlU+IXHdqlNM6QTPHQUZ9w9T2j7NnWtycsaDWx0xR9mq99Zmng9y6xgcTpZ+H1cW+/QDT7mcH9ayszDs4Wq1tw5ixnFBW9eGjwiHHoau+hJr3Ux/OuFEgoS5TW2BedRzkmUM08L2r7F+XnO8Sxjte3xO1cEgnowTjmdZ9S1IcoFS3dsi9W2h2NbTOKCk6ogyyv5bNKSla7Hq7sDGVTmNadRzjyv6JUVt04jsqpCGSNWth0fx5ITW5rXuK7NPKsYdj3OoxJlDOdRQce3CRyXyhiKuubCIOT9oymTuGCl7eFYCseyOZ6JzfE4kZDvSVry3evr1Mbw7uGc++cRh+OUlY7P1WHIJC756d6Y339+HY2kQLm2uGuO4pwXt7rklebV3f4nPveHN+qFuKofupS1ZpqWvH80Y9j28F37CzO/+rKban1VyH/N60mFYn+SstXz+fG9ER1f2Bwd33lih1lVprGx+niFrs1ZlNMNHL5/85i6huc2Oryw0V2yN/7k/aOlcdIif/IkyviP7x8D8PJWl1qL8dbd84S2Z+N7Nt0m+mtRtP7onaNHUuEXj//uwQQLi1b4MRa80vLJS01RGQrkqL7aDnAdteRdP+usYHHc1hpspXhrb8qN4zm/9/wag5a3TJlZDPkep3guuui/vnXG9z88wXNsrq21KatHQzeexC7a6j/5ZNQPXeZZxfEs5XQukWtZVdP3XcZxTqk1RxPxGg9Dl0lccOs04uuXVtHaEGUVa52AbugyaHscT3NGcUk3dOm3POaTFGMa6mMvIMpL7pzGQuX0HZSBstK8vz/DsiD0XdqeTVpUnM0087xi2JZcS9e2iPOKuKg4mxc4toWtFBu9EKUU4yjn1qmYWlUaLq21sZWirA2hpzibl0RZyVYvwFKwN06azTDn5a0uH+xPMEo2sXcOJA2o47l8eDznuY0eoSvXRVULY+WNRuW8uK4eLqQLs7WNXsDBJKUfelS1xrIUR9MUG5inwkj6IsyvfhtMtT53IVdKXQT+B2ATycX618aY/+7zPu5v63q8UCglVqOLI/fDaTZP6jCLWnMeFY8U/bSoOZymrHUGXFrp4DVMgcWqas3bezP+3pVVeoFADT+5N+a59TZW4553Os/xm2zPCyst7pzHJHnJpWHrETrgVj/g5e3eI9a0C+jlcYZCXomVq6UUr+8OHimu7x5MGcUF86wiyStCz6EXONiWeoTBsrjBf3J/TK21YKwtn9cv9Ll3FvGfbpzwT1/fXt50i8Hm0yieB5OMFzd7SyHM/VHCpccGrw/TTCdJwR+9cyR+I21vGYm2N045nIpC1VJK7G3jnP1JRnujcR/0XSotYqh24OKWNfdHCV+/tEqtNbZSbPUCnCYZ52Sa4SjFejvgcJLgWopZpbl9FtMNPCZJzjSrWW05zLOKKK8IPZt24JKXNXmlUUY+z1lWUWmhltrKYlIUJGVNFWlKrVGVottsOHll6IcOriMOl5O0oJ0ItFFps0x6shTklSYt5dr6f/7NHVY7Pps9n7O0oCg0rqOYpaIL6AYOZ1HONS1MqVoborzitd0Bb+1NKGo5LTmW4gd3RkvIL/RsMSdjQJRVrLYdbp/G+K5Ny7WhCcn+/bXOI4Epv+z6bTDV+iI68gr4PxtjfqKU6gI/Vkp9zxjz3hfw2L+V6+FC8dP7Y17Z7nPjaM7JPGpCe+WY/i/e3Hmkwzyd54zjnMCz8WyLoqp5a29M2aS6LAKIj6YZt84ifnDnnOsbXRxb0Qsc9sYJlZbiFLgWZ0nBdi9Aa8PhJCNwxQskrzTb/ZA/eOXKJ3xgnoY17/RDPjqJeDCKGbZ9hh3xVnn4tYHcJFFW8f0PT3l+vcP5PMNSFmlZ4Nnws/vj5cbxcKdkAWdRSVLU9EKPju/yys6A0yh7xOfjca7/4jEWnV6UV1xaaaEaky54ugf24ncnacFa26eohZFjW9APXG6fRTy33iFwHWqTkxSaCwPxh4nyqmFoOER5yTyt8N2PTyuWsri42sKxFb5rM81KsroGrYSlMhY3wNC1OI1zfnT3TOx5Q5eO71LqknXP52iaYbQmLSVyr7Y1W2HA8UzMrZK8AqXIipq0qKhBQrHTksSqmGUynPQsRVLVaC3/X7xqKnzPRhlNy9gEnkOcl7x3OMW1oOW5dHyHt/amZEXdJBZ5FE3QBMBGP6Df8rg3iuj6HleGbdE2NMrSUVTwvfcOuTRss9ULyauaKMq5ut7hLMrRaALX4R++uMn3b56yH6cNZTZY3hefl0f+28BP/9yF3BhzCBw2/z1XSr0P7AK/dYX8V4GjzdKSwLVASdCvAVzbJikq+qG79HI+nWckpeblnT5rnUcpgJNE1J4gsMOP7p7TCRwUiltnEe8fztjo+gSOxVqT5h7niklS0PUd6XB3e9w8iZilJW9eWnmkmD5OA3scvhD/bQn9naYFH57MUafwT17dIvSc5WtbrFEsBlJncUHgOniO3bAkKnZWgifaj3YDl3ujhI7vcDbPaA875FXNast7hKXx8Gf11t5kqcC8uia47iwtGDf+5gC+Y3Ea5U/0wL5zFqM1RFnN4VSyJ7uBy/4o5VrDHvId4d0fTjNqo1ltB4yTCteS7EzPtugHDnfOIlFQXuiL/0xecRrldIOctXaApRR1ZaiN4e55gsIwSyXQw1YK17JIi4o4r+Tca8TnvKw1SVmDAaUU2hiOZzkWCpTCdizirMQ0drieLXms2nMotcEYOTGcxTlZoRl2fDzXgkSGplkh720v9NjqhkzSmj6KRBscR+CzTiD4f5LXZLnG80Q3EBcV/9tvXeQ719aWwR+nUSYe5Wsy3ziczuREUtbN5trMfBqHxq9d3FpCaq6jGIQuNTJA/ndv7XN52OGl7V/Mv/zx9dvAT/9CMXKl1BXg68APnvC9/xb4bwEuXbr0RT7tr2U93B06lnrE0+QXUSI+vnqhy7sHU1ZaPtt9SQ3Kyopai8tdrY0IaBT83b0xP3tQ8Z1ra7y4Jdi5MYa/uHnKW3sTKm14d3/KVj8kKSrOE+H2uspintUMhz5lZZikJb3AIbQcfMfmeJYySUv6ocerO4Nlh/s0vva19Q6juFjCF73QpawM90cJz613eWW7zywtmKQVG71PwkOjpODaepsPjyMGLQ+D0O+macE/eH64LMwPd0qiUJxS1DVlrcjKail9f/yGW7zu/UkqkWcKfr43YRwXzLKKkwdjvn5xZTkYfjwMYbEOJinH05RByyUtBQJadNvio9Iir2pO51I4L660SAqR8F9caXEwTUV80/Hpt0BrTct3eO9wSidwmCYlZ/OcD48iXFtxZa3TODhmjXukhDy4rsU0E5fCbuByOkvRgGUUoWfjOhaWBUWp0drCQmMwFGVNVtgYI0Is37HpBi4rHY/jaUKcCx5uKWgHYi+blBV5DbUxaG2oDbRcC4yhHdhM04q60vie+Oo8GCcSPuE7tD2XrCipaovAt3lpu893rq0tvVAA/t1bByR5xYfHER3f5mSeN0Ea9fJ9952PAyYWUOQfvXNEL3A5mxfiotnxsRU8GCfsDILP5XT522Cq9YUVcqVUB/ifgP+TMWb2+PeNMf8a+NcA3/rWt8zj3/9NWk/qvBfdYa3N0tOk7dn81c0zfnZ/8sSC/iwd/Grb4539Kb5jM2iKoqXghc0uP30wxnMs7p3FeK4ECJ9HOafzA/75GztsD8LG0MosFZeCuxvGaYUN7I2yJulFOjjftXgx7HA4y5ow6ARbSdTcoOVy7zwiq8Iljv8k7HAhOFmsP79xwijOlwHCIB3caZQBLBOMFjeJbVlcWGmDUhzNpGg5Fry41cWxLVq+1TzGo1TAb19d4/sfCf3SsRSXmuCEx4vw4nVXNfQCm6SsOZllKEsxCFwmxrA/SZjnEq7wNA/sKKuwlKLfkoi7s3nG3jghqzTTtMKyFKezjJunEWjYWQ1xG5FKnJc4tuKNC/3ltdJqZhoH44yzeY5lWXQDh1FcAIqLqy1hoqQluSUMn9CxqI1GG/BdC200cV7Rch1KrTGlYTX0GCcFlmdhNBR1Lc6FlkXgWcyTmrQQfnqUV9S6Zp5KBFDggmPZy+COopKgCscSD3NlDMpSVLXhcJzSbTlklcTxtTzNRtdnnheYQtMJbH7nyiqBZzOJc/ptgX7+7396k1GcEXoOh9MMG0VRyyZS1ZpLqy1sC7KybnjwxSMBE4OWt5zN/OzBmKNpJlx932XVs1nvBk/Fs5/lHvyi7DP+c64vpJArpVykiP+/jDH/8xfxmP+51tMm2EkhXdaiiNcGDqcplYENz+ZP3jvmf3n7kN2VkDcuDJ6JlbFIT7m21mEcF0wbbu+3rw7JSs3+OKEfegSuzfGsoKhqXNsiLWv+/MYxl9c63D6T378ybDcUNsXxTJz3VkKXaSpDrrbvoDBM04oXNjq0ApeTeUZWGdY7LmvdgLbnME1yosZb42mc93cPJo/cGL3Q5YOjOesdXyCDec40K+j4MpR7bbe/hIfWuwHfvrrKWZRzYdCirvUyBWd30OJknuFYFv/9X94mLWqMYWne1W+5/M7VVfqhuxwOP+nGXLzuji94/9lcWB5lDZs9SU+yLYui0mxvBEs738cfpxNIgk9W1rRc6WZdy2JtENALHH54e8TeJKXj2VxqZgmjuGB7EOIoC9cWr5WFuvM8zvn//t0DfnD7jMB1GLQkkac2Bs8yHEwy2oFAHq5t4zsyJE3LirqWBKSseW2rLRfbVkSFllg+7eJ7NvOklKmVMaz1AmylKEpNVtW0fIeiqhglhqqWThxb4bsWSSKF1SARb5UGZQyuLcZZlgWlNvRDl7QwjGOxva0qTakN3UAgutN5TuhbDDs+6x2PDw6nTNISreGtB+dMkhKNRNzZluJkXrA/TvmXX9/FscQm17Y+mTS02NS1UTy/0UUpOZW5tvVUPPsXYaM8Plt5+DG+DLTEL4K1ooD/HnjfGPN/+/wv6T/veloXejLLSRuHwF7gcn8UU9TiN3LvPGKe1byy1SPKK947mPLWgwkvPzTkq7Vhf5Jy+yzmG41/yeK5rgzbHM8yMOBairvnEZayuLre4WCcyU3o2bQ8m6ys8FybB+OUTuBxabVFx3fYn6S8uNXlxc0eN472GUUFaVahLIVChChpWfPCZk+8xw1cW2tzOs+J84p7ZzGurWj7zlIu/Th2OM9K3tqfNrj6o8Il24KjacooKSkqMbCK8orRXDaGK2ttXt7uNdBBvoRnsipcpuC4jiKJasZJQi9wCR2b43nGe4dTrlUduoHzmUX84de9Mwi5cTRnkpS0PBvHYsmg2ZsklLXmYsOTf9INvjMIqWrDRydzxklBXupGQFWSFjX90MN3LfJSowCMFNjaQFxUPLfRWcr7HVuJYCeVeL9ZVnLeyPSl281wLZthx5WBtzasN3bFudaNUAsWM9njuOTSashrax1O5hkd3ybONdOsRBlJWaqqmgrB6JWSYnUeKbQuMcY0A15FWmoagS9ayz5QA54C21ZM44LYUdQGPjqNaTkO53FOVJTYymKz6/OtK0OivGwG0Q47g4BRVFDWBteyOZyljJOSpKjwHYu75wnbvYD1jjB79scp24OQ5ze7T/xcF/CHY0vnrpT8+/Kw/VQ8+/OyUb5MtMQvoiP/B8D/EXhbKfWz5mv/F2PMv/8CHvvXvp42we4Ewqd2LOkETuY5h5OUUmsmSYlriV+J59pkPd1kIAr/93CSNh4gBtuCK6tiPJQUNYPQ5aPTOdoYxnFBqTUnUc53rg7ZHYQcT3OmSckglOJb1LDdcdnpBWz2A1SDE6ZFzcEk4WiWU9ca11YkpYQ5hK6FMoZbpyKscG3FP39zl3vnMXvjFJbxDkIf7AZyWTyOHd4+E4fCa+sdorxq1J05J7Ocb18d8j/9ZI9RXDShzwKPWLbNOwdTLq4KU+RxeObxFJz79eIUIj+31QupjbymWhtanrOEaJ52Uy1ed8tzeGGzw9E05TQqeH23x/MbIru3EF/3h1/T4zf4Ip5v2Ba2xU8ejLl7FrPRDem2XWaZbOpzZDB9FhVs9sTL/R++uMYsKzmZZ+yPE87mOaDIas08rZk1mHelJUoPFHFRU9SG3UHA6bzEdRUtz6bKNZmRLhpLYI+8qjieZvQCh7SqsZRFUWscS4acpdaMk4r1nk9WaAaNaGmt4xE1ea8GwcLzUmiiCqEaKkB8MQFjZKiqDQfjDMuGQShOkC3HZp7XPJjE/PWtU7a6oisIHIu/+vCUoAnwnmUVs1z+XoVw32sUSSkNSsez2RpIctWTwingY/jDttRygL046TwNz/40NsqzdNpfJlriF8Fa+Uvks/+tWE+l1zUeE3IhjZnEJWCoNGAMgeewN8lY67q8sNnhZJby7sGM7X6LH9w5l6xCT26OBXc5yir2Jwkns4yO7zLY8IizirSsiPKSrPRZ7wS8sz9mnJastz22Bz5FWbM7CFGKJj2+pu3Z3DuPOZxmlBp2V1u4lsUkKWSjcWC7J1SwuKy5dx4T5RUtzxLBhTYcTlKOphkfnYg/yyiWzeZkltMJxPDo9d0BwNIpca3tcxZLnuWVYZuzmSTTDAKhQu6NEgaWDA4XHi1POwrPUglhbnkf3ySCmdZ8dDLn1Z3BM91UD2OeWVnzD66vMU1LNroBoWtzHhe4FktO+NNe073zmNBz2JukZGVMlJakleb+KKbbcvAda2k54FgWGz2f3UFIJ3DZHoR0Mof7owTLsqgNhK7iznkqcnxLURtDWtS4lsWg5TJoOewOWlwYtDieptw6S3AdjVMpnNrCGE1thODkWLL13jmLsJXF7iAkcSrcymKeVxgjG3Nda8q65vpGm1FcYVkL7F94/rWGQkNdGDxHMWgLXVI3MgSlED8XA7Ytat4ol8LsIJuH7wqGvjdJeOdgwqs7faZZiWtbZKUWdlHo0Qkc5nlJUQrMk+Y1qSehG8FjHjGL9XjBfeOCzKIWX2v51lPx7Kfdy0rxTJ32l4mW+JWy87H1aRPsQcvjuy9s8MaFAf/X//ABxazCscCxLY5nOWgoKulCNnshlTb8+P6IujYEgQySLq91sJXIlJWCv/ronLyo8VwZdgaehQ386QcnWOqUS6ttfvfaOu8dzhgnBa/s9LAti0prXCxWWx790ONgnJCUmqSssJXcrP3QwbEVgS8KzcvDDhvdgNN5xn9875ii1vzOlVWivFrKoF/f7XOelPybn+zx2u6AQegyjgtunUR0AimCN48j5nnFIJQh5LDt0fIcTuY5/bYEMqtmbw+cgrw2j6TAPO0o3AtdXNuSAmPEu3qSiBFUN5RO/OH1aTfVk/jkb+1NePdgwvE0Y6v/qNve469pkhT87P6EtY5P6LX56Dgi9BzaZc0oLmWDarlMkopuEGBb0PZc3j+as90PG0OxknGcc2VNUpVuHM+ZpBVlVdEOXDyllkNG17UJbbkdA9fGsRW2JUKaqoZKy8+FjvDO86LGGINBMUlzeR8MtAKHlZZLlFWkRc15XNL2bSZpxXrHY1aUeI5FqG00UBU1EgMNShlcW8npM6+ptKGqNEaD7YA2EJfgKOEqVGiMgbo2zcnO4CiLD0/mDAKXwzijNIZ5WhJ3hM8uUXyWPKcyjR+PR16Zp7KPnpYI9Fnrafeybaln6rS/TLTErwr5Y2vRzS1ueoPi+kbnEz9zbaPDVi/g3aMZSV6hMbRdi3lWcRbnXFpt8dx6hz/78ISWa6MNrPcCTmcS/DDNSoYdD200nm2htWZ/HBMVmroW06rnNjocTTOysubrl1ewlcK25fUcTqTj9x0L15bh3e4gIC0q2q5DVkkh0bWhHwpFbbsfcDRN8RxLfMsdi5/eH3N9s8truwMCVzD4qDl2/9mNE9KyIq00eSH0MINhpe2z2w9Ics3JLOIfvrghpkhIZFucVVi24myeMc1L6lpT1FJ4Po3adXWtzf444eZJxCTO8RyxOvVdUb6ezvNHotF+0Zuq1oZXdwa8uKl5e3/Cz/cmvLHbf6Id6p2zmNW2h1JwPs/pBi5dhFGEksCNeVqy1vU5mecM2y4t3+HvP7dGUdWMmk5yoxNQac0ozhjHObURY7JJXNIJHC4MQrSBkyhju+vT8ixGcc4P7o3JigKtFZWWjrjUQjdF1WgEPgmUK4O/QtKg6qxcWuZaCuq6ZqMjc4CDWUreZLCihMFjK/nbtZbXNc9KfMcm9GwwCqVglpUUlRTyRdGv5cfxLena87qmKkXZGU0q/DWbVuAwTQqKqmJ/VrPWCXAthaUsklLjOTL32emHZGX1VPbRp82ZPg2rfhob5ecPJs/UFHyZaIlfFfKnrMVN/zQ89vpGh++9d8ylQciVYZt39iaczgv6oZjsv3lR8Lw3L6yw1vG5cTRjf5zQ8YWFcG+UiMTbUsRljaUUo7hknpV0AvHMyEpNx7fwbEXS+GZEhUiz754nTLOK0FGsti3GScFWL+BomhOZkgurMig7mclAzXMcbh7PcRuaY8d3GHZ83tqbcusk4vmNDh+OYk7nOedxzk4v4MbxnHFSEro2l4ctzuICW4Frl0x9h37gsdn3mWUl/ZbLpdU22hj+7MYJx7OclZbLelfer6LUDec3fOpReNDy+L3r65zMMyZJgUHx3HqH5ze6pEXNnfOYTuD8UjfV40XhjQsr3D6NuHE84/kNwVofDvqYpSVX19p8eDzneJZhjKgRK635/evrvH8442iWMWi5SwOoV7b71HoxHC3JypqTecr4sCDKa1qey6yJPJPO16XX8phnBVfXOg07KWOclMRpQV7V+K6DNgrLAqMl4DpQwobJak1a1WgNGsE3LQNJLsNX31XU2nA4E65L27PIS1FdurZayuZtS4REdcNaETAetDGUNcvH10ghf9hZXVlgUFSVbrB5g63EsXCjK/DfStunrMWHvLI006Sg7UsnblmKkyjnd58bAjxiTHYwSbm4ItqKhXmWBH7I6fazBo9Pw8GftdP+MtESv5SF/FdNCXqWIcflYZs4rxhFNY5tcXm1xWY/4JWtHpallkOYV3Z6/ODOiHvjhKLUxLmE+O72Q+aZiGqqGgwis660wXMs1gchaGE/TJIClGKelzw4T/jp/REbHclMPJvnXFhpkxQ1UVHx0naXo1lOlleiRPQ9jDGstiWAVymLvVHKKzs9Dicpvm1xNM2YZCU7g5Bh2yUuKt4+mHE6a2h7WnPrNMJ1bNquOPO9cWHASstbKjSHHY9Xdnr86fvH9EKxq51lJXFjNXB9s4vnWJ95JB60PJ5b7y69zxer4ztklfCjf5mb6nG8sxsI3vpgnDxxiGpbCse22F1p8eP7YyptaLs2vdDnzpmETHz90spSmPXXt065eTzn/ijBsRVlVTNPS94difBmrSu2wdoYYkuxFrrM8pIHo4SsrLk2tOm3PEoNx7NUiqpRDZ/bIivEWREQuMMyKBRJJiyYpbNOLQVdWaL0LGpNnJVooxnHojRdaXuErs0oLcQ6t7ZQisb/xZAWUrRtQNnCYgEp4LaSzlw2dPl66NnYQF5rktzgeUrcM+OCuBBIxxihzRYNXVFnFUcm48E4ZbUt4rN392f0Ww6uJdmr46QgcOyleZZoFAzdwP3MweOnwTK/SKf9NFrib9r60hXyXwcl6LOGHAv+90tbXcZxQVpJF/KtnT5xKR4bnmOx1e/w1t4EbTTTWDycaw1fuzCg0pqfPchwbYfdFY+75wmh52BbEvzQ9Vz2RgmjOCf0HU7mGZ5js9LxeevBhP1JxitbXYZtjw+OZwxClyQ3/N71VdanKQ9GCUWlcWz42oVVpmnBX32U4xjDsOPxwdGM9W7AlbUQ21KM04JZUi67yUlakpcVtm1hSrP0U/Fci7TUGGCaVhxMJE3o5e0u46TgylqH9w5mdAOX7X6IZ9t8cDSj1pp5w0//rI330wbOz4KNPutjns4l/b0fOgSuTVKI4KqsNJv9kIurIdOk5M0LAx6MUlBwbb3DjcMpZ1HOmxc/fi2rLY8/vXHCesdnlkmKz2rHZ5ZWTNKYotT4nmK142HFBWeJhDBcXm0TeGIx+6O7I17Z7tHzXfotl8NJxiiW96wy4BgZNmoDRWmwlEEbKOuP9XU1DePEQJoLrTAuZEhqIQPLogbHhrqGsq6XuLXWhrI2WA1+YsyyOQearr8Z2lq2hW8LB76qDHWTgtT2HOaZGHg5lgxQzxMRIQWOjWNbKANRUzy1AW0kWWoUFVwctnhtp49tKSpteL9RwUZZtTSDW/j/fNqM5HEBX5TXODZLU7ZFp/1gnCzpr3fOYq6u8SvpuH/VzeeXrpD/KihBj7/JSvHEm/4syvnzGyccTTPWOj6v7Q6W7A1jRMxgWbDdFzbEuwdT9sYpg9Djxa0eaaG5fTZnnhVcHHbQBlCNFLo2y+l9UpTcG8XkhYQiW7YFRuHbFnvjBICsqLk/Ttno+vRDt/E5EbvaeVrJAM22qLRpjrkB/+S1bd7Zn3A0zVBGsd7xOZyJ2900yTmcJHiOTVbUOECuLPKqarIVLSpjiLOKrUGIrg0/fnAGSrHT9/ngcM44ydnqBTy30cFSCs+xGSc57x/MOZ0XXN/sMIqKZcr60y7kXwU2+fhjns5z3tmf4NmKlmvzk/uTpeFXO7CZHIvb4TQtaHsuz21I8ag1DNo+jm0tvdABVts+SV4RuTKzAInRW+/65GWNZStc20IbQ8sTafpaN+DCsE1VG05mOY6leP9wimNZ1BoWbsVKNSIdwFHgOha61BQfG1ziNJ3yAgJp5pEY5OsGwcwdpUjykqvDFusdj/eP5hgMgQ2lckjLmsCCpNB8PJ5uXgdgWWBZirquieuatufSCR2GbY95UdIPPNIqxlYWjiWUxqwocRxhuBggLiXo27IsvIY2OYpzVkKHqjacRTmXhx02uz6jpMBzLDQabWxe3Oo94kH/tBnJLC1xLMWHxxGBa9ML5JT4s/uTpQL76prc+2sd/zMprZ9n/Tqazy9dIf+iKUFPepOnaYmiZL2hqy1u+td2BU744GhOnJe8tN3nxa0eB5OUk3nG/VHCH7y8xXrXF8/qD094fr1L4IqM+N4opu053DqNsC3xg7YtoSMGrsV6t808KzmcCLfbVrA7CAk9m62+zzv7M86jnMARg/44r7mVRlgKbp3EKDRJqWl5NhhDXMrJoOvbrLQ85lnNt6+t8Vc3T6mMYZwWrIQeN49n3GuyOV/d7qMsKcKu2xyHS42lDGdzjdUNOI8L/s1P90DBd55box947I9T0rJmHBe8ujvg/nlMVtV8eDzDUzI0W2l5T7SNfXw9CZvc6neazfYXD6x+0mOeRTmvNZ7Y7x1MOZlnBJ4tflTGoqxqplnBWid4JNwD4GSWic1vUS2vjzvnMetd8R5Zawf0QoftbouDccJWP2BvlnE8TShqodtZCjzLcDTN2Or5OLZCl1oEQxbEeUXg29SVpjQGFxkwai0bdFk3wHWzqodML8R4TTrvRbFXgGlsaG0FH51GXFptcWk1xLEllk5XFb4tPHRtHnrw5vctRPGZ1xoLGgtaCw2cpQWuZVFpeGGjyyQRC4IoL3FtG88WnrzRQn1URjB/z1cN1VERFcKUSZY7lPDoH4ZD0qJuwrXLpQL0SWvhYfSwbYRSog14kikb/Op44r+O5/nSFfIvmhL0pDd5oxuQP4THLm76BWNi2PaImrizBRf5aJrRD11CzybKKz46mXPzOObuecqLGz0CTzFLSx6ME/bGCfO05Mpam5O5yMP/wXNr2LbFD2+PyCtNP3R5ZafPsO3xw7vn3Dyek+TiqlcZ6d5nqagoHdsir3UTuSbYrmcrur5DWht+/GDM7kqbwFX8+O6IvNb4jsVK6HE8y4hS4RMbbfjweEZR1RgjOH8v8IiVGEZVWlNWInZKioqW63I8y+kFHt3AxVJCQ5unJUWl+fGDCaMo58JKyGbPZ7Xtk5X1U21j4ZOno69dHADPxvv9rPUw3vnnN05YaXnU2vDuwYxpVtLzbarK0PIdtvsBx9OM3UHrE14xlgV/8Momo7jgwViG1leHHS6uhPyPP7pPXtX0wpC6MmhjsCwlql3HwrYtaq0JPLGu7RcVxgTS5TbGVr674FTXWK5F3xJBTlbKYLOo9GcKN7JaIJbFsgGjIdcG34Gyqjmc5rQ9i9CVf9LSgBHcfSHVV8jG4Nhgaggd2RBankPoSU5onFe0fZuW53Beata6PsOuT9f3+PB4Sj90KbTBsWuSXARNNQbPlo2z36hhi0pjjCF0JQx6mpW8utNbfnbX1jufiCpchGs/SRT2/Q9PWOv4GCOU1qyseWGz+0RTtsX6VfDEfx3P86Ur5F/0sftpb3JW1ks8dnHTL9bOIOSDo5K9ScoslbDiqtasd1r87MGYtNDMspKtfsD984S390f0Qg8QTu1618dxbG4cR2htuLTa4p2DGRs9n42+x+VhSFEbLq62OJtnovpMSgyK9Z7LJK0Yx5mYQoUORWWIspo4k4JsW4KVpkHNetdHGcV/+uCQXuiz2ff5r1/c5E/eP+a9gym1FuZDlFXYtnR8fuNJ3gs8Alex0nLZn6RLtZ/WkFeGbgB3T2PSoubiSggYBm2PP37viEkT4rzZ8ahrTVrUxEVFy7WXtrFPs8h9vGBnpeY8yqm0WYZptDznc3U0vdDldJ6zP0npBxIjFhca14aVtkfQUEkXKtcFFXWnH9L2bd7Znza2rLkoHT2bbhDwz9/Y5Y/ePeSDgznrPU84yy2X9DgiL0W5uYBK4qLm1omcqKxG2Xtx2MZWMIpsorKm71j0fI/a1OyNBLBOCv0pf5ksg1wDi7WAXFqO6B58x8axDJZtERc1dhMAUmrByFX90MDTEjzdAHkF3cCi6ztkpSYr5ARR1oY4r+iGHi3fFggtLrEbNWtalIBC2Ya6EqxeTj/yXvQDl9C18F0L37WpjeZC41u0WKO44PXd/iNN3CJc+0misDcvrXD/PGGWlXR8EbnZlnqiKdti/Sp44r+O5/nSFfLPSwl6uHgoBbdO55Q1y/SXbuB+4k1WCt7anyxj0nYGIZdX2/z4/pjStRm2XV7a7uHaFqNRwtEsY6cvhlB3jSGtDJNRQujZBL5D27UIXHmcvVHM0Szj+maPV7b7fHQScTxPsZTi5tGMThPp1Q8cYSA0Ks6iFM8PWykKXVNWgj8u/kHDOKlICpGRTxLoZQK1KKV4dbfPn31wzINRSlGJ61ypNaUyeMZgjPRi2kBlDIOWR1ZU1I2daaklazJwbNJcNoKzOMeyFOtdn9WWy2lUMs9LWp7D/VEMSnFlKCHCq23vE13299474uqw88jpKMoq/vzDE97Y7dMLPk5QemGzS1bWT/iEn21dXWvz43sjHKW4ut6mMjKM3ej6zNKCorQYtPylIvRbl4dLiO3KWoe9ccI4znlrfypYsFK8vN3jd64O+V9/4wL/6cYJV9c6nM4yDIZxUlBrLUwcDRpDx7fJKsO9ccqw5bK9EqKNZtL4sYRa0fU9amPwbQdNjkGKq2WxVF8ur9OH/v0QxL5crhK2iWUpPNcWx8G8ZJrVXB62aK3aHE5yQtcIs6TWVBqySoamntiUE2UaVEle1RS1fD2rNO3AIXQsHKVY63i8ezAFBRaKsnE6tBQUQNg4PxptSIqafujy2oUBKy0XrQ2uZX3CTfRgkpLkFXFRPxKJ+LTO9o0Lg08wkh5u+n5dPPFfx/N86Qo5/PKUoMc9xd/an5KVNa6tiLKSD45KLjfWqIs3eZIUzFLJK+wF4lfx870JF1ZCXt3pcbFJmxE/lTPeP5pzMEn5wJ2RljWeLarEvBRYYtjx8BybWiPhA8oiqwoOpykfHM4ojaTqhI3PxtFc0tjXugFbgc3d80RuIC22tJ3QZZRKco6lpHuCpnNGcFWUTPtnWc7xzOb//cN7fPvqEM+2mt9R9EIZdFVNiLDvSgpRPxQHwH7gUBtDz3No+w5ZmZKXoLA4T9KGDaHp+fJ9SSuq+PBohjbg2TansxQF/B++c5lRXHwC0qq1hE48LPpZhFAoZT0SPHDnLObV3f7nuoa2+yFJXtFveWwPQraaHM1JUvLiVo+2Z3MwychLzc4gZJwIDPbO/mTJwZ6lEprcDz32Jyl/9sGxRM4FLr5jkVeav7t7jmMp0lzL52PAd21s2+ZrW2I4NknFr0X8d+SEt9r2yEpxh3Qci15gM89qOTnVH8Mei7Uo5I8XcEsuAbSBtIRAGbJCNu9pWlABB5OEtudSaWFNlXmB61hQG4JGjGQ7FgNX0oYmjX+4BeRN9260XDv5OGG1JWHik6RooCWZ3WhtcG0j/HMDtqP4xsVVXtruopRw0t/Y7ZOVmu+9d8R2P2RnELLa9pbReo5tces04qf3x1xea/HSVo/H16Jhe9hi4nENw6+LJ/7reJ4vZSF/lrWQZH90Mn9EnbkoHjeOZgxCD0JDUWvanji6nUU5f/iaDFB+en/MT+6P8R2La2tdZk2oQNcXF75usDieJ9w4mjNPxUUwK4VDvNLyKesajAh8Wk22YpxXKGVhYYiLEs+y8GyLrIE4jDGczHM2uyFFZXhpq8fRNOXmacrhJKMXuGitCR25UdwGg7WUwrYMNaB0cyw20gk5SqiPaaEpqox39qacJyWz5vksTzaOzFKUqcazLGxHkZaGo2mOb1nkWuNbFp6r6QaWHKm1YRgE7A5CPjqNOI9ykZ/bhryq2ej5nEcF3dDl2lqbNy+uUDUF8HFIK3AUb+9PH+m4RolAVB+dzKm1aWwHLPYnKStt+f1flsq1MwgpKs1L233mzczjPC5wbVjvCnd5re0tI97Ssmaz63N/dM7zG10+PJ4x7ATM0hLXgjgTE7S/vHnGNy4PuH0a8c7+hINZjkJTarCaIIfK1BSVFKU0ryV707M5meW0XJt+4BJ4NistsW29cSR/v2tB6DoUVi2YdrM8Sxg1D59RHPUx40XSgWjYMJpZWkiXbAT3LiqFUrXEsGWSSmSMoahqdOM2KZmymij7+FkMLLv1qtacxxkd32GrFzAI5Z6qa3AtG9+xiYqq2VikiegELkfTDCzDd66ukRQ1f3bjhKSoCV0L11KsdXy+994R652A/UnC8VlMp2kWHoxTtvvhI8ESDzdsC3fLpKg+xYv8V88T/1U/z29lIZ8kBd+/ecreWPBPMLx3MGWaVvze82sARHlNr8E/80rz4lav8ceQY9rDWZF5UfPH7x8RpRUow4WVEK8JKv6LGyfMcrHm/OlpjO9atD2bcVqSlhUtz+bBKKHlOczzirSsWev4DAKbaVqQFYbrm20urITUGrQPd88jzuY5aanp+w61Jb1WlmvKWjNOctq+I1imsXBtg2OJd4WtBM+saYZUSgZoaVmRFcI+GbZ9Dmcp2hjWu36jxMto4+AohVGGQmsC12KWlPi2YJYSsiydaF4bQs9m3XcIHJtKS4xYYeDuWUTLs6mNInAU/ZbLq9s93ry0sjwKP8kidxQXGMUjeaSzTE5MO4OAWVpxMs84nkmAcpJX/PDOiB/fG/EHr2x9Il/0s9bDR96O73BxtcWwI7i279jkpaao9fIUMIpz5pklzo5G+NmOEsvYUSJy+rZnLQd5p1FOWmmUkdBoz266VwO6gF5HotrSsmboOXR8j61e3QQrSFF/aavHzeM5986Fi64VpIWEWriWsEhcCwLfYpp+jLUoGkm9EdqiBZQGwsAizfUj3uNZBbqu6QYOZSmfrbEEBmj7FmleMUlLilovC/tiEKoQd02tNVnZnBI8UY8WtaHtCQSYVTVZo0K1LXBNTWYptlyHaVZy/CBnuxdyMMk4mmVLRedff3ROUmqOGhV0x3eIfIdSS2j5iivePg/j5F8m18Ivav1WFvI7ZzHTtGTQhDIApKXmYBrxR+8e8caFAZYyTdK8DNDkZwQbf/hCsCzpEu+ex3R8l42ux92zmFFU0Gkc/v70xgnnUU5W1GwPQtKyxrVEVl/WYl27uxpwNMloew5lWZN7FhvdkNAT3+6i0uSV5sOTCEcpdgchZ3HB/iim7bsYYzhPclzLQiuwlUVeG9qehbJstgYhJ9OURnODhTANLAVJWckN39AJD6YZq22fK8OQ43lObcSatKg1s6zCtW06nkNeGdY6AZFXMk9rtgcBLU+8rd+8uMIHRzNmmTjleY5Qz07mOR3PZSX0mozMmu9eX6cTOHx4HOHYku35CYvc04jAc/j9nUFz8pE8Uq0F+ml5bnPC0RSVZp6VVBoR4KQF33vvmH/1zQu/8KzkSUfvhReH+JlL2JVnC+VzmpbsDlrcOZszinLiomKnH+LYLq6l+PAkYrXlklaSZdr2bGzbkkGvZ4OqyRvfkmmaE+cB612Pja5PXIpHjUbRC23A4ecPxiRlLScwT05IRcMwAjl1eZ5I5BcslcUGXvMxLbHZeyhyqeBO4z9ujBT40oCuG2xcG1qBC8aQFiWxhAmRl/WySXAaXMeyISvFPMtxZOPACJ6tkE3GNLRJx2qGrhrqhuieVZrQFVbJn7x/LHz8ouRkLhmfw7bHndM5cS4WD6/tDB4KlpBN/nEGyJfJtfCLWr+VhXyWlkIjCwUwjouKw2lK6FpUtSbKSuKiJq8KWp7F67sDkqJaDiB+/mDyyIVwOMsIXenOLWURuA4t3+E/vH3ENCno+jYYn+M65VbDuuiHrnhQ1JqNbsh6J2CeVQSO4jQqyQpNEQjLpDI1o7jgznlMpQ3Dtke/JZvQ0TQjbcKNS60bPq/iPMlRwMRWOJbFWsfHdixsLYbStgWDVkBR1+Rlja41ukb44Y4EAx9OU0otRSora0qtwYK2a1MZwzQuMbVQ6FxbfE9un0Zs9AO2+gEHU1HFGQxHM6FfWkbgkP2pBAWs2x7H85y1rr8MKX5vf8rxLGOzFyxpYUVdy0Az9NhGKJ3GGP7iwxO+vjvgcJoxyyqKWjNsexQ1n4iV+2UCAxZH79N5xjwr+fkDEUxVtWGjF/DiVo+PTua8fzjDsSxe2ely+yzBdWx6vkNaVYySgpZn0++GSwYGsJyD6EZqXzfTSQsapgecRzm90CGpDHvnMRdX21y93OGjk4iTqKRuim5tOdS6bnDkj5dtgWNZzHOJhrObAlssxER8TENcDEhrA74tNroAbvO9USqpTrYtsFlaGVAK3zZCe2yYKzYs23mtP5bsdzw5mRXaUBUlm52QeV7S7Dl4roXX+GIqJQ3OJM7pt2SeMIpKqtrQCR0Zwjcw5LzQDEKbSVzywzsjgVu6QRMs0f0EOeEXYYl8WRKAPmv9VhbyXujiNHLeoMlatBAZ9VZfuspJVhHlBZ7jL42TFrzkhy8EY6DlOuhabhRLKS6stqiqmh8dThl0fZSG87ggLUVqnJc1upnIayP49DQrQBvBNbV4N++PBed1bMXpTGxtd3oB2kiXow20PcXhtBK5tF440BlcR+IgklyIZtpkWFh0AoU2iqKuKOsKW1nUTYhu2YDnEntWM8s0hTZ4SkyZWp5DUpQczzNqI256cdunF7g8v95BoZYnnV7g0vZdbCvDwtD2HbFcNYphx8exwXMUcVYxTUuitOTSWote6GID7x/OWG35WBZ8/dIKvSb44OGVljXr3QDHtpaeJh3f5mf3Rww7/vLn8qpmteU90c/6SetJrnp744RO6vLG7oCqNryzP+Fq0WEci4zfd22+fW2V03lON3D4Z6/v8MO750RZxYfHcwBW2i7fbK9yNM04n+fM8lLsZrXBUuC7Dk6tsS2D0RqnkbiP44LTeQ7A3iQl8Bzavs1mz+f+aUwNtFzFJDXUtdy0C7jEtkUYtiiWtiWQwzytqWBpnqVQaK0f2QQCRyAztJHQkgoSI3az8UIeakwTviEznEUxtyzoBjaTpG5ojYqWL2Zsx7MMrQ3zomrgFRttjCQ7tVyMltNn13NwHRttFFFe4rmKwLNxm5NPrWvBt0vNLHV4ZavHKMn50b0Rv/f8OpeHLW6fRYyinDcvrSxx8mdliXyZEoA+a/1WFvKra232xolg5MYVJ7qqwrEsOoEryjcLNnstfvfacPlBP/z7iwuh7dm4CiaVbhJlhNlwNssYZaUYZNlW03GJ4ZUcgx0spQgcD9+3SYsa33G4P46ZpGIG5DniS+65FoPQIy00p3HO8+tdtIGbxzNOo4KiAs9pbEkbzNNUYFsy/a+aIWboSVq6No2wI6+xVI1jKVylMEY8rZNCIJ9ai4VuquVoPE0lEq1uCk9RSReZNaeVn+2NsRSkZckP7pxz/zym1nCaFGz3A+paBlRRUaELw2lcSsGyJMMxLWp2BwGV0eyNEv7m9hlXGm7vGxcGT7z5vvPckNun0fLrKy0JwRBrW7Ps6C+ttp+Zl/v40ftgkgojqRajqY1ewJW8w0/uj9FaZgprXZ+DSUpWafqBuBi+tNWjrDW95pp6ZafP/jhhf5xwHlf4lkVpy6zBGBnc5aW870UFgRKYKMpLxnHFhdUAozU3jyL2pwmerbBdi7qqySsZdFYPQRu2JRBb0TQZClF9tv2FnzgfKzBtizivyIqqoQIaHFvjWoq4Mni2wrUMtuPg25AVmlrr5rkMcVEvYRqLhVzfwrdriYXzBLvp+C6nKifXNTRwUtuTxqisDJdXQpKiYm+cMUpLOr7h0rBFnIt/fjdwifOKldDjJJaufKXt0QscjmYZr+32CD0H11bcOJqz2vb45uVVHNtaFmEQT5WHbagfL86TpOCP3jlikpaPUI/hy4mlf+kK+bOmYv/+9fUla6WsJcT32lqblZbPzdM5s6TgxSZjc8FWOJkJY+VhuhAKkU9rwywvOJqljKOC1Y7H1dUW47RkGheklfhHKAxrXTH+WW27fP/mGVlRE2UllTaczwtKRFk3oyZwhNc7z0pWWx6TpOKDozn90GaSliijGHYaauBDHZWtPh40pYVuvKclQssoTVULZUwpGvMi8b4AGfTqhuGQNl3wgpscZaJcDDwHZdUoo8ThMMrZXQmxULx/NOOFDdjuB9w5jYjyiqrWkv5eGbK6xrMsQtehshAxUF4xyyo2+yF75wmBa0sxUPCz+2PeuDB4KkWr38wtxom87//771zhB3dGnEYZqy2PYcfnTmMJACx9rZ92nTx+9I7yqqFiam4czYjyirMoZ9DymtOHQ1KKNHxvkrA7COmHHl+7IF472mhcW7zcx2nB5WGbKK8EV641WVHz/tGMNK+bTVY+P9koS/JKXC/vj1Khe2bCla5raIU23cBjlhRLm1kHYaDYTUdv0ZhlKWkmziLxk194h5eVJnRsLDkMstbxiPOCvDQU2tCyJXHItW0sx2Kt5UnKUG04i3Oy6lE4p2YxJNVYtqLl2LiW1SiaZ3JtacNq2218YwyWUvgOjBPhn19ZazGJSwZtFwtFLxBF9HovIM6FVOBZNkGgGHZ98T/XQs19fiCQzbevDR8Zlt88jvjee0cYY7i63uHV7Y+95h+vId+/ecq7B1OhYCYlJ7OcNy8NPpWX/pu8vlSF/BdNxf7uCxt894UN/uLDE949mNHyGrOdXDcfcM2No3kTWeZxFuePPN5iV/Zsm7/86IT9SUZe1Ay7gl8/t97hf3nnEGNEmVcZTVoY+qFmHBfcO5tjKcVq2+feuQTvLg7/C553VkFZVWB9rHI7iUrS0sFvbpCyqhsTKqBAHsMSbFgBaVFQ1osuyyxNkmwFgacoKoOFotCavPok/xjk5qwbZKNtKcpa0w1coqLiWqdFK3C5ut7hZ/fHXBm2ySvDPM+Z59Lx3z2P2egG2LZhOi1ITI3vOQxbPiOVE+diw3o4SSlqyRI9jTRuozL8o3eO2OoHS1n+w5/nk6hbFxu/loNJupTILz1ubp6iEArhk66Tx4/ejqU4nmf4ttin9gKXj45neK7Nasvl9mnEz/cm5JVw9dEQtwTTv7TaJsrl9FA3842TKhdnyWGbj05m3DyZ0/YcdG3wsJjEJVaTLOU073VVSQgERj4/1Qwry6rG8SU5CQXK1ihU49OjUNpgu4qqNDL4tBWVMVSVfP62Eh/1aVbJUNqWE5uyLZTWy03AUXKya2nDKC7ExsGyaAcOaeOBYhq+uCe0cLK8xlZgXANNExOXNaaW0wLGoBrhk+8InfEsylnrCHzZD13GsQjJjIGNnuDe/dBFG+iGLlH6Mdvq4mqbvNKstn1mebUMh5hnJT97MOb+KOHeWUw/dJmlU+Zpze8+N/yECvitvQl741TEdsgp7Hie8dHJnBe3er+RCUCftb5UhfyXpRUZA2/s9pcDs25g0w8DjmYZzzWmVlkpwQ2Pf+gLP5NvXBrytYuGv7s7AsSt7iwquLDS4mCcMis0vq3Y6EnK+0/ujdgaBLyx22eUlBhE+mzVD6kv+Vg6jRZHw8C1qSrNtC7o+i5RJnxfu2mvlAWukSxLhWGalsvBWY0cvbuuwnVsPNeiqDRVLQNJZQmP+bME3kVt8JRg8F3PldQiSrxm86vrxpJWKfK6JnQVtRbY452DCZU2tFzxFFnAKXvjlM1uKIpaxPZ0rRswiguirOJgmvLyVg/HUeyNE37/+vozpL/I99c6/kN2pRWHk5Rh1+Py8GPtwMPXyeMCjUvDFuNEYCXfEXsC35Xgg/Oo4K8+OsV3HTq+TVHKYFcD/4+/vsPuaou/d2WVV3f6jGL52UlaELo2e2OxSS0rQ+BYOB2PoqyZ5SVo8F0ZOtdGLTfgotKNgEoRBqKInKQFW72QV3f7jOKM87hgf5JhGjylVoauLza0caFRpinOgFZSpBciWKOgqGTDQH5daK8GlNJkWvjdvqUa7rkC18a1hfGktWDdIvzVdAOPvK6bsAdDaNu4vtUkD9UYauyi4lK/RTu0ubhqcW29zZ+8f4xqaJ4YQ1Fp2r2ACjlhTPMST1lcXA3ldKMs1joCgViWBLssTlUHk5Szec4kLrBtxUojpLp1GrHe9Xjz4sojXfZHJ3P6gehA7p1F+I6i7dl8dBJxcbX1G5kA9FnrS1XIf1la0WKQthiYzbOSn+9NKCuNZyuysmom4O1P+I4fTTN+cn/MIHRIS4PT0AotZfj5g4mo1TAMOz790MGzxLp10HLpBS4PxsKLLSuzhBIq83HSysMCjmlWkxTCTMhqKOuCauGdUgv04dqKti+ez5OyXLIHbAtCi0bkYeh5NpaCcXPELyrxpF4EA8CTlYALL+u6NlS1ZqsfkFUV2/0Of3vrjDSv2E9KqlqqyGrbJS2FX/zB0QxHWWilqY0MrHq+gzYWr11Y4WsXetw6mTNLNW3fJs5LzqOCSVqwqYKl4GRvnPLW3oTvvrDxTNeE2JXOG7tSl7tVxN3TiNd2Bkvc8/Hr5PEuf56JMdgsq+j4Nt++OmRvkvD2/oSVtgsoskKz2vGZpjnnkXC8d/shf3PrnLf3prx+oU/Ls/Eci9MoZ2+cch7nBK5NURuyqqYXuGy0DbPmc1FIfmRWyWBPLbrxpi3vNsEegWsx7Hji521ZTfOhxYPHGC6utNgfJ4zTkrz8mIroNbxD19Liw6NkjhJ6LklUCKvFagy1ao3nys9sr7Q4TwraroMdKBGrFTW2MvieiLJcx2Kz67M/zmh5tpheAW3fwXMsokLR8W1macUsrzAK3tjtcH+UUBuI45KWb/CaGUrRHAmNgufXxAl0kpb4tmJnJWSallwatri23qEfuvz0/pgoq/jgcMZb+xMxgmsF1Fo+77LW3B+ln+iyTXMmbXsul9c6nM0llcmxrC/loBO+ZIX802hFn4adP36Uti3FhZUQ31GcxTnDts/lYZtu4HIyyziLcv7dWwccTlPWOwFtz+LBJMVCEXoSx9ULPdq+TVHKBL7tSifebdn4boDC8GCckpXVcnhYGbEX9flYGPLw0ojt6IJvm9fgO8KUWbBVVts+tmWx0vJ4MIpFLajkd1qeQ8t1GGclcVFiYeHaiqoyy0GV74CuWAYNPO5W4jTD07IW8yzHVmyEAeMkZ5J87BpnCw+SSWooSoPvWqSVxkbERCttyYNMq5qu0vz9a0P+8LUt/uqjM+6eRZzORdS0N0pxLIusNCRlTdtz6BuXj07mj6SlP20e0gtd3t1f2JUuTmoutm1xMEl5ceuzvavhY5Xnw9eWbSm+f/OUqpZueWsQUFSaNNdM0oJJv6TWIpTCwA9unROXNVdWW6y0XH50Z8w0LWh5LV7c6nE8TYmKitB1ME16j6413cDBKRV5WWOa2UdlEAtbx2DbYjzlOzZQNjx6cSjcHQSUtWZeVJRaYzdSzsCRKDejBTpYZGwWpcGyDeTlEt5bCVziTE52xhi6bZ9OKIKzWVqCsnEsi5YrFgGaCk87DFpywrAsESn1Q4HiqsaJ03EUU63p+y6v7vTohi73xwkns6JhwzSBzkWFmyt0ahiEHv/VixuUtWGSFpxHOdOsZKMb8t0X1nBsi9unEV+/tNK4IR5R1DKjEEhGzLs8x8JzLMr6k4yV6xsd3juYChvHtdnsBQSuxSs7/S9lEYdH4/d+49fVtfaS722MWf73woCpqDQrLW+Z5zdpOrDFUXphS+s5Fr9/fZ3/zTcu8sJml4urLTq+w+3TiD9+74g7ZzH/4Z0D/urmKf/+7QNcu8EJK815nPP8lnBXW66D5Vj0AofAk+5Eaxle3jlPGcU5AmzK8XVh9q+N/LfdHHsfXsIX/xh+qSrISoOjBC7R2pCW8nc7jkOTs0tdy9BukktWZF7IE1q2Ra3A92SgZFnWsgV/vIgrpINb6dgM2w5bXZ+u73JpJSQpNbatiPKa0LMIG+/uWVoT+rY4+1U1ugmUiHLNStuj4zpc3+gsh8g7gxDXsQh9h7KWGLPVlsdKy+Nsni3fhaTQn/qZPnxNjOJCjvtGrF47vsNqy+M8zh+5Th4P9/3Ma6us2O4FXBiErHc88lJz82QuYRO+jdbSEdu2DPo+OJoRpSUPxikXhx2ub3V4dWfQBCzAxWGLbuBIULdn0/VsAs/BUhZpUbPV85dB3QCeK3h3L3CZ5xX/80/3eXtvyjguJLtSwYWVFrU2HE0ystJQmxpHCTxmKwm3KJphv+uIgKfSEJcGAwRKYDrbsVnvh2z1Q17fGfD8eqdh64jUfpJIhqiuoe05bPd92q7AJ0GjS3BdG1cpCi3S/iSrlurYW6cxUVZyFhXUGGaNck3cJQyTRE5vIAV8vefz5sUVnltv89xah+1BQC/0ltDqnbO4cUMc8E9e2+a59bbMIGzhqOdVTa0Nl4etT3TZl5swjw8OZ7x7MGGelVxYaT3itPhlW1+qjvxp5jPPgp0/aWA2SYolTSkpag7HKZvdgFGaczDOm4IlDonSJRmmScl646tdaUNgW7Rdj8NpTuDZdDybs6ggLsrlgCctNJ6j2GjJ8THKKlzHLCljuno6bl0jF3tRa9JKbszNfoA2ikEI51HdyOahrCS6q+NZGISL3gtcPMtikpQUql52+48vBwhcOXImuRYaWZSz2jbcG2tCx6KsFG3PLB30Au0winKirKQUyJyqEpw/LytmWc2VYchLW/1HTkff//CUi4MWgWtT1jUPzmO2BwFJLl7ld89jtDYMQpdr652Gx/zkecig5fH8ZoefP5gQ5zUrLZfXdwfYluIsyp/ZpOhJ11Y/dPn2tTV+cPuMXuCyP02Wc4FX1vtMk5JKayZ7JaEnAQt5WXO38dtZhCz3QofQU8zSmq1ewOW1DpdXWvzd3TOO5wVlw3gqa8OFlTYPRjGurQhcW8zLnCbp3hjyuiYvjGwkxnB/lKCUoh+4ZFXFNLVwHYWrNFluKIy4Hirk+nBs8O2P5zKWDVlV4yjFeZQzbPmcxymdoENlDB3PZpaJd/5pnGO3PEZpyTSrCHyHF7Z6jJOCru/w/tF8OeC0LAtlDP3AwbKkgfroVHE+y8CS92UR+tzxZHOzbKF+WsDZPKM97DBNhdKqFEtGUduzafkScr7S8lBK8d3rG3zv/UP2xxl5UfHahQGv7PT4w9e2P0E7vH0a8fJ2n81ezigpsCyL1x9zWvyyrS9VIYcnF+RZOnkqdv40yOVhBsyLmz3+4uYp98dy9NXAoO1Ra01c1qR5zbAToIxmre1x4zgiKWo8RzHsBOS15vlNZ5mUk+QVXd9FK6graHsiBrEtaDkuu/2Qbmhz6zQRwyVbNwrPT/69hiaqS4gBWLZCaQVK/DJc26IqheNrml9YHMO7vsNKy+U0KpbwS9kwsWwAJSwG17bIGnzTtS1ats1K22WcSlpRr+WJn0gsQ7zQd6gqTa5rlCWOer4lwQkoUfZ5luCqnm1xPMsfMTXqBDY3jme4tmDag47P3igFpZlmBVv9cNlx/uzBmLYvwqzFDbzgCi+M0d7em5IUNVeGLVqew62ziAsr4fIU8MteW39+44SrazI3eXt/woMJvLQpLBUJNBY20PE8Y7XjE7oOcSn5jwsmxFrL4/KwzUrbl0LZ8TiPcg4mCVFes90PSIv6/9/em/5Icub5fZ8n7oiMvCvr7uquPtgkh8cMh7OzMzurtbS71q4kS4K8ECRbNiS9EARYtmwYECzrHzAgw7AACQYEWX4jwTKwXq8F6xiN1qvdtaSZ3eFwyCE5bDbZd1XXmXfGHfH4xROZXdXsJvsostnN/ALETDeLWRGREb94nt/ve2AYKYMgxncNlUhkqx64LFSkW5pLKo5avXu2mvNEWcHuKGS56tLyTQ4mBW1NDR+DpEDoYBalPF5X7TJZgDA1DCGRomSwILB1SHKB0GAS51zZG1FxDGUfkast5DjKqbs65zsVBEpg1wsSoiRnremyuVChKCSX90fUy/vfNnU0IXBNg71hoGyPM/VddScp3UlKrOcsVR00oawYBnFKnEsWqxk5BQW6apeUjKJhmDCIUp5bqs5arb5jsLngK99/Kdlc8KnYHy9vRxd9U5fNIMnoTpKH9ur5IuFECrkQ4leAv4OqD/9ASvk/nMTnPiju1zufJBm//satWaJIlisFX901ubw3xjY0Fn2HrX5InEo6VYvtMs+yXbUYhQXdccJC1VZFVNORsmB3GOKaakUwKlsZVdtE1wS+a1J3lUHWIEwZpCmWJpjEKYYQuLaBRHLtIMDUS8/tOFUPy30wbbOYgMwlu5MQ1zAYRumUfIDGkZW2pPTIRnmpFEr2nOdF6S9zx9a0KJThUlEonw0bQZLn2IZDw1Ey/jCLcXVBmBZkuRrYGYaGlgksTSPXCuqesqGdJBm6ENR9i8Wqhe8YbPUm/J3f+oB2xUZKqR4YqXjINw8DVmqO4mrHyjvjKyt1hlHKIEzZG0a4tsGFTnX2AE/bK2/e6LHVD2cy++1ByIJvz9wpH3eFddSH/nS7Qqdqz1ooUVKQZpkaYLoGLU9R5kxNyfK744hJlGNpKd86t4CpC5KsYG8UYWgal3ZHLFVdskJ5feukuJbqVbc9m2GSUrV1XNOmFyQcotwVqxZMskJFwZk6jmkxTlQmaMM1GYUZFdsiy8E2JHEpJnAtjTgpyKSKmvNMZU9M6RNeq1g0DR3f1svwEKkyYoMEWSi/dSEkoyinYheYQsO1lFe+8lQRKlmp5qiVuyjFS4VUrSRTrcwvLFXZH8f4loVtmuT5RHnW2wYrDYfTLY8bvQlpnhNlGd862+ajvTGeeYdRJBFslkV3yhG/vDssIxdjlmo2uZS45sfDR55VH5bHLuRCCB34e8AvA7eAPxBC/FMp5XuP+9kPintJcvdGEdcPJmilF/ftwZBJlJIUkrpjKke9ussPrh6y2nBpeAZZnjMK1bCmGyg70XGcqZQdVLiDYeicDmP6oWI4VCydpmsxjlM+2InJpJJI26XCLsoKBpOEOJPUXJ2qozynw0S1PaqOYpTEaTbrWQtU/1zKO0IPszSVLgrV24yyfNZ3l4Ijyj41yIyyHF3X1KCuKKiYBoeRUnYi1c9YutriDssIOYEkTjMkgq3+RKn5MvBtgVUOEpOiwCr7u0GsDLZavk2QKk8XUR7HIMjIZcw4VgZiL6zWmOgpe+OY3aEy7doeRDMa3C+9sMwHu2MsHYZRymrD5ae3h2rHkcljD/DVgwmgXoJZrhSuw1ANlYM445W1Otndk+QHxHQHt90PubI/JiskS1WbJCvY6ofcPAzwHIPNdoUwkRwGEV7piePZOi+sNPlob4wQguWag2cbvHWrz0rDZa3h8PJanfduDzB1wSRK2Z/EJLnysun4NrquFhHJsGCp5rI7jAnSTFE1xxFBmpEUBZqmYejKDmHKvfbKfrvv6Ir9pKlWWJgW6nqURi1CSqSmZPta2U/uTpLy3sxmrpiHEyVICzM1hNUQaAjGYYZtqXtLCkHLd3h+pc6VgzG3egEt12IcZwR5NvMG6k4SbENRA9eaXrmSL3BtHV0XLFVt2hWb/VHMSt3l66ebM9bSP3t7+xij6HS7OhPunO34/NZPd/gXP9lB1wTnOz41x+LK/oRRlLK54M92cPD5pQJ93jiJFfnPAB9KKa8ACCH+CfCngM+tkN+vv5kVzFJsHFPn/dtDJJK6bWKZOjd7gQpcCBN82+SNbh9TF3THCZM0x7cMmhWL7WGk/LirNqahjLeiNKfumDQrJoaumCxV18DRdXZHMf00wTY1glipHS1DCYM0DYbjDENXvVZyqDvKZlbmd5zrOEITVHJsNfhCU25vlq7hGAX9QPmw6CjpflxmNea5GvxVLJ0gU0HMUKa5lMIOTSgJfpqB5Qgc3SAvVOjA4URxg01DDc56E3U+SNgdRASOia1rrNRtNE1jux9RdQT9IEeWxVmTkv1RhGUa7I2U6CNKciVcEhrbvaDM8Eyo2AYHY+XuWHVNVhsucZbTmyg2xamWy8Xl6jHlXdOz0ITkw70JVduk4Zr0g5S3twazrMeHwVTxNwhTru1PkKKksRVSnZMUXFyukRUFgyij7li8WK9j6gI0gcwlt3oTklwlMW20fQZhgq5pjKMUUQZp54XEtQyCRJmFgUDXNJJc8upyFd3Q2B2qot2p2Xi2ck9My4KrBpcC37KQSGxDw7d01XYT4Bo63zrX5o1rfQZpQsUxmEQ5Qi+IUonn6mhFodwMpWq7BLFKmbItHdvQmMQ5UZqUL/0CzzAIC4ljKFqfnmtlzCH0w5gfXFG73k7N4cyCzr95f08ponNl0eCYBusNj+1+yM+eXeD8orJAMHWNsws+661KaS+hVMBv3+xTdVQr9F6MoiDJEAKu7I+xDJ31pkeaK4pk1YOqraw5OtFxVefnlQr0eeMkCvkacPPIn28B37z7h4QQfwX4KwAbGxsn8GuPQyn27siyP9wfM4kTNNSw6HAQYZQr1DjPOeV6vHkQsD8My7g3ySBUtCXXMvBK69hx6RJkuwLH1LiyHzCK1aolSDKGYUKO8kmuOyqWq+Ya9GQ5mRfg2QJNaOwMAnRNxzYFRS5m3imurWMaAssQ2IZWSt5L17uiZLwIxWGvlFtfQxOcXvB5++aAJJPHBpi6WnjhWwadqsWVw3TGGY/zO14ZU1dDS1c9bd816AcZlgl5UnqaZ6CJgkwqnrEoeetZVlB1LfqTFDSBjqAXxKSZekFYomAcZ+QShMjoBQlprlR54yjlVjfEdwzyomBnGPEH17q8vFZnbxRzGCi2iW3otH3B6XZlpjo9unoKZyoX9T9pXuBaOuJjutUHw1Tx13AtNCEwdZXb2anafP10ix/KQwSC55arXNoZ4pQv1INJwmbb451bA3pBSt01SLKcf3/1ECElzy356EJnEKb8dGfIrW6AkHCmpdgTvUnEMJK4ZsH+JEEWBWGaUS9MPFPndNPjpztDxk5eKhstsmIaYqxe3LeHMec6Pn/+Zza4PYjJi4KqOyJINfJM7UINw8LWU0xDMIoybFMrKYkFiYQ0k8RFRqdiqUKJahN5jhJcVXUdTSgWja4p7r0KzBBcDkYs+jbnFn2CJOdnz7XpT1IGUYquCRqeScuz0DSPUaTCmn3b4JX1Omc7Pq2KwyTJuLY/IkoylpaqdMcJb1zvqr54lLFZtremxVfXxGxX5ts6o0gxxHvjiLbvEMU5vnO8xH0eaT1PAp/bsFNK+feBvw/w+uuvP9qT9gm4W75vG1rpVGdiGUqVpiMohJI4741iWhWTG90JYZyRFpLFqs3eOMY1dS4u1RgnGTuDkIZjcnsU8dHhhCwvqJThxHFekGUFjmny/JJPnBVcO5yw2nQxdcFWpnofcUkBEyXVKorVltQxDZqOzs4wRkOAUH3UUt1MAbi22gIL1GoLIfB0A9dR+YrIO+yXMlNWeW4ASaG211NFpSygyFXcmDHN/0JgGMyogXkxYW+YzUqhRA1hBarnSfm7pC7pTxIcSydIVK8+zZQlgJQQJ4qvXC8DMHrjhN5E+c0Mg5i259CoWHy4G+PZyizpRjfg3KLP3lCFR5xb9BnHynhJSrhyMGat9AwH1SMP04LzixW2hxHDIOG55RrnO/4jtVamij/H1HEttTvxbYOb3YDXNlozmXzVMbm4XFOKwklMw7XK/q7Pv79ywLWDgIZnoWspgyBjux+xXHfZGUTolP1nUycpCgwdmp5NlheEecFHe2MWazaGrpMUkv1RzChKOdWsKK8bIbh+OGGrHxIkOaahzyiXizWbX3pRpVtdPZgwijKuHIxndEVZttOGUabsCEr15iS+swzQUAPvSZwhNFhsOYpdguRMu8LeMGIYZriWwUGQoAEd3yLJVZDGSsPFMXXOLlSQC4KaYyLKtKFhlHFhscKl3SFnOyqb1TF1upOYKM3YHyr7YNs0qLsWN7oBSVZw4zDA0AW/9+E+X1mtcX6xesw33rdVy3KaQDQIM1oVON3xWG24H/ueP69UoM8TJ1HIt4BTR/68Xv7d54q7KYhnF3wu3R4hUb7HWSYxDUHbddXWV9fYG2VUHZOvrjd5e6unhn26WmXFacEwSDgcqu1jxdYJ44I0z6k5Gos1FcxrOGoFvdWPcAydZsVGFjCKcsJMydnjcjlZFCjVna62daYuiKaKP9fEEIJeqAITTEMVTsvQSBLlTZKVjoW5LhlnKcmRqK8CtXLXdcVDN3WwdR1dlDanqBt4oudICpJU0vQsnlv2uXE4YRhltNMcz1TcZpJsNgydpg1Nf5uGomIamiAvJGkOji7wK0rxN45SbFNxxdWMIkfTpJKnBymHk5R2xcEzdRZrNrrQkLIgSAu+dqrJBzvqe3ttozWLYBtFKQUc4wR/baPJ3jCmHyacW/BnDnZBks2S0h8GU8UfQKdqc707ASlnvPKGp1wOgySbJQrZpkbdNXn7Zp+lmkPbV0XZt01u9jJ0Q9EgD4OEjaaHayo631rToztOcE2Des3EMTXeuTXEs5SJk2PpDMOEumtx9XCCZahgiwVfeaRrwOX9CQuuyeJilVbFZBRlM0XstFB1xwlb/YAbXSVOm8Q5P90dEqbFLORYE3dM01zTwDA0KkInSAtaVZuKZRDEGTuDmDMLFYRQQ8OFqsX1w7B081SCoWGodiTv3R6xWLUV0aDiEGcFvq1j6Bpf22jxtdJ29s0bPXzboDuJ2RmGVCyDb262VWCJVL4vWSF5Za1BI0wYhOmMeTbtd682XJUcNYioljOrqmuwUnc/UTvwLOEkCvkfABeEEJuoAv7ngP/kBD73oXD3NLrqmHznwgL/5tIenq3z+maTXqlMPBzFTJKUKMnZaLl0ajbnE1/Z3tYtDkYh3SBifxRj26ov3PQcxuGkdH2TrJQrt/1hqIQXRQ6ZZBymFJpGP0yomDqpXir4johw9EL1dy1T0HBsHEPQHWdolo5r5tiOQYHimxfF1CZX4NnKOyXNc1zdIEO1PKZhAgUgczUYVfa6gu1+RFA6ZclCEmYZnYqNdAQrdWUqhYSf7gzZHUcqKCAvZlmP98qpL4AgBZ0CQ1OsCzTBcsNlEqYMUS+nimOW7AedSZSpAuHoWJoyR6rYOjXXpOPbaGjYpqDqKC95ZeYxXf2q4myV6qc3b/RmdNKpze00Kf1oQMjD4pjiz9JZrtlcO5iwVHewDI3vXOgAzLbl02tkG0odOIkzbnUDTrc9giRnEKYcTFI6FYvtfsjVfMwwyjnVdJX3u5khZcGFRZ9LO2M0HSaJmk0YmskgSAhTFTjsmTqGodHwTLYHIVGe8+qpOqdbyk8mydTi5AdXu9zuR+yP1PAVVJjIesMlKXK2eyFfXa9zaWfM9YMxaOBaGkmZ8qMJNVcyhaBi6tQdg07NYbXm8M62WkknqeSjvT11jwnFZmp5NllRcDBWPXnHUKKxD3fHnGrdobZu94Nj3uHTNoehC17baLHg2yzWHLavR6oHj0qvEkKUASLxjIky7Xd7lsGr6w3e1YZcORiz3rT5ymqNV55ybvjD4LELuZQyE0L8NeC7qBf7P5RSvvvYR3YXPs2+9l7T6IZn8asvr1B1TIZhytmyUfzh3ghL19ns+DMflKbvEKYqXWcY6cSpxLV1PFNXk/mKwf5IJ0gKao7O+UWf92+PsAwdz1K0woPxlM2i6FSGIYgyiaHrpcWsxNFVPmSS57imhWVqrLoeWR6o1baU9MOYJCvl8nmGa+nUXGXnGmc5ozhVx1yx0cKkHJqpc1MrcY0cJZO2TeU8N2U3OqbOKM5plkEMHyQTfFvnhZUqN3shO+MQTWhoQq22DXFvfjuUhl8S7PJnR6XvScU0SYui5L1T7gZM1pouG60K1w8nXD8M4NaAi0tV+kFCkkteXK7x7z46IMlyFqs2QZIdG0gt1/2PuV9e2R9ztuPTnSSznudy3S/vlf5Dpb68st5gGKb0g5RBmKi+cNmPf/NGl1GUzux21d/1sA0dzzJYa3qzvnlvkrBQdZS3eUWttnVd0J2o65MWyofc0DXSQmXGvrhWIytyPtyfULWVQRcYXNsP2Fjw0HWNlbpL07OIs5w3rqUqjLgXqOIvlD3th7eHdCoWK3WXa4cTfnitS9O30UE5IMqCV9ebvLSq+v/9IKYo1C7R1NXANM0kmiFYrDp4ls7Nw4B3bg1ouCaOoRMlKRVbZxRlRGlKmGSM4wxNU2yXqm2wXPdY8FVw9eEkxjJ01houX1mtHfMOP9rmmK7QgySjYmlc3ktxTY2KZXD9cEw/yPAdne1+eOS/VS+CKM35xmaLP/uNU1+a4n0UJ9Ijl1L+c+Cfn8Rn3QsPYl97v2n0vUxwpiEGRUG5dY8wdcGLKzX2xjErZQQZQq1mZC7pRykNT/V7VxouWV6QFDn9MMUqpfPDIEUIJb92yi20JgTIsqghaPsWaa4EQHYp74/SgvWmxzvbfZJSn2+WBTDLgSTHsTLCRA2oTKER5ClCaFi6TlDKsqeRW46j0/ErfLSnkmtUNKlUYh5DL6lgGmEiSQvFM19veqSZ+plRlGFoAs0QaMCwzA2bNiumHVWJMlzKpMQrTYqqjgVIGp7LesvjRjdgEmW0qzbNisXhJMHQBM8vVbEsxdwppGTBt4gypZY1NMH+KEbTVD9XIDm/WOX64b0VvN1Jckwk9KipLw3P4jsXOsfoh6auLARA8O72kEGoinl3kvBvL++zVHNYa3qzvrkQ8NbNPgWwVHWo2Drv3h5hCA3JdPgmGIaK7vfCQg1DF7Q85dhXsXUqthLiBHGurBVsa9ZuGEYZHd/mpbU6W4MQRzeouTq2bvDjW32arsVvX1KWy0GcUrENhlHGRsvjcByxWLW51Qt4abXOzW5ImKRYuo1nl/L6PCfNlfHV+Y5q/2S5ZLXhsNrweOtWn9Ol4+PvX+1yMElYqDhomihFZQLL0JBILi7XeG6pyh9c7/LSav3YIgvurdKdFmbPNiiKgkGYcbOr/OtrrolnWtweRFw/VBL96cLubuvjLxueCmXng0vwH2wa3fAsznZ8vv/RoUrWMTUcy8BzDL61pCLNfv/aIf0g5WAU06o4nG5V2GhWuNmb8MJKnawoZu6AaS7Z6k4QmkbVsRgESnadS0AqS9esUENP09Q4v1QlKwo0NOUVrqn09YppkOYSKQWFVEVNE6rnPQxTXFO9IIIsI8sUXQxUb1MW4FlqJZ2VDoJJhkoFAqW4zCVhosyMKrZOkhekiZKS3+gFdEp14jBUw19NU74pdqYoXEIILFNjHJWxdyj2TCYluVSWunGWU7MNznYq+I7B7lAjM3UqpaFXfxySFDlLbYevn25xcbmmAoaTHFNXrn62obEzDPnhtR7/4YvLM6bCD64c8PXTrWPf5d1ijsdNUD+6QuxNlIXD1IxLCMHtfsTucJeX1+qzdsqlnSEXl2uqMK8oFsblvTFZqbBteianGi7745iDSYIhlIOlEALTENweRKS5Es5cXKyS5AW3ehGebdDxLYSQbPVDLi5XATX0LaSk49tUbANTaOiGGpR/dDDGtQwVXJEXdLsBlqnRqarV8c1uQN1NcUwdz9ZZa3gEac5STZlv9YOEXMIvv7DEtcOAfqR2f5O0oJACTZN0g5j1ZoWff67DmW5Ab5JwsxfQ9iyiNOej/THuIOJc6VIokDPv8Pt9b3df/83S/fDfvL+PEAJDF2Tl4mqxavO993Z4ea1xz5f1s5LD+TB4Kgr5g6qx7jeNvvuLbVUsruyPObNQYb3pcvVwQnccs1J3GIUZvSCm5VrcOJgQpAX9yYibvQmOofPHX16hFyb88FqfOJW0Koay9jR1dE1jECQYusYgjEkyRe8rpA5C0PEdPFNN6s+0a4zijDhTA8ZrhwGmpaMlGWmufJ8NTQ0B9dJWtlAfg2+ZUKi0lbyUXwtUYIZrK9lGUmrxs0L1zF3boMgVh9wUSkCjCYEmBLqAOFGh1Mq9rowR05Ujn2PrpFkx6x2nqUpqqbqqrTB14zMNFavlWTpX9gNeP9PkG6db7I5irh+MZwZJjqHjWfqMUZDmBXvDiHOL1Vmgcpyq1kMvSFisqZzVlm9z9WDCq6fufMd3izk+6V55mAd8GKakuXpdXdoZsDOMZtFyzy/XZ+2UH9/scThOuNUL2FzwaXjmrJee55LrhxMGQYqpCSqWQZRknF/yKQrJtW7A2cLjl55fmiXZHIyVQGij5dKsKGGNUYY2XN4dEqQFAjkzy/pgd4xjChquzTjOGUUpnqmTFVJlvBaQpgXbvYgoy5WvixB8sDtkEKQYukanaiuVqKFRdyzqpXAtSnPaFYtBoAzE4jRToQ5hxnPPV6k6Jit1l//9B9cVvVHXOBgrZ0PfNviNp5CX7wAANBhJREFUH91kc8FnsaqEPlNJ/L2+t7tx9WDC5oJPbyMlSjLCVKLraqcQp0oMd/RlPY4yvvvODr5jfCxo5GnN4XwYPBWF/HFSsadF++hW+3vv7bLZrhx5EHQWfJu3bvQJ0pwz7QpRmrLScLlyMEYTSgW6VHP43cv7bLQrrDRcqo5yhRvFKRKhRD2owjEM1cOHFPiO8ptolmnhFUvn3a0BF5ar5EVBzTXxHYMLHZ+KpXP9YEKmqYchiJV4SNOUTNrQNGWelBUlLU8SZcUsoCLJCvJcmWhN7VBlDmmQlTalqJT1vMA09JnyUxdq0CalkmIXEsXiKZ37ypQ4JrGKJZNCORdObVPVoNJis1Pl6sGEQZjwu5f3WWu46JrG6Y6PLKOLhCb45ubCzCvc1DWSvMA27jBNBlFK2zMZx3fGrZvtCm9c732sd350sHm/e0UIHqrlUnNVn//SjgqrcC2DNMvZG8S0KxGjqMx6Kil9o0itFiUq+3QYphSyYHOhQj9MuHYwYbHm8K1zC8RZwaX9IaeaHq+sN2fX4esbLd67PeBgFGPqise+0fI4v1hlECS8ebPHi7MV/4ir++NZG3CrHzAKE7JcCX2GsXIenMay9cMUXZOkmaK+Ukw9dnKqtoGlaxQSRnGi6Jz9EEfXGQQJWSl3X6w5anht6Lyzrdo4P9kaKIFTuZA51XKZRDm7g5i6Z7Jcs1lvVnhnq8+Z2CfJcrqBEkn98otL9312P9ofcXGpRse3SXOVyKUojCndICnbXQqjKOV6d0KaqxamIQQ3ugGupT/VOZwPg6eikD9OKvb33tths+0fe3vnRUF3EiMCUfpV6Eip1HO6JhiGqcrIzNX2tT9JGcUZDEMGQUpSL2h6SuYfZQW1XFKzDd7aHpbxWbL00BDUKhZpVtD0TTUYddTgcrPj0/Ft4lwVi+eXa+wNI6I0Z0sPsMt2jGkoKXbHtzkYJ2gaCCmUwrRQftOuqRNn+SyAohCqkFuGEvQIlPCoyJV9ALIogyGUz0ZeSDzbIM0KHF1TFruaktWYpgoMrjrKbx0BSZZglXTIRCvoBymOqbjCe4OIm90AmRdouoZoKJ+Z3iRC03Q22xUuLPnUPSW4CtOchmfSqdoMw4Saa5XnohLVffvOltzQNb660ZjZEd+rfXa/e2UqHnnQlsvmQoX/7/I+3UlS/qxa4bZ8NcTc7ocANCsWzYqNqSvVZ5BkfP+jQ84sVJQNQT/kpTWVBekYOnXXwjDUtf3O+Tsvs1GUlnxqxfaRUu3IhFD3o106IaZ5wXY/JExyorTgYBxzeXdEx3fwXVOZmeWUIQ8FnqVRyEK1AkvPnaWai2UIanlOP8zwHZNTLY8kzdkaSHRNsYd0AVe7YyqWwfllH9vQ2R1FrNQc/u3lQ860PQopOdfxudkLON+u4FkG24MAocHFpSoS5Wi4mfj86EaXlbpHy7NoVWyu7I9nnjh3P7uWrvP21oBzCz5b5bWWslCtPE2jVbFn39V2P0QD2hXlhaPuoWLmR/8seKl8Gp6KQv6g/e+3b/XZ6oXqhrWNUuAC3cnxbV3Ls+gGKbahUyuVX3GmmBxZLhlEGQ3X4Ho3YBKn6EJxhQdByiRO6U4Smp7JrV6EqSnbzsMgoeWaOIZge5gor+cy8DbKVbZhdxLz3FKb84tVbENTniJ1l9/u73LzcEKc5eyPklLNWSiv7jKdZjqA3BtG+LZJ3TGJ84LDcQJS0RtdSyfXCoJEiYSm1rICoHRHnAqBikJQFAVxpkIhTA2EoRLZ665JWqgQiyjNMbWcOC/U8M5SNgS9IFEycdsgSlWoQSHh8v4YUxNouvI+L1BtGyngVMPhF19YUtv0LCcqd1XfudDh5fUG33tvdxao/Op6g6v7Y5qeNSv49xteP8i9MhWPHMUnPeDTOcruMCptejXWmw6tis21gzGHkwRLF6WRU87ptupfZ3nBW7fUrqHqKKuBi8s1Xlmr8/7uiM1OhZprstHyMHS1AxlFKZd2RohSdGMZOr/9/i7PLVdZ9G12RxEf7IzYXFDFNM0LPtwbo2uCQZQiyrSetYbDrW6EZ2qECeiWQZZLHFO5VU5psOcXlff6R3sT6o4yz5qULZyvnWqwUHUYRxm3h0p9GqY5/SBRL/ryJdqqWAhNMBinbLQ8LF3j9iDkbMdnUHr+11wTv3QgjNOclbrHt88tzK5xkGSzF+nHdCAdn7dv9dgbRzy35M/an1/daHK6XeHK/ni2MzucJJiaCgfZ7ofEWY5t6DPP82fBS+XT8FQUcvh0NVY/SPjxjR4Lvk3NUTS9H9/o0QtiruyPCEo5t5SoyXymGAJRue2O0pyzHaXMOxzF0PbojiKKQrJUd/EtkzDJcS2dG92AqlPDNTVudifsjRLWWi6nWga7g5iOD0VhEqaZ8lPRBP0gnRXlrCi4uTdhf5zwO5f3uLo3YX8U4ZoGnqWz1nQZRSlSCjzTRDME272ANJdKjm9JbEMwigtMXSClwNEVh7oo7gh3clCtDO7EuqVFTt21ccoCJESG0CSFFBilJe0gSqnaBkkZFuEYKs4uzSS3wxBDV62ZJM+RKC/sIJe0Smm3usaS9ZpFkedqkGzpnO1UWaw5M074UTMjgOeWfD7cGzGMMy7UHf70a+vHaIUPKqW+173yKGZJqw2X51dq6EKb9e6jNONMp0K9DKYukFxcVv3iUaR8Xqq2OSu400GooWu8ttH8GLsGYKsXECYZh5OYhapDEGecanmESc7eOObq/gTD0BS/2xA0PZswzuiWebJLvk2ttBXwTDWz2B7GCCGZRhG6rql88DXBD691ObdYLSmH6iW+VFM2sutNj5Zv0apY/OaPbnFxuapEOXnBrX7I2QWfD/ZGKm6t1Alc3htxvuPz0d6YvVGErmmcaftogtkc5O52CBx/kd5LB/LyWoP3d0dkheQrq/VjM426a85e1g3XVK6XjslqQ/mWx2lOxTYeS1fwNOGpKeSfhqsHE1q+jRBqO5oXsDuKyHNFZ/tgZ6h6tS1PDUps5X3x4e6YxarNxeUqu8OQ5ZrL2bbP7WEIAmqepVR2UVK2XdQy93Y/nBXMmmPgGhptz+bmYYBlaCU/V1J1DCxdZxBlZXLQmL1hiOeYHA4jbvVDBqEyQoryHJlIap5Ju2Jxux+R5DqioORVq+10lOYYVZumB8MQekFOxRLkuk5KQVEmsesoGqNEsVosQ1EgVWiDwThREXIVx6A3jpGApWuqjzlJqXuSqqNaVA3HUuG+pVlLxTZUKICt064ov/JxlFHkEt3QWGu6LNVcbg9CNE2y2qiwWFXb4buHj9v9kNuDiM12hddPt2er75PEo5glbS5UuNULlFBMmoDq0a43vdlA880bvfK+UDOB3kQ58vXDhKZnYxsqmmyt6R77XUd3DtcOJyqkuulhGRrvbPXJsoJaxWTT8qk4BnXHYBDlIAWDIKUfp2RZxmrdI5eS7X5Iw1W70OW6o0zXioKdQUJRSHRDUPcsJnFGkhV0RzGnGpWSAQM7g4CfPdtB09R5Xz2Y8NJaQxVaIQiTDGscc3l/pBwzy3ZP27fJCkleSF45VWfBd5TyM8rYaFXUnCdRv7MXJrxxvYdvq0G3ronZi/ReL9q7X35HcT/+uW8bbLQqXD1UFEbL0J4JL5VPwzNTyIdhyma7wge7YwDluqfrBHnGeqvCKEwVAyVMeXmtwUcHY+qOya+8tMyPrvf5zbe22GxX+LlzHVYaakXsGMrJMM0ktq2z0rTI8oLdQUiaF0RZwWrTo8gLbg8jmp7NZqfCwUT1O3VNwzVNNA3avsVi1VGeJ6bBKErZHaniiVAMEV1XYpH+JOVnzrYYRhkSyZLvkklJksU0PI1cKu/r1YaL0JRHyzQF3bN0jLwgStXgR0qVpF6xDVxLw9Q0gjRlHOekhXppDIMEy9QpsoJCSAxDR2Y5aemgiFTMkrzkoqcyp8jVbsC3VAbqc0tVRpEyx7J0jWbFol2xyPOCKMlY8O3Z6mx/FHO9O+atmz1avk2eFxgCbnQnswHVOMo+kWL2sHgUs6SGZ/HzFzq8favPh3sjJIIXV+vHFINf22jy9q0+P77RJ0xyXl6rY+hKFKTYR1BQ3PO4p8Xog50RYZJhGDrXD8a4pkFIzv4woWJG2KZOmBSloVrKrV5Iy7WotipUHYPrhxMWqhY6qpW2N4r41ZdWeON6j95E6RyEANtSdMaDUcyNfsBCzeYXLnRUQMUwxDTE7NyGYR/HVK0fz1ICpe44IkwLznZ8tnohN0vVa81RO67/9GfPzM5x+pKeKmAXq8pXPSPno72Q37/a5VTL5U9+dQ14PFfCu7/blm/x9TPPfvE+imemkNdckyQruLhcZbsfcjhJ8B2d55arSKnEGaBWCsMope6YDCJFMTuz4BGlGZ5pzAYrW/2QxarN1YMRbV/xhfdGihmy3PA40/YI04KKbfDR3lj1SnMl7BlFI+quhS6g6pmM45SGZ7Jcd2l4Jq+sN/n+R4cMgoSaZ1GxTLoTdcPnmSTUFeNgsWax04tZqFlIlHpyqzfBcwxMTXCuU2V/nCiRUSbZSTKitEAIZW06Ex7ZGhstxR7JC0mlMFiuOQzDhNvDmEGQUnMN5V9tGkzSHF3TQSpu8jSJPc4KxaDJNbCUt7SUECY5W33lk1F3Lb652SLOcq7sT9B1wbnlGheXa/i2wd4w4p2tPghBL0i4tDNhZxDw0mqDjQVvNqDqTuKPUczg3sPJB6UVPopZUsOz+EPPLc68se/176uOyTfPtrnZDUjzAsfUaVYsTF1wqqVW2Z9UVHzHYBKnbPWC8mdNRocplN72YVowjFRASbNsX7UrFnGac67jc6rpcasXsDuM+cUXlhhFGaeaHoMwLXvJOVXHwNY1pVNAstmusFp3GScZy3WXtabyqZkepwrVGNBwLWqOwYd7I6JMJd7LQrDe8tjpB9zsKZrp88u1Y+d49Fq/eaOHvaDTDFN+cPWQvJC0Kxa2oR8beD6OK+GzaIT1MHhmCvlR34XnlqpEWc44SrmwWJ0NQEDg2zrjOMcqPZgbroljGjQ8U+UQlrFeqw0P0GiU9ppJXuAUGs8v1XjrZpetfshaw1NhFXnB5oJHVigflrpn4lvKDS7LcoJYRZ4FScZqXXkrV2wDXdNUoG0hKWRBkiner4xz3tsZ8NX1hmrLhKmyHTU0Njs+k0Sp/n54rUu7YuEYGu/fHpIWEqGp+LG6a9KPMnRd0nQtRe/TBIs1Sw1CywQlS9ewdME4UnFxFUtRAQ1No+oYajVeSBaqlgqW1gVRUmCbGp5lcDiK2RlFRFnBLzzX4eyCOr6Vusurp5oz06LpA3owjtns+PzuB3tEaYFn61Rsk/f3hhRCquEgn95TneJxlJxHP+NxBCTT/u5qw+XHN3uMooA0yxnH+WwAOL1H7yVYqToGjuGzPThAKy2RX16vc/1wQm+SMklybFMnKgp6k4ymZ/Lccq089pS2b7He9Pj2+QX+0HOLvHmjx/4oppCS0wsVbh4G6Lqm5jCWwTiWLFbhYBJjaYLtQcgfubjIMEyPnZc4YpUWlD3n9aan4t8S2GhV0HSNC4vVTwwunl6fm92A8x0fxzRmbohFAd99Z4fluvOlEe98Fnh4i7gvKKZv9Ck1baPlsd700DVlDjUIE/phwkrdwShTaDxbxUcB1MoVtJQFvUmClKoXXXMMlqsOLyzXaXg2a02P51fqhGmBKFWNC76Frmk8t1TldMvlhZUapzsV0rxgue5xdsHDMnQ+uD2m6hh8tDciL5R3dncS0w1iTF0jzylDAzSVBTmMeHWtyf44UvmNUiXFNxyTV0/VaXrKXXB7GKFpgrMLHgsVB0PTiHOJqQmqlo6hC8ZxzlLVxtA1ruyV6T859MKEDJBC8cmTPKcoV23N0mGu4Vp0qjY1T5kXgURDpRD1woTlmsNKzaEo4N3tAUUBVceYFdPpd/MLFxfxHYMPdkZs9xUXWyLoVC0KCYeTmINRxFs3e6rwh8kdvjb3Hk4eZTtMQ5qnKesPgmmYxLvbA67uT3h3e8DvXd6fxck9CKb9XWDWKpskJdMjVyHSSVbw5o0e1w8nvHmjR5IVND2LJCsYhClhmnGmXeHMgsdSzaVim3z7XIe0TOupOzoLno0Q8O1znTIHU6dVMXlhpcZa050V082Fskds6ry20eS5JV9lf+aSMM5YbzpIJL1xQj/MqFjKtfLotZUSXl5rYOqKXVW1FdOmWbH55mabqqPajkmmWi1HWypv3ujxO5f2ePNGj365ug7TnHGczZ63OCvQhFQeKmWhn16jh7n2cyg8Myty+Pj2arryidKcF1frAGSFZKPlzcyRojRDlCZR39xc4FpX3Vjv74w4t6j450HJKBiGqWKPZAWnFlxe22gwijL2hoo7vTeMqXsGgyDj8v6Q55ZrCARJljOJExCSf/nubRZ9ZbtZcwzCxKQfpkxi5SI4NQlSAiKLbpiwUvOwdMHNnurNn1+sEiQ5dpmZWHMMhkFKL0iJs4Kqp2hfk0jlhi6Wgo9JnGOXdqi7wxhDV8OmIMpUAk5pG5AjlXdMlOKZBqtNm0IK4ryg7VnspHkZH1YaP+WS/XFMIZVT3U+TIb0g+Zj7XD9IuD0I2RmGQMH+KKUfZqw2XFaqNqMkY2eoVux/+OIiVw/GvHWrzytr9Zny8e6e6eNmMB4Nk/Bcjbi0N5jawT4IprvBrV5I0zVpeTZJWnB+0cezTG4PIi6WK+gpx1yJ0UYz5WbLt9hoe/z4hpobTCl3aw2XVkWtZg1DcK7uk0vFlLlyMKZAWR3fbRRWcwwEMEkKXjvd4j+4uMQ/fWuLOM1Ya1X4cHdEkCS0hMUgTI5d236QsDOI6IcJ7YrNc0tVWIK3b/UoJNzqB7QrFk3P5HTbn7VH4N6iq7Md9TOGJmbPW5Qq3YMmBO2KOXsJw7Mv3vks8EwV8rvxSX2z64cTfuunO/zk1oDFmsNrG010TRDGOb/84jKHkxhZSMYlj/xmNyjNnRyaFZM4U0wBNcBz1GrD1LA0HcPX+NH1jDMtDctU4coLvkUQZ+yOIjzTVDFlpVyyH6VoKBMthFrpupZigXRqDpsLissb54rHK4DuJKMXqOFTrXTLG0ZKvp2mBaMiVSvBvKAfpehCCSZqZTh0L0iw0ZE51Eo5tmMas/DdYZShSSXV1xAs1mycSAUGdIMYkUsG4wzPUi+MaVSbocF4mBOmOf/uowP+2Murs2t+9WBCx3f4/bCLEDrtqsYwzLi6P+K1000aFYvnlqu8uq4KStUxubI/5tLukK9ttI4FSkzbEkLwWBmMR8MkQLlD1qXJh3ujBy7kd/q7YzQ0fEdlaTY9xdKZ8pldU2d/FLHedPlgd6SMoByTKM34cHfEf/7tTU63K3z/o0PevNGjO0n55tkWqw2PC4sqlcgu2UG6JlhruMecGI8W0GmizvMrajZ0aWeIY+nKanYU0/Yt+pOM/bEK9piuqqcr6gXfZpJkjKOU928nnG77rDc99kcxSV7QrtjH/N+P5qjez9QsSgt+54M9LF3jXKdCd6K+w6PhD18G8c5ngWe6kN8P/SDhyv6Yr6w2uLhU4+rBhEs7I6qOwUtrDRZrDp2qUuXVHJPLe4o+VnctBEo8VHct3r09VIkmhkbVNpBlDFjVMdnqB+yPEnQ9nRlBVSwd3zGxdI1+oCKwRpGyIFXGVwUIDVEaUEFIs2Jh1RyuHU6o2yaFlGwNQlqeSZQZZJlyxKu6Kp7ONg2GYYIudOJC9ZyDJCfOVKF2Sj6ia+oIIJA5Wi7RygDcVsXG1DX2h0omjlAiou1ewPmlGofjmMWqy1Y/QGgaYZJRCBVoEaeS7kQNdju+ze99sMe3zy3MVuXDkvf8ynqDd7cHHI4TWp6JECa6psKqp+nooAr5K+uKAnc0iOBYwQpTJCmLVeeRMhiPhkkc/Vs5Y94/GFQxbx3Jlhwem8uAesF0qg5XDyalonhqxqXR8m3evtUnLyRnFiq8sFLj7a0+Vw/GM/Wnpgne3xlh6HB2sTJrXU2vydECutmucPVwjO8YpWgmpuaonVmSF+Q5rDQMNM3lzILH9z865NrBhJ1BNPMEdy19Rhw4GMf80ZeWeetmn6ZnlS02haPF95N2R46p8csvLJftxIRBoMKRp+c3vUbPunjns8Az0yN/GBztq9Zci1dPNfnm2TYS1Y+EO3Fe3zjTwjHVoOhWNyArCk4v+CqAVgqyQnIwSegFKdcPA/79RweMopRvn1soMzZN1hoOk0TR+V5db2BqKt/ycBJzexgSp3KW31kAjmUgJWRFQdu3VMiChLpnsj2IcUzBZsenahlIUZAXBZoUvLbRYNG3Wag6VG3lHX2q6bLWUGyZzbZLL0jQNDELjXYNxU7J8gJDCIIkZ6Fq8/MXO3RqDr5j8NxSjYpjMQwTorQoV5ESVxdkucQSgjDJ0Q0VHXZ+sUrNUZYEVw8mswL84f6Yn2wNqHsmP3e+w9c2mni2ScU2Wao5fOtce6Z2nOLog32vfnin6lB3zdlsZCo0etCB2YVFn2GkWmxSSqJUsZouLPoPfV9tLlRmApS75zLTv//Zc+1ZgLL6fUrhutmucHlvfOz8zi74SATvbPd5//YAWRb5nzu3SH4kym7qjHkUnarNSt2dXZeGa/HqehOBYnCdaXvUHAsNldWpeO9W2VYc8KMbXT7YVTbIL63WWK47x1J57vUd1VyT/VHMpZ0hb1zvcWlnyP4oplaKdzxL+bU8v1Ln2+c6fOdCh/1xVIrI5OwafVlSfU4SX8oV+f36qsr46s42farUW6g4dHxbxZ6Vg/xBlOHYGkLAwThGBxqeQT/MuLQzYqPl8csvLfPO1pCdYUzVNjjdrrDW9IjznB/f7DOK1RZZ1yDNwdQFrqER56r3fKY09srIWanZ7AwjpaaMBT+8dsggSGn7Du2KRXei+ppSSFbrDmlWkEsIM4llaKw1XHZHMTvDCKtMW3csjbZp0g0SpBQYhjqfG2XeY8MzOVupsFRzkAi6QYxtCPphim9rxClYvk5vEiMLqDkWq2X6zShKOdtR5kv9QPmVPL9U5drBhPe3Bzy/WufMgsdy3WajVaHlW5/KJb7f9xal+T1FIw+Cu8MkTF1jvel9jIXxIMyWoxS6u+cynn1HmPLVjQY3ugHDKMO3dU63q+ilw+HRglx1TF5Zq/Nb7++wVPNoV8yPtTO+tmHdU0wzjUC7W0l6uu3RmyT0EmUbsd6s4JgCv8zWdE2NS4djal7OhU6VOMv5yVZ/di6f9B0NwpTfvbRH3TVnStetXsCffm2daweTj313napNlOWf6J0zx4PhS1nIj97400zIw4lqJeyNotk2/crBGIHk62eafLg3Ym8YYeoaW72APC+wXXXDrdZddgYhQZxTdwwEkquHY37t66f49rmFWYjF9cMxgyAmjHM6Zc9ZFij1o6lCITIpEQW0fAPX0lmue0gk720PuNULcAydUZQSZ8rnfFrIVHo8eKaOb+lcGyc0XZPVmqJPpkXB88s+aaEGmXFWYJvK3a7hWTQ9i2bFpGIZbPdD3t8Z4hoaL6zWsAyNl9frfP/KAdu9gHOLNTpVmysHY5Cw0nA4HMVIAUGSEyQpS3Xlencwjlnw7VmR+fkLHX7v8j7vbPXpVB2MUu16fmn5U7nEjyKz/zQ0joRJ3K9IPwzF8UH4zK+sN5RRWRlPNy2G5xer91Q3rjUrfON0i3Gcsd0PGccjKpaOZytm0IOIaabXVtcEP77RY61ls9mu8M72kELKY31qQ9cIZ4HM4lib6ZO+o6Nq0FE8NeOq0J0kD/SymePR8aUs5NMbfxxlXD9UNrWmrnG6VVH+KKWhU5wVvLzWoOZa+LbB5b0RNw4nDKKE71zoUBTK8H+hdGLbLmX7N3sB/TDjN350k/OL1VkUWZS5jKNpdJvJYtUmyaFZUQnygzABdFbqNit1m2+caatgXE2ZdnV8h/1RxKD0yi5yE9FUmZFRmuNYGkFaEGaSTtUiTgu6UcJixSZOcyZpxstrDYIkR0PgWoJBqFgTUaYsV/dHEQWKutiqqNQaiYqn++qpJjcOA8ZxRsM1Wa459IO09LexsAz1OcNIFcSrh2NqjnlslblSd/nO+QV++9IeVce4pxPe/Qrh46j/PgmfVnwfN6zi3r/v48UQuOf5XVj02R/F3OhOZgPSYZgwiNKPZV9+0sp2Km56Zb0xe3Ed9SkBlTI1FRNNdwyvrNXJjrRy7ne9hmE6U4NO4ZgawzDl1VONz+S7m0PhmSrkD6fwa/Ldd3ZICz62ZZ32Wadq0SkqZY+v4VozF7fvvrPDwSShXbE41fT4cH/E4Thn0bdwDGMWD/bzFzrHGAZZIekHCZd3x8Sp4pRbukPdMznTrtAtg6JtQ+PSzohbvQBKD5mao9gFwzjjxkHAn339FJf3J3imwHNM8kxiGsoPQwrUZ0kJaYFnGjRckzQvOCxdHF1T5/YwZBgm7AwjarbBxmkPQ1d+5U3XnPmF/Jmvr6v4r0IySTJeXKmjaxqmLqg5Jt+/csCkTJxP8ju84DMLPsMwZRxn3O6HnO1U+NbZzuzaHm0V3A+Pq/57VDwuxfFeuF8xvF+B//U3bqn81EJyc2/EIMw4veDOaJIPo2y8n0+Ja+oYmiBMJd86d8diN0gyPPvTx2lCwE+2+tRda2ZcN23LPKnv7suCZ6aQP6zCr+FZLNcdXlip3XcCf7+V+4Jvzz77j760PPu9Nw4nHI6T2XbYLYdWgzA9VqQ2Fyos1Wze2R7imjp5UWAXGjEFF5aqKg80z/nt9/c4s+Cq/rUG/TBjXJpJuZaOluUkecHtQaiUhFLgWgZ74whD0/AslfoiZU4vSLE0jXGSst1PWS0HoLomuLw3Is4KhkHG7WGIqQts0+CUr0ILkkzO/ELgDtWtYulM4ow0l5xuKwXtRstDaMqYq2qbnO/4XNobcXl3xMXlGp5l0A1Sqq5qa02LxYMUxicV4fVZtHTuh/sV5JW6w83DgDevHGLoOusNBw3Bj2/0Hyst/u4Cu9FWq3FdE8fsgx+OBTR9nu7VlpkX7s8Czwxr5VEUfkIoQcgb17tc2hkyilL2RzE7g4jfubTH1YMJZzs+B+OYtADfMXl+uTaLHrt6MDmmKN0dRlAGBVdKH2bb0MgyeUz+3PAszi9W+ZkzLaquyTguWKjabLZdrh9MuD1Q5v0qfEGxVXRNUcFMXSX4pLlEFxp2yQxpV21kISkKZUk6LgdNuyMVVhCnBS3fxjNULNbvXd7nVi8ky5XV6fXDCT/Z7tMdJyRZwcEo4np3Qqeqwi8qljk79un5erZBJpXAyrcNDicJhVRb8Cm9rllRDoB111SDVkPj5bUaftmLn+LTCuP0RX1UEfl5qQCPslGeFLui6hhsDUNOt3yeW/QxdZ2bvQDH1B5YxXo/HFXevrLeoO6avLvd5w+ud4mz/IFZQFLCK2t15Y4YZZi64JW1OvJuduccJ45nZkX+sNvffpAoD5M4o+6YJFnOD64eEqc53zjTnq3qr+wrLu4nrdyPrjTe3Rqga3d+Ls4KDEN8rEhJCX/44iJ/5PklfnSjy84w4tr+mMkw4dyCj20YSCSGrtPwTBZ8E9fSQUpsXcexBFLCQsXi9c02Hd/m0LOIM7WKOt32mcQJH+xNeHGlim+ZdMdqCLXVm2AZytLXNDTefH9AWvLMTU25E3YnKbeHEXVH7XAWfP/YDmcakPv2rT6X98Zc744xdTjdUmkxR6XYtmHwwnKVUZxxcbnGKEp5f2eoRFcPuOo76T71w+Bx2wIntZOIk4xa1TiWFGIb2sc8Uh4Wx+2EVd7lo9gJ3zGuq83+7kHbMnM8Hp6ZQv6w29+rBxMWqw7tMo5rHOdESU67as3ShKaftTeMH+izNxcqbPUCbvUCJXlHJbisN92Prd6OHu+FxaqSxJcPpGPqjOKU5aqLbcD1gwlxJnlppc7uKMLQBZauYeiKtfLKep1RlDFJMpoVA9tQUvMwtRlEORJFZfMsne0bPcZxjlsUvLXVpz9J2RuFeJaBpkk0ISikSg+yCqEsUDWBW9rVTgvn0VbWN063CNOcvVFEmGYfk2JvtJQtsF+2UaqOyemWYrQ8aGH8LPrUD4NHbQuchKkXqBf/i6t1docxkzjDs3TOLymrhsdp8Rw9viDOHivv8rMaRj8KnlQb7knhmSnkD3sTTQuDEIKLy+qG/aEsEHcp+lxTx3eM2crkkz674Vm8vN5gbxTxk+0Brqnx+pn2MXXjvY53aoZ/4zCg5pq0KzZNr8reKEJKNdwspFTBD6ZGkOQYGlQdi6WaPWMhOIYqbMMoQxMqNzRJc273c8y2xuEkYRilaEjyXImM+kEMoOK5qhqG1LE0QT/KuLBY5cJSjSjN2e6HrNQd3tkeMAzTmQLw6Ap5sap66jXX5Mc3ejimgW0IupOUnUHIt851ZitwTYM/+tLyAz9cn2ef+iRxUjuJmmtyqllBSkqlsK4UvGUQxEkc3yTJHyvv8osy0Dypl+fThGemkD/MTTQ1BXp/Z0S7Ys0YK6aucbcye8p1naamfNJnH5X+f9rW9O7jbfkWf/E7mzMTp+4kASkZJSqIuFWx2RmG6ELQ9AwOxymTJOP84kJpc5vy7naflq+oi1cOJggkF5dr7I5CtvsRu6NQDT9tVfwMTcPUNQqpQp6FLpQrnVb2uMv2iG1o7I4iDsYRvqN8Yt7fGTGJ02Mrtymn/RcuLnK6XeF77+0SZQVrDZezHZ+9kTLqUjmWD/dQfZFWew+Dk9pJbC5U6AcJG60K3UnM/jgqk+gf/GX4acfn28Yj5V1+0Va/T7IN96TwWIVcCPG3gf8ISICPgL8kpeyfwHE9Ej5p+3t3rNhi1WYSp8oUaCfldKtCwzORMKNiHS0WD7K1ftgb6F6fWXdN3r7V57s/uY3QQBeCzQUVyLs7jLg1CFjyHVZLV7wkK/jNH93ipbUGXz/d4urBhO++u0PVNlhuuEgkp5oekzTncBKxWlMBtZ6toaGKOWis1h00TSUNpbnkbN3FtfRSug79SUKjYnF2wUcIQbtiMS7FVNMdzdGHvjtJeHmtfmwFvVRz7pnV+aDQNcG7230kgguL/lOxwjqpncTRF7+hC84vVWmVit5rBxNqrjn788MU1KPHt9pwHzrv8ou4+n3SbbgngcedQnwPeElK+QrwAfA3H/+QTh5HGQ+qDwiH45j1VgXfUZzqg3HMdy50+PkLnYf27Zh+/r+9vM+Nw8kxD23X1B9qGDUVbfzaNzZ4frnOV9bqtCoOB6OEumOx6Dss1hxeWmvQqtjKvc81y52CSn43NMXaWaoqf4ztfsSCZ3Gh41NzDJoVi7prY1k6ay2XF1drtHyVVq8LZaX7/HKNl9Zq5IXkYBxTcw1eWavPVt+rDZcCZgPLu5kc9/L/eNhrcff1tQ2d10+3eWm1fsxr5IuMk2S8HGWXbC6oJPkpi6c7TvjNH92iO04eitVz9PimLb5MMsu7/LT7/1H94O/lW35S+CQ/mGcVj7Uil1L+qyN//D7wa493OJ8NjvcBi7IPqJzzLi7XkFIqYyFvmsP44CuJoyuSpZqKhLu0M5olqz/qDbS5UOH3PthnwbeRUjKIUgyhtr9ZrkRKdvnCOdOuMIrVjbvdD2lXbMKsQAjVkmFR5TguN1ykhMWay94wwrF1mp6Pa+r8zqV9LizVWK07gGB3FHHe8rmwVJ21lY6Koz5tYHmSPe23b/XZ6odkObPg3mmx+KJvlT+rvvHdu79ekMxe6FN67PTnHkZk9bB5l4+y+v2sV/FPaxvucXCSPfK/DPwf9/uXQoi/AvwVgI2NjRP8tZ8MddN0S59oA02UPiOGzrBcOT/O2/roA7XW9Li0M0Qgubw7RNc1uuOYr5YWrA9zkzbuMleq2joNz+JgEs9cV+NyNXaUEbI3ignTjNvDCM/UWGm4ZViFzl/42TOzY97uh4yjDN9RvPJffXmF7X5EN4hpejY/d36BUy1v1gbZXPi4fPzowPJOn7Q/2+Zf2R8f+/lHeZj6QcKPb/RLGwCdOCu4tDPiuSVfBUPf9bNfpF7tFJ+FEObuAjqOM2VUFd+5Jg/aTnic43uUF/b9WpBv3+pTdczH/v6+KEPXzxNCfgpbXwjxr4Hle/yrvyWl/L/Ln/lbwOvAn5Gf9oHA66+/Ln/4wx8+wuE+HKZv/q1eiAqb1+gFMUiBZyn/64222lo+6Grg7mKx3Q851fRmHPNRlHJ5d8hbN/u8drrF5kJllm7zsCuOoyuXLC94e2tAlOZYuqBiGRSosIhrB2NeWmvgmBr/8r0dirzgdLtSOvqlnG57PL9Su29Qwj97e5vdQYhrGSpEOssJk4ylussff2X12PFMz31KqZdSCauGYUrnLk/wqcfM4zyYb97ozbj5U//uKM3JZcFXVusfc/e724jqaeijPwqmrcJpEby0M2RcvtCnPO6jdhOfFR7luv/Opb2PeZoPw4Q3rnf55tmFL8X396gQQrwhpXz97r//1BW5lPKXPuWD/yLwJ4BffJAi/nli+uY/2/FVQoopaLgW4zgjk3LWB3zQt/W9toS3ByGOoc+451XHRNc1Xjvd4tVTxx+gh20FHF1ZRGnOV1bVAzqKstlqerXh8tWNBt2J2nmcarpkuaTqmCz4DsNQiYBAPUB3F9R+kMz8YDq+xULVoWIZxGnOOMrucTzWxx7et7f6jKOUVsU+Ftk1TYZ5HAzDlM2Fyswb2zZ0pCzojuNjfeYvG1Ph7vZB07PY6gWcalUeSVr/qHiU1e+9VvFXDyZKefwl+f5OGo/LWvkV4G8AvyClDE7mkE4Ox7niNbb7IaMoRdcF//Fr6w/9pr9Xsdhsq2zFaRJLmOZ0xzFfP9069t8+6tT8Qbe9p9uV2flO7U6HUQplFFqaSbqTmPd3Rrxxvcsvv7hM3TV580YPq/RBD+KCa9GYlYaLEOA797497r4OWRkXd5TBclIsgaNqwek5GZrgq3et1L5sTIV79bb/9GvrdCfJ595OeNjWzL162N1JwtdPH3/pPMvf30njcXvkfxewge+V26TvSyn/6mMf1Qnh6JtfJf7ccTd8lBv8XsXibnP8aQF8Z3twLNfw85iaT893eq4Ab93s0fCUTa5j6nR8m2GY8L33dnluSYVLrzc9BmHCMMwZRAX9IOGrp5q0/Htfo7uvg2/rJFnBOL6zgj+p850+9J5l8NxSdbbSvDv44WkVDD0O7lVAT7e/+Ok691rFf3Wj8YnpUHN8Mh6LfiilPC+lPCWl/Gr5zxemiMPJmx3dj9Y0Ncd/9ZQKDDjd8jF1rQyuHbA3jD4Xk6V7nW93kmDrWmlipSOEoOZa5EXBh3sjXFOxQDQhWK7bvLxaw3dMtE9QDN59HVYbrmLVlI55J2kqNX3oP40S+kUwtprjwXGUSvm1jSavrDfm399j4Jl2s3nQIvCg+LRicSyXcFkVxLRQUXCfx9DmXuf71Y0GUaYCoqeIs5yWZ82i7ab5pKaucTCJabjmJx7v3ddB1wTrTZeNtnci1/l+5zV96O/1uSf9Xc/x+WL+/T0ePpW18lng82KtfBb4JIrb/abxl3aHnOtUH4q5cT+GyMMq+PpBwq+/cROjXIlP0482WhVMQ9wzcuxBRVBfRKrfHHM8y7gfa2VeyE8Qd1PCRlHKW7f6VG2DV9YbD1wo+0HC713eV3FbYcrtfkizYvPNzRZRWvDOVp+X1hp0qvYDfeb1wwnfe2+XvChm0WqaxoxRMi/Ic8zxdOCR6YdzPDjunsZPw5vPdvxjtLxPo1RNjbMarkWc5rimwSBM+Wh/jGcZD63gO92u8GtfX79vwX5QxsF8FT7HHF9MzAv5CeLuafw0vHnqTwIPRqn6cG9E3TFxTJ0wLag4BmZWcKMbst70SgXfHYbIg3zm46oLv4jmSHPMMYfCvJCfMI4WzLvDm+HBKFUq51C1vDxLJ82K8o8S39aPSfIf9DMfF182wc0cczxNeKZZK08ad7M79oYRP9kasN0PP9Hx7cKizzBKidKMBd9mHKf0woRTTZemZzEoedyfJ03rJN0M55hjjpPFvJB/hjhKqbrZC7h6OGazXeFU0/tEm9FX1husNz3yQpLmBatNl5W6w0LVmSn4Wr71udK0vozWoHPM8bRg3lr5jHG01XJ3NBrcuzXR8Cy+c6HziYPFz1vB92W0Bp1jjqcF80L+OeFhvUA+C+vTx8GX0Rp0jjmeFswL+eeEZ8EL5Iv2cpljjjkU5j3yzwn3kvfvjSJGUfqZxF3NMcccXx7MC/nnhLu9JOIsR6D8tR8mY3GOOeaY427MWyufI462JqZhwnNe9hxzzPG4mK/InxDmvOw55pjjpDAv5E8Ic172HHPMcVKYF/InhHkQwhxzzHFSmBfyJ4S5kf4cc8xxUpgPO58g5rzsOeaY4yQwX5HPMcccczzlmBfyOeaYY46nHPNCPsccc8zxlGNeyOeYY445nnLMC/kcc8wxx1MOIaX8/H+pEPvA9c/9Fx/HAnDwhI/hs8D8vJ4uzM/r6cEX4ZxOSyk7d//lEynkXwQIIX4opXz9SR/HSWN+Xk8X5uf19OCLfE7z1socc8wxx1OOeSGfY4455njK8WUu5H//SR/AZ4T5eT1dmJ/X04Mv7Dl9aXvkc8wxxxzPCr7MK/I55phjjmcC80I+xxxzzPGU40tdyIUQf1sI8b4Q4m0hxP8lhGg86WN6HAghfkUIcUkI8aEQ4r970sfzuBBCnBJC/LYQ4j0hxLtCiL/+pI/pJCGE0IUQbwoh/p8nfSwnBSFEQwjx6+Vz9VMhxLee9DGdBIQQ/015D74jhPjfhRDOkz6mo/hSF3Lge8BLUspXgA+Av/mEj+eRIYTQgb8H/CrwIvDnhRAvPtmjemxkwH8rpXwR+Fngv3gGzuko/jrw0yd9ECeMvwP8Synl88CrPAPnJ4RYA/4r4HUp5UuADvy5J3tUx/GlLuRSyn8lpczKP34fWH+Sx/OY+BngQynlFSllAvwT4E894WN6LEgpb0spf1T+/xGqKKw92aM6GQgh1oE/DvyDJ30sJwUhRB34Q8D/CiClTKSU/Sd6UCcHA3CFEAbgAdtP+HiO4UtdyO/CXwb+xZM+iMfAGnDzyJ9v8YwUPQAhxBnga8APnvChnBT+Z+BvAMUTPo6TxCawD/xvZcvoHwghnvrsQinlFvA/AjeA28BASvmvnuxRHcczX8iFEP+67Gvd/c+fOvIzfwu1jf/HT+5I57gfhBA+8H8C/7WUcvikj+dxIYT4E8CelPKNJ30sJwwDeA34X6SUXwMmwLMwq2midrebwCpQEUL8hSd7VMfxzEe9SSl/6ZP+vRDiLwJ/AvhF+XST6reAU0f+vF7+3VMNIYSJKuL/WEr5G0/6eE4IPwf8SSHEHwMcoCaE+EdSyi9UcXgE3AJuSSmnu6Zf5xko5MAvAVellPsAQojfAL4N/KMnelRH8MyvyD8JQohfQW1v/6SUMnjSx/OY+AP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- "text/plain": [ - "
" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/Clustering_18_0.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "def gaussian_points(dim=2, n_points=1000, mean_vector=np.array([0, 0]),\n", - " sample_variance=1):\n", - " \"\"\"\n", - " Very simple custom function to generate gaussian distributed point clusters\n", - " with variable dimension, number of points, means in each direction\n", - " (must match dim) and sample variance.\n", - "\n", - " Inputs:\n", - " dim (int)\n", - " n_points (int)\n", - " mean_vector (np.array) (where index 0 is x, index 1 is y etc.)\n", - " sample_variance (float)\n", - "\n", - " Returns:\n", - " data (np.array): with dimensions (dim x n_points)\n", - " \"\"\"\n", - "\n", - " mean_matrix = np.zeros(dim) + mean_vector\n", - " covariance_matrix = np.eye(dim) * sample_variance\n", - " data = np.random.multivariate_normal(mean_matrix, covariance_matrix,\n", - " n_points)\n", - " return data\n", - "\n", - "\n", - "\n", - "def generate_simple_clustering_dataset(dim=2, n_points=1000, plotting=True,\n", - " return_data=True):\n", - " \"\"\"\n", - " Toy model to illustrate k-means clustering\n", - " \"\"\"\n", - "\n", - " data1 = gaussian_points(mean_vector=np.array([5, 5]))\n", - " data2 = gaussian_points()\n", - " data3 = gaussian_points(mean_vector=np.array([1, 4.5]))\n", - " data4 = gaussian_points(mean_vector=np.array([5, 1]))\n", - " data = np.concatenate((data1, data2, data3, data4), axis=0)\n", - "\n", - " if plotting:\n", - " fig, ax = plt.subplots()\n", - " ax.scatter(data[:, 0], data[:, 1], alpha=0.2)\n", - " ax.set_title('Toy Model Dataset')\n", - " plt.show()\n", - "\n", - "\n", - " if return_data:\n", - " return data\n", - "\n", - "\n", - "data = generate_simple_clustering_dataset()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now that we are our, albeit very simple, dataset we are ready to start\n", - "implementing the k-means algorithm." - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": {}, - "outputs": [], - "source": [ - "n_samples, dimensions = data.shape\n", - "n_clusters = 4\n", - "\n", - "# we randomly initialize our centroids\n", - "np.random.seed(2021)\n", - "centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :]\n", - "distances = np.zeros((n_samples, n_clusters))\n", - "\n", - "# first we need to calculate the distance to each centroid from our data\n", - "for k in range(n_clusters):\n", - " for n in range(n_samples):\n", - " dist = 0\n", - " for d in range(dimensions):\n", - " dist += np.abs(data[n, d] - centroids[k, d])**2\n", - " distances[n, k] = dist\n", - "\n", - "# we initialize an array to keep track of to which cluster each point belongs\n", - "# the way we set it up here the index tracks which point and the value which\n", - "# cluster the point belongs to\n", - "cluster_labels = np.zeros(n_samples, dtype='int')\n", - "\n", - "# next we loop through our samples and for every point assign it to the cluster\n", - "# to which it has the smallest distance to\n", - "for n in range(n_samples):\n", - " # tracking variables (all of this is basically just an argmin)\n", - " smallest = 1e10\n", - " smallest_row_index = 1e10\n", - " for k in range(n_clusters):\n", - " if distances[n, k] < smallest:\n", - " smallest = distances[n, k]\n", - " smallest_row_index = k\n", - "\n", - " cluster_labels[n] = smallest_row_index" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let's plot and see" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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60gMDnzs+g+/YdzzendZ0N1ni7dbiOxabw+S+eOZbP/OFVsCnDnfYGMT0Y6FWljshS60AC8UrV/vv2ceD3n3c6/x9kv3cp4H8cUG0B8M1uPw12HoDkqE8nsfClX9IuJ8g/bjd0r7fYPxBi2n3G7g6NY9m4PL9p4RuaIXeHS+Od1vT3aiRVuiyPUo5vzFkEOf0JhmvrPT56lvbDOPinhfl233m802fw906rdDl9EKTmbqPUgqlYKbuvWcfDxp473X+Psl+7lNq5XHAPjdeX4B0DMkI4tdh5iRYNiw8/aEd+n5ufx+nW9oPQgN90GLag0jq7pd2uNea7iRLPDlX56UrPbKiZG+cYimL0HVIfc23Lm4ziDO0UTR8m0PtgOSWAHqnz/zMQoNXrvaYa/gYIwF1bRAzW/f4ztXeTTTNg+rf7+f8fVL93KcZ+eOAfW68uQiLz0LQhLKAyTYc/vyHyo/fTxb5ON3SfhAa6IPeeTxIxni/2ftaP+bq7oSXrvQ4vzFklOT3taZOzeNQOyDKSgoDoedwYrbObM1lbZCwMUhoBQ55aTi3OkCpm19/p8/8uSMdXjjWRRvYHCWsDWKW2yGd0MN3rPfcweyfkx95cuE+tfKfzIz7Xphm5I8DkoGoVACCFgTPwLyBeO9Dlx6enKvz6juXqCfXqOkxkdUgC47y5JmTB9s8aGb1UeBOBdoPIu970DuPO63hYWXv/ShjfZDgKGiFHmmhOb8x4thMjZnGvY+x3AlZ6UWcmmugqkidlYbQs4kOLiIGhXnPa+/2mT93pCMeLb2YduCglEWSlzy51MS21AN1ht76N/RJzbjvhWkgfxwQtIUL927okPuQufF9dBjzGesCa8qiZ5o0Vc5nrAs0mAeuX0QepS/Y3eiTD0ID3Su43hiIlJIMfr4ZvC8lz/1cHC/tTDg5W+fq3oS0KPEdmzQvuLQ75nMnjt7X+3npSo9hnB1cCLJC8/RSm6w0DJOchu9wcq7BO1tjjOGmAHunz/z62sdYWDQCxfHZJs3AZRhnvLY2eCADrWnGfW9MA/njgJknhCMHUankMeSTD5UbP8DeBRrNNmdnb7iIZNGHLnn8ILgbb/x+W7/h7sH11kB0brXPOMkPCn7vpznlxkB5/SLRv6nVfr7pE3o2a/2YYZJT9x1qvnPfF4uffGaRX39jg+1xykzN5fhsHW00nzsxQzNwWR/EfO2dbRTQ8h2K0tCPsvukQWbICn3w3kdJzqurfRqBe0+3w/2M/tLOmM8cm/nEtNq/X0wD+eOA2oxw4XsXhE4J2hLEP4pAeiOtsw83lHU8orgbffJBaaA7ZaG3BqKihFbgstaPeXLJvWkNd8Pt6AXgttmqbSnivKQZuAfHiLICz7n/0tfx2Tp/7HNHD4652A4Yxjm2pRjGGV97Zxtdap5ablNouLoXcWymVl0UuSsVcutF8+L2GIM6oHJuDNYGxVOLTYCDpiLfsbCwDpQ40+z8zpgG8scFtZmPJwMO2jDelMCdjsFvSGCvz9202cM2K/og+7sXffJ+aKAHba1v+DZZoRmnxcFj26OUnXHKV89vPVBjlW2p295h7Lfaw4PfXdyIW8/H/nt9bW2AAp5abtPwr1NPe5OU3UnKS1d6lFozUzUE9aOMU/MN9ibZwXna/70XZWRlyXOH2zQD9z3B2nEUr672ee5I9wbXRUUjeH93M580TAP544x935X+CmQj8JvQPvJw/VfCWXj3NyDogN8S/Xr/Gnz6jx9s8kH4zQfJQO83I/sg9Mmd1niv9dx68VjuhHx3pU/TF2fB7VHKa6t9PnW48559gASpl6/u4dk2p+YbjNOCtX7M7iRlc5Dy40/f3BUZujZJXn4oReb9wL7Wj+lNMi5sj2kHHvNNn5pnc3VvQj/OWGgGpLlmfTDCtkacnm9wYXvMp294jxe3xzdp1/fGGe9u7/HSlT3KEhZaPifn6pyaa/DdlT4Xt8cH7f1pUXJ8tnnwft+P38wnBdNA/rhiX1seD2D1JShScH0oS4h7D0+WGO/CoRfk33Qk0sfuCfmdU8CD6atvLQgO4pyFWwqCd8pA7zcje9gqmrtz7vLvWj9mfRBzcrZB4Fpc2p0wTHJ8R3GtFzFOCj51uMNCK7hpH+dW+pTaUPMcLCySvOCfv3yN3UlG3XFY7gRMsuIgW73RhVCpu1Mb++d6rR8zTgoagcNyJ7yvu5t9RUzNs0mygjgruLxbMFNz6cc5GMVb68MDrbnnWHz93W3OLDTv+LnN1D1+9bV1hklBWRocW3F1b0LDdzi90OS5w23Obw4xKDSGJ5ea93RdnEIwDeSPOu7kdrh3AXQJay+D5UCzA8lEgvqpH3l4xchkAI0F0bDvw5ibOPL7lfS9pyC40meUFszeUhB8bW3AF47P3HN/d8PDVNHc6f1d60X0o4ya53C0WyNwbN5YHxJlBYc7IV96Yg7Htg7oj8C1OL8xZJwWNHyHQ+2Ad7bGfGq5LYHcgnc3JqzsxXiOIqzZXN2L8F2bt9ZHfPvyLoHnUPcd5uoBsw0Xz7EZRgWDJCMrNUe7IYutkGbgMIhzaq7D5Z0Je5OMOCtYbId8/Z1tFlsBxkAjcGgGct5vVKVc2pkw3/A5tyKe4nGWMYhzLluKE3M1ru3GuI5NK5QC6DDJGcQ5hTbvOU/7n9veJGOm4ZEVhpWkRGOYCb2Du4+jM7WDwub9ui5OIZgG8kcZ0R5c/DLEfShzsF3oX4VTPyYBNtoFU4q2HAV+XbLxaAfsh/TR3of08X4lfe8pCGpD+zYFQYV5pDpF7/T+xknBXMM/eDz0bMZpTpyVBK59cHEaJwUvXdllnJTMNnwOd0LyUvPqap9Cc1OT0c4oox/l5GXJICpxLIPvOjiWIsoK5mybUZQzjAuu9SzOLjTpxzlZrlntT8gLDQZW+4ZSi03s1jChGbgo4PzGkNCz2ejHnFlssjNOGSY5UVpwqBPiWBZff2eHfpRhjGGm4WFbOdujlMC1Ob3QICsMk7SgaSmK0lRr0xgDjnVz59CNn9swzikKsCxVHTtBKcWVnQlRJsH8hWMdgEeuL+FRxzSQP8pYexl61yDsQFAT+qR3TR4P2rD5uvxb5OB4UGby+2QP5p98OGuYeQIufBmS/vWLSdCBJ37sYJP75aTfWxB0yIqScXq9gzHOS04vNImygnFSsDdJ2YsybMviJ59Z5OPAnd5fI3DeM98yzTWerchLzfmNIYe7NVb2JkRpSejZlKXh8s6YQ50aBkXdsw4uEuO0JCtL0rxEKYVnQ5RrhklCO3Q5Ptvg+GydrCj5ztUe89V4toVmwG6UUmjDha2xFBPTgmeWWnzj4i5HuyGeY7MzTrEsBQYGcYFj27y53mdzGHO4HaJUjEJRlFKk7dY8BlGOZSmeWmqRFYYkL0lyTVJoiigj8GziTGMpizOLTRqBS5QVt/07aIUugyTDQtEIHTzHYqU3oZfktGounzvexbGtg9rBfv1gintjGsg/KryfgRDb52VbV3hV3ABMWx5//k/Aha9Im/54W57z29A6DLYl+39Ya1J3//1+OenbFQTPrfRoBO5Nt9CfOdZlEOf8+hsblBpmah4zdZ+L22PaofuRZ2Z3en+XdiYH72dfaVHzJLAHrli5/tobm/iORZRpTs43iFLNIMnoRxlfPDVLP84PLloXt0eS+Xo2rcClGbgkw4Ss0MSZZr4plrCFMQzjjLwscWyLumezshtRFNJuvzVK2JtknJipMU4LNoYJ2sBaPyJ0HdaSjKwwREXJzjAh8Gxc1+KdzTHPLrdxbIu3NsaM0oK8hDjPOTHTwLJAG8VC0+PJxQZvb40Zxjmn5+u0ax6eY/OTzyyyN8l4a2PIWxsDBnHJ197Z4vMnZnl2uY1lKSZZgVsoLGWIM81MzcW1bdYHCcud8MAyYapQuX9MA/lHgf3CpFsX6V4ey+/3KkjmEYw25d8iBicEtyYFR4Dmgjxfm4E8gXQIZgnO/oF7XyRut6aLX5ZsG3M9sO9dgMYSzJy6/tobGoJuVZ08f7Rz36ZHtqU40q3RCl16UYZSYFuK717rszFIODnbOCgOgmikb/yC30nx8rBnNt5JenhyjoP3sz/fsuE7oESit9aP2B3FPDHfxG94rPUTziw0OT5bY5jkOLbFcidkpu7xP72yytYwJStKXNeiKA1JUVCUJbal8V2XlV6MZ1tsjWJGVYD2bIutYYqlwLUsFtoBWitspXh1bQDKME5ybNtiEOdEWUmUFuSlZpRmRKmmETgErs3WKKV3YQfXUigMjuWRZDlbwxRQKKM4u1hnuVsjygrOLDSJi5J3tyccndH80c8epR26vLE25KvntyhKzeFuiK1sfvOtLYZxznOHO1zrTdgapiR5SWk0Rzs1unWXvDSc3xhxdrHxHpOuKe6OaSD/KPB+BkJEe1J9mmxBFoPjQjIGLxQt99rLMHMaOidguCIab8sRRcnsqZv3c6di6Y1r0oXQNkFftlk/J5QKFhz5/M1rqxqCHlR2eLvM9ktn5m/qjNRaguB3ru5R9xw+fbhDaQzjtKTuWdR8h89UQxb2j+1YitdXB/za6xvUPIdnDrWYb/ofWAZ5P63j++9H5lvavHCsyzgp+PU31tmZZGBZ+K5NoQ1Xt0aMkpxnDjWrjF1oh3MrfVxb8anD7YOhD/0ow3NdnlnusNKPqLk2xmiu7Ey42otxHYXnWFgo1voRzcDlUJXNaqM51A651pvwmaMzXOtN2BlnHGqHrA9icmMoAReFBnbGKbvjjMVWQJYXYNsUWjNIciZpSc2zSXMNxrAzSUnzkmbgyWcYOMzWPH7giTku7064sjvh1ZUhtqXwHJfepGCp4zIT+lzamfBDp0POLDR5/kiXq7sTfEfOzXwrrLTjcGl3wrPLH779xPcSpoH8o8D9dEfeGnCTISx9CkZrwk2jwHHACeTx7fNw7IuglJhowXvUJAdZty6lMLr5Olz7Fpz9X793TcOV6rh92H5Tsv/aPPQuwdorcPgzVVGVg2Ln+7F1vVtnpNZwdW9C4NostUKu9SL+yYtXOb3QYL7hEWU2XlIcBN2a51BqczANHgODKOXq3oTQsw+ka/eTxd8pWN+vTezJuTpff2eb19b6XN6eMExEudIOPV5b7XN8ts4T83Wu7EZ852qfZ5dbdCuP7nMrA2ZqLg3fY5QWGGC+6ZGXmlpg88JR0WWv9CJeXxvi2grfsTnUDim1IS00eWnoBB4N38FSMM5y8tKw3Al4+lCLb13exUKR5CUqyqSBJxEJqEJmZSZFiTbgVtz8zsaQVuix2PKxlKIoStYHCd+91qcdumij6Y1KbMuw2o9Ick3Ns5lkOZaS4u9+o9Biy2dnlGLM9ULm5jDh6EzIzjhjtR9RFgbLBqUUP/Xc8p2/T1O8B9NA/n7woHz3vZQft6VeXoIj3wftY1CP5TE3lADbWJQAe6d97q/v2ouSaZc51GbFzzwZwNv/C8w/ffPr07Fst/uONBaFHQnktargtPsuLH/mJp+X4bX7dxK8n87IvYkoIwLXoeE7rO5NKA1obYizku1RyhefmDvYT7fmHQTxwHUotMFSFoFrHyhhblzP/XZO3to6fqQTcm0vukk2eLtbfwPsjjM8x6buy0Wm0Ibjs3XSQmNZiiMzNeq+Q7vmcbRbI85LVnsRgdNgthHw5GKLa3sRK72IojQsNgOW2wGXdiIA6p5NYFsEnkMr9MjLknboshdl5EZzeqHOaj/BtuCZQ00macH2KEUBmdYoy6LmOWSlpoEhrIqgZamxLUVgKZSlDoYitwIHy7KYq3skackwzdidJLx4aZekKGkEDsdnGvzW29skeclSJ0QZSAuD54iiJc41UaqpBw6tqsaxf2HdG2dEWckoKTAY8gK6tQcPSw+7s/hxwzSQPyjeD999L9Or21EvtVl5vLkgqhQ3EB7ccaVl3ii48tuy3cwToibJJ9A8dH19ypJMPI8lMCtVtdxvy3HyyfU1FSlsv3VdlRKPYPMtacWfOyuyxlt8Xlphj2y8Sye+hp0NKL02/fAorfrMHRt/HEvx+tqAr729xQvHujx3pHPQ9ffWxpD5hnDik6ygEXpyOz9OMYCl4O2NEbalaAYO51b6vLUxZLbuM98KcG0FSLY6TMSP+0b5252y6xt166Mk53cv7LKyF9GPM0LX4dtml8+fnMV3LC5sj/nWpV2eOtTkxFz9oB19Y5Aw1/A51K7RCqTQ+c7WkNW9iCcW6gySguMzNSxL4VqKrORAnrjUDvn2lR5LrYB26LDQ8unUXZ5dbjFJS/71q2tkpSFKS0ZZye4k5kirRla67I2F/56tu6RZyb9+bZ2yNNiOxdFujX6cozU0A+kw9W3FdpyBUoQV5ZNrg2NZ2AosZeE7NqVnkLOpiJOCgVLUPItSw8peTIlBKUWei+944NgMkpy0KDm72GZ3VxwZm4EDRrEXp3z+ePcmj/p998VadQeWFpokLw+8XO632Dl1TpwOlnhw3Bh0lZJ/3bo8fifsm145ngRDx7s58N9uFufMExKEwxkpdsZ9+dfyYP27Mh3oyPfJtivfgiIRzvzK12HrTehflkBeJOA1hGsHCdj1GcDcvCbbhdYRyfh33oH1l2GyI+8xj0Qdk46ury/a44nRi3Rf//uoze+icUjTCGf9JebsyU3jx67uRqz0YgZRztubY2xlMdfwubobHQwaODlXx7YshrHol/tRTt1zmGsG1H2H+UbAYiugF2dc2h6z1peCX9N3idKcdzZG2JZFw3cYxhl1z37PgIs7DYbY160DnFsZ8OZ6n1yXLLR8MIaVXsSb6wMu74wpK7/uUVLwL7+zwt44o1vz6McZV/YmWEqoDoDAsdkeJ5zfGOFYiieXWhgDIN2QIBcO2+LgYvf62pAXL+8ROBbPHelwaXvM9iSjF+cYNHXfxmhYG6as7UVoY6i7NqfmmrTrLnGuKYGlVkBRarZGGYM4Y5KVnFpo8v2nZmnXfNJC7nKMgYZnU2pDL8rJSimG7oxTXNuu7jwMpZY7iqzUzLdDWoFHzbVxHMU4zVkdRERJzvogYS9KmW96WAp6cU4jsPi9Ty3wA0/McW6lz3/39Yv891+/wLmVPo4FO5OM71zdY60fcbgTMt/0H2hM4CdpZuyd8FAycqVUB/i7wKeQO8w/Y4z53Yex70cO79cN8G6mV7ejXmwXDn9OeOk8ue6lko6kZX6/0/LwZ0VFUiSw9y5EfajPQ5FJ8O6vSJRwAvFKURY0T8oxb11TFsOF34TeFYST96WoGu0JL59NZL8XvgwKWnEfe/k4O5OMbPMNvMVnOHZ4kdX1t6nNPH9T449rKX79jXXqvken5jDX8Cmq1vR9e9nDnZCvvr2FZ1t4jsXx2RrnN0YstQMKo7m2E5PkJY6y8Byb5490eHdrJE0urk3Ns7AtxVubI450ayy2gwMt8stXe7y7PcZ3LE7NNQA4d23Ad1d7JHnJO5sjvnhqntdX+/iOhWvZzNQDsiLhSDfk7c0xnz3epe07HG3WuLIbcaQT0IsyFloBs3WfcXUX0JtkbI4SPFvx9KEWk6ygXnmuOJZikOQ8f6QDwHdX+ry6MmB9mDBTEz666TsH2voLO2OUgXGSM04LbGXRCVx6cc7mOOXUnMNTS21QBs+20NqQFXJe96KMnVFKt+biOxZ1z2GSlsw2PDaGCeMkpxl4dGseNU+z2ovY0wbXsmj5LkrBYvv655QXYpAVuDaDSU7oOeRlSW9cgBFeXymLKzsTlAULzYB/+/PHDpQ5r670WelFtAIXULx0pceV3YizCw1OHZshLUpW+xG2pe5rMMY+PsiwkO8VPCxq5W8D/9oY88eUUh5Qu9cLHlt8GEMe7kS97GftR75wfdt3/w0oRwYw77sRNg/DznlYeh4asxJsTQmjLcDAaAdsWzLzEz8E1m105kFbAvjsaehdhnQicsYyk3+VA6svV9n+NeHpW8vU/SZ1ZwBuH6KXYf5LZOM9wsXrma+l4FovohcVHO7WKEp4Z2vMqfn6Ta3uM3WPn35umUs7E1b7MZ5jcWq+jtaGdzdHeI7N5453We0nXN4Z86nDbT5zbIbTC03e2Rzy3Wt9Pnt8hp9+bvmgNX4Q51zcHlPzHJ5abPLqap9vXNwlygre2RwRuBYnZ+sMkpwvn99ia5hweKbGYiskdG0C10IbB88W6ZxS6iBL3W+8GSU5k6zg7c0hrmNzqBXi2NKJeXapxaFWyNY44eVrPWqejGGTNdn8zrvbDKICpQwbg5jtUcIPn1nAcxTnVvoMo5ys1BgjbfBZbvAcxWwtoB7Y1DyHJ5cajNOS3iRFodibJKz0YrTWpFpXckPDUitgcySdnHFWEDoOaa7ZyBI82+b4bJ3tcUro2TiWQmvD5e2IRuDgWOBaNpNEpIuWBZOkxFTZumMrCg1zDZd26LA1SNgbZ+yMZC3rg4huzWeh6RO4EnbyQtPwbXYnGd26Xw3GKLm0O+FzJ+6/Gehxmhn7YeEDB3KlVBv4YeBPAxhjMuB791L4YQx5uB+/8f0C5torsHdZArjlSCv+5nnJ2LEk405HQoVMtiUIh23Rn+cTafGvvQBr37m5ULvfwdlflTVoLbSLWxNa5/JXwW3A7BNyjN41ydBtRy4QeST7jnrYwfO8vNqnKMXONUo1Ralp1mQGpAKyvGBlT5QOgzjnc8e6/Du/74e49O75g7d87Ikn+a/+h1/j9bUBZ5aaHO7UqHsO/SgnszgoaDYDF9u2+OzxGZ4/enMA+MaFXU7M1Q++5M8d6fLV81u8tjaQQmK3RujY1DyXJM8py4DQsbCV+HzUPYf1fsRCyyfJi4OxZcdmaoySHGUpzm8MCVybJ+abbI1SXl0d0K07KEvx7taIdzZHpEXJ5ijlc8e6tAOXd7ZGvLE6wHIU46yg6ds4SjFOC/7nV9f5gy8sc25lwGzT5+2tEbvjDNsC17YoSg2eYbkTopTCUor5hsdaLxK72MJQFJrcaJKsYBTn9JO84vxtJlmJZSlyo7GrC1NRGmxLRr2ZrGScSBNRPXA40QjxbYeabxOlOVuDhHFeMEoKitKgtfSgZUWJ71oM4oJ+UqDigl97fZNnD7cZJyWrvT4/cva6i2NeakLXZqbhk5clF7ZHYKBbf7AA/LDdLh9HPIyM/CSwDfw9pdTzwEvAf2iMuYmgUkr9HPBzAMeOHXsIh/2Y8DCGPNyqeglnxU3wdiqYaE/ojdWXpLCZZzBahySU7dIJ7L4NnWOVzHBDMvGdd0HnUqxcfFoy6smW8N/Hvnj9InRjoXbuLLzzb8AOgKTa345k5aN1OPL9YLQEcpNKdp705YLit6C1RJxlWON3yM0z+M05skLz5saQJMvZGeecu9pntuFxqBVi2SWuBZ6t+JN/4EtsXHn3ptN09cJ5/vK/9/v53//f/iVzDZ+gsm5tBA7GOOxOUoZxxqXdCS9d3uOFox1GSX4gOwxdm+1RwtOHWgf7bAYuh9ohb28OOTlXx1JSJnJtRVooFtsBrqVI8py0UIDh5FydZw+3WevHzDT8qmFF89pqn9C3aXgOoLAsi88dn6EXb7A1TDk6U+PanljRGuSi8LW3t6n7Dk8tNfE9m+2qCWgzLwlsuYCM05zffGubuYbL549LwW53lJLkhpprEbgOzcAlKzUn5+riRohhZRCTFiXjNMcgXZNKgWNZGK1Z68U0Q4em76KUQmtN4FnkhSbRmrQosZUiLqW5SBuhxC5sTzg22yD0HFzHITcpUV7iWhaubTAYjFFoA1d2JmijK3rH5sruhM1hSuBa1EOX1UHM2QW3OucWWVFS92xKDU/MNzFGow0PfSze9zoeRiB3gM8Cf84Y802l1N8G/iLwl2/cyBjzC8AvAHz+859/7zTXxwm347vvV5J4q+plvCl+34deEJfBG4MryP/710QKGO2KdFC5kgnrEuqzUuQsEgn2o01I+zDYkHUsPy8cebQDW+ehTOHiVwBTFTJTuPJNmDkOexU3Hs5AMRHlis4gq6SJqy+JxtxrgWXLujAS3Cd7MNliPPM56s0un2v0edtZ5OpexOWdMYM459hMjflGg2v9mLe3xnzhRJenDkmAvDWI7+Od82/ywrEOV/cihklBw7d54WiXOCu5uifT42fqHs8f7WAMnN8YHdifxnnJfDN4z2234yhaNZ8oLWgE8mXPS4MxhjOLLb74xCzfuLDL9ihhvhnwxSdmOT5bv0mJsxelrPSkWOs5Np863OInnj7EMMmpuTa7wNYwY3ciHYx74wwDeI5FmJaEnkuSaTZHCaFjoQ0k2pCVJa3AI8sLbMvjrY0RpTYstUN2JxmWJQoU37FZH8Qc7YZgDF84OUuWG35zmFSzN8XmNnSkAOg5UtQ1wCST8zGKc5JC49s2uSrJC4PjKSwDudFoDZO0xBjDOE5pBzavrQ1I80LMsSxFK/CkcJrnlc2CHNtxLKJM/i0piDLD9iRBFyVRWlJzbTQG17ZExui5gBSKn1xq3XVI8+3wMN0uH0c8jEC+AqwYY75Z/f5LSCD/5OBBJIm3Sg3jvUrutysFzBu7PkG21YUE3PWXpZTsuiJH7F+WwFxbgMkmlAXkI1G22A4Ukawj6Mhz4w2hWbbekHXGexB0YbxeOSfuQvckbL4KXiBUTJZIMdXxJKAXqVAqWoPJxf/c9mU94QxO7y1Cv47JN1mef4E31weEnn2go05KUV6UWtOPM97eHHGLYd57MElFK73QDA5unS0Lltohri1mUUoJRVF3bVZ7Ecdm60RZwRefmOXi9hiAotQH/Pts3WVzmGCMwXVsBlUmtx+0j8/W37OO/WDx6kqfX/zWVbKy5Ei3RlJqzm+OaYd7dOsexsCRthRI41zjWIpJXoKBTs0l15oL2yKjNNoQ55K9RlmBrRSB71D3RMpSaoNnKWoND6MgSmSbvNTM1Hw0QlGU2hDnBXXf40gX1oYJDgqjIMnl+ZprM84KoUBsm5pfyTB9i3FuUMqQ5BoFpCXY1b4V8O72hGu9hKyQc28QKmWUSmZvK3FCzMqSUgO5xnJtlJF9WEYReDajtKDUJZPMMN8M+NEn5/nq21u8sdNnmJS0ag62pXhi/s5t+p90zfjt8IEDuTFmQyl1TSn1pDHmPPDjwBsffGmPER6kBf9W1Us6FlriRmmfG0L/CgzXKy34jlAlbh1aSzBYk4DqBNIBarswWJXXWVW23j4s+xysgrEkMLs1yayTIYw2RFs+rLL9MpXf7UTUMaMtiLchS0W94vhgCtG0p0PJ5muzwo3rssraU0JjMUiGWJlIBEsjDTZLLR/fsZhkctutjWG9H7PWjHHsu0fyV672sSzYaaYstAKWOyFL7Qa//PIKcw2fVuCSFqU0sSjYHCacXmwe3F63Q5dzK31eudpjpuHzpdNzJLnmpatiM6CyktOLTX7v04u3DeC34v/32ga51hgUhRGViGdZvL054qmlJhpwLDEFq8U2K71YulBLTZSW1AOpFYyTgpm6z84kJcoMRollrW9bzNQ86q7DUiek0IbtYYILuI4FRmxkZ+oe2+OMumvz3ZU+F7bH7I1j0tJQ5iUFwnnbFoS+xyTL0aXGdxSF1kRFznw9pCw1BhvlKvpRRlaAK4dh3168LKEo97N8hVIWlg1x5bXeCBzqvss4zkhLTWkMRkOGXAiMgbmWT7fm0q0FzDY8HEux0osxRp4/OlMDIw6Ou5OMzx2/mePuR1n1OfaZqcvEomt7EV97e5sXjnUOehJufc2dgv730gXhYalW/hzwjyvFykXgZx/Sfh8PPIgk8VbVi9+QwLpvhAUw3oLhmgRqLAi7oiSZOSUOh9YGYMBtQhlJgNY5ZFqCa5nLa7NIaJigJTx2uyF0SBYLFVPuSmv+zBkJ0Ec+B6M3Jd1SRjJ7K7+ucc9TCeYauTBkE7lbyCaiijEG2w2pbb/KoHGGN+MB28OUJNd0ai5zjYBmUfJKf0IvymkFHqu9SPjjxRNMNi+/53R1lk+SliVnZ5toJHOdqXt848IuO+OMJNcsd6UI6jtiqmUri5ev7jFK8oMvdzNw+f5TcwcUSyuEHzm7gOdY922Xuv/F//blXSZxzlwroOGL10tvkhKNS5phl1Pz0ujSDm1GCfQnGZaSgNqLcwpjCBxFXpbUfJfPHu2yPowZxOJD3gykUWeYZpghzDU88tKw1o9IsgLH92j6Dostn/VBwtfe2SYtNJ5jM1N3GaeaUZKT5CWuIw1Sk7TEsixm6x7KsujFKZ6lmKQ5dd9hqV7DUuJMWBrh1tNCjC73eVAN+LYiLwyea3CUYlRK0LctxaFWwKYSNcowyVEYLMtCIRcEC01ZanbGCZaCQZzxlfNbnFlo4tg2CoXrWvilZm9y+6Ekq/2YuYb46Hz1/BZPLDSlJ2EvotTmJl79bo1C8MHGCT5qeCiB3BjzCvD5e233PYsHkSTeqnoJZyS4dk9IWpLHogefeUIy3Su/LTSG15SCY21W5nL6bVGlZKWoWPJEsnSvId2gRgun3j4kr3E8uTgYI1m4MWCHFe0SQ9aXNny3BmGLH/k/foXfupQeLPvHTlr85p9ZBJQEebchjox5Akp8TjAatz5HZoWojVdxln6cVtCgFThc2JxU2Z6mF+W0A5eZpss7WxM6dYc/89/8En/3z/9R4q0rB8fsLJ/kz/23v4zvWGwOY3YnGVFW8NrqAM+xObPQ4N3tMe9sDFnuhlzanrDSi/j+k7NsDzPeWl/ju9cG/MwLy+/RGo+SnNVexOYwAbhnNnZjUKj7DqO0OPD0ti2Lsop28w3h1CdZweWdiE7doxs6pKUhygpmQpdWTfj7oiyYb/gstkPmWj7r/YT1YcwoKfjMsY4MgBrGXNySwRCubWEUDNKMAocXL/fo1jxKLfRFWmqirCRwbM4sNtkcJgyTnLwwpEWJ1GRtsqKg6bmM0ozCMszUPRaq4Oi7DloXMigCCB0LS0FWaiwld1eWAs+xKQqDZ0uHbMv3aAUOWgesDyMpHucaWxlmax6OYzFMxGb3jfUhe+OMRuCQ5AXfvLTDTN0jKw0N32W+4WIMnLvWpxlcn1hU8xyKElqBzdYooeE7jBL5XIdJ8R7727v55Oz//n7HCT5qUMZ89HXHz3/+8+bb3/72R37cDw03cuS304HfbvvbqVYGK0KHbL0pLfhpX2gO25MsfeM1KV7GI+G1R+tiMatsed1oDbx61aHpCn0yd1r2n0ew8apcFOKe0C+2KxeFbCI+5kUEls8P/fXf4HcuDN+z7B87afObPzsnx/NrcnEZrkoQt11oLoMyDKwWhRty8Ymf5V/0T+NYis1BwvmtMWmhqbkOZ5bqGANpodHaCBdsW4S+ze44ZbbuEXoulgXjtMAYQytwqfk2k1Tz6cMtXNtmlOS8tTFktR/hWhZLrZBu3cV3bAyQ5gVLnZCzi018R3TX+0MgFIa67xzw6XfLxvY7VWuew//0yirfvLRLlpd4riIvDVmhOTbT4AdOz9CbZDSqpp59ad7vXthlkpX4tqKsGoMsS3zLfcei1KJLr7kW/Ypyubg1ZJJr4rRgnOUErkualyRZCUqadPYVOqOkwLXFzbAoNa6lGMU5tqNwlEWUF2Ql+BYURoZ6FKWm0AaDohU4LLUDxmnJMM7Yi3LQQuVYFmBAa01eQt0Xv5akKKVTVilc22KuJjx+WpRobXBsi+1xKkE+tBilmlGU0ql7KCwC1yIpNHFacrQbstAJGEYFcVFwbKbG80c6B59NlJUc7dZ4e3PEIM55dbVf3TnCpw93aIcuZxeb9KKMH3lSJI5fPb9Ft+ah1HXqzhhz0Ch0p+f2X/8oQin1kjHmPUnz1GvlfnE3VcqDShJvq3rpSIB1AjBvSPasFPgdya5nz0gQTiYS9PcDaJFCNhQlyb4nStIXymbx02J6NVwRfbguJYA3FuRCoywhP5uHhCKxfQg7tw3iAF++VApXHrZkLbU5ufdOxpLxWxqckNiew3Mcmhd+hef9z7BazjBWy8zVPeq+i0aL0CXXuBZc7U3IC8NyO8CxRde8OkjwrAzXAaWkY9GzhW+eqbm8ujJgmIjqYq7uETjSObnSjxilLt2aT6fmYinxBwEOZmeu9qIDBcfhbu22w5Tf6464h4VFI3A4OhPiOXO8ujLg4vaYbt3j2Hzlhb4dMYhkNmW75vL25ogkl2y5WxMDKt9WBJ6DZ4syw7Zc8lKzM0rJtaZb9yhKzWwjINqbsDNOSXND7otM0q6anbIyFy04FkoJfeHZFsZY9KKU0oBbQKIliGugcg+giORcuI5k2VujTIqlnkNpDKoKklkhnHro22htYdKCwHNwbYVju2i977lS8u7uGEcpFloezdAjdB2eO9JhnBR06z5vrPeZpAWu7eA6iqIw5KUGS7MVpZQKJmmBMRqo3fTZbA1TafIJXL5zpYcxCtBgFBe2Rvzok4vvaQK6V6PQ91IT0TSQ3w/uR5Vytxb8+9n/W78i7fXZUALkYLXiuUcSzLdeh+XPQe+iqFt0IY066UD4b1WtwfHFsxxLJIJxD+Jh5dPiVu6Ivcoa10A2EMVKkUHQEL78bqjPy4WlHINOYPF5ORdFgtAuNrVyh7GZZzNxafgp83ULZ+9NhvYZLLfD7kTc7rLSkKQFk6ykW/MwGC7vRnRqDmcXmryyMmAwyDjUDvAci3e2hmwNU+aavgxwQAINWixYpQtTJGzb44SLu2MWGj6Hu8F77FMXWwGHu7WbNOc3DlPe502/9s42CvBsG0uJTHGcSsD7vU8tcGaxwaF2yLcu7VHzbd7dHDJOS97ZGrHQCjBGUeoSpQwamK97KEtcEgPXoh26+K4tczq1JkqknX9rlBI4FqVR4s+iqmIjNrmWeZy2grI0jNMcbTgwFvNt4bctQNkykV7f8jHu60HKAlwlfz5xqsmLjJlmgO/YjJMcpRS+o4hTOeByJ2S+5TNIC0ZRLvsuNJNMukddWzFMSmbqFq3A4eRcncs7E84u1rncG7Pc8am5Lr5rsdqLRQWLxWzdx0aRZEVVB5FpSOc3hgedszXfZhDnnJqvs9qPuLQdcXK+zuFOwNYowXXUTU1A92oU+l5qIpoG8vvB/ahS7qUjv9Pz0Z50VG68KsXF4apw2q1DUgRNBsJjZ1VX5t5FmH9K9ONoiEsI56Rppywk604nQvGYEixf1uv4cpGItXDbRQJZBpMBDDeEkmkuSnPR3RB0RFGTjWF7ADsXqn3H8lNmeJ0OedTHt0Ma4zcYJAWO1eS5YIffHIVyI1GWMhGn1HRDn5m6xyQr0UYTZyVrw4Sab+NYoszQRnTv7cBjvR9zqB0y1/QJHJu0KDg6I05/cV5WzwcSGPKSvSjnpOI99qlr/ZhxOqbh23Rr3nuGKdc8mUSPgVPzjaqDU9GpJr8XBrp1n+1xKvMn92L6UUFeFvTigt0o53DHJ0pkBqdLyWurAxzbohXYRKncEeyMU5qhS9OzWR8m7I5SQt9ha5BXOmxT0VCQl6V0eFpS284LmbC8H8iVgckNRcokl+dvLFreitzI864N+xKTduCgS41tWxgD3ZpNM/SYb/ms9WM2Rwl5XuLaNrnWWErh2Uo6RgvN7ljsAv7NW1voUjpJ5xs+fivk4s6EzVHCKBb7Ade2OTZTwxhDaUTx0ouyg45Z37FxHSmaipmazbPLHX7g1DzDJGecFGRl+R5q7F6NQt9LTUTTQH4/uJcq5V4Z+52enzkNb/8qbL4mGXY4I9n0eFP45qApgTOvHAwxkn1vvl5JD8dyr1xmkiXbDtSWZfveZeGy66FcGLbfgXgAxY7o0FFgMvlXuRL0ox5wgS+dqvP1i+91jvvSqWoNRSK0TJ4CpWT7bijqGq9GMLhKw+qyrAwX1AnKPON0uM1amuM5J0QqaBR1z6GhhGNe6cWUpUyZt22LcVoySUr6k4ysLKkFDmleEmlRQgSuRZyVdEIX13EoSsOhVoCNxTApyDUstQMC1yJ0r5t89qOMjUHCr76+zlzdY74ZsLI3ZmeScbhbo1NzK+/xkoZvHzgmNgOXJ5da4rqY5Ni24ieeXuTK7oRf/NZV3lwfkhUiSYzyAhu5W1jvp9Q9G8+12ItlvNtsXbhulMJC6JJj3foBX92PMslC85Ia4FqKUhmRchooSvAsKfqVVZu91qI0CSveOd9PwavX3KsSZriexe+NRfFS8xwOd0Ou7EzwXKFD9sapbKxlvy6gjUFrg22JoqXm2WyPUrbHKcutGp863GQQZyhlURhxewSYb4rqxrbgjbUBy+2w4t4LLu+OOT5TAxRpUfLkUhPbUkzS8ibLhUOERFmB51h3mUr1YENOHkdMA/mtuF3mfC9Vyr0y9ts9nw6lS7N3RQY+lKlkuo1FIILRqgxyGG2KquTQp6txbi6UIwnojUXh0ocrso+5M3IRqM1I4XOyLY1CdiCFTKORm2q/olaoOkSRC4kpYbDK1/7sEX7wFzb53Xf7B2/3R076fOU//QIMrgllUxSArhQrVYeo6sv7KnJCHxyr5KmZEooCa7zJbHKZw1aDtHUCd3aOQZyxN8nISo0uNMaxqwAKrucwSHLGWY5nKzDSHJMXJaFrEXoOTd+ringFu+OUubpPVmq+78QMNc8mKTRN3+XThzsU2hwoT671JnRqLlf2Ir5ztcdTSy0+tdzm2l7Mr7+5ydNLLbo1j7TQrPcTjs6K/FKCucvWMOHK3phffnmFwLMPjpXkBWV5fdKOrWBcirFUOpGhyk91W8w2fVb7MWkuGbrvSmt+w3dZ6gQ4tnRfBo5NVPmgFLdEYgMygq1Q2C4UVSFyX22yn4E/yOTLKj6LAiYXS4Ek08S5ZpJnKAuSTLJmz7WF59bCyqn9PSjFOMkpjaIZ2Mw0XGYbIWacAIbNQYExiqPdkFboMdvw6CcFcVYyTkuePtSiGTi8cq3HuWsDFto+c/WAd7ekw1WcJG3mb2gO2xoltEOXr57f+lj04I+CHn0ayG/E3TLnvaqF/HZGWffK2G98PhlK4F1/XbTfRl8f+NA5LvSHG4jiJItEn33y90HniEz8ycZCn8R7gCXZ9uwJWHhWaJFSQ2hLMXK0cb0rFKr755pw4slIntOFRBynIWvJI/Ab/M7f+Bkpau5elIw/mwiFs1n1epWpHP+AfS3kG21ZYDm4+Zg4WMDJejxV7vKObpF4s4SmYDF5g1cmZ8i8FlletYBnop/Ocs0kg2ZoCB2L2LKwbYVrKzo1j3GSU3MrygNo+g4l4l8+yQqOdGv4jnzBF5sBLxzrygQg3zoYJ3dlZyLact8lykou7ExwbYtr/RhdlryuDZ+tXjdTd0lzTZQVlXdLKh4rns1cw2ell2BbFoFjkWYytNhSkjVrJSrPmisFxTgteXdnTC/KaAYOeWWSBQ5RXsrxjHSwXtwa0YtSBrHGQb6oRXWm9/3LA8cmLyp3w+rPyFZyk6arj1shNMz+Y3eDQl4PUguYpAVZqfEceX1WGLHLtSyyUrzMi1K8UTSVY4Qvd1haa1zblRtGDLMNMR07teCyNUrIq4lEzx3p8sbaALuuONytcWKuTm8io+FQioWGzysre6Sl4VinxqFuUN09lDK2rqKNfMc+COwfpR78URlqMQ3kN+JOmXW8e3dVyp0ydhSsvAg7b4sipLEoGa0TSmONW5Ni5mRHioh+s2qyiWHhrKhLhmvCnwctuXiUWtYwuCbHbywLJXLxy0J3+B249k25eFTabooJB19rx678yzOgqJQvllA1upR9BDMQ78BkQ4qpRSYNQ/1LcieQR4jTopI2/X3oBAjAtXDzhNb4Elm6x8BuYy+8wIzTItkIGI0LFt1rvF6epV4NgnAti3boYdslSeWsJ+3zomNOi5LFlmRhWVGgDcw2XFqBxzgt+f2f7hK6DhuDmCgrCFybum9jW+qgiPXda332JimtmscgyXl7e4SFnNLX1gYopXj2UIvdccab60POLjX5vpOz9OMcz7HoRRk745RPHe5wrRdRGnFgzLWmMHLqHKUojaKwKldAhWTqGoLAIi81V3sRS60QbQyBK49FiRT5NgYp/UqiN0hKAjujsibHRoK4ZUHNl3b3mi9DLvabbkwVVKsO/wN4FZ2R6us0i83NGbtvS2OP+M5AUmqivKTpu3j29clGw0T6NbuhW3Hhsi/HhjiTc+FacqEJXPEnt5RhbyINT6U2hJ4DBoZJTi/K2B1nXOvH9KKU5Y4oVtb6Ca+sSM+AZys2hynzLZ9aZRr2mWNdXr7ak4lG1ezWcSo+MLal+OEbnBY/LLyfubUfBqaB/Ea836ERt7O2HW/IN8YJYP4ZWH9FstnuSTnrti2BWi1IIw6l6MOjPgeUxWBFMubhqtAZvSuw9ZZk8uGsRKDt1yVT9prS6NO7gvDYuWynxTsbbInlRSoVMQokozZALm3/+8Mn3FDOhdbVD7JdOryBnrGq/98CnUOuwa3h6gI32sRWO9TqXTZP/AyH4oDLWUFHif9JkpeEvsPhdohtK6woY7Hpsj6QgBv6Wia6x9KAYyt45nAb27J49lCLZuCyPU5ZbMqIONtSnF1sHjT7eI51UMTaHyfXDT1evtLDVmAriySXotupuYZ0mipFMtJkhaYTehybrR00pby60sdSIlXbGIi9gGNbeJaFE7iUxjAeFni2ohBJBpalaHgWeSGZqjaGvbEUNDGKwLWZZAVP1ltc3h5T823mmyG745xRnB0E232Hd2MgdB3agQyMiLKSYVxQ1YPxlPxpFEDNlSKkaN2v8zOKm4O4a8nouUKb638VhcYYaQaqO8KRayNOh54t/1coPAd81ybJNa5jcCr9+V6UcbisMYgSBlVhM3TlPPSinFKbytGyoBm4LFZe5S9d7fH8kTZHujV6kwyFwncV7ZpPt+azN0kPbB3W+jE7o4S3N0e0Q49DnRBLwStXe7dt2X/YeFSGWlj33uSTg4Ftc273DX575xzn+u8yyMfXM+vVb0tmGs7Iv6vfFioGruvIiwSufUNGsY22JEP3amJUtfwZSdmGq9Ksc6wa8ODUJChrLfrvdCBZ+XhLaBhVbbPybbkjSEfCd+sCkp6kYdlYDLSSvhQ0436VvSdVxlwFawDKKgDbojZRrvxf2XI3YrtCpbh1uVvIRrKPpA9pVI24c7nzjbotF4Q8kczd5Fg6xxpdw73yWxy2e3z+cIBbn6Fb89AGHBRxLrzyIMp5a2NMmotX9VIzwGiqLD3nUCvAtS0Wmj7HZuo8udRivuGTFmLD2vAlW5up+8w3A4ZxzqWdyU3j5DaHCa3ApRm6RJnGsiwWmz6TNOO19SG9KMPokq1RzJff2iTOioOGoMVWwCQtuLg1Zq0vnZPr/RTftfBdm7w02MrCsiwcW0lgrZqT6r6D79t4jnXAYytLOHXLshgmObpSoNhVOm0ri/3EukSe0xrGcc7WMKmOp7At8ByFY8k+bbvKyjFMMk1eGnJzcza+v1+FXAAKLTJFU/msgFgMK6hshGUQhVKGmu9IcdWT95lkJa3AoeGJnNK1LGZDnyt7Y/FtV6KEcSxLagKIrHJjEJNrODXXwLYUV3Yn5EVZDb42nF5ocLgTMtcI6YQevmOxFwkX3Y8yLm5P+PblPoOoYGuU8vbmiDjXzDT8j2TU275W/UZ8HHr0aSCvMEgHnNMxWTaijUOmc87tvMEgqgYVu3UGSnNucIHfHr7LuXiLweZ3b96JKWVKz7EvSjFx75IEYxBqZPl5ab5ZeEZMrdpHpYtz/WXZ5uSPSfDbH4rcvyYt+luviwrF9qG5BMqr5IapfOPiPbkAqP2vp+a2gdYYDkpaTijbup5k+5aS+/Z92WJjXuge25ci674PeZ5Wx7hx9uV+SHBk31lc0SwWWC4KjRdt4w8v8MzeVygGG1xlifmWT8O32ZlURc+iZLYZoI3MxbQtmSzUj2V+Z8OzmW362JUr4BvrQ6Ks4FA7oB/LbMpD7YCtYcJrq33mGv7BsOd/8DuXOLfS5/tPzhBlBa5tsdgI+PzxLvMNH9dRXN6N6QQuRsPqIGFnnGMM/NJ3VtC64rnzku9c63PuWp+dUSzTezCEnqhNjJEpOcYgjTVUPLIW/+26a+PYFklesDFM2Bkn9KMcrTUb/RhbKbJcc3l3TFpqkkLjqOtnuTASpLEsHEuxNYrZm+TEhfDaWSnqExT4noVjWwdSxX1Y1c72XSctwKouHGVZ9Y1Vrfe+a+M7FqM458rOhJ1RSpaXjJOC3iQnzkV6mJcidXQci5mGz1I3ZLkb4FgWTy02Ob3QxKq6WQ0wzEo825LOSiN3XDvVXcoLRzuMk5K1foxjKXYmKRd3R/QmKa+vD8iKkpNzdc6t9Cm0Js4LAs9GKdgeplzaHnNytv5Acz/fL07O7XeeChV465zYjwpTaqXCleEVwsYCYdCFwQphNgavzpXmHM+lCQNKzvUvEjo+bbdBUqac236V55Y/S9tvv5dfb8xKQXG4AsEz8lhtVjLb0YZowtfPSYZ95PuFZtk6J4F0tAG7FyoaxKkqW03xTgk6MHMMxjvCs+/z3Wqf6rhLSUspuQjoQgqqyVB+d/yqoDmWwJ0NoThSNR5p4fC1BlKhXvYliKYEHHk9lujToVqDc2C25aKAnFa0gjF7nDY9EnedV5Pv58qoA8ZQ80JGSYGlFA3PoRfleI5NaSTQuJYEpfVhiqUgHKW0ai4LTZ9G4PDscguQrHKfxw49m/MbIwJXWvN/+51d6r7N0W6NvGXYGiW0QpcvnZnjGxd2KLVmvlVDoUjHJaEnHZM7o4S3NobVLb4MQ/BccXI8OuPS6brsTTJ822JzlJEUUrh1bJc4K0SWaAyFLjFGMYwL8irjLTX4bkmSCaktjVKipY6zssqU5bJpVdm6ArqByyjLGcXCj7tK5ID7n74CklT8URz74BJ78GNuyM4tJW6NRcnB9qEnnbKgSEtNnuQURgY7lwXkOidwROqoNfgOBJ5FlosLpWcpNoYJjqUIA4dDocvWIGE3znEsC601w6QkLzS5Y7E9lrsLbRKUgk7dZ77hc3l3gm1L0dMYkXQuNKVZ6N2tEYvNgGShZHeSUZRQ922UErqr5n/4eeqjMtRiGsgrjLKRBGQnPAi8gTEMUpEhXtk7T+j4hLb8EYUGCGe4MrzCc/PPca33Li9GG+zkA+bcNl9oHOVoPJAgGA8k0Ee74mmy8apk22Uu2ff2eQmsWSTa8MmWXARsH2wt3xzTlP1MtqvCZCGZcpnItzJoyWvvphg2FbetbKFvLFtSMMtUHaGdigcvpaU/bEO0LZ2m+4/nlTd5kQKu/F9Zwo1btqxrX++OqbYzuFiUJqagzpwa81neoD3a40r2w8St41gWDOOCJC+JspI4K5mpu4SOoqc1o3HO4W4d31Fsj1PWk5LPnuhyosqIbuRD9z023t6UIF4aw8YwodCG440a41Sadp5dbrM2iLm4PWFnIiZOO+NMAmXNI/Rs1kcpR7s19qIM37bJSoVj2xzp1ljpiZGXaytp+7fg9HyDK3sRQTXIYUcpfFuz2PLZnWRsjZKbPiENJDnYqpSLlgUtzyEuDb1cmpEce7+IarCqguY4l67LUoPnUFkRaAqRp5MVEpwtS7zFi5v/Em5ag+8Ih05VsPVd8VfJcoNyYCb00cawMypp+nI+hcrSMnpOaZqhtOuLGgXG1c3ioW5IP8qZrXtERUmaa1KlKY3w7w1fBl9M0uLAQmG+aTh7qEHNszjUCWX4djU2cLkTHgydMJXI8uiMzHX1HRttRKJ4py7ND0Mq+Cjo0afUSoWm1yQpk5seS8qEpteEmScYJXsEpa4cChMoYoLuKUbZiGvDa/xK73XifMyiP0OsM35l9xzXai3p1lz5luzwyPdV9rFVB2ZtVnThozV4+9dg8y0xwyozMcpyfEmtHF9ek40BJdx8MgLXF9Msu8qybVuULwe4kf6oWva9QPxXbK+yya0oE8uFZFvuEILOdbdEreVioTMJzkFH3hNUXi2ZBOsylwtP0BTdus4rjlzL60yJRYavSozjEpqYup7wE853SfMSrRVZYYiykklasNAS3rsX5Whj8F2bYSR897iygt2dpFzbi3h7c8yvvrZBvyowtUKX7VHKWxtDLm6PeXWlXxUuXQLXxnNsTsw1eGdzRDNwePpQk07osdgMqHvC/UaZcPJZVvJDT8wSpzlRljNJC5Qx7ExSDndCHKVIC804LUgyTTNwWWoFso9c0w1cPnu8y0zTZ5gWBxlv9YkA1zswo0wMsVzHkklCrtBLlhHaBCQwKwV5NZcTIC/kMavix3WlGqm5lvy93HKTduOXXnh6oVU8Vwp1jm1Tc1zqnoPWikkhShOtINMiG7TVvkpGDMDqvsgmbRviwlBozanFBkdnalgKLmxPqoKzKDxsFHONgMV2wGzdIy01hTEkecmzy20Od6/b6j53uMPnjnd5ckmK26ErjVpnFhoMkxxbwfHZOqWRu7Hjs7Xbyv/2pYJZoenWPLJC8/LV3sHfzeOMaUZe4XjrOOe2zwEQ2AFJmRDnMWfmz4DfpnnosyT9y4TpULosZ06SOC5Ny+PFjRdptY/TGm+Dzmm5NSgzXhxd4ujSD17PoIcr0pWZjuUn6QstkgwlgOpSsmXLEWWJbVddlwaKsRQibaeSIibSVm8HMNsRlYlyhJbBA6oAqqtCjLKF6gja8hrblfcRD+S1RSxB2/YkKPevyPOtZVHg7F9YTBWg7aDK8Kso4dagfqiSQ2SyzzKTYnGZAZocl7oZE+bvEltNeladp/Rl9iY5/UkPyxLTJ8tSuJW3SYymKAyObRgXmiiXwt5802e1l/DG2gBLiRFW3bf50pl5HEvx62+sszlKMVrTj0t8R/FDp+dJq8ntWVGy1An4wSfmOb8x5PRCnfV+QsOHnQhGcUGh4fc+tcDphSbb44TdUSYDEwzM1HxCz6ZTd2lVEspBXBDnYnqVVJ2bTd9irlHZrLo2FsKzexZVO7uh5HrB0bVtVvoRKKFwjBERkAaUDBkSbxSlQRl8R7JvrQ013xVdPqIiUUCpNK4lBc/9kpwGHHWdK9elaN/THCJKnLQk9cvKgEtokjjTB9OMEsqDNTuWjJ6zKy7baEVgW9UouYLQsSiM4UinxpFuSKkNvmPxzUt7FKVmqRWSFiU132GS5JUcNWdvkqIUzNW9O5pbnZwTHrwf5eSl5nAn5NnlFl86M3/bLPtRkQp+GJgG8gptv81z889xZXiFQTqg6TU5M39G6Bbg+MKnOacMuOF7Av031r7BYn1RJIDjTcgmNNwmm7bc+lHmMuvSCWUY8nhLAq5tVRmuD04mNIbXFpVKEcs3LmyLpttyxNEwTyXzdYwEac+G9jGw1qW9P96RVvl8IlQNjgR4xxF1jGWL6sVrSoBVXJ//qWw5XpFxnYIZC49eqwsd4+9fRAKhW/y2jJcrtQx+nntaUkOdwziqsnH53QYsUkqgoQs+xXfZy5s8qS/yrnVK/ERKQ8Nz6EfSxKSUBRZMUo3vKZqBWxk6aXzb8NLVnviNAJd3x7xyrUdZQivw6Ec521EmPVKew1sbQxxb8cLRLq+uDvAdi/MbQ1651jsYnzZMSuYaHp1q4s33VcXR5XbIoXZIzXX4jTfXsS0xcFIoSuD5o12SvGRjmDBKSmqeTZnkXNmNyQrDZ4/P8MRCg3c2x4wqe9tSm4PgqhB+WlmQJAZLleQlN0kPlZJiJ0oGPIxTg2Mr6oFcEDRSJO7UPFxHhiqnRUk/yqU8UnWbujaEvk1ZyBi9UpkDd8TqL5ZRqnGVxrUVG4OCqiZ5G/Mtg6UsLMRKVxtD6Is3Si9KSbKCuBqwHHougyhjuRPy6eU2F3fH7FQTjaKsxFWKbhhQGMmeXzjaxXXUAeVyq7lVp+bxpTPz902VPCpSwQ8D00B+A/aD+d2eu12gnwvnGOdjWl5LpvgA42zInF1lwOvnJIi7gTQF7Q9BDuYBWxp23ACcumS5zSUYb8s2RSwZuVMNobAnVVHTCN2hHOhdkHFtC09JkE3HMlxislN1dlaWtaaESU/uEBafln1e/V0pnGKEXon2qmJoWE0dKmUAha46N9NhZbqVXh//ZvabwT1I9uQ9NQ7JvXcZU04y0j2bqG/IRpZEi4aCBVAB/BnnV/l/WX+UDXWIUhsKY1BaxpQpNJ5loW1DzXGYrXuUwM44I3MU47wgtG2Upbi2G9OPC84uNMgKzZGZGifnm6z0IraHCb4Ddc+uPEFKokx41brncGV3gjFwuBPQqftEac7x2RqFNtR8iy+dmQcke1vu1hjFBYc6IQtNn+VOKHNBFfzYU4ucuzbgGxd3mG8EPLXU5OpewqsrfWwFozinLK+3w4M04viOTVqUFRcuBc4beewSsG8okI5TXQVXQ5obPMfiucMdjs2ErPZirvViNocJRalRSknhFm5QBFnYjiWB3FSipX3xkbnBhAtpbNovzu4HepCMPnRtbAVxNevTs62qziF3DCMjDomXdiJaocvhTih3SkZTd216k5xW4NKteYySkrQsOFR1dVoWPHekc3DebywmgrgX7gfw54/eWzN+L1vbxxnTQP4AuFOg/8LSF/iVi78CQMNtMM7HDNMhZxbO8M/33mJn9cvM1Rb4Qus0R50azJ8RXjyPJOBmM5I9d45C2pMgWpuRwqZlyfSg+pxY0haFdJoqVenExmIqrSzYfhs6J8VfpUjkojHerAjUoOrQTIX/3n5LPFIw8rupujp1IQE8NZJJdw+LfjzalQuHW5fjlingS4Zve9ebn9JIArwuKO0u8YZDfG2MySWIl0qTTyzMuk3xpk3asjg0s8XPzH6VfzTz0zgKujubtMqU2A8ZzS2T+PWqaKfZHKacmK0J1ZKI7CwvTdW3qoizkqcWm+SlZpwULHc8Ts83aPg2s3Vf7GHdCUmhGUQSRAxyGjzHohW6hK5FlELo2ZyYq3Nld8L/+J0VFDLN/tR8g3c3R7RCR1wWLUWn5jJOCy7ujHljbUDdt5ht+JxdamGpMa+v99kepSy0g4o6yCULR/hs37UwaMZZSc2TbDor3xvMoVKeGPAVuK6iLMWAzLYUzdBD92LagcP6QJPkBoOh6VvYlsJzHbnzyUuSLMOxRa3iOABCWyWZOegM1VQfcXHQBiZ2uUDoWxTakJaatuvQDF1KrUkLQ5oXuJYMuvAcuXAOIsXeuE+n7uFZUgNYbHpkGhxLcazrU5qSV671+dEnFzg137gp294P1u+3Lf5etraPM6aB/CHgaOsoP3Xqp3hx40U2J5vMhXOcWTjDS1sv0fJbLM49y3i0yq/sfJufmvssR0/8Hsl+N15nkE+4EjYY6TFNq+D47BnaxpKgefYnYbIrWf1kSxQmlFJQzCYS3BWVvnyx8i8fw/EfhNUXxSdFOdVsz7iawVkVKJ2aBOT+teuF0qBTdXRWnZ9uQBlr0rUR5bjEdhV+c4Rds2X9+6oUZVUDKyrJpVuntFpEI00yLEjHA6JNH10olJGimRiRKFTPYDsFJ8waX8peJ3U89rTNYtTjzG6P2trLvLx4ljcPnWXkBTQCj7zU5IVo7UxVcJNAbogKiKuCYT/JOYR4htQ8GX+2PU4PlC7d0GWtHzNMco7P1UhyzZW9iKeXW3z6cJvdSc4/+eYVfNdmsekTZyVffmuTQ+0azy63uLI34V+dW+NoN+TkfAMF7I5T3t4YYNkWW8OUvDAMElG2RFnJfNMjrqxvPBtcpchKQ2DAsS2M1tRcm0QpkvLulldl5X/i2JCUYsC1Pog5MdtgY5Cw3K6xNU7Fi6WUqfWF0QwrF8aFlkNWaPYikQRaFkhvizQGuZVeX/Tf4oFeIjSP5nrnZ4yh6Usz0N4kxULJWDgjU4LqnktWKPKyICvEDCxsWOxG6cHM0yQrWelFJFlJM3Q5OiN3ObcL1vfiuu+kTHlUpIIfBqaB/C4YpAOuDK8wykY0vSbHW8cPOPNbt+ulPRbri5zunuZ46zi/ceU3aPktoVtmT9JSBkzJiyrnqOXA3GkGYYdza79LaDdoY0jKjN91DC1TYJqzNHXMcUvRrs9Lh2W0LTx5bVa6Nxvzknmn1ag2XUgTUjoRD5bxplAp+y36ZdWWXxRC3RzIBZHsfL8F3w3IxtB7WzG5ugdaYwc2XttDmQLbU5iyKoS1SvylOv4TZ7FPfpZye5V0Lyd6Yx1sn3hTU0xqoDSm0Ji8rC4+LrgaVYDVK2iOhvzE1X9DbrlkYR1qdfbac4yN4rntd/HSiN2wQ9OzyGsN9lSLHduv2tIl4Bhj8FyLizsTnphvMIoyru1N8B2LhVbANy7ucHK+fjBWLS81pxeaXNweYVsWu+OEmbrH4U4NWykmacTGMJFeKMROoBv6FKVhc5jgVs0udd9hb5zxztaIq7sRtiPNKcOk4JWVHkc6NXzbIskK1vsSnH1HMuFUydzLduDQT3IsS1rus0LfZJQFN9MaqnpOl3Ihy8uSTGnSvKQ/SdFGOi5tpSRzzuUCp43Gt0XeUmgotWTzaWWCpbXBc0QJY5QRYVI1yGIf+21lpRa+3bGkc3Wx5cuQC23ICpnB5NvyHJamLA3aCC/v2IpxVhJ6NpO4YC9KcSwLz7bR2vDPvr3Cjz25wFxDrBf2g/W5lT7vbI2xEDfK5U54oGTpRdk9s/VHQSr4YWA6s/MOGKQDzm2fI7yluPnc/HM3BfM7bffy5sscbx/HUpXYK5ughxtsTtb4ubP/Fsw8wbndN8guf5UwS8DxGSd9XtcTrPEOnTyj57pklJzKS2qWS3P7AseVR3t/Gk9ttuKuU1j6tHi2ZJEUVxXghpS9Hmkvp0w0ds3DbylsO5HXeTVRlegS/CbZXsRk3SLqOcQ7LmWhsBxVyQuR+icGy1Eoz0AJuvCwGgHKdXHn2liNJm6QkU58jNUgfutNTBpRTko4mOWcIxp0yJXQ7Knto8qSMgxxgpC02WEzaLMSdJnLRwRFRi9oUZ4+y8hvMEo1v2Uv0HMCCi0cby0e8+xkndP9FU57GaY9w/ricSYnznD0xCGGcY5nW6wPE+qewyQr6ARiZ2swXN2Z0K17OLbF4U6NXpQTpzlGSSB5d2tMK3BwHYu65/D80Q7dmsfmKGGtF/PilV08y2a5E/DW5giA0LbwPAvXstkeJgyyXJpgCkNWXSAaviJ0HcZJQSsUZ8din0e/oclnH/vyQcmapWZeGrEAcG3FKMnxbeHAs6wkKQyujdBIBpQy1H2Pmmczzgp0qdmdpGINYFnM1DyG1dQhy4JxLBcfY67TK7aCeuBwfCaUIdQWKMsiSnJyYyRoYw7oLs+RVTs21D0JvIU2pJXKZrEVoJQMbF5oebi2zZnFBr/v2UMH73sYZ7x0pcdMw8NC7A+SvOTJpZbQRtUx9mer7mPfr/wzxz4aXfmHienMzgfEleEVQjckdKQ7MXRCxtmY37jyGyzWFw8y9Nttt49xPsaxHHbiHZIioVQlh5eehyNfAGDUU7RP/tiBouVi7x229ja4km3ypKrTri2xOV5lpxjx46VN5jc4lw54Dmg7dVGETHbkG3bxyxB2KHWddDOhnETooEG2ClqFFKlF2c8wGryZAMfJKHKLMm9hIVlyOfKw64Z05KJLG5OW6MKgc6B05QLhyo21QqECF53bECU4bZuijFFbGf7ROey5JfLVTYqJgYRbjLGvF5eUBmPZuIGP0QZfadAZetxnwWjaOsHkBUWjhTfbIpwJ2d3p0+vM8eloyLfcBklWMmMSnuld4Iloh5kipj3bZsnL6PSukIaG488d5twwZufVd1nKYuxWk9rSEb5ycYBCScGt7kk3aSmZXzt0GSYF2miWOzJGbm+S0ao5WMD6ICbOCnqVha4xCqUMo1Tog2FSkBU5815AK3RY60sFMy2NzAGpzsEkNWR5Lq3rcY6jLFwfRsnN3uJwXb2i9wufRhg326LSc4uMLys0tpZgCrJ9lJUstgPGqdQWfMcGDDvjktB1qAcunz/eJS5K3t4Yia69KLAssRcotcYxHChgirJka5TS8F1OL9T47rU+lrJoVnM/HaXYGafSB1B1gSZZie84uJbMAQ1dm9V+RJJrQs+iGdiUpSErC1Z60U3fyUu7E2bqHqfmGgcdu75jcXF7zOFueOBweb/KlEfFgvZhYBrI74CDTs8K42zMynCF3OScnTlLUiac2z5HVEQcbhy+6bWBHXCsdYxLg0uMshEtr4XWmlE2InRCBumAtt+WJiSdUXaPc3H923yt9xq5UnTmnsXKJryZbjGrLIxls6ENZ8IupEOuhE2eK/T14RbpCLAoYxhvJJSRRR43mbwVoXSIIkVX6Z/RkLwLyvNwGw5WLSQdpyhTYnJQGaQTR9Qq5b550n7gdSvvLRvjgImpCqku2gSY3Qi72yVZT/AcQz6u5BdUXi7lrf2FDk5ZQOCKIDkIyKMIspzANngzDmE2ETPFMiHbXiGLhrRqTVzPxusu8npp0QldPj3aYcEr6aSw1GngRBGkJd44xhoOIe4xG2uGwQxmdo7ROMJ/4zU6zcP43S6LrYC4LEmzgigtKUqN5yiSXGaLXtqekOcl46rjtOY7FIXm/OaIF452xTXQElXKICnQpaEVOgeTc4ax2KuGjo2lhC9XxrDvt1RoCFxLRp1VZLRdZds3qkb2jSgtbnaDD20J5FoLvRRnGgdFWQpVsq9CGccFudbsTTIWmx5xVpKXmmOzdVqhmGL5jiUWt1pTlmKapYzBtS2MNtV7VTKaTRuGac4oKfBdl8AROqdUio1+jGVJR2bDc9AmJ/B9UbrYStSzjsV8I6Bdc7AtaWDq1F2iVKyML26PyYqSvSpz/rEnF6oBH00Z1ZcYNPqgMHphe4Rn25yabxzMY72TMuV7SVc+DeR3wH6n536GvT5ZR1mKGXdG5FzV49vR9k3bgXSEnuqcou7WeXX7VXaTXQDaQZtLg0v84lu/yBeXv0jX7/L67uusjdcYuD7NzilWx6s03ZC+W+fKxjUuFQmLRuGGc0Rug1XdIaZgnFs863Zozz8LF78Ctke0qkh6Bp06xFsFOkkpjA8J2J6D5Qn/qGoGk+Zk/QI7NxhVo0wz7MBBR2kVIYq7dvvLc1VOaRR6NAFjKPojlJei/C1MWQgfr/V1O77ilmCulEgm9uc+1muYKAKlcD0bg02xt0e+t4VVq2FbAY12A5UOUceO8zw2zZ11Tm2+TdjfZT4dEgxSCELoNPGHfZw8h7pH2ZznZN5nNa4z0BbLjTrPqxHfyZu0y5j59VUGO3ukfo006GIIODnXYLU3YWuS0/JtlmseNVeKhDuTjN1xyiDKyEtN3bMZxQWuJTK/rWGM79h0ag5xpmkEkuHXLVUNTS6lOafiShQGz7bQSpMUcvNjWRalMjhIJN4P/AYJ9Dd6pmSFxgLqnodvlyhLYeciVbWVEut5o1EYSq2ZpEK7HJ2p8dljncqvxJDkmt1JzjDJD5qGCmMILQttKeqOTVkZeinEqfHSTkzgWgzigmGSYlsyhs63bbo1F9e2mXHEu8YYCZqBZzNT8+hNJEgvtwNmmz5prmmFLs8st3j52h6H2jVmah5qTnFpZ0wzcA+mNUVZQVqUB4XRJxdbnFsdcG6lx6cPd3Bs647KlO8lXfk0kN8Bt3Z69pIernJZqi8dbFOUBTvxDtvRNt2gy7HWMdIy5crwCkv1JbaiLU53T/PW7lvEZczWeIuW32KST9hL9lgbr7EyWmEr3mJttIZne1hYbMfbXMiGENRI45KgLPjdbIcnXY+u36KZjnjLtRkbl8+sDAnWPezAYrIypMw1qAaUA3SqMUUGuZEvcCl5nN1pUiYD5D5fg1VgxilFWtne+S4H0orbwUKkiU7l1bL/oxREEaYsyba2II7BF28atJYg7nnyb71eGXFRdboU6F5P5ol6Hna9jhmP0XkOeYZrNG6RE7iQbw4xgx4nT57g2SMxO/6EIMhQ8Q5Jr0/ebEsGuLaC5QWYRp1yNMQ9/ARplLAQ9/AOHWO25hLt7NHOYxqvnqeWRSR7E5aaIWc6Y641FOlej+XJiG3lUXaOsXB4hkma05vkPH2oycWdMbtRzt445Wi3TjNw2R2lrA9T6p7HUtvDoCiN5sRcg4s7Y5KswLLEx1shHuDGgGXbNH2bSVbiGhlGYSnxB/cdm6Q0eErjuzaDuCQ3wpFrA1GODNwIbRpV8Xe27rPUVKwNYorSoCyZEYpl4SiFYyue6NQIPYeiaqKq+TZfO79FWhSS9TvVRwsH3iZCi7j4hcFog9GaYVoSOAEoQ6n3C51SvAxdl3EmJluObVHzbBZbfjU1SPMHP3OE33pni6zybzk2U+MLJ2YrN8sanz7cZq0fU2jD+iDBsvp88eTcgXzQttRNmfXzRzpc3Bnz1uaIzx7r3lGZ8r2kK39ogVwpZQPfBlaNMT/1sPb7ceHWBqC212Y2nKXhNQChWt7YfYO23z7gyr+5/k0CJ+DJmSeZC+dYH6/zq1d+leX6MtqI73U/7TMfzrM2XiMpE7aiLU61T7Eb7aLRPNF9gm+tfYtRMWIxXMRr1EjLnCza5u14myN+hyP2DEFvRL41Zr014vSJZzHDHuk4w3YLimFCPjGiHS8TIVJTg8kKsCx0klfdlhqtjARWy5Jeb9+GUlWzQW+xAXUcSadcF9JUgv7+8Il9s4/9DDzLJAKYqmJm79MrEryV76FsB72/3YGJSC6/NxoY35PgPokgDFG2jUkSXKVwGjW8tSs40YCjJ0+zF9bJPZdaOsGUGZocFcd06iHe8jzDJMOnJHJdFq0CQpdoOEYpxR/YPEf81luM6038xgy1mkdje4Uj8Q4vtk6w6dZom5LO9kVypyDb2uZkETNXLHL27BNcK0WKWXcl4xzFBafm6xzphFUBUTxDRknOcjukFThc2BmLn7g2TJKCwLNwLUVhDL6jsC2bJNPUfYesKEmqdv164KKNtNSr/aYddZ2G8WybRuBQJiKFtKt5p5aFDLHwHQJHpv0MKlmipRS51iw1fEoberE4T9Y8JfNULRnubCtYbIlpVVYaPMswTguUsXAsTZQXaG1ohjIMO6wGNudavFrqvoPnOCR5gW3beLbNYiukFTr8wRcO4zv2TUH10u6YoOq+DVybpVaAZ8OF7TGHO7UDa4BvXtxFKakPzDekQeu5wx16UXbbAuc+vpd05Q8zI/8PgTeB1kPc58eKGxuA9tUpcRET2AFv7r7J+nidpcYSm9EmJ9onAMmw5mvSBejaLlprojxCG41ruZS6xLEcVser+JbPuZ1zvLj+Ip7j0fbb1Nwanu3Rpo1SiuPt42xH2wzzMSWaY+0T2MpmvLOOXYuIWm3U0hdQSR/7O9tkOxHgonyHMkor/VpFjitbAnlvJEHZ8yTgGiMt+GnKwXwwy4IwvD5hwHEkyGoNQQC1mgTtIIAkuR6MwxAVhpiikIC//3hUFa6UgjDEbrUl9usadqNBORgAYNKUcjxGTyZYWsuaALTGZBmW64LvUwwGcPEi5uRJPOsis5ai6DbJ+nWK0Rg7mYApsbMUL/Q4dPwIhz2fOC/Z1QGhySkGPRYcTfPym2SeIczGHEpKRukineEuwfaEM5/q0ml2qM/MUOzs4L71Iu90jtKYm+N426N24RzNEhpvXWSUlDz19Fm+XVsiDWuURuNYkoEGrnh6z9VdCgNRWtKt+7hKMfDygwEQcVZScy1yrejWPDxbMTAydadTczFIy7rvVNx51eQrrfcWTV9sjxeaXuXlndOvGn98x8a1FZ5jE6VyrNB1mKm7oA3n1kYstwMsoOa5pFpjGfBti6KQocyBq1gfSIGzUdUJdtKMmmuLIsaSEXCF1hijsG1FzZO6wKFOyAtHu0Rpwbs7EzaHMQstn1PzDdqh+56galsWaSFGZIEroaruezx7uEMzcMTIS1eySm1IKo/zUVJwbKbGTOPuPPf3kq78oQRypdQR4H8D/OfAn38Y+3zUcGOGvjpe5crwCifbJ+kEHQbJgPO75xnlUtg83j5Ow2tgMDzRfoKNyQauLbdry41l0jKll/S4OrpKYAVoS/yqN8YbaKMxyrBQX6Bb6/JE5wnSImXot8hNzlZQw7d9DtsJw0OHqNWPACW0DmE//cNw/gJmMKAcbMm3XFmVpOGGbNp1Ub4vwVGBcj0wBlMFbuX7GKMrMXGVsds2BAFWvY5dr2GURbG9jdVuQ72Osm3Ic4zW6CSpsnSuZ+2WJcfeP24cUxYFluti6nXsRgOdiMWr0hpVlnLh8KomJgBj0GWJqrJ2Ywz5+jp6MkFp0WfYs3MYrbHCEKO1NCqlGaYocZa6BNs7nDg8z5lDh+iHJdvnL5CX4NVrnKp5WJMxRe8K4zIn9wJmA4v6aJPa4TZ+TVM0LBxGzA8G1DIPttepxQmHl4+zM0450lvF9kZ81z3LxcTm5EKdJ+Ya1WzKnJlGQK51VSDVxGl5UJwsdMZ80yNwHdb7EaXWZMqmEbj4NYtT83UKDa+vD8hLjTZiGbuvbCm1NBUlRck4lTvA5XZA6FisDWMmaUrHeEyynElSYilFnBXEvs3hTohl22AMi22flV4k/uu2hTFG3CPrHk8vtTnaLXhtbcj2OMWzLTqBC5bYH8SZlvmkrkMrcHFtRSNwKQrDs8ttLu9M2BzFdGseP/P8EZJc8+tvbHCoHdIMnIOhyq3Q5SefWeSXX16l4YtWPi00SV5ydrHB+c0hzy53uLYXcbhbY2MQo9jnvV0u7Y753Imj9/xef6/oyh9WRv63gL8ANO+0gVLq54CfAzh27NhDOuyHi1sbgrq+3HJtR9s03SYoiPOYzWhTilKV0+C7vXc53T1N3a3TCTpoNIfqh9icbJKWKRYWO/EODg5HWkfoJT1G2YiszNiL9zg7c5bV4arMd9SatEyJi5iu18XGJikS1qwhc4lFvLTEdzHU0oJmb0DR20P3B9c5aYAwxHJddJ6jKpmfCqXRwglrGMuC8Rh7dhZdFJT9viTijoMKAkySSEatwG41cZcPY9Vq5M0myrLQcYyq1yk3NihHI+HG63WUZaF8D727JwFcKaxaDT0eU6YpxDFlGGJXQdik4l1u+T7KlQufrRTleCyZfS5Uj7EsWcfiEuVwCFlGEU0w2mBZFlajgfJ8KZpqjXvkiKwzSej+4T+E3ekAEL3yCkeOLZGTka+tYdkWeVGgoojOzAxOq4WjYqK0x/C1l0nijMAUPHvmEFupJl+7hrOxSlFvQRDw7HyH8WhCZ5xwthjwwueeI81FceE5Nn/2R57gm5f2cBTV8Awpqp1tNplkmlZgk+71SC5eYGEwIg/rDGcXKMI2wyTjWi/mUMvnzHydd7YmjJO8mrijKEtNs3JhzAqNYylagQxmTgopJNtK0Y8zUbdUlI9BoUvNICmZb7hsjlKaoceyEcXO7jhjnIrLpe9YjLOCTs2jETjMORatwGFnlLExignrPkfaHqmW2wTHUvTjgp1xxun5Oq5tMVN3hQayxedltR/jKEWUFsw1fKKsuEn+98KxDlf3IoZJQcO3OT7bPFh36NqM05JuzcN3Rae/O0k50g2p+c4jl1l/mJr1DxzIlVI/BWwZY15SSv3onbYzxvwC8AsgDUEf9LgfNm5s9Gn7bXbjXX579bd5evZpFIq52hyX+5fxbO9AsWIpi07QAQXr43U6QYf18Tqfmv0Umc4Y52MG6YCzM2d5p/8Oc7U5lFJ0/A5RHmErmW4y482wbq2zG+3iKAeF4snukyw2FhkkMu1dHVnGuTTAzTWetshffpX+hTcJ4gxlVRMFajUJqlmGcRyU62JKjeV5+MeOk6+vU45GQmO0WngzM6JDKUu8Q9KIUY7H5Gtr4HlYnS4my0m++13sVhM8n3I0QtVCzO4uTruNlefoqvBp1WoSaB1xX7QcRzLlssQkiQRc3yffWJcgXK+jHAfLri4gto3Jc6wgwFgKk6QS5IMA/+yTWL4PRYGyLErLxsQT7CUpRjvtNhw+DGWJOzuLd+YMZb93EMT3oQDvyBF0HGMmE/RIaCf/0BL55hbZZIKlFJ0kQacp4ac+hT/bop4VbK9q8rzENZqTs3VCz5HO0NGQJw8FrM/UGMY5pxebB1/aq3sRUVoQ5ZpaOuHZrEc7ijk31Bw+dggrWucbRQHtJgu9LT536UWsIGStNsPVo0/yxNJhxu9eZqnfZ8/yedfrsk0g03yUDHYojSLPNZZVUsYZuZFRbq6j0LkMRpYuS0OcF3SV0BRpYZhveGSlwbcVV/fE6qClhFM3GFZ7EVuDlIZnkxSawHU5tehxerGBa1ssd0N6Uc7KbsT2KMWyRcu+OkxIL+9R82xm6h6H2iGvrvZZ7tTwHZlXejv533NHOpTaUPOcm3jsMwsN4ryk4dukhRavmXbA0ZkaR2dqB81Bjwo+bM36w8jIfwj4GaXUHwACoKWU+kfGmD/5EPb9seHWRp9+2qflt+infZRS7EV7GGO4MLjAidYJ6m6dZ+ee5VTnFOvjdSlidk7xU6d+il4qGfepzqmDNv+rw6vspXsoo0hICJyAtEjJtUx+eXrmaTYnmyRlwunOadp+m8AJaHktRtmIHb3DzNMnKEuLvd/+OtaVdfzhkMyyqbfaFL2eZNppKjSEbWO7LugSu9NBhSHe4cPkOzsoz8NuNjFVUDRZRjEaQvL/Z++/YmXL0vxO7LfX9jt8xInj3fU2XaWp7qpusrurqWY3R0MKGPTIQJAgAQPoTYAECDOP0szLCBIg6oEQMQIEgZQESOQAQw7NdPnqqiyTmZX33rzennv8CW+330sPKyLuuS7zlkt2JnMVEpkVJ8w+ESe+9a3/9zch2aTgYprg+0gpyeKYrN1Byyt/kazXRzNNtJyHYRqktkXaaJJOTwS2DaMRslol7XbVbWmKllOxasb8AtlwqLrmJEErlpBhiByNVOGu19FzOUBizM0RP3qEJiWG6yLOnkVLEkS7Tbi1hXPxIpk/hjAiiyKM5WXS8RgZBOjFp8c35uIio5/9HE3X1cYxuV7D80jCUG1Aug79PjKKyIZD/Dt3MMolnFKZhWqOcOChlxTzA4AoQjNNivUq9aJG2NknPeqjd4qkGxssl12iJON81ifc/pD+MKBr56ibOU7e/YhubRGDjFO7tzm5e5fAsOjpBmYc8nu3/5rXi6/zYK7MdSrYvQGv9Xa4XV3Hd3L0g0Rh7JYyyJKZZBSrKDhDNzCEPgl0UiyZNJWEaUp/HFNwLMaRxun5PDvtMbvdANsQrFU8mqOYcRjjmga6rrPXCxAa2IbgsK/YJBeXCryxXuHvXFzkl4873C0Mud8Y0BvHZDLE1gWNgbI1ODmXp+JZ3DroIzSNXhBTsA0GQUzeNp6i/70MxwblfljxLLbaI8JYWejO1/J/IweWv2vO+m9cyKWU/ynwnwJMOvL/7Re9iMPzgqBRPCJv5mn4DaSU+IlPyS7Rj/qkMiVv5jlZPkneyrNaXOVk+eRsULrG81jdn278Kf/Vtf8KgUAXOuN4TCIT3l18F13ouLpL2SnT8lskMqHqVLnfu08mMwpWgWE8pCMSrtPiVKcPi3PQ6DGWMXYUqqJsGMi5Oej1ZlRAvT6H+9rrqhgWCtg5TxXp0Rjr9CnivX0FpaQmwnXJehOYxjAQlkWWJKrARRFaGKJZlvqZ62IuLJIcHCANE7G6qjDvKCLz/QntMFbYvBAgNKQfYNTrWCdOIAcDoqMjtMlmkqUpMo7Qczms1RXc11/HXFgg3ttHs23SdgetUEBkGebiIsZggCgUQAiEZRG32iClOlHs72NUq+S/8fuz9z/tdiFN0Stlsigi6/WID49wLpzHmKsTfPIJUtcxq1WywQBjZQW5t0d6eMj4ylXszQ3MhQWyRJ0u0iBQOHW/j7WyjFEuM75yVXHfyxVkEDC+cpWNsxe4etjF+OsfYFkmpZUy7jjgNSuifTim3Gtyatgi19pn7HgIw2Rh1MafW2A+NCm29zn7++doxE2OhiG2LTkVd2nMzxFnY2xDTBwdJYd9peQVQmAZGoZQrorjICXnmFiWoD0O6Pox80nK6fkynVGMZQhFFTR0bKFhaBJDCPw4ozUcESQpaapcDQuOwXzeZrfrk04G42GS8dHjDmGcUvQsKp66fbnk0vUjOuOQMFHhyt1RiGvpzOWK3D4YvHBI+TIce1rgg0SFQecdg2re+hsps/9dc9a/4pG/ZD0rCMqZOYbxED/2WcotYeomd9p3CNKAzM+Yd+fJmTn8xJ8FTnya6dZKYYWL1Yt8cPgB3bCLQFCwCrSDNikpOTOHZ3h4hkcv6vGDnR+wVlhjs7SJoRmM4hFhGiIf75MZgqEcY3sazkASEmNZlsKIx2MoFrFXV9F0HaNcViZY5TLm8rKCMopF4v199EqVwY9+hFYozDjdmmkifR/GY8TcHEmjoZ5TaEhjQh/MMmQcEx8dIYUajpnlssLkgaTdUgrEQGHgJAm665GMRqRhSHJ0pLph1SaStNsYxSJarYZw3RnMEx+oMA6jUiG6exe/20VzVSftnjtH4Y/+iNGHHyr+eZIg45i01UTLMqKtR0Rra8itLdJ+n+TwEL02R/7dd4n29wlu38HZ3MRcW8M+dYrw7l20KFKbwOIijEYK3x+PMWo1ktEI57XX0SsV0HXigwMk4F68gHvpEuHWFsLzEK76+9Em/3aO9riYxexLydh0cXXB8nIFR6ZYwy69nYecT2KMo/tkmk7ieYTlKrV4xHLZhaHq7FYrHmYwQhy1ybW3eagnnIwSusOQ2MkTLa6Q5i36QQxSzqh59xvDSRxcRs42ydkehi7o+hH3DofUCzYSjUvLJe43RrTGMf0gxdQ1dtoj9MmmoAllNdAZxcQpzBcs0hT+xZU9/taZOo4uVDc+Cim6Js1hRMGRuKbA1HVu7Pep5WzQJGXXouvHCKG98pASnhT4T6MY/k1Zv2vO+m+1kEspvw98/7f5nP+u1rOCoLJdZm+4hy504ixmf7hP0Szyev11ojTiQfcBe8M9FnOLKh4OnsLYp5L+qenW9eZ1hBD86eafYukWD7sP+c7WdzgaHXGidIJUpjTGDUpWibpXJ8syDGFwt32X5cIyNbfGh/sfYh91MCs6lUOfcK5AIciID3tYUkfzPKSuY1jWxJBDR6YJ/vXrmPPzJJ02zqnTIATuhQuMr11D+gGiXEa3LDXENE3kaKSKei6H1ushh0PQdGSno6CHCZYd3riBVqlg1RX90pqvg2mhOTbZeIzmeSTbO6RxALquiqDvkwYB0vcRnqfwf9tGr1RUp9xp41+/gbm5iXvpElm3S/T4MebiEsbiItkEOgI1iHXOnGbwwx9CmiEsG+fd99DLZaLHjzn6R/8I5+IFzOVlksND0tEY5+xZnDNnFfMll0eORuiFArmvvUXw4CFyfx85CTrWNA33/Hn0QkE9vtWk+Kd/+hzuDpBeu4ZefrrAaI5D2u3gAusnlpFpqnB+IB2FiMYh7v1brAFJyWMUJWThCAKDmhFiGSbYOQDyiU9ytIMwNMordcK9beI4JSitYusZ1t594oUTCM+m7Fh4jk6SKqzZEEJli5ZcVisujYFPL1DvYd41CIcRwwnvXUqJLTTlAimhYAsyTWPgJ1imCpHIOxaGEAyiGNfQlfy/5NIYqWi8UZRMsHdFJRRC47WVEuvVHOM4pTlQp4JxmLJR+5vXTf821u+as/5VR/6S9awgqOJUZp7jtzq38AyPpfwSnuFhCIOLtYss5hZncMq0iD9rprXV3+L1+uvc796nYBWwdZudwQ4/3fspiUwYhSOO/CNKdglTmIRpSM7M0Q7a9MM+g2jA4+Fj6m4dqUkOGBLlDLI85GOdYd6i1DGINZP8wgLG2hrCMEiODsnCCJEvYDkumucRbW+jSYl1/jxxu4VeKmFurJONxhjlsupqt7cRKyuQxE8Kt2k8SfwNJ3Hptq0Kuj8m832stTWslRWigwNEoYjueooOWK8jgoC02ST1fTAMrPl5xbDRNLIkwZibU1i975N2Ouimhba7C5cuEbdaZEGItbyMc0ZtmFkYkg4G9L/7PdzLlzFXV0kPj0iDAGHbSN8n2tpSaTdjn/jhI8KHDzHm5hC6jvvmm4hcjmw4nGDxYJ86Rdxukw76pJ02oljEqJRxTp8CoWOsrmDU6y8s4gB6sYgMAjTXJR0MSPb3STod9FIRY2kJzbYJr11DZhmaaZK222RhiF4sIMc+FpNczVoZqWkwatEqlsgWV7CaDap3PiE7OkRzbHS9Ts61CUYtLu9eJ6gtMmfDG0dX+CC3jFVep7ZUIWh3WdrbJe0PyNUqzG2cZj/JEELnzdU8qYR63qE7itnp+qxUXDqjiKNBoLjlzmQQjcQxBbqmYRpKpWmbOkeDgItLBXp+xMl6nuWxw3ZrzNCPWF/PUfJMzszn0SYslemQMlfLsy4ljWGgTh2vsL5oroW/a876V4X8U9aLEoGKdpGG36DqVLENmzANCZOQU+VTDKLB7H7PYuygOvteqIQv2kR4szPc4TuPv8MwGqpNwTUIkxBsGEUjUiPl48OPqbgVDv1DxsmYbtDFMzwEgtFikfyjPp2lMnaaJxxEjFdrpPPzVL3l2etkmVTd5+XLpL0e8e4uum0T7u4p6XUSY508Re7NtxhfuTIb/MkkQS8WKf35PyDe2yf45BOC8fhJIYeZctOoVRGFIsI0SXpd0k6H6OFDtEIBc2mJtNMhHY2QUYTUNPRSSRkvHRyQRRGmZaEJTXXi/b6CZlwXo1olbbUUrBPHGPP1aaTN7Br9Tz5Bd130cklh7GGI7nkkjYZ6D+IYLZ8n3t7GPnkSY2WF5OAA/8YNzJMnMcpl/L19zNVVxQDRDazFJYRlMf7gA2QcY6ysgNDJggBrbfW54Wna7RJOoBtNCNJeH811iR4/Vkwiw1CwzN4e4eNt9FqNNAiI7txBJgnOyRP4UURmKOWsputkYUgyGGACa994h0O3gv/RB3ijDmfPrdE1XIY3b1HXTeJSjgoGYdKh14oZFCr87cslwv4e8VhnPjhEP1/nygOY6++R/9EdouoGpTPnmS9WMXWNrdaYauYjjh4xJ0N6hovmlugXcnjRGPfoEfnYZ2A43DUr4FZwTEGcpliGRiVns1ZVQ70Tc3k8UyfJMtZqHmfm87y+WubqTpfdzpit1piSa7JUdknTjCiRDIKYH9w++tTiPGWAZBm0RyG3DgZ8uNXm71xcZKOWm93nb1qh/11y1r8q5L/iKtklXpt7jZ3BDsNoSM7MsVZZQxc6nvBm93sWYwdlplWwFNX+ZOkkvzz6JR8dfgQSckaOcTrGFCae6TEIByQywdANzlTO0It6GLpBf9gnkQkNv0HOzPGII7ZrIYudLqG1xuLpGsNxRskpsiGqRNvbCMtCFAroxaLyJ3/4UEEJ1Sr6YEA6HJANhiS9Pt47b2OtrhI9eogMQozlJcxKFRlFBLdvqy5a44lIxzDUsBTIwgi9pJGMhmTb25hzc0hNQ4YB4a1baKapBqBZhu44iEqFtN1GZhn28jJaoYCMIsLtbYxSCT2fR5hKwuheuIBZq2HWavh3bivxCpCNx0QPHijV5+YGxAnZcAhxTDZVo05VpmGE7nkIy1KQUa9LNhrS+Sf/BHNtFffsOTTLJO121NA05+FtbmKdPMnwpz8levAA7cIF7LVVEDr2xsbss0273eeGmyCJtx8jkxi9WsVaWkJKCLa2yLo9NNNAd13M1VXMhQXSQR/dcRQLaGJZoBkGRrGItbiIPe6zsruF895rZJ1VsiRh3rYZNreVQdlCmc7uAWa1zFw1o1zOUX/3NP1On4OPruJLgdvZ4++O22zZBZL8HCejHlnrEXE1x+nTywTtLsbj+ySaRm6uxklHo7pzwLa0WNq6jZOEkCSMYtgwLBrld0jSEvWCzZlckTjJWFnwqBds1qoejYFL0VU+6AXHpOfH9PyYVCq4oTUMubbTpV6wWa/msA19Bj28jJ73sDkiy+Bxe4xj6tTzNn0/4q9uHPIfvb0K8KWxp33V9VUh/zXWpblLpDJ9Lkxiio3D8xj7s/e5NHeJa81rhGmIpVvEE18Tx3AwhEEv7DGfm6dsl7F1W2HkGPiZz0puhSAN2O5vEyYhoujy0Ito2SNetxZYuHHAuLVN9cQSi9ULiL0mhmkgqlV0y1Ic7knRzIIAs1qFoiDe2WH8/k8xV1fR5+povR4i5yFyOfwbN2fJMAQTKf9EpPNfHx3xDxsNjtKEmq7zv6rP8z8+fZqo2VT89dFIwSqgCmoUkuVyZM2mKsDFAkyGqvapU6QTKqLI5bAW5pUz4/IyyWiEtbqKMJSxVhYExLu7ZHGMMTeHWSqh2TZGpQq6TvjwkaI16gLNcYhbLezFRXUKaLdJOl2Meh3NdbFPKihFcx3y3/iGGlZathq2ui6lP/07RI8ekUUherWKvbHxFKzyouGmUZ8nbjTIvfd1lWI0GBDevYsMQ3VKWVxCBgH26irSNBFJjOY4iiXkuogwJI0ihG1jLC4ghE6ws4PIF3BOnya8e1cNkQ2DeGcHfTSmbBiYngBhYJRU0+DpMLd7H+vCRcJ7W2SDAZdEn/7qSfoypdNtUr/yI3L6m8zvHzIq5vj9hQqNQQiGzpxnUPrgfdJiFScOCaREyJRLqzUWxD53F8/QwOKtimBl1KJ/8wED0yVbWUUa7lPF+a9uHHCilueN1TJ7XR9D11mpekRJyuZc7pXoeSrzNMQxdRxTbehF16IxDHjYHM0e/2Wwp33V9VUh/zXWs/h5wSpwpn7mKSjl2ftoaOiazifNT2YMlnlvnpJVohN0SGRCzakBcDQ+QiJZzi1jCkUli7KII/8IS1j04z5dv6uUlpqOH/uUnTK60LkS3+cPvvYmxa0R3eYR0g7Z+FvfZO7EOcJHW4T37iF1HS2OSbtdjLLCYNNGQ3W3oxHR4QHOmTNY62ukQUg26KuBn23PaHvKqUnnv242+T8cHhBMqGfNNOW/PDxAA/6HGxuKppjLKQbNaATFIhgmWberREP5PJphKlGS64IucF9/jbTTwbt0CX1uDk0ThHfvkA6G6K5D8c/+jLTfI7h3j7Tfxz57BmtxkXhvXw0/J52/WakgTp2EJCE+PMKs11UKUa9H0u1iLi4qywHXRTgqoSbp92fwCLpBvL1NOh6jex7mxgakCd4bbzz3N5H2++jlCtlwQDS1DfA8pB/MsPJkfx/NcRCuiyZB2DYZqA2130cUClgnToJpET98SBoqeqZz+jSaZaHZNqJaJX70COe0GlIHN24Q3rkDgByNSHo90nYb7913EZ6H//HHDH/xC5Jmg2QwRJsMmUUQUH5wk/r8PPLUGZrtAYP+mOX9+/ROnifvGLiWzl7Pxxz2WM/rjHM6sZ1nyfOoOwIpNPwk5ULW489eO4d55ybC81g8u4YMAu49vI115iLexGjOswzSCRxyvlji3KI6yUkp+eGdI1xTf+o9fRk9r+ia3DoYUM/bs9vCJFXJRr5qiL4s9rSvur4q5L/mehF+/rL7vCgO7v2992mMGlyoXeBe5x5Nv6nw9kmHvp5fpxf2iNKImlujOW5yMDzA1V1iYnpRj5yZI2/l6Ud9bN3G1mwGyUCpNL9xkSOZUqme41BYLNQ30EslksYRuuuQDkforkOWpsR37hD3+wjDQGoaWd8n3tnFqFaxN9YZ/fgnaIUChm0TmyapaSq4Ikn4vzYbsyI+XaGU/N8aDf4n58+TjscKiTFNMtdFkxLhuqrrnsj1hW2jFYug6wQ3blL4wz9A1udx33wLmcT4n1xHlCt43/wDhb+323hvvE7+m99UeH4UI1wXkcuroWK7DXGM9/bXMOrzAGTDAeMbN4gePcK5eBG9Pkdy1FAwz9ycunDLQg76qogLQXD9OnqphMjnyaKI+Pp1nIsXXvhZ68UiabNBcOeOmgNMXCCFZZE0Ghj1OslohJhY9IKGDEOyOCZ6+BA9l0PYFmQp9uYm9voawx/8EE1C2u4gKhWirS2yXo9k/4DxBx+g5fMk3a7ytokjjLk5rHqdaHub8Ucf4V68QOoHZKMhaJoa+NoqwV5MjMrMlRU822R9fR7nzDJ+1MKPejT0OaIk4eRcnlrHxCyeQA5H6KUamjYJlej1sM+tYroS/WgP+cyJZKhZFI72lLH67g4MBywHGkelOiw9aXr8OKVecF6ZnndiLseHW236fkTRtWb+LOvV3Oz+XxZ72lddXxXyz2G9KA7uYHRAK2yRpAkVt4Kf+oxGI0zN5Gz1LFWnyuH4kDRN+ejoIyp2hYJTwNAM/MQnb+WxhU3VqZKzchiaQStoUXfqrBXXsAYh7mEP5/4thnZG+u46erlM8U//FOF5BPcfENy/T3zvHpk/RrdtpdocjVTn3ekQ3LmrBo6FAmmzAWjq2O95M1fCw2eDIibrKE2Y5sFOcXEsC71awV7fILx9G5mmyojLMBCapuCVOMK/fQdzcYHg5g2ysY9eKmFtbqIXnlj5hFtbeOUy9sYG4ysKwhL5PObaGnqtRtJoEHc6RLu7CqJZWiL37nvYy8sYS0uMP/5YQRb1OsKbzDYmqky9WFQ4u/qNX/jvZ5e9sUH7Jz8hPjhUylnDJBuPwTNBF2iWUtVKmc06+uD+fcLrNxQb5rXLpM0WaXuH5PAIzXXR5+ZIjo4IHz9GdNpw4iSaaaEvLSreepaqU9V8HeKEpNXCcmys1RWSg0Oi/QOshXkyL4eME8zlZdJuh/jhQzXgrVbRpCQLApx1BfmZGxskP32fNdtQpzDTJCPFOnWK8bVP1MlqwqdH19Eti+TwkOCTTzDmFzCXl2efU66YI9rbwx70VdBHsUQ56RM+ust4pYxbrcxoeL93qsaDhnrPP4ueV/Ys/s7FRf7qxiGNYUDVs1iv5hBCFXngS2NP+6rrq0L+OawXxcY97j/GEhZnqmc4GB0QZzFCExiawWJuEQ2NilOhH/TRNI2KXeHy3GW6YZf94T6ZzGgHSuK/6q3S8BsIBG8vvI01CBG3HlCbO0nkWnixZHzlKt4br6OXy+S/8Q3Vne/uEmeTiFzTRAhB1u+jSYm0LLJ+j+D6dczVNeWFUsiTSQlTWGS+zryuc5imz/3OdV1X1DrLQkZPjrTW0hLW2hrGwgLBjRtorkPa66sutttBLxQxymWsU6fRsgyyEcZcjfhgn/D+PUQuh7m4SBb4M5ZI5o/VSSOfx1hawp7fIHz4EE3Xldozigju3MVaW8VYWsJ74w3sjQ2GP3mfaHf3OVWmvbHB+No13MuXCB88IHrwQF376oryYnnB0icQlSgWZ/4w9qlTIATxwQGlP/uzJ5vOxEtGEzrWiRPYJzaVYtVxwLbJwmAGPcnxmCyOkElCWmrjnD+PJjR1gkKDOCFut9ELBYRpolnKnExmmfK0MU1VwAd9suFA2cMX8sg0QXg5kn4PMwyI9/YxloAwANebeJ1raBKMuToIgffaZUbXrpHt7pAFIXohT/df/WuckyeRpgGtFtlohH3mDHqhwJKj8aDXR5Yr2LZDmKTgOnzt7BL9w106To5i7HNi1MBt+LiWy26uTsd0P5Oet1HL8R+9vfpSZsqXxZ72VddXhfxzWFMGS5qlHIwOuNO+wzAaUnNr5Mwcp8qnWM2vcq97j8a4QZIlOKbD6fxpDkYHbOgb2KZNySlhGzbDaEjDb/C1ha8RpiGDeEDNrXF57jJBGnDv5l9jpRpH/ZCVbJl3l95DJPqsi9XLZdxLl/CvXEUr5Al++TFJp0MWRWjTbtuyIJ8DIZBxjLW2Snj7DmapRCQlDAZkwxH/i/l5/k8HB+q2ybI0jb9cWsLe2CBpt9E8F00Tyj9Fgl6rYRoG0YMHqvNNU/SFeeR4jMjl0QsFtCwluHOH+OCQ3re/jVEsKPikWEKv1fDeeH3GEjFXVpFBQDYeY29sEG5tYW1sEG9vI6NI2QiEIeGjLSpvvQUw2dB+n/H162puwBNVpl4uK6ik3YY0wzl5Ug1XBwOig0PSbveF/HHhuuiVKrrjzG5Lg2D233q5jPfG6wqD73bIohD38iWS/QOFndtKOJX2+hjz82TDIfbGBv4EEsmiCHNxgejRFtH9+wgvp7r/VkuJt3Rd0UorFYzFRTRdqO57ZQXheiQ91RTo+QJamkEUohkmSa+PNEziVgsZxzgXLmBORF0Ame+rIW8+jxuGBLduK8+bnR3MiWOmnssR7exgra4S7+2hra/jZTEbG/MceS6DICZnG2zWckpl2u1gOz79n3wPmaXElQpupcLpYDxrOD5rfRqd78tiT/uq66tC/jmsjeIG7++9z95wj4JVIJEJnunRj/p0gy4lp4RE4hou5yrnyNk5CmaBKI1wdAfLshgnY3b6OyQyAQ3Kdpk/2fgT5ty5GSOm5tT4t4/+LeXEJc47SJmxP9pnFA3JeQuKVjdZ4dYWeqWi1JY7uyT9/iTFR+GnaBqapsyuszhGmzgkCsdBK5WUU+BoxP9yZZn+z3/O/2tvj06WURCCP9/Y4H//X/4fCe7eIR2N0fN57LVVRD7P6OpV4v19nEsXsV+7THT7Dsakc85MCw1JtLuLf/u2OtoL5XueDgYIz8UYjoh3dxG2Rf4b33xOAj8dVBpzcwjHIdnfJx0OFaMk5z1VIPRymcI3v0nhm9987jOzNzbofPRLtEkAh5wEcFiTjcJ7QaGxT58muHFTqUAnj8n6/adwdb1cxiuXSbtdoocP6X/nOyQHh5gb6zhra8jRCC3nKfqilApWqtWQUmItLxPcuIkol0lHY4XxJwlZqLx19HwevVrBKJdnuH9w5w7+gwdY1RrW8vKM0hg9eoReKaNXa0QPH8CjR9gXLyCEUGKlO3dmQ159cRGyJ5CQtT7ZJINQaQGSBC1JcE6dJO524SjBPn0K78zr6FtblCYzjOnKfB+EoP+976MZBnqhAlFEtL2jGoaXvL9frZevrwr557BKdomiVaRn9oiyiLyZZ6WwAlLZ5epCx9AM3l18l7XCGt/b/h6toEXFrvDe0nu8v/c+hmaQkREmIUESsF5Y5277LjvmDidLyqDrO1vfYT43T33ZRItSyDLi/T0ebP8bCstvY1QqjK9cIe33CR88wFxaRh4coM/X0dtt4v19pO9jriyTDkeqkLouWadD3Gqh1+sIy8IoFDBWVxELCwQffch/9j/7n/O/88fKUXF5hfLf+wusjQ0GP/oRerkyEyUB6PV5/KtXyAYDsl4P59JFkm6X+OEjskGfTKpgCZHLYc7NMbp2DWHbyso2SZWsvVAg+OQ6hT/5FsAT5eRopIaFp08rt8NCYYbXZr6vcOpXXHq5jLm0SDYaK8Wn52GcOaOGhMc2xOPLu3SJrNcn7fXUxmMYytsGGPzoRyCEgnAGA6KHD8kGQ8XntG3ircdk/QFEMdb8AtGDB6pgx7Hqxsc+WRiqz2g8Iu0PFAyWy6E5DmapRLy3h33yBPb5czinTikYxXPp/Zt/S1YoYq6voxcK+NeuIapVsjjCEAJcD+E4hNdvkPZ6DL7zXWWTUK1iLCyg7+zgvf019V5P2DnpeKxooxNzttT3MdbWMJIU7523Z0XfhtkMY+ptnzQaJM0G4b17mPPzyu/H8xBA3Ok8JfZ61XVcjKUXi8/RQz/r51/09VUh/xXWp5lgfdaSSC7NXULTNIbRkLudu9iGja7pnK2exY99Ls1domSX+Pun//5Tr2MJi/vd+4yTMVW7imM4VNwKOSPHanEVP/YBaPpNFnILxMsG7i/vYhx10D2LgRyQHDVIdnYVa2KujmbbRFtbWJubiL09nEuXMEpFot1dklYbLa9wVMJQDSUNg6TZRFgW1vw8zuXLjD78QBlQtduISgVrZRUZx/R/8APsEydIDg+RSaqKxP37RPfuER0eQpKQ+/3fw6jVCG7fIZuELMsgVAXCdWd+5EQRFAoYlQpZHKEX8ohCgeTwQMEpSYJ/9SrpcKjMvRwb3cshcjnFEZ8Uj2w8xjvz6SyjZ5e5tDRjxExX3GiQtppqk3pBQRClInHjCAC9Wp95vmRxTHD9urqP66iouiRVw0HHIXjwQNn2zs9jViqk/T5SSmQUYa6s4OQLjK9eJWk0MOfmEFGMpoFmqRAOc76OXshDkio7BAl6oYD35luKtmhayDQlbbUw5uaIHj+GKFIhImlKEvhkfkAahYCGiGMkKCvgXI5kdYXxlSuEDx6g2TaaBkaxSHxwCFGEcBw1P9CfFkrp5TL25gajX/yCaHtbMWykRKaZMmcLAqLHj7HW15WdQauJc+rUr/Q5vUiMdXwm9Fk//zKsrwr5K65ngyaeNcH6rFWwCvz9/97f58EvH8xuW39nnf/8//Gfq6HnMR76s9TGQTTgVOUUmqZxt3OXJE2wDIthNHzKw2XOnWMYDykWimR5h2xgE4djcrkSRqEKUpJ0uxj1eZzNTcbXPiHa3kYCye4uSa+PtbxCGgRkgwFpv49Zn4NEIlyHtD8gHQ4V1/reXUY//wXmvKL3ydEI/+OPEaUieq+Hc+YMab/P8K9/rFSUmqbw30k3O/rBD0nGYzTDIG42VUxcmiDyBdWRaRpZv49erZL6Y2Qcg9BBE8ggwFpZJRuP1QaxvY30fbIowlpaJD7Yxz57dqbS1ItFdcz/Fb+0xxkxmuOQNJsEN27gXrzwXEEAZsXC+9rbyCDA/+QTtVG6LvH2NnqphAYEDx6o7jznwHiMffYsxsoK0dYWwrbQLIvS3/0z4lZbec5MBpfC8zCXlzHrddJeD1EoKt58lhFv76AvLZJ2uqSjEcmdO6o4CkHxb/9twkdKsOTfuoU2HKr3NucpK4HhkPjwUM0mNIG1sKCCNsIAkfPQcjnGH3+MtbaOfeYs/ocfEB0cQqY2aTKp4KAkofgnf/x8J/xoC702hz4aI/f2yNIUoavXzUxTQWCNhoL6nlHMvsp6mdPkFKL5rJ9/GdZXhfwV14sohNPbP4tPDvCX3/pLHlx78NRtjz94zH/xP/ovuHX11qc+9rjcfxSPZvh5zlRUq6mHy7uL7/IvH/xLAOwso32iwii2+cOVP4C7B2j5vOIUAyJfwN7coP+d76AlCWkQYNbrKpAiSZBhqGh4YaS+AEGAe+ECWagEQuO9Pcx6HaNWI4siZLut2B/tFs7Zc4R376HnCxj1OtG9e6RRhFmtkAldJQeNRoqWOB6rgaSuo7kuWpoiTBO9pDY1c3mZbGuL1PcRnqsKnWGS/+M/wnvjdQbf/S5Jq6k6v+UlNN0g6fbQHj6k9Gd/9uof8AvWs8PJtNXEvXhhxk0/XhCAp4pFliREOzvEBwe4584RN5sYk00PQEzggzQIMAHNMLA3N3EuXpgNbDXDUPh+Pk98eIh98uSsq9Uch8z3SQd9hONiLi8rFkvOUxbDUTxzZwTA0PFvXCfa3SULfMylRYSpBqjZaASTcA0xyXU1HIcsDNFKJVX0J5bCpAmaZaNPumnhqbzW3JtvzgbFx9e0iMbb2+rxhoHuOGrTzqQSljkO8dERolh8biN4lTWFe46vqdPkq/z8y7C+KuSvuD7LBOuz1u1rt3+l24+v43L/nJHjaHxEO2hTc5UStGyXqTgV1oprM4fGpu5TifN8beUPqPQlw+vXlc93rYq1uIixuKRw77k5nPUNxJ3bCtoYDpXl7ASXTntdcJxJoXWwV1dJB33CO3eUV0q/r5SeE3ZLFoZkvR7ZaETa7RIfHpKliYJG0lQlEcFseCjjGGHbylcllyNuNJTbn6apIj4YYJ84MfMlMapVrM1N8l//umJsODbW5gmMSXhz0mzO1I0vY5f8Kms6nARmmP/xdbwgTH82leILzyXzFfyTtFoqicnzsNZWyUYjwgcPIcsYT7pq98KFGVTjlctwTEE6vnKF4MZNFUL8ySfIICBuNhG5nAr2sG2yKMK5dBnd9bBPniS4c4fhT39KvH+AvbmB97W3sdY36P/rf411+jTRg4cqD1XTMObmlE98tUrSbKJJCZal/N6jGO/EJgDRvgr20D01kHVOnUJUKoh8/sV2vscx9Xwe3XFUcpSU2KdOEm49Rs952G++8VJL4M/8jI45TU7X8USoz/r5l2F9VchfcX2WCdbvch2X+0sk24Nt1gprMyjlQfcBl+cu8+PoxxSsAt/a+Bb5+T9hfOUqaWPA8Oc/V2Ibw0DTDfo//BH5994lbbfVH7NlqS9QkiLKZcYffIBeKqmiHieIMMRYXSXr9RWzRGhKSGOYRI2G+sJPirJRnwPLYvSLnyNyOZKjhuJ5j8fomoZ16jTJ4aEKZtZ1VbRBwSlJgijkEaYJUUThj/8I79IlgJcOqoxKhbjRJBkMVAGaODFq+dxvjIM+OyDThPjUgvCsFN+szREdKLsCY3FR+agvLSnKYatNdO+e4n3rOubyEiLnveRKFMwT3rlLenSEubGB1ukonN33wbZAaHhvvIVerZE0jtQmnc+TjRR8FW3v4LguZr2Off488d4uej6HlvMw19fU5zQYqH8mMwXNthGGjn3hAtbKCoDaKCeQj16tqk3q8WNkGDy18UyXJgSjX/yC4Pp1ZJpizs2RDvoKYooiNNvGWlpS2oZX+JxeNLR8FgJ7dibyWT//MqyvCvkrrs8ywfpdr+O4ecWp0A27DOMhGhq2btMO2hSsArdat3h/931eq7/GhQubRP/0n6nCOzeHs7mpMO52m+DuXbz33iPa3oYowpybI9x6jKZpmIuLypa2UsE6sUly1FAdWqmIdeIESbuNU58nvH9fqTyFQJuIYfRajfD+PWSWYcQxGAaa7UCSEI1GGMMheB66aSDbHcVXThLVUcYxZrmClstR+nt/8RQt8GVYpvPaa8iPfkm8s4NMU3TDQCsUcM6fQ3jer42DvmhAlvb6gMSoz5PFMfHWFmmng/vG61irq4SPFMQSNZuzomitrpDFEZqhY8zXcS9eUNBV4FP88z+fwTSgmDUvu16V6LSE0VglunmTpNHEXFlRHHHPU6pUoWxv414Po1TG2twkvH9fUTu7XYY//zlGtYYwTeRggHP5tZl5WtrpED64T/jwEUapCNUKRj6PKFdwz55T9r2+r2CYideOWa8jJv47wc2byrfm2EabdrvEe/tEd+8iSiXSdpvw0SPSwUCFinS6WOvK//7X/Uymm/VxCOzZmcizENmvOzP5m7y+KuSvuF7FKOvT1oXLF7j5yc0X3v6rrEE0YM6do+4pwca9zj1MYdKP+tzv3sc2bKpule3BNmk+ZdPUyL3xJmJiOys8lwxJsren/r+uE038wOVoSNrtIioVzIVF3Avn0efqJM2m8vnotDHm5rBPnSTe21eUwCsfkzZb6IsLOGuvQZKQNlskrRbZaIy1uQGZJG42sTyPpNPBPnEC+9w5dNti9IsPyKIQHAfbtsEwcS9cIP/1r7/S+zGl/CXdLqZpIkwTkc/jnDr1G+GgL3YzrJNFIVkU4l+5gl6t4r71JpphEj7awt7cINzZId7ZRngezrmzoBtqOLu2hlGtzGh5U5jmWZMtw/NIJxj51PNFA2SWEVy/gSZ0jKVltFyebDgkevQIa20N+9w5ksYRWi6Hns/jXLqk6Jeep+idBweQpoj1DbXJRDHR7i4yCDAqFeyLF1Ws3Zwq9OlorDZhzyVuNCh8608Ubi8lUtOwFhaU1/reHuNPPsGc3I8kYdztzQqnTBLsc+eQgwGxEETdrvJaT1PMtTWE6xLt7zO+fv2FfP7P+kymt3tvvPGpG/ZxiOzLuL4q5L/CehWjrJetG9ducPG1i08V8wuXL3Dj2o1f6XmehXhG8QhTmPixjyUsjkZHjJMxmcxYyC3QyYE3HEKxiPTHhFuPkUmCsbiEsGyYyLwZ+6DriPkFdM/FPnUSzbJIux2MaoXcW2/iX79OuL1NNsnO1IsF3NdeI+n2MJeW0HM5xQ1eWlL8bU1TIcmui2YY6vZiAefMmZmcPvf1rxPt7BDcuwcoUY33gqHZy9ZUoZmNlfOfMfH9FvkCme8/hYO+Kpc47XYZf/wxCIGez8+eT3McCNRz5t597ylaIkDS7aLn8xT/+I+JtnfQ9ImYKAyJtrbIvfXmk+suFom3HjH+5DqkqfI08X3VlT98iHP+PNjOjLLoXr5EMhyQNppkvZ5yMMznodlk/MtfknY6mIsLzP3lX5J0u8hIuQAaS0v4t28rjxzLJLp/n2hvT80uxmOcS5cQxSLhjeuMr14lCwL0nIeeyyuuehAovjvMNqG03SbpdkkaR4R372JVq4rqmaRPiXrSfl+pPstltEoFE0iOjsjiGFEoYFSryDgm6fXwr137zEL+78PQ8tddXxXyz3FNi/ZxPvrVxtVfiY/+LMRjaAaDeICmaRyMD7B1G1MzyUTGTn+H9FydxV+qgWzcac+UgN57qhBlaYK5tIR9YpPgzl1FBRuN8K9eQ8/lcN94fUYHS3t9ssEQUSzOvEnsEycoX7pE//vfJ4uiibqwSnz/PmmSkPpjDF1XwhHTRAJ6TUW5TQuptbFB/jO+xM+uZ4ty7t13FbSRpYR7e6SdDprQKf7JH8/u/ypc4tn9LBspBMQJwZ27OGfPgG4o6f4LCopMYvwb15Hj8cQ8aol0MCAbDhQO7T2tKjXKZTr/3/+f2gwsi+ix2mD1ep2k15tsBGJGWYwPDjBKJaLtHUXFjCLSRoM0CNDn5pSnjG7gX7+uwp8nMI/IK8uDqNlEGw6hkEdGIUa1StJskPV6ykhrd08JwAwDTRNkvo9wXWWbsLb+tElZt4e1uoaUUs0OHAeRzxEfHSqL4HYL9/x5jKUl5ag5wcIBsl4PTBN9ms9qWcp4q93+zM/834eh5a+7virkn/P6Tfnoz0I8q4VV+lGfG80bTJMfYhmzVlgjyRJ6cy7l/+APGP3iF6TX9zAWl/Dee2+WTC/jGIlGtL+PzDKirS2Cx48Ruo7+xhuE29vIJAVDVxmVlYqSvY+V9F6USlgbG3hvvomMYmSSEN69C6dOEf30p5CkZOMx5sYGei6HubhA1ulgrq39VvHr8NEWRrXK6MMPIU0xyhX0alVxmEulV+YST+9nbW6qEAjHQbNtgkePsJaW8c5MIINjBSUbDvA/ua6Gi7pOcPeuCow4fRr71GlVHJ9RlSbdLqJcIm13iG/fRjgOxsoKcjCAfA4xEQnZG+sEOzvEjx+j2TZyPFInpUYDGQQqTMNxIAyx19dJ+gpmOo4Jm4sLirqZL5AeHpLs7ZMlCZppEe/tgdCUf/v8vFLAouLzsihUWaXz8wrm4Wm8Od7aUjz2SQarMTeHKJVJm02ig0OcCxeQUcToww9nFgIySSaFP6/cMZNEecgc83aZfsbHN2qjXCYbDvGvXEWvVDA3NpQ18pdsaPnrrq8K+ee8XsZHv968Tt7KP6cafZGa9PjKW3nWCmts97fZH+1j6iZrhTV0TSfOYnJuDmtjA2tjA/vMmeeUipppokmItreJth6TdDrKUrZUIjk8VDmcpzz8G9fxvvY2QtNmsncp5exYO3UTjA8OFPzSbkOa4pw9gwxDjMUFrHodHOVFbv2W8WuA4PYt3EuXnvP1mBWEVziWT+8nNA37zJmnpP/T7v1Z2Xnw6BEA+twc6cMH6vThuiQHByqIemWF/Dd+/6nXiff3kXGMWS6rTjjLSB4/Vu+NoZOtriGjkODGTeKjIyQgJCQHhxhzNaz1dZWOFIYIXYDjkPT7aJE9K7qghFrZeMz4w4/QcjmyVou410PoOqKkov/SST6oUakouGNyaks7KvLO/+UvSVtN5Gik3CWnTJFyBc3LEdy5g5BSKX8ndFJrYwP/1i01TJeSpNNRm0YSYxXXkVlGOh4pr5VyCfe1y08+g2c26qTZZPjXP1amZm+9Sbi1hf/xL3HfeONLpc78TdZXhfxzXi/ioydpwrXmNd5efPupLn2zuMmj/qOnuvf3994HYM6bm932qP+IS3OXOFs9Szfs0vSb+LGvckQjj17Yo2SXCJcqbP/su4yNDC9fYUGv4BSLZKMx0b17SN0AKcmyDJGmqlsfj1XHxxN63bOp8Gm3C6C40bs7iqGxvoaWzym/jmoVYaqEm2ko8m9yJH5ZUU4aDexz55+7fcpUeJVj+fH7Tb1azIlPy8tYEDIMcS5dIj04QC9XMIpFokaDdH8fZEaUJviloorJyzL0YpHk6AjdyxFvb6uCHwQKftB1ssGAwfe/j7AtsuFIiYaWl9EtC31+nmzQJ241EaUSWhAgHAd9bo748WOS0VB9hoMBZBmjq9fIhoqaqceRirPTdYVfD4ck4zEyidEME3NhQV2HpinrhdEQU9NIIuW5Ezx6hGvbjLs9MHTszQ2i7R30nIfMpEpTunFD+cH3+2SDPnptTomVbGWvm/T7M1teY66muu1iEXdCM4XnN+p4d5es32f4s5/jnDuLs7kJp04/9Zn8+76+KuSf83oRH/1x/zFlu/xcl/6Lg1+wXlp/6vZpbNxace2p+0ZJhNAEZbvMIBzguA4yk1Td6pNNIdrCPb9Jfq9N1O3wwOlz/p0/xry7rYpLr6+CmvN5hK6TdrtYiwtK+HH6tJLYD4fPpcKPr1wFQ0cmCe658+pLCyqcYvsxei5HmqRoYahoievrynJ2fmNm4vUqRkbT4/bU78PZ3FSSflRRNur1lxbr41ximcSqCLfbuG+88ZRw6GWc4xdd63T4pxeLyCgmGo+VdWyrRdJsKhOsXI54a4v4qIFRKeO+/Y66786u8mRJM1J/rAKXTVPJ1A2TrNMlaQZq+FgqIvs9qNVwL1xQoqnhgLTZJB0OMQoFgocP1e+uafQfbyuBj66jFwqkowkcMxhiVqsKj5aSuNNRaUW2jTaFOJaWVMh2t4tRKqIZJpgm6dERfhSRdntYmxtk3S65b/4BVpoS3LpJ/HibaH8fcwJ7ZP0+/tVr5L7xDeQkKUkzTTAMsnYb58wZZeN7TAQ1+5yPbdTpYEB4755KkEpisl6f/q3b6LUqwrZ/a+ZXX3RTra8K+ee8XsRH3xvtMe/Oc+XoCjkzx2JukZyZo+k3OVs9+9Tjk+z5RB5Hd2j5LQpmgfd33yeRCZulTU5XT5O38viJzw93foghDBKZkFvJsbS4gbXXYPvn32e5JbHPn8dOUuJmg2jrMTLnIWxPGRuNxzM/kf63v/1UKvyUHeLfuA5CRz9WREW5jBmFk2CJmKjdUt25Zc54169qZDQ9bsssQxM6wdVrhHfvkf+9rysa33j8ZODJ88KPaRc9/NnPVHSdaWJubiKj+KnXfRHn2J7feO5ahz95H71UVB7lR0eqcI/Hytq1WCRtT6wODg6Ve2NFBWjEDx/gvvGm8k8ZjTBqNfR8QakfDw7UppLPo5dKKoVHSiXBn19QAq2tLeJ2G+E6SG0yZL11C3SBsB2YKFyzZhM5kdYDij0zUe0mzSYYOkIIzEnBTzodwgcPlMdKlmKtr6EJtTlrtk3aainYZ2EeTRNEu3tYW49I2h2cs+fwbyg2VtxqIa9fV6eFfJ7g/n3cEyeenOqSBK2Qx7l8mbTbecKEOVZIp2ZrZr1Osr+PKBTQkgQpBPHhoRKS9QeIleJvxfzqy2Cq9RsXck3T1oD/J7CAGrf9Yynl/+U3fd4v63pRKLNruEhNUrCUh8q9zj1WC6szr/Hj3XucxbT99lNFP0gCDkeH1OZqrBRWsHSLKH2SypOkCdeb13lr4S0KRoG02+Xg1scszZ0g9nTEQBlXCcvEXt/AXF5Rgyzfx15be8pDw1hYwD53/ilr2umX9FmGAlGkBn1Cx7106ani6t+6RdrukA0HykfFddUX1tCfYrBMv+Djjz+GNEPGMXqlgnP5MtHjxwx++CNKf/fPnhTiyWDzZcKPeG8f+8yZmRAm3t5+bvB6nHOcdrv0v/1tFfZQqWBMhsTR7i7awb5K+xFCpe10uiS7u4rtMZklyEmnKzwPaZqE2zuqkCeJUnQuLKLphsoEPTxEaJoSZx0ezJSk0dYWRj6vrAcGA/RKGTkYKtm77Sinykn+p5jkgiaDvpLcJwkIQdbtqiI9YbwghMr87CmqoigW0Q2DtNkk6fdJvR6666rrZDJLEZNwkMBHatD+f/9/lKf54iJpp6N863WddDRScE8+r4ayGxvqddMUORziXriAf/2TmZ+PJnRGH3yAzFL0SkW5Qd64ARcvkoxGGNUq4cOH6I6N5noKc+/1yG1uohnGb2x+9WUw1fptdOQJ8L+RUn6kaVoB+FDTtL+SUv5qBOl/j9ZxPvrVxlXOVc9xt3OXxqhBIhMymTGMh/zFib/gUf8RoLrupt+kE3RwDAdTmERpxPXmdZI04WztLK7hkrNyHI4Pedx7zAcHH3CqfApd6BTtIrvDXVKZMv+wR96yOUjabLIAaapk5Z6rEu/DEGNxkeKf/DHWM050L8OazcVFwvsPiHd20GtVjGpNFYEkxjp/4akvSTYcMPzxj7FPniRutRFCkIx9TMPEv3J1tnEc75QQgrjZggk3XM/lcM6fJ2k1n/L5eFb4oZ7jyqzTS8cjrJVVRX2bpNin7Taa8XSC+5PHXiXp9dCrNbI4VowcIRDFItHDhzgnTyo2SZoix2PM1VVlyzsaQRCooj4aKfMr60lijabrmKtrYOhojo3s95FRSCpRZlu7uxjlsppJNBoMP/pIWfN6noKq4gTT81SHmqnXRmgzPxsOD8G2VYSepqmgiiBQqsxCAcZjssBXCU2OA4aB9P0ZDJK122RCDVE1KdFyikmTjscEN2+CYSp9QC5PeO3ahH+uEotkFCnrYymVn3m5RPh4GyOfx1xfJ9M0GAzVXKHdpv/d72GurmIuLEAUkQw72BvrpK0mZCma61D4W3/I8Cc/Id3dhSTBWF5SNr6/BR75l4Gf/hsXcinlPrA/+e+Bpmk3gRXgS1fIfxM/8petQTTA0SfHThWUiClMgiSgaBfZLG4qEyy/SZAEnKuco+bVVPJPPCJn5uhlPeZclQgjNMHHBx/PnBEf9h5yp3OHOXcO27CpOTUqvQH7OYN8u0ctzUOujH3pEtGDB8j+APeN158qps/SwJ6FL5LGEaBhra+r0Ip79wm1B5S+9S20nKfw2mMr7nTQJtxh3XEUdhpFJO021vLyC+1H9Xye+PE2Wi5H2mwi1tchitCnxl3PrLTbZXz9+kyBaW9sqGzQXp8018WoqC+uZlmkrRb26ec9sMMtxUuXwyHhwQF6qYReKBDv7GGdOKHuZFlk47EKQ05VR5n1emiTtCUsC61UItpSNEjn8iWSxpEaNDYbJIUiRq2mZg5JSpamivIJxL2eyu4UAm1SaKPxSP2tSKlOPlFE5iuPbzRdbcqNBuhC5arqumLdaJPgbNNEWCZZakMcq8cFgYqkiyK0alWdsKbZpL6v+OLlsqIn9npQKsHYR5gmRqWi2E2VCqnvowUBmmWRJQnS9yn/g39A7ve+Try/P/mdW0+FaQd3D9R7FYbqlGfbCFBumQsLFF97Tc0sshRMS5mvJQnECb1/82+x1pWq9TdZXwZ++m8VI9c0bRN4C/jZC372nwD/CcD6+vpv82U/l3Wc/61r+lOeJtNAiF9nFawCN1s3qdgVFnOLAIRpSJZlXG9eJ5UpNbeG0AQfHX2E3/R5d/FdzlSUx4uUkp/s/YQbzRskMuFm6ybzuXn8xKc1bjHnzWEKk2E0pOpUSbKEfa1PNSgwNzJwChbJRMhhFAs4F87POtyX8bXtzQ2SbncGX4hSCRnFJNvb2CdP4pw/TzYYkPT7OJNh6fEvSdrpYG9uEt67B6UyQkoVYjAYYG5szArz8U7JWloiuHVbFfw4xghDsiDAWlt97gs3ve54bw+9NofQNPyrV0m6XcX5vnoV7/XXEeXyC8MQpivZ3yc6PFLcaD8gG/skgyGMRiqoeW1VXU+joQROqytk4/HsuuK9fYhjzGoVWSqpQu+4+LduKwFPv0/aahHcu4cwDOz1dTLfJ2m1kOMR2VhFomEYil8+gUOiRgOybOZIOfVvJ4pmwctIIAzJbFsVaylVkczlMCsVwugQxmOYQC+4LjgO0vdJw1DdnmXqH8+DLEPk82S9njppmRZGrUa8s6M46fk8Vj5HGoST39PG3Fgn93tfVzj4BAvv/Zt/QzZSPvK656kTVbVKejzU2rJmARPTmUX/29/GKBRIms1ZSlRmGEQ7u5iLi7+R0+WXwVTrt1bINU3LA/8M+F9LKZ9rkaSU/xj4xwDvvPOOfPbnf5PWizrvKf87zdKZp4lruvx076dca1x7YUF/lQ6+Yle41bqFpVsU7SIFq4Cu6Zwqn+Ja4xqmbrI92MYSFlES0Q26NB82+bsn/y6LuUVafgspJcNkSMEs0A27aFKjG3bRhc7+cB9Q2Lqpm9i6TeHUGew7+5S7MeFgH6HroBuIUpnw8TZWGMIbb7wUO1SCkydOd4Mf/UjBE5MAYUCl+LSaSqM0VqZI0y+JJnTMlVVFRzs4VCIUIbBPn1ZJ8BMXwOOdksgXyL37DoP330dDQxq6KqQvCCKYXrdM04mXuk/SaKoOtVBEyj7h/j7GaIjwci/1wE6HQ+XyVywqc6hmk2xvjzQM0Qd90AXJUYPo/n0ywFpennTOY5LRCGHoOJcvqZzTiY+KsbREdHhI2mio9z2fVyHPmoa1ukrSbKqiFkUgxyoEO03VP5alOtPRCDxvEmYtVfhGp6MyV7NMddqmYptojq3CtMNQhXVoGlGSwFDJ7qewCrGS9DOxF0bX1T8TS+EsTUn2dhHFEtL30TKJliQYc3OkwwGZpiG8HM7b76DbtoKjSiXigwOO3n9fnb68HMnBPlLo6trTFJkmWKtraEIo//NJyPXxgAm9XJ7NZvwrV9QwNE3RXRetWsGoz78Uz34VNsqXwVTrt1LINU0zUUX8n0op//lv4zn/Xa2XKS/HyZiV/Ar3B6qIpzLlaHREIhPmzDl+sP0D/urRX7GcX+bS3CXWCmvPccCfVXD2wh6P+o/YLG3SDtoMogGjeMQ7C+8QpiF7oz1KVglLWDTGDWIZo2s6QRLwk92fsFZYY6u3xUZpg7XiGoNogK7pHI2PGCZDylaZXtDDT/wZ1NKP+syXT5Ne9Mg3jiAM0Ws1JfP2PNJ+X3lU82LsMItjwhvXn/5iFIuEd+6i12qKFTEJS9byObLBAPfCBUa/+AVJo4FRr5N7550ZjCKTdJaCY66szGCB5scfTzBbOTPv0ktl8m+/o+h4Ez72i76Y0+vWPY8sipRQJZdDJglmvU7meaDryCjCOnVKeaSUSs89j8jnSUdjVWBcF1EsIBsmZq2Gli8w/sUHJHt7aPmcwruRira3uKR8awwDYdk4p0+DbpC0W3T/+T9n9IsP0BwHo1hU15GmSE0jPjhQiseJ94q0bbUhTeP2ggAZhoAqPsI0yUYjdMchm+DnSbfLtEsy5+ZUEY5ixRzK5VR0X6/3pBNHDTLlePykmE8H2VMoxnUVUyRJEcUiBD5pt0s6GKjuPI5n+HjaaJB5HkathlmtEty+Q9rrIaVk9POfK81BlmGurYEQpI0Gye4epf/wP1SD0lYLdP25zXW6qQNYp1RalgxDMI2X4tm/ChvlZaZaXxRa4m+DtaIB/3fgppTy//ybX9K/2/Uy5WVj3CBIA0bxiLyVZ2ewQ5RGDOIB2/1thvGQs5WzDOMht9q3uNa8xvnq+dnj0yxlb7jHo94j3ph/46kuf7W4yuH4EABLWDwePEZogo3iBgejA/zUx7VcXOkSpAG2YbM92CZn5FgprJAzc+wP9zlTOcPZylnude/R9bv4sY/Q1JdVIonSiNOl05SdMqmVUn99A323QTYakWxvI3QdLZdTnuI8jx2mgwHB9euIQv6pL4a9uaEobEdHJO22MkLqdtFHQ4attrJz3TyBfe68wtTb7Rk8Y4XhLAVHs0zSZoOk21Ov7biKv3zrNs6JEK1Q+Mwifvy6jaUlwrt3lf2s66qEeCGU09/eHjKJMVZWX/oFN5eWIEkIHjxQdMIwRJvAMTIIEMUiluOQhYFSwwKiUlGy89EIczIIDR49QuiGcpAcDBWXut8nbLfVcDBNFfXQMBDVioq9kxl6raY67TieYNzpJO5OU3OCtVXc8+eJm03MQoF0PFYQjKYpumGscj31fJ5MSjVPaLXI0nRWpJFSFUh9MuydyOaZ5LSi6woXH42gUkY+fIBuOySdjhqkCoG5ME/ua2+RDkfopSK6l8NcXiKZXLtmGGQHB6TttkoVskyirS3MxUX0+hwyToh3dzHf/hr26VMv/Fyn8IemK6teoSlbAWf9zEvx7N+UjfJFoiX+NjrybwL/U+CapmkfT277z6SU/+q38Nyf+3pZElDeyuPHPoZmECYhjXGDo9ERcRbTDbsYwuBgeIBpmIRpyNH4iIJZYM6b42B0wIf7H5KSoqOzVlyjF6pOuWgVedh9iJSSbtAlyRIafoN3Ft9hKbfE0fiIftCnZJdISUmyhJpXY8FbYCG3gIbG3nAPP/E5GB5w5B+RpAmGMPBTH01qeIaHJjUedh9SsAoYusFfnPgL8t3HjB/uqg5uglMTRQq/5XnsMJpI0Z3NTeRoSLS/T9JukzSOyL39Np3/5r8habdJOx10yyIDNDT8W7ex1tYUU+RZeOaZFJwoTTFKpRmFUVtQrBqtUIAkRfNU4vunfamm1y08TyXhHB2SNZtYFy9inzqlGDqTYITj1/TsF9wolxn+9Y8xKlXMtXWCKx8Tb20pH+58Hq3fR8vnJ5BNgaTVxqjXQdfJ/+EfKuvWRoNob1eFXmgCGQaK1dLvzyALYVkgpUqijxOMyclE2DbS8xSzJAwhScHQVNENQ+LDI5XbGQQzaiCGobrVOFang3odRiNEuazoftWq4oUPBqpoZ5mCU3RdFe4Jzj6DVSbYOnFMun9AKgRJqYjmeQoqG42It3cY/fRnmAsLGIU8musw+vFPFO4ex2rI2e+r04emkSVqI5HToO18HmNxAXNp6SnI7viawh9jQ8e/cgWqVewz6qTzMjz709gor9Jpf5Foib8N1spfA9pn3vELsl6WBLSYW2SjuIGu6VxrXqMfTgZyMgXAMRx2h7vUvBqnK6dpjBvc6txiIb/AhwcfIjSBrdtIJDuDHVYLqwyjIXvDPRp+g5yZo+gUGUUjwjRkHI8J0oCaW+NG8wa9sEfNrbHoLRKlEcu5ZTQ0RvGIcTLG1V22B9scjg5JSVnKL2HqJr2gRyfqYOgGK94KRatIkAQKd49HCNfFWFxUSS/7+yQHB8rqdHVVHdP9MVHjSA26ohD3spJST50S9WpN2Zq228q0qdnEyOcVPzifJ97ZQdN1ov19nDNqg3jpUbjfV3zjwjEvGMuaqfvci5de6Ut1HPMk8Cn8/u8rHni9rqT8nQ4Yxsw47GXXFO7soLmukohHEelwqHjdOzs4hQJiYlErcp5SudbrWCsr6Lkc1tISaT5PuLOjCmQmwbUUP388BtOANFMDR8NQOHCxiLW0hLW0RHR0RPzoodrQprRBmc1CHTAM5Yr48CHoOtbKCmKsYCA5GqmNebJRyCzDXllBttsqkm+S0cl0qDkt5JYF5bK6pix78kb4vnpNIVTS1GiMnJwsCJWXvIxj4t1dguvXsS9eJO0rlbD0fWQUKegqrzzUZRQp9pHvI3xfnZRs+6Xso+MF17t0Ce/SpdltwvNeime/jI2CEK/mhPkFoiV+pex8Zn1aElDJLvGNlW9wae4S//Cjf8jR8AihCXSh0xw3kUiSNCHJEua9eZIs4ePDj8myDNu0SbKE1eIqOjrdoIumafxs72cEiYJLinYR27AxMfn+9vfRNZ3V/CrvLb3HnfYdumGX89XzM4WmiUnZKVOwC7OuPEgCNKkKfEEU0IVOzsgpc63iGvVcnfa4zQ+2f8DRTsjrJ96idDgkuHMXo1TEuXyJtNOh+y//W5yLFxGlEnG3S/zggcKMWy3C+/dJhyOMYhGtUMCoVBCeR9JsYJSKGHP1mWAodRwlVhmNZu/xy47CerE4oyLKLCVuNhUsYlsYUz/wY+vTvlQv5JNfv0544zrx0SHmwgLy2Mj92WtKu11FXazNIdbXiR48QHgeWaDMpKKdHcxyWQVaFApK+OQ6BHfuTJ5biVbSbhdnY51xq0V457b6faJIZW1aQoloNA3NMp8IqxwHYegKRoljVWyTBFBBxdOh8VRUJHs9gm4XpFSd8sQpkPGYtN1W84FeT5l6TXjjIk3JsuwJVi5VELJmWYr5Mh6rTSNJ1Aaim6q4j8dPYJhJ9iZpQrK7q2x/hSC4fx+jUCA82IckJR300evzKmUKZhi8JuXMl5woein76MWJQC/u3I+vl7FRNENXJ7vPaAq+SLTErwr5M2sq1rnevM6t1i2klJwqn3ruPpvFTebdeW51buEnPpnMcHSHQTyg43dYKaxwoniCv979azxTsTBqXo3WuMU4GdOP+szZc6SkWLpFKhWG7kc+sYyJs5iTpZMcjY8IsoDX519H13SEJjhVPsXB6IBbnVtYwsLQDAxhsJRfIkgCHNMhzEK6YZc0SymYBUxhsugt0hg1MISBpVtkBcHH7Wv8nnVSuQbatgpBGA7RMsnwRz8k8wP1BQjDWTqMUS6hLy2T+j5po4H5h3/4pMjaNnI8JhOCrNkkGQ7V8TqKkBM89mVHYXtjQ3V19++TdntKmJJlGJai0KXNxlPRaL/ylypJcS9ewjl7Fv+T6/iffKKCFV5ghxpubaFXqwhNI2m1lP+MzKHAIknW6xKMlHdJ3GigV8pYnkf+6++RRhFJp0P0aAtzroZMU5J2h7jTVcVQ08g6HSgW1cA3y0gaDcx5lbqTdLuMP/iQ1Pcn1z2xZUjV6U9OOeFJogaYQqiuGeV2KE3zSQedJJhzNdIgIDtU5lzS99XnWCyQCoH0/VkHLodDVcwd58lz9HpqQ5lc++xaJgHNoJHF8Ywtk47H6KaJ4XoqVzSMSHd30efm1Eatq9g4YZponouxtIQMwpeyj6YFN0sS4r09Oo8e4r355mcOHl/GRhlfu4Z4habgi0RLFP+uL+Bv6kplyvnaed5aeAvLsLjauEov7M1+fqp8iqbfZLWwytuLb7NYWCQjI2/myVt5Xqu/Rskt8Vr9Nd5bem82kIxlTJAE7PR3uN6+rjD3LCTNUjphh1bYIs1ScmaOKI3QNA1LWIzj8Qz3HsUjtgfb9MM+w2iI1CTdsMtCboG8lccUJieKJ1jNr5I385ScEmWnzL3OPTphhzALcUwHa30DgoDm/kNkEhPcv4f/y49UunoQENy+jX/lCtHWFpptqeHWRKUo+32E42CfPImcDP+stTXcU6eIHj9m/POfE7XbGLUaxuIihCHJ7g6aZb50WKTSfr4xC1OQUuKeOEHu3Xexz58nfKRsA6SUKk1nPH4h//tF6ylxUaGI99plRCFPePcOWRSiGTrja9eUCrTbVaEZGxtkQUDcaJA0m4SPH5O22xS++U2sjQ2EYSIqFdw338RcXMSepPok2zvE2zsKhmm2GH34IdlwiOEpeTko3FnP51VIh65jbW4qimOjQXJ0RDoeK0gly57QAaV8wixxXQWfTIvw9H5SqqIeBGqYmaYkB4cknQ6ZH5AFyrhMMwwl2smyJx32pAOXUaTgFt9XRXxKf5zCLceB1Glhn242k3/H+/tq8xcCUa2qPE/HURt9FKF7Hnp9Dk3oxK0m1qmTgJqTDH70I8ZXrhBPQqxhYp519+6s65965EydN1/4HX4JDn6cATNdL3TCnGwEmmWSdjuf+rf773pp8vj58nNa77zzjvzggw9+7cf/LhSWx9fVxlWiLHoKJ/cTFaU2ldZv97f5R1f+0cx1MGflCNOQ89Xz6JrOudo5/Nin5tT44PAD7nfvE6XqvoNoQN7OMwyH9MM+iUxmdMYwCZnLzbGcX1Z+31lKnMVsljaxhc3OYAcE1J06aNDyW1yeu0wraOHoSrp/OD7ET3wEgkxmZGRU3Ap9v48udII04FztHEESQHfIhQ8bnGvZePNLZFlKsrNDtLevOs1JTBuapoZynocmM/J/8AcYlSrSNMnabezTpzGqVQY//CHx7g5Rs4kcDNF0ndLf+wvs02cmX4TPPhJP8yyP+7lIKYl3dzCXln4tKtinPadwPcR0eDfpujB0hGWT9bp0/9W/hiRG8zz0QhGEwKhWMebrOGeUqdnwZz+FNCPa2VF0viBQMNTODtlwiF6roZmm4qKPx4hqlWzQR9hKhGOeOIG9vg5ZSvTgIcHDh0ppmcs9kddPr91QvH9AFVzDUJAHPBlSTlSlxPGka2b233qlguY4ZJ2OOiVNfFBIU/X4iVkXQqlDn6IlGsaTztyYXIPnPREkBT546pQgDEPF7dWUXYPQdbI4JvNVrKDueSpwolLGPnESa3VFmW2Zpoq+63Swz1/ArNcJ79xRwiMA08A5c1bpD17yN3Uclnmqm56Yv73sZ38Ti/TxpWnah1LKd569/QsHrfymCTuvsl7GXJl25FP+97nKOdpBmzANEQjeWngLP/aJ0ghLWCwUF7jeuk4mM7qBEuhkMuPS3CUSmXDNv4au6SzmFtkebuMaLkITuKZL3syzO9yl7bdxDZem38QUJhW3wvXmdQ4GB5ypnqHslLnbvUvRKuJnPm8sv0HNrbE73CVKIgxhcKl+iV7Y4+fjnyM0Qc2pca99jzl3jpXFDdI1nT06WGYf/dZDjMMOZrML44BsMtDKggBzcQHdslRRQSPp94j3VX6nff4cab+nAgVu3MDI5RHzC4p+d/u2GrpN1HufeSR+mZ/Lp7AaPmu96DnTZkOlvxeLignj+8TNFjKJMet1rLV1kn4P57XXFAYMWCdOEN65Q9xqzYoCgF6pMPrBD9Hn5sj6fRVhVq2iDweKRx1FKpSjXFb86XYboghzfUN5vgyHjD/8UGV1Tjr17OAAOcG+Z3TACWVwVtizbAa5qF8qfdK9T+/j++p2oYqvjCI0Y8JfT5JJ4dcgkU8Gn9Pue9ppwxOWi+M8GY5OX38ir9fyeeSEkSNBqTQ7HUXJdBzEtPiPRurUMRnGamlG0mphr61hX7iAEDpJkhLeuomezyuBlWWRhSHOulI1f9qMZCYKSxLiu3dJx2M0XWds6BS++c0Z5JLs7szor+HWFjb8Tor575qP/oUr5C/jeW/1t37tYORnO3wN7TnmSstv0fJb/Hj3x8pp0K1xvnaee5172IaNlJK230ZogoXcAgC32rfYG+1RNIucrZzFT3y2BlsMoyGrxVUkEk3TyGRGkiXoQsfBIYgVqyRMQ4IkwNDUx2TrNrvDXTIy/NRnZ7hD3a1TskusF9bphl2+v/N9RvEITWoYukGSJVSHVepenW+tf4ubrZscjg/R0Ki5NQ5GBxhhlwOvx+ajAxakQRaEhI6BHUxEF5qGME1IlO+IubSIzFL8jz8GKTEXFwlu3yHrdlWXeuokmtDRTJOk28G/dY+k2cI+fZqk3SGdpKy/7A/5d4FNPvucabOBf+MmmmmhOS7+lSvEu7uqW83lyLoqyDntdhG5PNbm5izNXi+XwTRmXugAZkVZ6TIcqo4SJZ4yanMKD9YNxVSRICZGWKJWw1xZgTQlaDTAMPDv3FG+K8eLsxBPMPFpMZ/SBqdrWryPF1ZQ/54OJYWhOOwTSEqfqxPevq1+5jiqy/d99fxT+f6za0pNnA5gPU+ZeVWrSpxUKBBOOu6p38yMpx6GZFPoR4jZZiAnPjtmqYSMY7JWC319HXN+XmWvWqbyvJEZztkzT3nQv2xGkvb7oBtE9+7NouWyMMS/cmUW7m2D8levzc3+zn4XPPHPg4/+hSvkn9Ut/6rrRR1+P+pDpFJ4pl7fN1o3uFC7QMkucadzh3E85kz1DKcrpzkYHdAYN9gd7PJH639Eza0RpAE/2fsJJ0onsA2bOW+O7cE2nuHxqPcIXegYGOi6zu5wF0d3qDt1BvGAg9EBMpPoQmc5v4xruix4C9xo3aDttxWzRZiMkzGPeo/QUBxxNAjiAMdwQINRPMLSLQpmgbJTJkxC3ll8h5/u/5RUpnTDLiW7xC73MA4e85EjOb94lo1tEI5NXDex+kpNKCfDS2N+HtFu0/0X/wINjdx77yFKJZK9PTLfJ+608S5eJHq8DUFAcPeeGnChOp0X2cY+u17qCb61RXrt2q/V0Tz7nEmrhXPxIlmnQ3DzJsnR0cReQKqhbpKQDAaYc3NYG5tPxcfFjQbRo0fqaD/ZFMJHW+i1KtHuHmatBsUC1tIi0d4e5sIi8cG+Uj2G4RPIwjRJDg4wFxYUHBOGSnyj64oi6DhPCubxQn1cUj9dxws/PLnP8WKfZapL13XCBw8w19aw19ZAF8QHh4rFYlmKh/7s82koFk2aKsohym8eXUfLMmXRMCnwzunTJL0emi7IRiPSSaAEx9WjoFg2kxNGhkqYyrJMJSKh7GI018F7443ZRpz6AdHunsLGdZ3iH//Riz/vYpHgxs2nbSM0DarVF5qywe+OJ/55vM4XrpC/jOddsAqf8qiXrxd1+HPeHFGi4JFe2KPlt7hQu0DdUwGxFafCKB5xMDqYGV0djY7IW3kcw2EUj7jfvc/97n0e9x5zqnoKW7cZRAN2BjvsjfYYxAPWC+u0/BZFu8h7S++hC50P9z8kTENKVomztbPUnBofHX7E/e59xskYUzdJpRIWDYMhURapgWka0gt7CE0gdIGFRcEuECQBHx99zGJhEVd3+fjoY+I0xjIsSnaJpt9kOz/ma8OUbk5yY3Qfx4mZG0g0TSijIk1TQ7WJ6i86apD5Y4TjkhwdYRcKM2FMtLtLOhiSRZES0LTaWCsrGIuLyrkuDF9qGwsv4A2/9hrAb6WjOU5JnGLmcZLg37xJMhig5/NoaYJwPYyFBeLDI+zlZdVpc9wrRlD8kz8m6XZJdneIDg6xNjcxV1bo/PN/phSghRWyJEVmUhU4iZK766oYZo5D1u8rMy0U6yBLUwXJOA7xVLCj62iuO5PoP9eJv2yF4ZMhJqhCKifQiW0r3cDhAZrrIcolhOuQBcGTASo86eqPP34S2aZ5rrLY7feVb0suh+Z50G6jV6uYtaqCK+7dVwEcUYw0Jh3/FLqZwCyiWJz9XlqaIiaRgFm/j3Pxwuyzszc36H/3ezPfcrNSmYVrv0gUNvrxTxQ+P/m9syDAPnP6haZs0/W74Il/Hq/zhSvkn8bz/nXWyzr8MAlnUM2Pd3/81H2Wckvcbd9lb7jHIBygCY1EJqx6q1xtXCVIAobRkHl3nt3hLjebN8lbeTQ0hvGQOWcOA4N73XtkMmOtsMbN1k3qXp26V2e9sE4sY1YLq8oUK5MzM6w5d45e0KMdtYnTmLJVJsoihvGQUTwiI0NHJyMjSAPm3Dk0TeNHj39E0S4yn5vnD1f+kB/s/IDb49ukMqVh+HxcH7PU17CTlMaCg0ZAzSyhpSZ2pUK0twtoyj5VSghCyClcMRuPsVZXZ9TE/ne+o4IMhECfn1fhChOWiea6M9vYl1nkPluws8BXIp4kQUzENsLzfqOORi8WSZpNkv199GJRYfjj8WwYKBxHYaeFAhJUAhJgLi4i8nn8mzcntqxNRLmM7jqIep3yn/85/e98l/D2baWqNFRqkvR9BTFM5fYA4zHBw4fK91sIFa23uqo69lZLdbCepwpdms48cGaDzU9bU1x9uqZ0wYnPjGbbIHSFlY/Gs4AMmSZPTgzHii1pokRNUkIuh57LI/3xk8I8yXeVk009abeg23vi9TJl10yfd3rCADRNUxqCCUdes21IU6yVFbxjWZ5Jt4t7+fILw7VfJApz33hdWQIPB4hcTmHruqH87fn8eOKfx+t84Qr5swk7BaswE+u8yjqOh2toPOo9IsoiKk6FpdwSeSv/XIevoakAB5mQM3Ms5ZZYLa5y9egqsRFTMSucqZzB1E1aQYvGqMFifpFEJmz1t2Yyftd08QwPx3CwLZtFe5G94R4HowNOVU5xrnKOB90HNP0mmqbxsPMQ13TJtIyiWSRMQ0bJCM/yCLOQkRwhNDHjnYPyVJmubtTFT3zyZp6u1mWUKKhF0zQu1C7ww50fsjfcI0xDtlcsDGJCM6U1D3/YAS1VlDaZJOilsurYkkQJY6JIcY5dlywISMdjklYLDWXWlE0NlPp9ZD5Psr2ND5jr6wjbxiiXn+uy+9/7PvbG+lNH0HQ4VAnqly+j5QsQRQR37mKfOU0W+L/235G9scH4lx+rgIfNTWSWqY2nXlfDyTBEL5dmilDva2+TNJsEN25gb6wT7uySdrsEn3wCus5AF9jnzpF/+23K//3/gMGP/lrx4o+OFPuo21Vd9VT2LiXk8xAExNvbiqa5sKDS5Xs9hadPfNiVF7f5NBVRiOfhFZi5Fc7ue/z26eN0Hd2c4uU+6WCAvb6OcByiwwM0G8VkmW4GQaDwc8dUhXs8JhFC3T51WpzYAwjbBl1gVKoqgm7KoJkKm4RQpwLHUbYLWUo2HqMXCjiXJ0lUWYpmGriXLj7VaSf7+yosejyebehaLv/Szta9dEkZfT3LTpnMWj4vnvjn8TpfuEIOTyfs/CrrWU/xG+0bBHGAKUxG8Yi77btKeanpsw6/F/boR32llLQKxGnM9dZ1lnPLnKudYyW/gqZpMz+VO507HIwOuNO+g5/52MImb+aJkog4jSk6RUxhkskMUyjsOEojDoeH3DXvkmQJcRbjGR62Yc88XapOlfncPI/7jwmTkCRNsHWbvJOn21cGWZqmIaQATVHrMjLiLEZDI0oj+kGfQ+OQf3b3n/H2wttYwkJDU+rUUoldfcxcK0F2R/iuScOzEbKA2/MxrIIa9OVyylUwCJCRws6TqYVqHKPl82g5D8txSXM5xf1NU6RpkjYaoGlU/+P/mKTbfR43TFPiTucp0U/abs/UhseDB8KtLdwLF37tvyG9XMZaXCAZjzEqZbLFBbSFeZKJ17i5MI/wPOL9/ZkBV9ZReP34xg1kpCh7yWCgfFvKZZLdPQZjH71YQBQLaLaFTGLGv/jgCa1v2o3bNkLXsS5fVkZQ3S6iVCLzfcxSSb2XlaoKdhACYZqkxaIKfNB1Vdynm8J0TdwMn7oNnhT36aDRdcnGPpojFcyQpkT7+ypvNU6UgjdNEUKoNJ+J1wqGoUIlfB+m/O1pYZ5sHOloRLazg1lWat+s13vyO7uuKuYTwZImJcIwsd98E/fcWaYEdee115FhSP9738daXMBYWsIol4kOVF6nZuiEDx/iX7mCub6O+4Jgielp77jFhLm09JSc//Oyr/08XucLWchfZfXCHteb13nQe/CUOnOKh98d3KVoFilaReI0xjM92kGbtt/mWxvfAhSf/MrRFSzdYqO0wTBS8EXeyFO0iuStPE2/yd5wj3udewyjIdv9bTUwpU/FqRCnMdKUeJaHZ3izrhpASEEQBxjCwNRN5a4YjZBScjg+pObUCNOQ05XTHI2PeNh9yP5on7yVRyKxhEU/6GMIA4mcKT8zqahj2uR/MTFCEyQkBHHAUXrEzfZNOkFHwTFZhmZqkPfYdiPidMCVMzZvDU3sQUDSOsDDhiBUKkjLQhQLZCNFYTMqFcylJaKHD4nbLZXMblpKdj0/T9ZqYRQKWCdP4L72OjJLX4wb2hbB9RtPdVxpt4u5uEB4/74KNygqOXy8v6+6ej6bzviyZSwtoUcx4uw50sFAdXydDpgG5lydaH9P8b0nEW9ZGKLPzRE/VgEa0b17GJUKcjhE6LpSNJZKDH/yPs7X3lLF5voN4oODJ13plOedpooGaFlk4xEip9gfSaOhREulEppjI8plrFOnCO7ceVJMPRfCCNJjopZpx34cTpkGToAq4qY5Y6Jko5EqxkkCrqsG2lNbhclmkUmpOm0poVjEKBbJokjJ/6dryjd3HIVDh6HazOcXVIB0uz27FjHB5tWHrSE1DS2fJz04IECSf+/rpOMxox/+UBmBOY6Cpmpz9L/3fcy5GtHeHslRA5HLobke8c4u1uLSU8ESx1kixsoq+qQDfqkX+efAHf9dv86XspD3wh7v773P3miPgqkgklvtW/SjPr+39HtgKEZHwSyApjri05XTSCln7Jdp5y40QZiGfP/x95WKUkpW8iuYuslaYY0f7/6YQTTA1m2u9a9hmRae6c1gDc/w2B3u4poug3BAkAXU7ToFu0Av6hEkAScrJ1nKLylxhinZ6m/R8TsESUDBLpBm6QzzjmVML+iRs3Lomg4amJjoQkdKiamZRDIiI0MiEQhc3WUUjwiTEB2dilPhYHBARkbNqaELnb3hHq7pomka+RDGZkQBE2MwYFSy0AcJnusq4CZLkWGE7rqIXE511RNbVC3LCB89mvl/a5Oi5Fy4oFRyk6Pws7hhNhyQtDuTgmMp5d61T8gGfTAtrKUlFQrcaBIfHWGfOUM2GjP+4EPGv/yY4h//0XP5op+1jh95RT6PubamhDuGjmbZZKGCDqbZnrLTUXCSYyM1TfGgDQPNNEgmfO84p9J0DNNSoqip4VQcq852QuckCJCFSWhyECIqFfR8DmN+XqUWGQZxo4lz9izBvXtEjx8/Se05lhxEOoFdHEepMKdrslnMqIsTuT6uqzrqKSURFESSJArqmUImE764ls+p36HXI4miJ/Fw0w1iWsSnRX+SUCTTRPnI5HLIiXtjFoZP8PGJaErMz5P2+8RHRxiLSyT7+8QHB5irqwgpGb3/U+TYJzk8VCKtXA7hDpFxrAK7q1WMev0pnPyL5Fr421pfykI+xc+LpjKhAqXMPBgd8N3t73KxdnEGN6AxC12YYuPHmSxCE1xvXufx4DE5M0fdrfN48Jh22FZyfDPPD3d+SDtoE8QBC/kF/MRX9MB4rPjh6CznljkaHeEJj1CG2JnNvDvPwBgwikbEaUycxtzr3kMIwVJuiVbYYre1S97IIzVJK2hhaiaZlilsPItxdRfP8jANk+a4SZCpLk0gMFCWpsN4SCYzLGERpAEHowPKTpmN4gaNcYM0Syk6RaIkYpSMCF2Tc12PJIqQ81UYuQysHmVrES2XQwO811/Hv3NXeXNnGfpEDp42m2j5nPJtGY/J+n3sv/WH6Pkcwb37CpNeX3sONwwePUK4Ls6FC8jBQHW3U3xYKJMjq1wmSmLFNR4MVEpMrUY2GND/7veo/A/+wSt35p929J56cVhLS8rlEVSx1HUV8ba0TPjokZK9j8dYCwtoholumkR37ynf7yAgPTpSTI4prDIZsk3pe/QHyNFIhTDMz5ONhmi6gkdEoYCeZYyvXn3ifjh1QnyWPz7ttI97ik+ZIdMOfco6mXbY0259yk8/bqI1tQ2WEjnyn0BCUfREATp9Pl1/kipkGEpYlGWE+wfqfTtOm5xe3+S6NKHsdnEciGOG3/seerWq4uYaDeLRCL1aJXr0iHQ8JtrbxblwYRYskYUhwng+WOKL5Fr421pfykI+iAYkWYJrqZ14nIw5Gh3h6A5Jkijr12hMV3ZxdIeLtYv4iT9jv3zS/OSp4enB+ABXV0VdExqO6ZAzcvzVo7+iH/XJm3mQcJQesdXdwjEcSnaJTGakWcpcbo6aV2MYD7F1m1bQIkxD4jQmk9nM4Gqrt0UqU6pulYJTwDIsDseHNAM1/EzSBDQ15e8EHTXt13RM3VSdtaYjEDPYpWgViWVMlEakaUqmZZi6iSlMBIKD0QFJlmDpFmEUEssYTWoMFvJ4uwnpoIdOhiRD2hbW5gmiR48w5usYi4voBwdkoyESSXxwoKbwmkbaapHu7mIuL6FZ86RHDeJJelB6cIB28ybJ/gHm0iJZFELgI8MQ59IllZozsZeVUjL66x/hvPa6Cl4YDsmiGK1SQYvj52Llfp3AgOnRO2kckQ6HjK9dU91fkmDU53HOniG4f5/w9m00Q8c9e45gawvNNBGFPFngE/V6yg64VEJOvMkBBQVNu2ZdPBHYTMUwcUzS6WCVSsjAJ9ndw1hZwdw8QXT/Pmmz+aSrnhbdKQ4+XdNCPhg8zQY5PgidMk+mz5FlalM4XuSFQHa7aJ6nMPlJ9Brw5L7HY+Cmzzdlw+i6MvCa8N7laAQTL/SZSnRqFTAZgGpCkHY6iFIJYdvEjYYarBcKRHt7ZEGAPhqRDYeIYpGk08X/4AMwTMy5OTW7OPN8sMSvwhL5oiQAfdb6UhbyglXAEAZRGmEbtmKBCI2yU2bemydn5pThVDjEdEzude9xsnRyJvM/zlWXSHJmbgZvGBgs55dJ05QPjz6kYldUt+y38BOfVKZESURqpzOvFEMYDMIBUkqCJEBmkt3BLnujPSpWBV3Tafkt0GDBXVDFPQ3JZIanexyGh2pTkCmZzFS3PVF7BmlAkKrnFLogp+eQUhLJiIQEQzPwpU+G2jBczcUzPZIsoR/3ZyEUQghcw8WPfe6lBziFjL/1WKIbY0SpRH3xpKLI9fvo5RIiX1A4pSaQmkDP55FxjEwSlTCkG2BapKMhstdD9vsYGxto+QLCMPDv3sWoVUHoig9eLKoB4rElgwCjXldp7WeVp4nIefhXrqJXq0/uGEWqC36Bn/WL1rNHb5kmRLt76L0+zuXLyCTFv3EDe2NE3O0SToQluXfeIW610AsF3D/7M8YTM6zw7l0kKojCe+st0oMDRW0cDFTHCepUMfHtZsIlxzRVOlCrpUKFpUTu7c0gK6NeJ9rZUUHLrquk78cfP2WiDIdPivIkB5Th8AlLZDIsnkEp0w1lKrE/FjAxw8lhAqGgHut5qpufFnMh1Ov0euqxrovueYh8nvjwUN1nNFQ01Ul4M4BeKKhBfBShFYpKMSwl2aCvfu9Jhmg6iaNLJrRNfTDAOncW2e0RfPQR4hvfwFpdVd16p4P7xusznPxVWSJfpASgz1pfykK+Udxgb7inMHIK9IM+YRqiC30mwTc0g3quzruL78646McfP+Wqe4aHIQyGsUqhR4NRNKLlt+iGXQQCXdeVLWyUYekWiUzwdA+BwHZsbN1Wpluaxe5ol37Ux9aV73graKkO3lSq0nbY5kT5BJnMuNe9R8tvEWWRev5UGWBlZEgpEVKgoZGQMEpHeJqnPNFlgiY1xvF4xkoxdAOZqMeM4zGxjJVPurDV767BIBzgGA4yk2wtCY62QnQtw8RnvTMmaFwDIcjGPv4HvyB+vA0yI2200OYXlDGS6yJHI7LMJ221lLueoWPUasjxGHN5Wdm67uww/NnPsTY3VKd76dILv3y5d98lfLQ1u92YRKNh208JPay11Vfm5T579I4mXHI58QY363WyjQ1GV64ovLtaVUPO/QMV61YsIQcDnHNnFcujUABNwz13jnh/j/HuLkm7rdwMp5AIqNzLKUwSReq9DEPS8VgNdZeXIcsUJr67q04ck45eTged0xi2aaLPcWhjYmQlcrmZiAkhEIWCMuwaj5+EREyfbyrSmaovdX2C5QcQJ08gk+OS/ckJQeg62WQz0GxlNSzyeWg2wfdJUXYEwtBJA7UJWOvrCibZ2VEeLPk81toa0WRQbHgeyWiEXiyQttoQhmiVCqJQID08wrl0Ed31EIZOeO8uerWK+9abaIY5K8IAGPqM+++cPv18aES3S//b31b00kpFDb4nJ6kvIpb+hSvkr+J8WLJL/P7y789YK0mWkMmMU8VTlJwSD7oP6Id9TldOM4pH7I/26QQdGuMG39r41lNcdQAdpcwbRAOOxkd0gy4Vu8JGYYNO0GEQDfAzH0MoTLpiVbhQu0DZKfP+3vsEicr6jLOYVtAiRTkaSiS2ZqMLnX6sWC69sMe9zj0KRkGlEGnMRD8JT7wvxMSBeCqK0tFJs5RES9AyjTiLSUjUpoVixRi68voYx2MSEiRyhqlrUt1vHI8RmiDKO9w9Cd6hxmtdk27cJF9bQtM0gjt34dQpxNIS8aNHKoeyWiMb9EmjWKXmmKbCydNUFeXhkLTfV4rJ3V2FiwoNTRP4V67iXrr0UoqWXio9ub1apfqXf8nogw9IWk30SgWjqhR+ZhjOWCzAS4/Mzw1aRyMwLTWDvHuHbDQibrWU73qprIpT4BPu7alsyaVlsmIR7/XX1HshJUJXyfBJt4e1vkY6GqnBXxSTBQHBrVvK9W9qYzul7aUp6eTf8e6uMssajVQoRJqC56IXCopJMu2ep34rxztzmOHiWav1BIaZ6AA0133Slc/NKchj2mG77qyoC8NAr1ZVks/0uSaKz9mavGY2NeXK51QXPRqR3r0745/rlYoSH6Wpkv1P0plkFGFvbpJ0uxilknr/cjn191GroU3eOyb4t1FTnu5MnBqNk0ukwwG5d9+bnarSwYDg/n363/suWiaxNjexz5+fec0fX2m3y/An7+PfvDXxA+qiNxq4r7+uqJdfQCz9C1XIfxXnw2mazzdWvsFPdn/Crc4tJcOXKCMqYRAkAXc7d3F0h6pTpR20n3q+KVfd1E3e33uf/eE+QRxQdarYus2J0gn+u8f/HZrU0FKl7gzSgKJZpBt0edx/jK7pVNyK8g+P+qTq0DpjlYQyJAqjWWEWmqAxbjAyR9iGjZVZhFmoOkXDRCZSFXSN2e88ZbOkaYqBMXsNHR3bsGf2uWESEvMCEQlKSDT9mSc8EplwtFKg2xxyeG4D8jlOpZuMr13DWl9XdLXBQDEsTJPw8WOVZWmYKhlnPEav15WZUrdLOhqhGQbJ/v6s802OGgjdQDg2/W9/G2NhYSbLfypB/QXULWttVbnX7e/PJPLGBDcd/uR9QGLU5194ZH726I1hkB4dolk2hmkh8gWye/fJJsKl6MED/OvXlYGYEGgSsvGY7PJlrLXVSQScT5alGNUKcSNFdz3sjQ0VknH3LlrOQ0tTpGUpOGLaUU9zMidWstkU5pjiz1GsPLiP49Kg/v+UyqjrqjBPn3M6tJzcR0420SmzSKSp4ocfFw9NHiMtS4mXgkBdaz7/pBs/7vUCqsALDdCUKnR62wQzl1KqGDfXxYhjZRvcbmNWq+hFFaSddnsk7TYaKMZOEKCVy+hZhl4oqgYgCNTPNzbIwgi9WiUdDp7yKvevXiXa3ibc3sYoFtXf4GCgvICeUQGPr18n2t1Fz3kwCSePGw30+/exz537G5kA9FnrC1XIf13nQ4nkYvUiB+MDBvGAnJmjZJc4HB3OTK3CNKTqVHFN96nnG0QD4jTmzfqbvFZ/jY8OPlLFdtSg5bdY9VbZH+/jZz6mblK362RkfHz0MfO5eS7PXaYbdJGaVF0zKXLyP1AFHSAlJUgDbF1Fwg3CAXkrzzAeEmexKvSa6sRNzJmYpx/2le84Gcnkf3mRxzTUfaIsIslU9y00gZBi9povW3EaY+om45wgrBZoaiOqQ0k0ZyFHI5IkIRsMnjAHLAtpGOjlEtGNm6oAeR5plqH7PsbiItHBAWaxSDJJeReGgVWfI+l1yXaGiP0D7HPnEIZBvLtH/hu//9npLxMeuV6bgzQhvHeXbDRSr1WtquAMnqefPSvQsNfWGLc7Sro9cePTbBu9WiVutxn97GdKOp7LqdDjoyMMmdH+p/8Ec2WF3Ntv454/T9LtMnr/p6S9LsJxiXZ3iHd3VMCy46LpBjIMnwQfT7HrSRbmUwNF04RiUdm/Tsy7nEuXSNpt4k6HZH//6YKfz6v/P4VOpvDL8X9PVnac9TJ1WjSVTZWMIjXsnGSCSkNHui66aZJOYSF4ok4tFp/Y5U4olpplKbtc3yeZOB1aKysY+TzW6irm5iaj739fvfYEdpJhiDk/j5ZlIDSVLGWamGuryOFI2T/UaphLS2hC4Jw+PTtVJfv7pM0maa83E2fJIFDGYLUazhtvPNVlh/fuIYpFjEKecOsxwrYRnkfw8CHm2trfyASgz1pfqEL+6zofFqwCURZxpqLUmsNoyPXWdeJMFawwDQmTkLXK2nO+44ejQz5ufEzRUhJ5QzcI4gAEXG9dx9EdUplSdsqU7TKGMOiGXcpOmaJdZGe4Q3PUJE5jNKGhZ/oM8tDQniqqvaiHiflEdh/EJFLRF1NNGWUZukHOzDGMh/SCnrLCRVPdt1CbQEpK0SiiodEJOwgEURaRySc+LC9bU9bLNNAiq8/zOA4oWkts3fo59cgn29+fsSL0SkUVpyAgvH1nxprQ0oSs18PI50ED78J5nNdfJ7h/Hzkcorse6WhE0mqT9fvo9boahEUR0e4u4+vXKXzzm5/5N5H2+2iGQXj3nsq6zBcg2CLaeox78eLM8vRZ+tlzmZ6DAdlorEIgPA/vnXeI9/YY37iuIIJJTN10qBq3OzhnzmAtLTP62c8Jrl9XsXGui2aYKlVod5eo3Ua31eYgg0D9jnNzZMOh6mKlfBIeMfVhmRZXTUMrFNQJxnPRa1U0U3miGK6r/Gt6PaUfWFkm3ttXroBT29ipKnMKxxyjAArHUZDDtKOXEqJEmXsJgb66StrpoHsuiAm2PxyqE8mkWAvDQJ+fJ9rbU37uUYQE9OmGOBqhFfKk/QHZaISmaWoz2tmZ2RYo8ZipHCOThExKtP9/e3/6I9eZ5/mhn+fs58SeGbmvJDNJkZREqRapVItnqqvKM/b0TNmDgTHeANsvBgZsX1/DgOHx/AMGfHFhA16AwVzfNx54Lu546Z6edndPTdd0V6uqu1SLKIpUiWsmc99ijzhx1scvnojDJMVVpESRiq9AiJkZjHjOiYzfec7v911SgXXqFEmrRdpoIi0La25WpVItzGMvL6GXSsqHp9PGv/ox/pUrCNPEqFQQw+FwHBNsbt53ly0A4XrYS4tEh4ekjTbC0F/IQSe8YIX8Yc6HD+ud32u0pWuK121rNrV+jTFnjIXKAnkrz0HvgJpf4w/X/pC97h5Vt0reyLPT2VG/iIbDXm+PolUkZ+TwE58wDsm5OZBqjY7uoEmNrfYWfuIr5aQUxKlikZjCJEqjrAUyxLC9oUtVbEMZYgtb8W0HDIIxZwxN0yibZTY7m2qwKVQh90wP13BpBa1s0GkII5sRSCS2ZiurUJJsJ38cJiYR6jbY1V0OqyZvbeY5cblO0mzQtQRuq5UVhKTZvOOoNywgUYRerCpZfhiiA7m336b4gx/Q/fM/p7+2RnKoFKDRMLR3kCWpeR6yWCS4fv2utPQHUcP0YhH/o4/QhmZLqB240HXCnR2c1Ud7VwOYMzPIMLrLkEkYBt1331XJNAPf9TQMSXo90nabdGqSME0V0yKVdN77BXS7mIuLSnX6q1+RtloY09M4K6eI9g9UOpDrKhuFUPmPaIUCUtcV3W/Yix5Eq8koUgU/CNFMixSh/LtbLSRysO5QFcsoukNVHKYDDYegw/76wBUxHe6o0xQqlTutGSmVm2Ahj9A1klYbYSjLXXI5aLeRgJQpenFSzUMGQ1u9VCLtdEjjWIVr6Dqi0cQoFrHOnsUsFok2N4kODrIevkxTZK9HPFCm6uUSuW9/B6KIpNlUjJ52G6taJfftbyMMk2BtHe/C68oN8cf/AhnFCMNAH/Tbk24HzbIza4F7GSvOyoryohcCHFdlpdoO7rmzL2QRhxeskD/I+XCqOPXQ3vn9jLbemX0HuKPgdHSH9eY6f7HzF5TtMjudHbpRF1MzWSwu0o/7WLpFw2+wUl7hdus2rukSE1O0i7i6y7g3TpKqwc7wouJYqo83HC4CDP9noKT1x4eY934dy5g0SXF1F0MziJNYccJ1xQmXqdo9SySdsEOUqExQTWjYhq1omGmIrdtYuqXMtQa1+347c0M3yBvKqbHqVcH0qATTiA8uYacaRr1FlLMxpa52fu02olxW6r9jdLe418NZXkbGMeapUxR/8AP0chljZgaxva3YLXEMuoFRLKmwhcNDtMVFBJD4/mNRw+ylJTrvvgvjVTSpWgN6Po+UkrhWe2Tg8/HnuZc1IwdtIQDZ65EEgbIKkBKR81RPu15Hm5sj9n3Cq1fRy2XE1hbO+fPqjqRaVRc7TcdaWCDa2iI+OkL3ckg7URc/TSPt99GmpkhbrTtRawP+tlEuE3c6tH73dxGDY9dsGxlFmLOzhDduqOI4VH4OvciHA9XhkHOYLhTHd4q8bSvpvO9n4cjOqVPo4+OEW1tg2QgpCW7dglgFMAvPQ69WlUNks6nOl+8jLAthGOp9DUNVpG0bEUdEa7cQJ08SD5hMSadzx8QrVndv5uwsSLU5sJeWsJaWCNfX0VqtAavkzoU4WFdEBPf8eZwzZ2jGsUpdchxF6QzVwN1cWvzE74w1P4//wQf4Vz8Gw8CcnMKam8U95rT4ouGFKuQPcj58nN75/Yy2mkETXej85ug3+JHPdnebCXeCZtBkp7ujXAoNm7XmGgBJmtAMmyo4Iu2TJAmWsHBstUt3DAfXcKn7dbpxF1M3ISUzzyqZJWIZ0427qognMvNGeVC7Y7hzjpJIDTWTiElvkpSUol2k3q8jhcw45nEcZwKlKInIW3kM3aAVtAgTFf123wKOgaVZaCh6Yt5UPjLj6Tib7Q1y4y5T+xFaziO2NIxQI3Vsdevb7d5pCxyToietFubCAu7p09kHaVh4jfn5QTEKCW9voE9PE/s+ol4n3NiAOCIsl3GWlxFCPFBmrZfL2KdO4X9wibTbxahUsM+dQxgGydHhY5sU3c/YSC8Vyb/1dTo/fw+9VCLZ3s7mAsOeuEwTkmZD3cpLSRIEJGtrWYHGMJSJ1sAW15iawlxaxF5YpPuLX6gCPOCGyyjEmp1V3HHLQrMslZlqWYi+Twpo/T5pGKDl1B1gtLWJ1DT0YgHp97PhM4OIvoxmCHf+btt3OSmmvg+mQVI7QitXiOs1tHxeKWddV6lzl5aIazWlJajXka0mwnWxT68qP5VCgfDjqww4OZnoSSsUEEIjqdWJuEV4dIQmhMqBNQxkmqrHxDFC1zEmJ5RQaHBRT9rtwR2MyBhFwvMwPA+Ry6GXK2hCkP/2t2n/8R+rgJMgwHn1VdwzZyj+4PufoB0Ga+s4r7yCXq+rBCIhcAepQS8qXqhCDvcvyA/rnT+o5XKcAbNSXuHdrXfZ6mzhhz6pSCnZJRISpfiMfcbsMYQQjLljXKtfU7xw3WLMHSNKI06VTlG2y2x0NvATn4JVIE1TYhnjGi6OcDB0AxOTmfwMBbPArdYt2mEbMzUVdfDYTnwIiSQhyXzGh/4qw1aNLnQiomyIqqG8YSzNImfmqNgVJYhCqUCHtEcNxUE3NTUU9WNfDWR1HUc4lJ2yuqD1dthIC5Qdk0qthrRsdD2HTKM7viG+D7aljLIgszZN+30VkrC3f5epkZbPE169CqaJXigopsfWltKQNpuY01MIy0YIjd7Fi4r+J2X2AWaQ25k0GvQuXya4fFllii4toXke4doa1twcxe9//4k+nPf2zds/+Qnm0jIFx8G/cgU2N7FWV5HdTmYcJYWmhp9jY+iOrVon+TxSCOL9fWUotriIPjauitPYGHGtRri9jez1sKanSQY87rTRQOTzWIuLarA6YICkYYgMI/RcDqHraK5H0u4g+33i3V2M6SnMcpk4rSMsS92BDIeeQzOtYTzbMb551lrRNMVxlyGappF0uwS3bqk2V7erPMoToeYHhQL2yZPZ+YobTUQ/wJybRywvkyYJ4Y0byGIRLU3RB1x4ZZG7h+44xP0+1vISyVGNpF4nDUP0iQnFeR+EVaRhiNbvK9phmip155BR1G4TttrYq3cGnno+j728rIy9APvEMlrO+8R7fFwMNnTZTH2fuNF4Yq+eLxKeSSEXQvxV4L8DdOAfSCn/62fxvI+LB/XOe1GP37n+OyQyoWJXiNOY7c42RavIjcYNLN2i6lXZ6ewQpRFVp8pedw8pJOPOOJ2ww1FwRNWpomuqgGqxxr6/j2u45DQ1dAzigLyZRwhBwSpQNsu04zbNqEnYD7GERS/uYSSGimEDbnduYwgD13CJo/gT/fJ7MSy+KSkHvYPMhEsOzfkRGRsmTVMMQ3HaD/wDYhmTt/JEaUQUK1vb4XOl8o6IKUkSpCYJUa2Ykl1it7PLRcfHMVwKVpdKHDFFUUWTDQN44xhRKKoP3TAhvVRSdETPI9zZZu9//B8xxsYQUmItLIBUfetwawtjahqRLyB7PYRl4Zw9S9pqk7RbxAeHaJ0u1qlT2Qc4GVio9i5+QLS9jT43j+77hDs7GNVx5VNSKj79DkvT6H/4ITJJsOcXMMfGibtdgmtXkb5qvdDvq/DmsTFIEjRdZZVGtRqy2yExTfInvgEDxkp8eIjQNfrXr2NMTCDiGKnr6s/AYEobH0e222gFtZOPGw2o1dRFIpcj9X11F2TbCMch7nQQQlMc9HYbrVJRLoOmmfmyiIHD4VDyLzwX2elmroxapYIYG0P3PNWJSVM0xybY3UUmiWKwCEHa7SpFr2mqYqhp6k7BMJCehzU1RTLYdWtJohg7AxaTZhiYKyvoh4foroc+YxOmKjDDyKlUJnNhkXBT3ZHJvk/u7bcIb95E83IZowgpsZaWVC7qgCMe3LhOtLtDfHColMVJgua6n7iDe1l9WJ66kAshdOB/AH4AbALvCSF+V0p55Wmf+3Fxv975Ye+Q9dY6uqYTxAF73b1MlFOySiq9Pj/NL3d+yXRhmoJdIEoj2pHyaan367i6Sy/qgaN8P0pWCdMxaUZNmv0mnaij+vJOiV7Y41r9GolMcHUX27DJGTlCPaTZbxLKkLyZp2AX6IU9/NAnb+fJaTn6cT/bVQ9hYGQqTh1diY0QyFQihSSIAxKpHj8s4AKBKcxM6OMJTzFPZIprudkAFFTLxhIWuqbTCTrourpQRXGEFJLt7jZ+5Cvqo5vjo1Mmeuzy9uWITs5iXOSRHeXSp8/MqJ5yvz+gA6jd8nDwJEwL55UzpN0e8cEB8d4+xliFcHcXBirw4ne/S3DjhlIgttuYMzP4H3+MME2V3nPsAzzsj2qelwlFhjzp1Pdxzr+qdpGfAkPvjWhnh+DWLdXjn5wiiSKinR2izU1EzsNcWlbqxVoN4XlEu7tonod75jTBrTVMIRBTU2ieR+/Sh5jTUxizM7jnz9H/zW8QhqESho6O1EDTNJUYRtcxq+PEUYQ5OUG4fwC9HubYGFG9TtLrKZXngCJojI2RhCFaqYTuuSS6jp7LqZ30UCB0nDYIig2DUENmUG2wRgMjnyfq9RBJgl6tKkvfwYBUSjlQ64us3SGTRNkYj1VwXjlDcOsW0dYWRrlE3PMVXXFgVysHvvJGpYI+O0vSaCjOuOth6Ab6xCT6+DjJ0RHW9AzuV97MWEvNP/iDuxhFxupqJtyxl5do/cmf0P6jP0LqBvaJE2jFEuGtNZJ2B2d5KbuDg88vFejzxrPYkb8FXJdS3gQQQvwj4IfA51bI79c7L1pFEpnQ8lvYhpLJX61dRSIpWkUsw2Krs6UCF4IWeSvPpYNLGMKgFtTwY5+claNkljjoHmAYamhoaIoF0k+V8KdslzGFyUF8QM7I4egOB8EBjZ4y5OpFPUIZYmomZaeMEIJW1MLQDcI4RAhB0VLOg8PetY6e7bBBFXUDQw1MNcU8GR5TM7hDQRwyXXR0kiQhJCRn5ujFPfqxUnCawsx68gJBmqRERFjCwjRNdWxxn37Qzy4MURqxqTU4eNXmwxnBV2/t85fXuuTyHk6phIlOvLur1H6tlmqS6joSiA6P0Gyb+OAAUqkCKQZilXB7h7TfJ64rDnd8dIQ0TYxiAXN6RjFEajW125udxT72AQbQyxUE0L95Ez2XUzvSVktRAc89eejEUPGXNJv019fRpFT92VgNBgUS5/QqaZIoymShgPHKK6rfi2JyBNvbmc+4tbBA2moiDZ2k3cGcFWiuq3aprkvi++oCpQk0Tc0X7NOn0QyDeHeXxO9jVKukrgu9nsrZrNXUWizrTsvJthQH3hw4GjoO7je+gf+rX6lk+3we2euSCKEk746jPFWGboYDCX48TO6xLHVnVfNVzzsZpOwMrG1lGJKa5mCIqxG1WnR//h7a2BjmRBWxtEjnT3+i3rfh89s21vwc8fY29ttvYZ08iX/xIrppYi4vYc7NKYqnTIl7XfoffKDaJUtL92UUpb6P0DQVDWhamHNzyCgeUB5V3Fxar5NUx+96jz+vVKDPG8+ikM8BG8e+3gTevvdBQoi/A/wdgMXFxWfwsnejZJdYKi5l/fCbjZt0A8VdtXSLvf7enSKc9JktzLJxuMFh55Cb9ZskJDT7TUzDVCwW00FIQTfpIhAU9SK2bnOrdYte2ENKiS98mu0mpJCzlcgokYnqjyepyvMUAs9QvisH3QM0TcPWlQeHbdpEMiJn5jB0QxVT3aQXKi8UE5Nk8B9SqT5zZg5SxfdeKC1w+eiysuM9fq4H2Zo5M0fVqbLWXkMKCRICgox3LpGZbN/QDHJWLvNW95OB0ZYMMBJ1EYmSiE4eCgUolGNe6RYo9RsU+7oaAnY6d9gSpqkyMONYeXc3mqrPOzZG2u3ib26qAmOaRHt7dH71K9zz50kPDgjqNWSSotkWolpVrZiB6dLx3ZPs91XSuvoFUy0e1xmcgydHpvgrlbI0oqTTQZ+o4r3xJl2pXi2/uqLsbR0HaZqktZpScV7+ULE4CkVkGNL7+c+RQmCvrCAMnaTZovXxVaLNDRBgL8wTDBwQ47CLcBKSWo04TUn9PnpJonse9sI8/atX0bs50jBULarBxTBNEjRdUyHYJ05Q+lt/i3BvD5kmBMWCunAmCaJQxDQMEstS/fh2+44b4VCqH0UqALmqfHGGwi0x8JbXCgVlq1Asgq4pYdNAfDRsFVmnTiL8Prm33yJpNIhbLaXeLRXRx8bQ5oUSk+VyaPk89vnzOCdOKD2C36O/tg5+H2NllbhWp/fr9zEKeeJWW/HHqxNZ8cXQs7sykc9Ds6XcN+vKyiHp99XF7hg+r1Sgzxuf27BTSvn3gb8P8LWvfU0+4uFPjHvl+5ZucdQ/Im/lsWKLftzPDKQEgsPeIRWrwka8gZ/6xHHMhDvBYf8Q27BZrazSDbvs+XuUrBL73X3WWmtESYRjOkSRYpHEaYxneKwUV4hkxHpzndn8LCYm28m2ci1MEgIZKHWmhCAOQIAjHEpGiX1/H20gDgliJcdHKnqgYziZqlNIdUtsWRY5PcdOZweZymx3baLsaYe88yAJEJpg0ptUO3ddqggvNHRN7fqFUB4rZavMVGGKjdYGB9FBdjcwNOUCiGRE3odX1gVmatD0G3iBTXDgY8apKgSuyzCtPfvw67rK82w0IE2Vx8bYGEa5RHTzFnouh5HLEW2o5J3oYJ/44ADn5AmSTldFy0lJuLaGOTub7Z56Fz9QisATJ0h2d0nbagBmnTypdv1PiKHiT7NtFZicJJDLEW5s4l14A2GaCAlavoBzepVwZ4ekdoRRKmHOzGAtLtD5+c+J1tbRSiXFymg2iXd2MKemiPf2FCc8lQjbQcYJwjAwy2U1n+j3CW7dwhwfR9iWGnIOck+tuXnMmVlymqB/+zbJlrJ5FZalqIuFAsbEBIXv/ZY6lvV1NVO4taYuLqap5imWhWy3kLatCnmSqH47DFwNFQc9bSt6oL6wgDbgl1uLC4oL32ohHBWirQ1yTrUwJDk6QkxPgethLS3DsjpXQqjfx7TTwTp5kuDaVezl5cztMK7VkUFAeHCgfMptG61YJNrYIA0CgkYdDIP2T3+Ge+YM9upK5hsvHEe1W3I5pGmRdDqq9VOpZLv5e/F5pQJ9nngWhXwLWDj29fzge58r7qUgLpeWud68jkwkuqbYGpZpqfzMNMQUJofRIXk7z2vF15TSU0ZYuoWlWQRJQCtsUfNraFLDMz38xM+yMku5Eu2wja7pdENlvOUaLhWnQkpKJ+7QT/okMsl8x4dcch0d11C5oWESEiQBBaOAYRo0+03iNMbEJCbG1E2iRImEEpkow6skpq/1CRLlFz3skWtooCmapC50JX4SOu2oTSpSym45SzmK0oiKU2GltMLt9m1l2pVUcA0XW7chQV2EBmyY4ess7STYfZ2+k3AwpjN7OyGNQ9BthOepbMnBIM7wPDUgG0jeE99HazRI6jWVxON6WBMToGukAuh28S68jn/tOiBxL7xB2mkT7uwgByZOxznB3oXXiQ/2iZtNrJMnsGZmFKthICz6NBju5I2JCZXKg5qPpL6vvNIR6rY+l8eaX0CzHfRSEf+DDzAmp5RCMYrR83n6W5tIw1CilloNa2FBnYdWC2tuTtH8XBd9agrTcehfvqxaTJ22SihqtdDKZcL1NTXY1DX08XHc06fxdZ30xk30Ugl7YgK9UiFtt/EvXyb/rW9lhSqu1Ym3tgg2N1UaUa9H/+OPET0lVNMMQ/XKB3RB4brouqFaE76PNT6m2kG+T7K3h7W4hAbEA9uAaH2dpNFQsXSuS9zpYBRL9D/+WPnWx8o8S4ahsrk1Tbw33sC7cCGzkbVyOdX/391D5HJ4X/sachBYktTrkMQ4518lLbVJOp1MGDbsdxszM0SHB8q7p1DAKJfRikXM6enMQO1lx7Mo5O8Bq0KIE6gC/reBf+sZPO8T4V4KYt7K843pb/BnW3+GZ3p8ZfIrNIIGEknNr9FLeviRz3xhnqpXZTleZrezS8EqUPNrNIIGtX4N21DKypJdotVWftdBEjCdm8YxHI56RwjtjmFWJ+gghaQRNPAsTyX/DAKQpZQZL7zklFT4g13EMRxq/Ro6Oo7hKMtamar8zlRiiIFfuHCVd0oSZ0EXQwm/QKiim0oM1OOFEOx0ldGXRNJNVdxbxamgaRpT3hRFp8gCC3zc+FilBUnloy4GA7Fh/ucQ84eSnVLC/FGP2ZqO3tZJDQOEhjk7q4p4u60GZIWCuu0dcKh100TL5zEGPHORyyEKBcxqVQ3vbFv5nBsGcpgkli/grKriLCwVVt27eDFTew5tbodJ6anvf+qe53HFn3BdzKkpZQY2OYmwTPLf/KZ6/we35SrkQSIsG31qirTbI9jYxF5aVKZazRbJ0RFGtUq4s0OwtobsdDBmZ9RFbyD8sU6dJNlLGQAAP+dJREFUIrx2LaP+aZqG5uXUQDAIkLqhZO+6jlYqE+3uIvqKK20vqD1UOujLd957j2hnR2V/DrjmwlI9ZAYDW++1V+lfvZYNjXFd1V5JUxWmPfA50R0HvVDEqFbRp6cJPrqCfeIEMgzpv/suYlD8Za+nskvjmOSoptZqO2i2Q//mTaz5eWSaIiyLaHf3Lu/wYZsDw0C8+Qb6eBVzYoLe3p7a+WsaWl61urRCgeToKGOiDPvdmufhvPoawW8+Iri1hjE3i3PuLN4Lzg1/Ejx1IZdSxkKI/xj4QxT98H+WUl5+6pXdg0fZ196PglhySvxg6QfkrTztsM0yywDcbN7E0iyWi8vs9faIkogxZyzb4XZDVfAc3cE1ldCjYlU40o/oxT0KVoGT5ZNcrV3F1O/01IcXgFbUwjVcLN1SwRJCORJKqQaVOTNHlKoWjaVbzBRmlHthqkQ9rX5L9aZRA9ackaNgFShZJYIkoBf0CGSghqdS0E/6hDLMduWWbiGFpBk2cXQnoxciwNZtemGPslOmHba50biBZ3icLZ9lo7NBvVfPOOZS3jH6Og4zlpS6oMUpMWBrNsSJ6n3quioMA+tSTFPZmJZKWLMzynv69gbh2hpIiX3mtKLXRRH2mTN0/+IvkGGohny+f9dAyp5c+oTaM1hbx15eIm40sp6nPamYLcmlS0+U+uKeP0/SbCr5e7ulIuwGPdbe+++TdDp458/jDVgQvYsX0Sy1E7VnZ1Xf3HWI6w3FQBFqZy9sW6k3a7U7sWxhqIakUYSMQuxXXiGJY9KbNxH5vMo8lZLw9m21k9c0lcpULpMGAd1f/wqnWlVJOr6vLj7j40S3bmKOV9GnZ4hu38b/1S/RKhXQdJV/mqR4r7+Gc84k6XaI6427cjSFUBYAQgiM2VnIeYSbGyRXLqOXBgrcfh/d84g7HZIgUKrQAQ1Qr46j5QsY01PKfjaOiI8OVbuqXMI++8pd3uHH2xzDHXrq+4o6WKshHAfN84hu31b99lyOaGcHLly4+0LQ9/G++lUqf/NvfmmK93E8kx65lPL3gd9/Fs91PzyOfe2D5Pv3s7g9Xz3PBwcfkMqUTthhp7uDoRmcqZzhyD9iOjdNEAVoQsPUTVJSlQFqF9GEytOM05gwCWkGTWzNJkkT5USoaeTNPI7u0AqVM6EUEiEFlmZRdsokMiFOY2xhU7ALhFHIXG6Oj2ofqf65VOwSKVX/uxf3cCOXvXgPW7PVcCuICYUKnPAT/87wUhi4psu4M86txi369FUWI+CaLpZpUe/V1fmJfXUBkQmz3qxqPxkW3aCrUoN0DU1otOM2oPrlO1WD774fE1qCQy3F0UysToLlOGoQV6lgoGhe1sKCKgKdLtb4OHq5QlJvKBn4mTPg2NAPEEi06jhpGCg1oW4QHx7cGagB9soK4ebmfUN140YjK65Pk/qil8vkv/nNu+iHmIZqDQD9Kx+RNlu4588pp8Of/hR9agp7djbrm6MJ+pc+VIEUk4p+GHz0EZppkIKS9iNIWi20fB7rzBmEaWCUKxAoxaaez6lkpV4PzbLQikXcQbsh6XSUE+K5cyTbO2i2jRjwzfuXLqGVS3T+5E+IdnfV7t7zEM0W9uIi8cEh5uQk4eYWzrmzhBsbJL7qs2ueh9B15dcSR2heDvPkCdLDI2SSYM7MYMzO4l+6hLWwgPvVr9L7xS9IazWV9CRA9gdeMbaFkBJn9TT2yiq9X/0S99z5u1gncH+V7rAwazlPSfxbTbqbm+iO6ptrnke0s0u4vq4u3oM7s3utj79seCGUnU8iwb9Xvn9vER8+drm4zHu779GLe9nOO2fmOFU+BcCv9n5FI2zg930qToWFwgJz+Tm2O9u8MvaK4meTMpmbJJEJW+0tECgv8qChdsipRGrKfCohoY/ya1koLBDLWLVG4li1TXR1fEODq6HSU5OaYtQETTzLQ2iCXqJCnaMoyga4Ukpsw1Z99zRkq7OlzK/iQZtECMI4pBf2MDRV7MMkJI5iNDS2OltUvSo5PUdXqjBoXVdKUhPV0hCaYH/RQlzsEGgxuq4MwHwnJTRSKu02URiiFQtYJ0+gFwqI/X00O0bP5dA9j2h7mzSOsRYX8d58A2f1NL0PL5H2/CxIV1gW8d4evV+/T+G3fivzGe+99x7um28Ax4yt7hFzPG2C+nF73LjRQNP0O2ZcQhDu7hLt7eG++irGpGqn9K9ewzm9qor5mVewT5ygf/06aRIrmf1YBdObJzo6JDk8UupMz1PtAssi3FUe7VJK7JUVFS69va16yhMTCCmJd3awV1cxgHBtDW0QEKF7nvIYH5huxTduqt28roys0npduQtOTiKThHBzA71YUjvdnIc9O6usDaanVAjGwHag8L3vEa6tkzYaygTL7ylXQU0nqtWw5+fIf/tbRJubxEc1JcwaG0P2A6IbN4k9T3G6S+XsfTqOB4lwhuc/WVoibXdo/+Qn6s7OMJFRDDLFnKjS+vG/GMjqP3mxfllyOJ8EL0Qhf1z72vvJ9+GTbZmKXWGttcZiaZHZ/CzrrXUaQYPp3DTtsK1saO0yG60N/Nin4TfYbG/iGA7/8tK/TCts8cu9X9KP+1TsitrJ6hamMGkFLSzdUt4mcYhu6FhCtTomrAkcw8ExHBYLi0oVmgR4hsd6ex1Lt1RxT5WdqKVbhLEKnUhkogRAGhS1Ip20Qz2oZzmiAkE/7uOaLqY0CVNFSUxIsl16kiSZW2IraKELXVnrohMkAZ2oQzfsKgfGQd89SiMcwyFO1YUnzee4tRRSqoPmFOh6HloQMrbdpmeZVFZX0ByXaP025ptv4n71q6R7ewS3bythCMpCVXNdrGHIchQR7+1hnzqVBSqnYaB2iPU6YmJCDeEqFVWQX30te2/vFXM8TLn3JB/wpNVSxdWQBNfWiXb3FOsjCLBfOYPmupizs/iXPiCu1Qi2trBPKIl4/pvfHPCvU6Lbt0nqDdB0dNdDej72qVOkqfqZlSQUv/tdhGEi/T7R4QEyDDHn5jAqFeXkpyt+eP/6dSW6giwyL7x2DWE7iHKJdGAdoHue4qcPAyaAcHsb+n2VRqQb6kLTaKgiX60iHBfNtBT1slhAy+WVcGZ8HNFsIvt9kiBEH6sgWy2clb+Eli8QT01R+///YzV4HLxfMk0x8nnqv/tPcJaXlLjo8BBzYuKB79u9CNbXsZaXcet11VoLgkG/PJ8lKh2/WCedDq0f/Qg9n1dBI0tLdzYAL2gO55PghSjkD7OvvRcPKtrH2zI/3vgxi8VFkjTheuN6lhB06eASfuyzUFygn/SZzk+z1lpDIJjJzTCZm+Sn2z9lobDATG6GvJXPFJ4A3ahLQoKnDxgTApDgWR6a0Kg4FUzdxDM9rtSusFJeIZUpeStP3spzqnyKnc4Om+1NIiI8w1Mcb2GgazqapmEJ1e8OkgDP9JCpJEiDzKY2iiMSkSgXxgF1ME1ToiDCEEbGcImTGGEIokS1VjShzLLgmCe5TBSDBTLaZjfu8ssT8C1fIyoYWK7GyZuQOha12SIzi6o/HTcadN79M4yZWYRpYC8t3rET0AS5r3/tjle4aao4siGvGZTgplJROZMDmEtL+O//+hO98+ODzQcp99C0J2q56MUiRBH+x8qoSXMcZBwTHhwoBWK7rcJ6JAjTUvMBqd70pNkkbTYhSTCXloib6gJiTk2Re/st5bl+9Rrm/Dzea69m58F94w3Ebz4iPjpSvXTTxKrM45w6RdRq0b94EefsWZzlZYIbN5SCdHqGJAwUFbKtApfTIFB/H9riSqmEWkJktEyZJGCpi4cYeLggBGmvh3PyJPHODsJWzBkhJcJxMCcmkGkCA+8ZrVymf/mKeu9sG03X0efmVAbp3h56qYg2OYk1N4d/5QpyaZEkDJVRlaZT/K3v3nXOj19og5s3sVdPY1arpHGsDNakJO20Sep11Yoa/rt2m/D2bWSs0pSErhNtbKA5zgudw/kkeCEK+YP636sTq3c97n699B9v/Jil4tJdbZlEJjT6DRpCqS9tw0Yi6cU9RdcL2xlHfNwepxk2aUdt6EIraBHlIopOEUMziBIV/pAzc1w5VGLWHj1FAdR0SlaJOI2p2BVMzcQ13GxYW3WrBEnA9fp1VsurHPgH9OM+O90dTN0kkQmWYaELnXF3nCP/SHmM6wamoVSYUsgszg2405ohwcS8K4kokpE6D6nihKdxqlwT0xTP8ggT1XMf8stBCYX6SZ+8mScViq++Vwn5+Ws2F7ZS4n5E3KyzPe4RO5J+TQUqpGmqPKpnBXS69Gt1NE1gLi3jnDqJXipnFrNGsUhSrSqfkEJhEEysZxzhITTTxL1wAWGZDxRzPEi5JwxdMUUes+ViLy3R+elPFbVu8FgZx2qIqQkVWSdQkvNKBc0wsE+fJvV9uu+9h7W0jDE2Rrizg3fuHLqXQ7guWqmsWDlA7hvvZEU8abdJajW1U83lIE5UcdU04nYb3bIQA+vacGeHxPdJ+32i2hHhtesY4+NKYBVF6k8wiFwbJNinA3dKPZfDmpxEWBZJGCIH4R/m/Lx67q0tNE0gCwU0XSe8fRvheTirq+rfHBygz0zT/dnPMJeWkGmCdfIE8eaWMgdzXbU+IbBXV1RE28Qk9lKX7sUPsKanMcoV9LExgrV15X0+aIccv9AK26Z/+TLW8jJyZ0dpngcBGSKK0cfGsvcq3tlRyUADsZk2CCiJd3ZUe+8l8FJ5FF6IQv64/e/Lh5fZ6exkhXU6N00iE+r9uvLWHqBiV6gHdSzdomCqD1KYhJTtsvJbCdsU7SIbrQ26YRdN1yjaRVr9Ft2wS61fo2gX2e5uKzdCCY2+SgWyNIt9f1/5m+hu1psWmkrrOVU5xcnSSSzDohN2mMpN8aebf8pme5MgCTjsH5KkCYlQLZGyVQahUo1MzWTf3ydv5SlYyhum1qtBqgq1Z3jEIs5UmRF3DLKOuyNquqItpjJVxVuzMDRDeXgnMXk7T5ImVKwK/aSPKUxCGTLhTGQ88wO9wU8mTQpmgX+9E2D4EZV+Sv3oBjlTR8NUt8JpSjLYGRrzixS++12k7yPDgLTvow9aEe7587T++MdZoLL32qsqhqtSudtT/BG3yA9S7vUuXVIBEMfwsA+4Xi5jnzhBvLevgojjGGt2Fq1SIdy4rXxILBNhKSqlsao2FWkU0bt0icT30fN5xW1fPY3z6qv0r15V6sRiEXNxAc1Us4ek3Sa4dg2EUA6Olk3nT/4Ee2UFvTpBvL9PcP065vKyer0oVr40QiNtqpAPYVmYMzPEW1uqePs95UY5tBWOIhXWkCRYp06S9nrEN28iymVSv6d64LqO98YFzGpV5ZDu7Smbgb5P0qgjwwAch7RWRxsbV/a0zboKm7Ys4p0dtBMn1O7fNDEKRUWBBJIwxJqeJvf2HdF36vvZhfTe2YazvEzv0ockh4dYKyuKr16v4154HWt+nmBtPbszi+tKMGTNzBDu7ChGkKXEQfByeKk8Ci9EIYcH97+HaAZNLh1eYswZU6KfJOTSwSXqfp215hp+7ANK1BImIWES4hmeUj8KQRAHLBWXeG/vPWq9GvOFeWq9GqlImbQnyRk5+nofx3LYbG/yivkKjuaw1dniwD9gNj/LfGGe/d4+Y3KMNE0VLTAJQUArbFGySiDUrnmzsclR/4g/2/wzbrVuUe/VsUwLT/eYyc/QjtoIKXAsBwODdtAmloopk6apYpfEXQxdtUts3caP/azFAneCI1LSLNw5SRPydj4LmWiFqlcepzG6rtOP+yov1MwTyIAgDbANmzRO1eN7LQxhEMmIMFB3AWurBc5d7VOwyoSxUnm6/QhjblYFETgOmuNgnzyBOTGRccK9Y2ZGAPbpVYLr10nbbayVFcqvvXYXrfBxpdT3U+59GrMkc2ZGsWt0PevdyyCAJVWMk24XkaZYq6tZ0n3/8mXlrz0ouMNBqDDuCGHgDrsGUAPgfp/46EgZZvV8zPl50qCPODxQVE3DQHY6pIaBqFRI/D6yXkdGiqqpFwpKJDPojw+9b/BU6LMoqiBjDAP/V7/GOnkSoenq7sBQw1CEhjk3r0ywymWi3/un2GfOEB0dIdKUcHsba2mR4PoNpcwsFiFN6V+7jn3qFMGNG8SHB0hdx15aVFmdgznIve0QuPtCeu9sQ8sXcF89T//qVSUIOnf2rpmGXirduViXihjjivZozUD/6jXFAPK8p9IVvEh4YQr5o7DeWqdsl7MszCRNOPAPMh73tfo1DM1gvjBPSkrOyGWeLFW3ykp5hQP/gOncNEvFJXa7uwhNUDSLuIZLJ+xgCFVQNTT2enuZcKZgqni3ilNhs72Jrdk04gaJTMgbeSzDoh22MYTB7eZtDroH5KwcR70jtrvbtIO2MryKQzSpUXSKjDlj7Hf3ieKIWMRUc1V6UQ/HcPATn3F9nLJVphW1aCZNXMNVzzEo9JlrojAyub9lWNngtGyX6cW9bI21oIZMJZZuEcQB9X6doixSsAr4iU/JKpGS0o27CCnI6TkS1L/dO1Ni1bSwr7aRcUJq6JgLc5hTk4Q7u6Bp2DMzGFV1V3Tv8DHa2SHa2cVeXsL7ylfveGk8Q3wasyR7aYloa5twawtZLKrkolYLa242Ewf1Ln6AMAziVovuu++S1OvYJ08ob/HKoEWwtoY1M3t3L//YnUN4e52k38eenUVYFr3LV0jjCLNYQltaRMvnEMUSsqUEaUmziWw1kVGEPj2NSFLlvlguK5rg1JTqFccx0f6+UnDqOubg4pMEgbpozM8TrK2hA8HOLvm330ZoGvbAYdI5d06xXoRQVEjTJBw4VKbtFrpjo1cqmHGETBPc115Vg1MhiFttrIV5RE4pRGUYkjSb9N5/X0nqZ2ZUPNvgQnq/C+29F7/jeBD/XOTyWAtqx67lPLVheAm8VB6Fl6aQt8M2S8UlrjeuA3DYO1R5lTJmrjBHK2zRj/s0gybnqudYb61TtIp8b/F7XDy4yO/f+n0Wi4u8Pfs207lpOmEHV3c58A8Ik5C8lWcyN0kkI/a7+5m0fio3RZIm7PX2qDgVlkvLShgUNjKan6ZplJ0yVbdKO2rj4tIO2xz0DlSeJ0LxtoUKSW70G3x9+uv0QlXMqk6VhIQwDXEMh1yaIyVlujCN1tWycyCRSvqfKBaKhqIlSqSib+quCo+O+3SjLmEaUrSKtMKWMvIaxNRZhqUuIAO1KpLMDtfTPRXwnCYYmoFjOfQNg/rbK7S9NvpRg1zfwDTLGGPjyCgm6SsXv+HuLD48JNq4Tf/Sh0paHqvMxXBjE8d1lcy+034oxexJ8WnMkhSv/B0VXnH9OhJwz529K03Gu/A6vcuX8S9eJO33cV49jzBMlaIThmrAmyb3XfewGAXXrpH6fTTTIFi/je44EAilzvQ8NMcG31f+K90O4dY2WqWi3BfzecLbGxhjY0hdmZfFh4eUfvB9/F//mqheV2IcIdTMYaJKdHBItLmJMVEl/+1vQT8g3t9TRW9wbMmlSxntUnNdhOcS1mrg+1jLy8Tb2wQbmxiTk4hiEcP1GPvbfzs7xmxw2agjNA2zOqESlaKI+OZNur/4BebCAuV/9V8Bns6V8BPv7dgYlTfffOmL93G8NIW8YBUI05DVyio73R1qQY28mWelsoJEqh65hHbUVowWs0ArbBGlEYulRcIkxDZsdjo7AOx0dqh6VdZb61TcCt1IGQtJJJO5SRaLixlz5GbjJo7mEKQBM/kZOlGHgllAExpFq0gn6lBxKkzmJiklJc5Xz/PL3V/SCBuU7BI5maMe1LNQiCAJ6Md9xt1xdnu7VL0qUkhSmbLZ2aRoFDE1k+XiMkf+EePuOGEaZpa5EjUAHXq82LrNQm4BXVN+LbERM+VN0Qyb7Pv7qpVi5UGDnJZTvuWGUnaGiQrGSIVqFRnCwNRMTN2kYldI0xQ/8rlq7tA8neP0donlya/i1AOCW2sIQ8c+dUq1F3J5ooMD+leuIAUktTr9q1eJd3exz59XAqJBYHJUr3+CYgb3H04+Lq3w05gl6eUyhW99K/PGvt/P9Xye3NffUoEIkWJwMBiAmgsLCMt8+AUjnyfp9gi2t5U4p1wm3thQvuGeR9r3B31nS134ej017OsHGMsnMOfmCLa3Sfb3yf/Wd5HtNsbcvFJC3ryl0oIKBTTLUkNPwFpcxJqeIe32MKemMObn0PP5O+vUNPqXB2rOfJ7w5g1EGKKZpgqinpsj2t0l2trC+8qbOGfO3HWMx8917+JFrOVljGZDBVSnCfr4GJpt3TXwfBpXwpfRCOtJ8NIU8iGzZRjdFsQB3ajLyfJJdru7qlctIWfm6EZdFW8W+ZTsErauFJbtsI2jO1w5vMJMbgaE6s33op5KvU8TVsurfHjwIbvdXWZyM5mJ1VJ5iSiNCBO1y81Vchz6h0RppHrZkdoJT+emCdMQ13AzT/IhRzyVqeqtR30+rn3M6xOvY2mKk96JO1i6xcniSfzYpxN1+PX+r6k4FRzd4eP6x8RprHjhQqdoFWlGTZJEtVFCVEGecCfUkDMN6UZKwWloBr1YHWPOyBEmyj89Z+UQCIIkYNaZJSFBlzp+6uMaaod/2D/M2DaL89+mML3M3n6P3Ow05ddey0yLhh/Q5OgQe2mR9rvvKqtW10Xk8/SvXYM0xZydgdXTj+ypDvE0Ss7jz/E0ApJhf9eamVE+MN0uaRQh2x3MvV3sEyfoQfa8976eVihg2zbRzg5S0zByObRXXyW+rYaqstdDc2zVnmg0MMpl7MFwNWk20MfHsebmsL/xDfLf+pZaw6FyErSWl4k2lXlXcHSE4Tik3S7p5CTR4SHCtgl3dij85b+kwjmG5/qe/6e+j/A8rJkZFVoRRcpe2NCxT608NLh4eH7izTbOqVOKfTNwQyRNaP3oRxhTU18a8c5ngZemkN/LbFkoLKhBnqYz5U1x5egKCDhXOJfxzF1T+aEAFOyCMqlC0ugrcVAYh5TsEiVRwjM8enGPmfwMYRKy1lxT4p0kZswdw8BgqbyELnQmvAnliNjZYTY3S9Eu0ot6XGtcY7WyylpjjUhGOJrDQe9AuR1qpmpjCJVkj4C93h5fnfoqf7b9Z0ogpFmZ3/lySfnEaGjsdnfRhc5ycZlO1KEdtZXDo2bi6i66ptOLeszn59HR+bj5MWW7TCpSWkFLGWUh1Q48DbOBacWpECcxnukx4U3Qi3r4sY9IlM1ukAS0ghYT3gQlu0SaprzX/YjFmUVkpcA35+70Noe7pdYf/AH+b35DvL2DME1wHIzxcaKtLcUvti16H14iXF9XrIh2O+MC3284+bRKzuNhEmkcoxkG0dY2+W++8/i7wUF/V0EoPnm3S9JqYUxOohVLyDCid/ED7OWlzORreOFJmi1AYi0vIYSmfMD7fay336b3F3+BPlFFCoGGilrz3n5b7dBtZdfgnHmFtNfLiqm9tET9V79G83J4b1zAd2yCGzcRQUCSppizs2hSkjSbaJaFubSIMMy7HCNlmuK+ep5od5e000bP5xHFEpppYpw9S3DlCtHBAcb4mGLi3NtSOX5RHJyftNvN6JbKzwWC2xsQx9hnXvnSiHc+C7w0hRw+yWwZioOCOOCVsVcASGSSFflW0FL+3wOK3ldnvspGa4Nm0OR6/TrLpWU1XIx86n6dVthiu7NNnMTMF+a5MHGBdtTmoHvAdnebw96hoikGLW42b7JSWUFDo9/v04tVv/tH6z9i3B3n0D+kaBXpx31aQYte0kNHx9ZtPN3D1EwqToV6v860N42pmWx2NrOgZz/2cQwVfpG38rTCFs2wSZAGFOyCuvMIu1mhNTUTP/YxNTMLubB0i6JVpBt1s7aOhqZERFIFYziGw7Q3rURFA+vbIAmURe9gkBynMbV+DYmkbJb5uPYxzX6T89Xzd1FEk0aDcHePeG8fKVSvPGk2MaenMSYnSTtt4r197KVl8t/5DuHaOv6HH+KcP49mmvftmT5tBuPxMAnddZVYZ2uL3uXLD2yn3Ithfzfc2VatlkqFNAzJnTyF8DyS3V2M06cBMo45SUxw/RpptwuGgVGpYC8s4F/8AL1SySh3xuws5liFYGMTzTQwT55EpCn26irh2trAVdD8hFGYXiwwtNzNXXiDwne+Q+Of/j4EAeb8nOrL93owNqbCjo+d26TRIN7bI242McbGcFZWYGWF3qUPSaXM5PhauYy9uJC1R4D73h0NL14YKrdUCqEufLqm+N+VSuY4CS+/eOezwEtVyO/FwyiLG60N/nTzT7l8dJlxZ5w3Jt/AEAZ+7PPdpe9S95X8vRN20IXOZmeTVKaMO+N4thLP7HZ3mc5NQw66cRdbt1X/WDO5eHCRhcKC4ovHHcbcMfzQ58A/wDZsKpaykhWIzFzL1mwltU/7ONKhHbSpulUWigsEcZD18QHq/TrNsIkf+eTNPDkzRytsIaQgjBRzpWyXiZJImXmhMeaOkfNylO0ya+GaKt6kmXLW1pUz43CWIIXycteFzrg3jh3aLBfVMDeRCQeBirfrxl083ePQP0RHp9vr0o/7/Hzn5/xg+QfZOQ/W1zGqVbrvvYcQGsb4OEmzSbC2hvuVN9HHKjinVnBfUxJ8o1Cgv7ZGcO2qYi9kgRJ3bGzRtKfKYDweJgGo2/5ikeD69ccu5Fl/d+0WUtMxcjkVAj3cpQ74zMJxiA8OVE/72nVlVZAvqFCFGzcZ+7f+Taz5ebrvvUf/4vvEjQa5r30Nc2YW+9Qp+levIWybuNvFNAwVsnHhTsjGXbv8VhtreRlnIIvvX7uKbtskcUR8cIg+XlUF+/BQpRAt32n79C5+oH7e7ZF2uvgfX8VeXMCamyU+PCSNVNLTcf/34zmq9zc1e52079N596cI08ReXiau15VN7rHwhy+DeOezwEtdyB+EZtBkrbXGmfEznCqf4nbrNtfr18mZOc6On2XCm6DqVtnp7lCwC9ys32QmP0PRKiKEkrXn7TxXjq4QpzGWbpG38kgpWamskLfybHe3OfKP0ISKdjM0g5yZI2/lMTRDFVeh0YpUkRVCZJRBgaDVbyEQlJ0yk94kt1u3VYRcmrLv71OyS5nXeTtSAqacmcM1XFpBS3Hj04CZwgx+6GeMF9dwkUKSM1TBjpKIftxH13SqXpUxawzDMDjqHWUxcEIIdru7nCyd5Cg4oupV2evsoQudbtIFCbquk0QJzbBJySpRdar8dPunvDXzVrYrT1otZBDgvvoqwUdKiq6Pjamh3sBz2zwWBKDlC7jnXyVp1O8KIjhesNJmkxSh7GI/ZQbjvbFwA+X9E0Evl/HeeCPLlpTXUClJkKlTZb+PMTGh2kGOk7FCpBDolQr+5cvIOMFaWsY+8wr9Dz9Uu91Ba0nomuJV6xr2yRNZC6J38eInCqi9rHboej6vLiC1mmKX2LbaFUuJNT2tuN7Ly3Tfe4/g9m3ivT2M8XHMiUmV3rOzo6Lojo4ofv/79C5dUspLceesHS++D7s70hyX4nf/sgqRqNdJmy3s5RPZ8Q3P0csu3vksoD36IS8fjrspFuwC5yfO89WZr4KAqqu4znkrz2plla9MfgXHcDjsHSrVaBozX5wnjuOM3njUP6LRV8ZaP9/5OZ2ww1vTb6FrOjkzx0xuhn7cJ4ojzlfPY2mWCq7wa+x19+gnfTSpkaaqN+0YjvI6SRLGnfGMr16yS+x19zB1xVhxTSW3T9IEIQUXJi4w7owz5o6RN/I4msNsbpbp/LTKNC0t0eg30KVOyS5h6IYKzhDKHMsUpuKoO+O8M/8OVbeanQfP9GiHbYIkUC2hVDk5SimVjD/uY2gGaZpyonwiu2Ctt9YHBfgiwc2b+FeuoJdK5N55B/fCBXTPQ8/nMCcn8N5+O1M7DnH8g328Hy6ECjI2JibRS8VMtq+ERo/fY3VWVrILjJQSGQQkrZZqJzwh7KUlFSjh+5jT08TNJkmziT49nQlTcl//OkmtRirVTCINAmS/j7m0RP/69buOz1peBqB35Qr+x1cVz39xkfw734T4jkd80mp9wl1Qr05gTU9l58UolfBefx0JGJOTWAsLaIM7mjQMiZtNZTPcbOF//DG9i+8T3LiOFOCcO6uGkcdSee73HunFIvHhIcHVq/Tef5/g6lW14y8Ws/fOmJjEPX2G/NvfIP/Nd9QO3/ezFKa01/vSpPo8S3wpd+QPclOUUt5lztUJO1ypXWHMGaPqVjNZ+7D1YBuqFXLkH6FresbJvla/xnxhnt9a/C0+OvqIvd4eeTOvzLbyM0RJxMXDi3TDrgqdEMqV0NRMHN0hjNWgcqGwoKKXZcKkO8leb48wDZGR5Nf7v6YdtKm4FSpWhVpYY7+3DwJmcjMZ77uf9LE0i+ncNPudffZ6e1hCuSy6hsuYMUatXwNA13QAttpbdKIOZatMuaTuCEC1c2zdph7UyZk5giTIvpZSknfyzOZncXWXbtTlRPEEncMdevsNleJy+jTh7Q36H3+Mc+YM9vIy5vQ01sI8+tjYI7nED+qHp33/vqKRx8G9YRLCNLHmZj/BwngcZstxCl3a93HPnVU7+yRG5LyMTudeuEB0eyNzKjRWV1XIBHfbveqFAs7587T/xY+xpqbRKxWMmRn0QuEuefuDVKvGzMwnlKTW0tJgN9wETcOcmxu0eHLq4ujYyoCq28M6dQrCEP/Dy7jnzgIP53snzSadP3sXvVhE5PMknQ7hzg7l3/5rBLdvf+K906sTWEHwUO+cER4PX8pCftxNsRN22O3uUuvXsDSLw94hVa+KozusNddAwhuTb3CzeZMD/wBDGGx3tkm44ww4k5thv7efpQcJIbjdus0PV37IWzNvZSEWG60NWmELP/GZcCdwdZdUpOx1FPtkKJVHQNks45ouU94UAB8dfcR2ZxtHczL7W1d3MxMxT1f+1q7ukjNy3G7dpuyUmXan8RPVWlkZUwHRQwWordlZK2TMHaNoF8kbeXa6O1yvXccxHFbHVrE0i3Pj53hv9z22O9ucLJ9kwpngVusWmtCYzk1T89Ww0499enGPqldltjBL/sYRWm58cNvvUnjnG7R/+jP8K1cwJqrouhLBFE+efCSX+NPI7B+F42ESDyrST0JxfBw+s3f+PL04yeLpsgSklZVPHJ9mmlhzc0rx2lWFMbhxHeF5GJ4HFy48lphmeG6FoauB6uws5tISwUcfIdP0jqUwgGGQDnbdwxaTvOd57vceBevruOfOEjcapN0OWj6HNT9H3Gg81sVmhE+PL2UhH3LOu2GXjdYGQhOYwmSuMEc/7hPGYTZcPDd+TrFArBw3GzfZaG3QDtu8M/sOiUzY6e4w5owhhWS3swvATnuHelDnn9z4J5wsnWS5uEw9UKk83bCLozsUrSJVt0qURpTtMpvtTZpBE1MzmXKnmMnN8ObUm8ozXIis73zoH9IIGsRJTGqlWQ++H/exNeW30k/7VHNVlWAUNRl3xonSCD/2OV89Ty/uKR8X06ETdtCERpAqr5mD3oGKeNN1Sk6Jfqw+0Dkrx2vV19hqb9GLehTtIlPeFK2wpfxt7LwKrR7w94t2kdut2ywFecT4nV2mMT1D7p1v0PnJT5QApVLBrFTuFoY8oBA+jfrvYXhU8X1aiuN9X+8+xRC47/E5KyskhweEG5vZgDRttwlb7U9kXz5sZ6uXy+S/9S3c8+fvXLiO+ZQAIKVyQmy1sjsG8/x5SOJHnq/7tXiE45C0WnivvfaZvHcjKLxUhfxRuZ5DDNks/3z9nxPJiIpZyfzF/djH0ixen3g9U4sO4ZkeE7kJSlaJt2beAuCfr/9zav0aFafC3PQctxq3OPKPqLpVbN3mN/Xf0ApbvDP7Tsag+eDgA6I0ohk2udG4QRirQaSpm5TsEov5RRWaIVS4xLX6NbY6KoEoJqZgF2j1W7TCFpvtTX64+kNuNm/i6i6e6WXy+XbYRiBohA1Uzhh4hkfJUoPSWr9Gxa7gmA67nV1aQYuDnvKBmSvMYWgGiUwo2SXWm+vM5Gf46yt/nXq/TixjelGPV8ZeQRc6hm5QsAr8cveXdKMuZatMmIb8tPUBK2mTE/lFzMMWabdLuLuLvbxM7q37O+E9CE+r/vu0eFqK4/3woGL4oAJf/z/+T4RhkMYx8cYGcaulLmwDmuSTKBsf6FPiOGAY4Pfx3norG0IqDxPvIc+oIDQN/8PLGKWSujAca8s8r/fuy4KXppA/Tq7ncZTsElO5KU6Pnb5rAn88eehBO/cxdyx77u8tfS973c3WJvV+HVM3mc3PKp43IrvADAv5UnGJKW+Kj+sfq/ZKomT0QRKwUlrJFKI/2fwJS/klpJAZ08UPfRWibNiIVBAmIXudPeI4ppcqU62D3gGGbuDpHpZukZJm3i/dqMtOd4eZ3Axlu4yu6dyo3yBMQtphm93uLqZu4hgOs94sYaycIiXyrguRa7p4hkc37hKlEatFZY0wV5hDQ8sCM7QTJ6j9+hqidZ3Z6VVc0yOt1UkHfipZuMRjFMbnFeH1WbR0HvhaDyjI5sw04e0N/A+USZcxO4sUAv/ixadKi7+3wNoLCyTNFsIw7rYPfoyd8/E2jOABbZlR4f5M8NKwVo4zUYQQSkJuuqy31h/4bwSCK4dXuLh/kev163TCDkf+EXvdPd7depf11nrmZxLJiJyZY3VslQlvInvu4e7e0iz2e/tIKTlROoFnqh2MpVvEaUw7bGevW7JLnCyf5CuTX1GK0rjLuDPOYmGRjdYGe9095csycBwsmSV0obPT27njZpimGMLA0q3MDyaVKalUOaLdqMt2Z5t9f58gDgiSgHFnHEd36EZdfrb9M7Y6W8RJjGu43G7f5srRFRpBgyiJOPQP2WhtZD4unuFlax8er2u6JGnCfGGenJmj3q8jU/WxtQ1b0S7LFYK8hVkoUmvvoZsmzrlz6Lmc8o4e4FGFcbhzlGGk+tQDpWTSaDzNr81j4Tgb5XmxK7RCgWh3F2txEXtlBc00ibe2EI6bcbg/LVQxv0DhO9/BPX8erVTEv3KZ3q9+iQyDx2cBpakSbxkGaaeDZhg458+ryLkRPlO8NDvyx831HKIZNGmFysOkYBYIk5Bf7v6SIAl4c+rNbFe/1lojb+UfunM/Ljz6zdFv0LQ718cwCTE04xOxdBLJt+e/zXcWvsMH+x+w19vjdvM2+/E+y8VlbM0mFSmmblJ0ioyH42r3ToqNjSbUa4y747w59SZVt0otqBHGIX7iM5+fpxf3uFG/wSvjr5Azc9SCGt2gmyUQLRQWMDWTS5uXiGWMozvous6Bf0A9qLPf2adoFanYFcbd8bvucF6feJ2l4hKXDy9zo3GDzfYmlmYxX5hnq7OVWR+ESUhRszHPnqETd7EnTivf7qtXiWu1x971Pes+9ZPgadsCz+JOQgAEAWLIKR/8LmqWeZdHyqfBcH3xzk6Wd/lp7IT1YhEZRpmKFR6/LTPC0+GlKeRPkusJagdf9apUnAo73R26kVIjjrljTHhKDTd8roPewWM991Jxie3ONtud7exn7ajNbG6WpeLdu7fj6z1ZPplJ4iUSR1fMlIncBJZmsdHaIEgDzoyd4dA/xNAMTN3MQpVfrb5KJ+rQi3qUvbIKbU5CFRIRtrM2h2M4XOxcVCrU1ObK0RXq/TqH/UMczcExHUSqXA+jJMI0TFWQBTiGg67pWYvoeCvrzak36Sd9DnuH+LGPIYzM+qCf9FmqLhD6HTwvDyhanbW4SHJ0+NiF8bPoUz8JPm1b4FmYeoHyPrHPvkK0v4/0e2iui33qJGnPf6oWz/H1xb3eU+VdflbD6E+D59WGe154aQr54+Z6DjHcwQtDsGqpx7wv3//E4xzdIWfl8CP/kc9dskucHz/PkX/ElcMruIYqcsfVjfdbb87MMV+YZ6O9QdEpUrbLVJwKR70jEEqclJM5cMAylGujoRkUzAKT3iTnq+dZb61j6Upo1I7UkDOWMUEcsBfvYWom9aCupPdSKk48yjQrTVJacUsJfITE1ExaYYtT5VOcKp8iiAN2ujtMe9N8dPgR7bDNXnePcXf8rizUqlcljJX746XDS9i6jaVZfOQ1KF7f48LCW9kOXGgaxe9//4mNqT6PPvWzxLO6k9CLRay5OUiliqyzVOAzuv5ULZ7j65O93lPlXX5RBprP6uL5IuGlKeSPm+sJqq2y193jav0qFecOY8XU7oQVD9FP+szkZlgqLj3yuY9L/y9MXsgK/uOst+JU+LfP/ttcPrrMdnebuq/8yYdeL2POGHvdPQxhULEr1IIafuxzonxC2dyGHT6uKVfDaW+a9fY6SFitrHLQO2C3u8tB7wBbs5GGVC6LwkDXdCQyC10O0xBNaDi6kxVpS7c48A+o+TVyZo6SXeJq/WqWWJS31E7b0R2COOBbc99iobDAjzd+TJAGVKqzjBWX2d0+wDswKIxPP/EH/Iu023sSPKs7CXtpiV6jibUwryTuR4dZEv3TFKfj69NyuU+Vd/lF2/0+zzbc88JTFXIhxH8D/HUgBG4A/76UsvEM1vWp8DCTrCFzZLe7y253l6pTpSd6dKMu12rXmC/OU7TVL6wf+5/YeT8qMxTuHrjCndbMccbKo9ZbtItcPrzMj9Z/lImElovLmLrJfm8/C7wYsk6iNOL3bv4eZ8fPcmHiArdbt/nj239MwSow4U0gkcwV5+hFPWp+janCFDvdHVzTzZKJJJIpbwqhiSwNaLm4jG0oJo2Ukma/Sdkts1xaRghBxVFhG7vdXVYsJWc/3m6qB3XOVc/d1Y7yq5PsaRZTjziPD4Sh41+5DChp/Yuww3pWdxLHd7sYBs6pUxjlMnGjoVSTxWL29ZMU1OPrs2Zmnjjv8ou4+33ebbjngadlrfwz4FUp5evAVeDvPv2Snj2G/dxhio6u6dSDOrP5WfJmnkhG1Pwa78y+wzuz72BpFs2gmfHJ77erv9/z//n2n7PZ2qQTdrKfObpzF2PlUSjZJb45903+tdV/jdNjpzk7fpayU6bm1yiYBcbcMSa9SV4Zf4WKU+Fm4yZFu0gjaFCwCyyVltA1ZYc74U5QtsvsdJRo6VTlFAW7wJg9RtkqYxs2s/lZzo2fo+JUlJ2v0HAMh5XKCufGzpGmqXptu8C5sXPZ7nsmN4NMpbKvlUrR6Ud+NgsYhnQcx5OeiyGyYmHZeF/5Ku6588hjXiNfZDxLxstxdom9pKxhhyyepFaj8Xv/lLhWfyJWz/H1DfMuZRzfybt8REG+n/+N5nmPZNIM/XfaP/mJcrN8huyjh/nBvKx4qh25lPKPjn3558DferrlfDY4vlPuxl3FUklD2lFbRcFJSTNoZgX7UTvv4zg+9Jv0JunGXa7VVYBE3so/dOD6MCwVl/jZ1s8Yc8eQUmZp93krr2T8kPXEF4oLdCJ18djp7lCxlWe4EMo98SQn2e/tM+VNkZIy4U5w0DvAMRwlCDIcfrL1E06WTzLtTSORHPqHuKbLqfKprK10XByVt/LMF+ep+bX7tpuedPj8MPQuXyba3kYmSRbcOywWX/Rb5c+qb3xv+2Aog0/rdcTExGO3E5427/LT7H4/6138i9qGexo8yx75fwD8/x70QyHE3wH+DsDi4uIzfNmHY1hoBSKLLguTUPmED3bOn7bAwN0XiZn8DNfr1xFCcKN+A13TaQQNXqu+dteF4nFQsku8NvEaG+0N2lFb9aadEjW/hhgYr4ZJSNku04k6ykccOPQPCZKAnc6OCoXITWd97H/jlX8jW/Nud5dO2CFnqQCKv7L0V9jp7tAIGpTtMm9Pv818cf4uEdO9w2Rd6Hxv6XuU7NInVLUVu8Jaa+2uxz9s+PwgJI0G/sWL6ONVNNclDUOCa9ewVlag73/isV+kXu0Qn4UQ5t4Cmna7yqiq282+97jthKdZ36dpHT2oh+1fvoyWzz/1+/dFGbp+nhBSPtx5WQjxI2D6Pj/6e1LK3xk85u8BXwP+pnzUEwJf+9rX5C9+8YtPsdwnw7CI73R2VAIJgnqgfrE90yNn5JgvzuNH/mO1UIbPebxg7XR3mMvPZbzeTtjhRv0GHx5+yBuTb7BYXMTQjSd6jXvX75oucRJzpXaFftTH1Ew8y0OmkjF3jPXWOmfHz+LoDj9a/xGpTFksLNIIG7TDNouFRVYrq3xz7pv3fZ0/XPtDDroHOKZzh7oY9ZnITfBXlv/KfY99eDGRyCwcY2g2NizaQ4+ZR1kmPAy9ixfxP/oITdMz/+40CCBJcM6d/YS7371GVC9CH/3ToHfxYuZ9Dio4Iu100XM57AGPO/X9QXvkszOl+jTnvf2Tn3zC0zxpt/B//T7e17/+pXj/Pi2EEL+UUn7t3u8/ckcupfz+I5743wN+G/je4xTxzxPD3fJSaYnr9evYhk3ZLtONuiRpojI7NeuB7JZ7cT8bgL3uHo7uUPXu+Jjrms4bk29wfuJuK9QHDT0fhOPMliAOeKWi4uraUZtu2CXv5pnOTfNa9TXqQZ0PDj5gvjBPnMbk7Txj3hjtsJ3deby79e4nCmozaPLx0cc0gyZjnrLr9QyPIAnoht37ruf4eXB0h8uHl+lGXdVnN0TWTqkH9Sc63vshGXiKBNeuq4GOZYGUJPX6XX3mLxtT4d72gVEu42/vKMOrJ5TWPw0+ze73frv4YH0dvVL50rx/zxpPy1r5q8B/AfwlKeXjS8A+Jxzniq9UVrJ2gi50/sbK33ji3eH9WCmLxUXWW+vkrFy2G20EDS5M3L0LepjK9GF4HLYMwAIL2fEO2SSdsIOQQjk6piGNfoOr9au8v/8+3134LkW7yAcHH2QBE37oczu8zVR+Ck1q5N38Y52HWMYUrMJdDJZPe7z3YqgWdE6vEu7skHbaYBi49+zUvmxMhfv1tsu//dcUa+Vzbic8aWvmfj3spFbDfePNux73Mr9/zxpP2yP/7wEb+GeD26Q/l1L+h0+9qmeE4wO3vJVnxVrJ3A2ftIjD/W0Aqq6yix0yXQSCvJnno9pHjDljTOemn2ro+SQYHu/wWAEuH1ymZJfYbG/i6A7jzjjtqM2PN37MSnkF13SZy8/RClq0w7YKce43eW3iNcacscc6DzkzR5REdKM7O/hndbzDD73medgrq9lO897ghxdVMPQ0uF8BtV6AdJ377eLdCxcemg41wsPxVPRDKeWKlHJBSvnG4M8XpoiDGtD5kY8f+/elyD0phoXyOPpJn+ncNK9PvM6r1VdJZMJcYQ5TmHSiDtdq15R0/Sle93Fxv+Ot9+tYmoWjO1msW8EqkMiEm82bOLoaiGpCYzI3ydnxs6o9JPQHrvfe8zCTm6EdtTGE8UzO83FkgQiPiHL7IhhbjfD4OE6l9C5cwDt/fvT+PQVeGvfD++G4U9+T8MIfhEddGIYthwlvgtWxVbVTlRFH/tFTve7j4n7H+9rEawRpkJlYgWK7VOxKFm2Xt/KsVFYwNZNav0bJKj10vfeeB13Tmc3NMl+Yfybn+V7c+6G/X7vgcQv+CF9MjN6/p8MjWSufBT4v1spngYeFV7y79a7qyR+bxreDNtcb1zlROvFEzI0HMUSG1L7HZYM0gya/c/130DVdBWUkKv1ovjCPqZkkMsmGlkO2yeOKoB4nxGOEEUZ4dngQa2VUyJ8hhurR4+HNl48ukzfynKuee+xC2Qya/Gz7ZzSDpjKo6u1RcSp8ZfIr9JM+Hx19xLnxc4y744/1nButDX688WMSmVCxK1ScCprQsiHqqCCPMMKLgU9NPxzh8XGvaGYY3rxUWsrCLuDRNMTLh8o4q2gWs6T6Zr/JzeZNPNPLJPlVr/pYz7lQXOCHKz98YMF+XIrgaBc+wghfTIwK+TPEvY6Gw/DmoT8JPB4t72bzJgWzgG3Y9JM+OStHGIdstjeZK8yRN/N3MUQe5zkfl8b4IDxplN4II4zw+WFUyJ8xjhfMe8Ob4fFoecfbXa7hEiURSJUKkzNzdKIOefPOxeHzoDY+qbPjCCOM8PnhpWatPG/cy+447B1y5fAKu91dPjj44IG76FPlU7TDdpaz2Y26NMMmc7k5ynaZVtCibJefOdXvYXiWboYjjDDCs8WokH+GOE4H3Opssd5aZ7G4yGx+ljANH1jMz1fPM5ufJU1T4jRmJjfDtDdN1asy5ozx2yd/m4pT+Uyofg/Cgzj0n/WdwAgjjPBojFornzGOt1rujUaD+7cmSnaJd2bfeehgcYGFz+kIFJ40Sm+EEUb4/DAq5J8T7ifvf9iQ8mmHk88aTxKlN8III3y+GBXyzwnPMmjheeGLdnEZYYQRFEY98s8J95P3H/YO6YQd3t1696HDzxFGGGGEh2FUyD8n3OuDEsaKlmgZyonxYcPPEUYYYYSHYdRa+RxxvDXxwcEHWIY14mWPMMIIT43Rjvw5YcTLHmGEEZ4VRoX8OWHEyx5hhBGeFUaF/DnhWYdejDDCCF9ejAr5c8KzDr0YYYQRvrwYDTufI0a87BFGGOFZYLQjH2GEEUZ4wTEq5COMMMIILzhGhXyEEUYY4QXHqJCPMMIII7zgGBXyEUYYYYQXHOJ4rNjn9qJCHADrn/sL340qcPic1/BZYHRcLxZGx/Xi4ItwTEtSyol7v/lcCvkXAUKIX0gpv/a81/GsMTquFwuj43px8EU+plFrZYQRRhjhBceokI8wwggjvOD4Mhfyv/+8F/AZYXRcLxZGx/Xi4At7TF/aHvkII4wwwsuCL/OOfIQRRhjhpcCokI8wwggjvOD4UhdyIcR/I4T4jRDiAyHE/yGEKD/vNT0NhBB/VQjxsRDiuhDiv3ze63laCCEWhBA/FkJcEUJcFkL8p897Tc8SQghdCPFrIcTvPe+1PCsIIcpCiH88+Fx9JIR453mv6VlACPGfDX4HPxRC/K9CCOfR/+rzw5e6kAP/DHhVSvk6cBX4u895PZ8aQggd+B+AfwU4B/ybQohzz3dVT40Y+M+llOeAbwD/0UtwTMfxnwIfPe9FPGP8d8AfSClfAS7wEhyfEGIO+H8AX5NSvgrowN9+vqu6G1/qQi6l/CMpZTz48s+B+ee5nqfEW8B1KeVNKWUI/CPgh895TU8FKeWOlPJXg7+3UUVh7vmu6tlACDEP/DXgHzzvtTwrCCFKwL8E/H8ApJShlLLxXBf17GAArhDCADxg+zmv5y58qQv5PfgPgP/reS/iKTAHbBz7epOXpOgBCCGWgTeBv3jOS3lW+G+B/wJIn/M6niVOAAfA/3fQMvoHQojc817U00JKuQX8v4DbwA7QlFL+0fNd1d146Qu5EOJHg77WvX9+eOwxfw91G/8Pn99KR3gQhBB54H8D/p9SytbzXs/TQgjx28C+lPKXz3stzxgG8BXgf5JSvgl0gZdhVlNB3d2eAGaBnBDi33m+q7obL33Um5Ty+w/7uRDi3wN+G/iefLFJ9VvAwrGv5wffe6EhhDBRRfwfSin/9+e9nmeEbwF/QwjxrwIOUBRC/C9Syi9UcfgU2AQ2pZTDu6Z/zEtQyIHvA7eklAcAQoj/Hfgm8L8811Udw0u/I38YhBB/FXV7+zeklL3nvZ6nxHvAqhDihBDCQg1jfvc5r+mpIIQQqH7rR1LK//fzXs+zgpTy70op56WUy6j36Y9fgiKOlHIX2BBCnBl863vAlee4pGeF28A3hBDe4Hfye3zBhrgv/Y78EfjvARv4Z+r94c+llP/h813Sp4OUMhZC/MfAH6Km6v+zlPLyc17W0+JbwL8LXBJCvD/43n8lpfz957ekER6B/wT4h4PNxE3g33/O63lqSCn/Qgjxj4FfoVqwv+YLJtcfSfRHGGGEEV5wfKlbKyOMMMIILwNGhXyEEUYY4QXHqJCPMMIII7zgGBXyEUYYYYQXHKNCPsIII4zwgmNUyEcYYYQRXnCMCvkII4wwwguO/xuGx8nRIA1gCgAAAABJRU5ErkJggg==\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/Clustering_22_0.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "fig = plt.figure()\n", - "ax = fig.add_subplot()\n", - "unique_cluster_labels = np.unique(cluster_labels)\n", - "for i in unique_cluster_labels:\n", - " ax.scatter(data[cluster_labels == i, 0],\n", - " data[cluster_labels == i, 1],\n", - " label = i,\n", - " alpha = 0.2)\n", - " ax.scatter(centroids[:, 0], centroids[:, 1], c='black')\n", - "\n", - "ax.set_title(\"First Grouping of Points to Centroids\")\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "So what do we have so far? We have 'picked' $k$ centroids at random from our\n", - "data points. There are other ways of more intelligently choosing their\n", - "initializations, however for our purposes randomly is fine. Then we have\n", - "initialized an array 'distances' which holds the information of the distance,\n", - "*or dissimilarity*, of every point to of our centroids. Finally, we have\n", - "initialized an array 'cluster_labels' which according to our distances array\n", - "holds the information of to which centroid every point is assigned. This was the\n", - "first pass of our algorithm. Essentially, all we need to do now is repeat the\n", - "distance and assignment steps above until we have reached a desired convergence\n", - "or a maximum amount of iterations." - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Converged at iteration 5\n", - "Runtime: 0.476909875869751 seconds\n" - ] - } - ], - "source": [ - "\n", - "max_iterations = 100\n", - "tolerance = 1e-8\n", - "start_time = time.time()\n", - "\n", - "for iteration in range(max_iterations):\n", - " prev_centroids = centroids.copy()\n", - " for k in range(n_clusters):\n", - " # this array will be used to update our centroid positions\n", - " vector_mean = np.zeros(dimensions)\n", - " mean_divisor = 0\n", - " for n in range(n_samples):\n", - " if cluster_labels[n] == k:\n", - " vector_mean += data[n, :]\n", - " mean_divisor += 1\n", - "\n", - " # update according to the k means\n", - " centroids[k, :] = vector_mean / mean_divisor\n", - "\n", - " # we find the dissimilarity\n", - " for k in range(n_clusters):\n", - " for n in range(n_samples):\n", - " dist = 0\n", - " for d in range(dimensions):\n", - " dist += np.abs(data[n, d] - centroids[k, d])**2\n", - " distances[n, k] = dist\n", - "\n", - " # assign each point\n", - " for n in range(n_samples):\n", - " smallest = 1e10\n", - " smallest_row_index = 1e10\n", - " for k in range(n_clusters):\n", - " if distances[n, k] < smallest:\n", - " smallest = distances[n, k]\n", - " smallest_row_index = k\n", - "\n", - " cluster_labels[n] = smallest_row_index\n", - "\n", - " # convergence criteria\n", - " centroid_difference = np.sum(np.abs(centroids - prev_centroids))\n", - " if centroid_difference < tolerance:\n", - " print(f'Converged at iteration {iteration}')\n", - " print(f'Runtime: {time.time() - start_time} seconds')\n", - " break\n", - "\n", - " elif iteration == max_iterations:\n", - " print(f'Did not converge in {max_iterations} iterations')\n", - " print(f'Runtime: {time.time() - start_time} seconds')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "And thats it! We now have an extremely barebones, un-optimized k-means\n", - "clustering implementation. Lets plot the final result" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": {}, - "outputs": [ - { - "data": { - "image/png": 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/Clustering_26_0.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "fig = plt.figure()\n", - "ax = fig.add_subplot()\n", - "unique_cluster_labels = np.unique(cluster_labels)\n", - "for i in unique_cluster_labels:\n", - " ax.scatter(data[cluster_labels == i, 0],\n", - " data[cluster_labels == i, 1],\n", - " label = i,\n", - " alpha = 0.2)\n", - " ax.scatter(centroids[:, 0], centroids[:, 1], c='black')\n", - "\n", - "ax.set_title(\"Final Result of K-means Clustering\")\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now there are a few glaring improvements to be done here. First of all is\n", - "organizing things into functions for better readability. Second is getting rid\n", - "of the small inefficiencies like manually calculating distances and argmin. And\n", - "finally, we need to optimize for better run-time. It's like we always say: the\n", - "best way of looping in Python is to not loop in Python. Let us tackle the first\n", - "two improvements." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Towards a More Numpythonic Code" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Converged at iteration: 5\n", - "Runtime: 0.41599512100219727 seconds\n" - ] - } - ], - "source": [ - "\n", - "def get_distances_to_clusters(data, centroids):\n", - " \"\"\"\n", - " Function that for each cluster finds the squared Euclidean distance\n", - " from every data point to the cluster centroid and returns a numpy array\n", - " containing the distances such that distance[i, j] means the distance between\n", - " the i-th point and the j-th centroid.\n", - " Inputs:\n", - " data (np.array): with dimensions (n_samples x dim)\n", - " centroids (np.array): with dimensions (n_clusters x dim)\n", - "\n", - " Returns:\n", - " distances (np.array): with dimensions (n_samples x n_clusters)\n", - " \"\"\"\n", - "\n", - " n_samples, dimensions = data.shape\n", - " n_clusters = centroids.shape[0]\n", - " distances = np.zeros((n_samples, n_clusters))\n", - " for k in range(n_clusters):\n", - " for i in range(n_samples):\n", - " dist = 0\n", - " for j in range(dimensions):\n", - " dist += np.abs(data[i, j] - centroids[k, j])**2\n", - " distances[i, k] = dist\n", - "\n", - " return distances\n", - "\n", - "\n", - "\n", - "def assign_points_to_clusters(distances):\n", - " \"\"\"\n", - " Function to assign each data point to the cluster to which it is the closest\n", - " based on the squared Euclidean distance from the get_distances_to_clusters\n", - " method.\n", - " Inputs:\n", - " distances (np.array): with dimensions (n_samples x n_clusters)\n", - "\n", - " Returns:\n", - " cluster_labels (np.array): with dimensions (n_samples)\n", - " \"\"\"\n", - " cluster_labels = np.argmin(distances, axis=1)\n", - "\n", - " return cluster_labels\n", - "\n", - "\n", - "\n", - "def k_means(data, n_clusters=4, max_iterations=100, tolerance=1e-8):\n", - " \"\"\"\n", - " Naive implementation of the k-means clustering algorithm. A short summary of\n", - " the algorithm is as follows: we randomly initialize k centroids / means.\n", - " Then we assign, using the squared Euclidean distance, every data-point to a\n", - " cluster. We then update the position of the k centroids / means, and repeat\n", - " until convergence or we reach our desired maximum iterations. The method\n", - " returns the cluster assignments of our data-points and a sequence of\n", - " centroids.\n", - " Inputs:\n", - " data (np.array): with dimesions (n_samples x dim)\n", - " n_clusters (int): hyperparameter which depends on dataset\n", - " max_iterations (int): hyperparameter which depends on dataset\n", - " tolerance (float): convergence measure\n", - "\n", - " Returns:\n", - " cluster_labels (np.array): with dimension (n_samples)\n", - " centroid_list (list): list of centroids (np.array)\n", - " with dimensions (n_clusters x dim)\n", - " \"\"\"\n", - "\n", - " samples, dimensions = data.shape\n", - " np.random.seed(2021)\n", - " centroids = data[np.random.choice(len(data), n_clusters, replace=False), :]\n", - " distances = get_distances_to_clusters(data, centroids)\n", - " cluster_labels = assign_points_to_clusters(distances)\n", - "\n", - " start_time = time.time()\n", - "\n", - " for iteration in range(max_iterations):\n", - " prev_centroids = centroids.copy()\n", - " for k in range(n_clusters):\n", - " vector_mean = np.zeros(dimensions)\n", - " mean_divisor = 0\n", - " for n in range(n_samples):\n", - " if cluster_labels[n] == k:\n", - " vector_mean += data[n, :]\n", - " mean_divisor += 1\n", - " # And update according to the new means\n", - " centroids[k, :] = vector_mean / mean_divisor\n", - "\n", - " distances = get_distances_to_clusters(data, centroids)\n", - " cluster_labels = assign_points_to_clusters(distances)\n", - "\n", - " centroid_difference = np.sum(np.abs(centroids - prev_centroids))\n", - " if centroid_difference < tolerance:\n", - " print(f'Converged at iteration: {iteration}')\n", - " print(f'Runtime: {time.time() - start_time} seconds')\n", - "\n", - " return cluster_labels, centroids\n", - "\n", - " print(f'Did not converge in {max_iterations} iterations')\n", - " print(f'Runtime: {time.time() - start_time} seconds')\n", - "\n", - " return cluster_labels, centroids\n", - "\n", - "\n", - "# quirk of numpy / Jupyter need to set seed again\n", - "cluster_labels, centroids = k_means(data)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Note**: the start of the timing is after the random initialization, and first\n", - "'cycle' of our algorithm. This is technically not the correct way to time it but\n", - "due to this being in a Jupyter notebook and the way it is structured this way of\n", - "comparing our algorithms will produce a more equal result. When timing code we\n", - "should always encapsulate our whole computation block.\n", - "\n", - "So we see an improvement from just switching to numpy's argmin function. There\n", - "is a very nice tool (or category of tools) called profilers. These can be\n", - "utilized to make clearer which improvements to our code we should care most\n", - "about here is an [excellent source](https://ipython-books.github.io/42-profiling-your-code-easily-with-cprofile-and-ipython/)\n", - "on the topic. Even before optimizing we can understand which parts of our code\n", - "will be taking the most of the run-time. It will be the longest Python loop,\n", - "i.e. the loop over all the samples. Nonetheless, let us do some profiling!" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Converged at iteration: 11\n", - "Runtime: 0.8284149169921875 seconds\n", - " " - ] - } - ], - "source": [ - "test_data = generate_simple_clustering_dataset(n_points=10000, plotting=False)\n", - "%prun -l 10 cluster_labels, centroids = k_means(test_data)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we can see the reason for profiling. We now know for certain a lot can be\n", - "gained just by vectorizing our distance function. Ideally we wish to perform\n", - "most of our loops in numpy, i.e. C. To do this we need our array shapes to match\n", - "and clever reshaping will let us do so." - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": {}, - "outputs": [], - "source": [ - "\n", - "def np_get_distances_to_clusters(data, centroids):\n", - " \"\"\"\n", - " Squared Euclidean distance between all data-points and every centroid. For\n", - " the function to work properly it needs data and centroids to be numpy\n", - " broadcastable. We sum along the dimension axis.\n", - " Inputs:\n", - " data (np.array): with dimensions (samples x 1 x dim)\n", - " centroids (np.array): with dimensions (1 x n_clusters x dim)\n", - "\n", - " Returns:\n", - " distances (np.array): with dimensions (samples x n_clusters)\n", - " \"\"\"\n", - "\n", - " distances = np.sum(np.abs((data - centroids))**2, axis=2)\n", - " return distances\n", - "\n", - "def np_assign_points_to_clusters(distances):\n", - " \"\"\"\n", - " Assigning each data-point to a cluster given an array distances containing\n", - " the squared Euclidean distance from every point to each centroid. We do\n", - " np.argmin along the cluster axis to find the closest cluster. Returns a\n", - " numpy array with corresponding labels.\n", - " Inputs:\n", - " distances (np.array): with dimensions (samples x n_clusters)\n", - "\n", - " Returns:\n", - " cluster_labels (np.array): with dimensions (samples x None)\n", - " \"\"\"\n", - " cluster_labels = np.argmin(distances, axis=1)\n", - " return cluster_labels\n", - "\n", - "\n", - "def np_k_means(data, n_clusters=4, max_iterations=100, tolerance=1e-8):\n", - " \"\"\"\n", - " Numpythonic implementation of the k-means clusting algorithm.\n", - " Inputs:\n", - " data (np.array): with dimesions (samples x dim)\n", - " n_clusters (int): hyperparameter which depends on dataset\n", - " max_iterations (int): hyperparameter which depends on dataset\n", - " tolerance (float): convergence measure\n", - " progression_plot (bool): activation flag for plotting\n", - " Returns:\n", - " cluster_labels (np.array): with dimension (samples)\n", - " centroid_list (list): list of centroids (np.array)\n", - " with dimensions (n_clusters x dim)\n", - " \"\"\"\n", - " n_samples, dimensions = data.shape\n", - " np.random.seed(2021)\n", - " centroids = data[np.random.choice(len(data), n_clusters, replace=False), :]\n", - "\n", - " distances = np_get_distances_to_clusters(np.reshape(data,\n", - " (n_samples, 1, dimensions)),\n", - " np.reshape(centroids,\n", - " (1, n_clusters, dimensions)))\n", - " cluster_labels = np_assign_points_to_clusters(distances)\n", - "\n", - " start_time = time.time()\n", - "\n", - " for iteration in range(max_iterations):\n", - " prev_centroids = centroids.copy()\n", - " for k in range(n_clusters):\n", - " points_in_cluster = data[cluster_labels == k]\n", - " mean_vector = np.mean(points_in_cluster, axis=0)\n", - " centroids[k] = mean_vector\n", - "\n", - " distances = np_get_distances_to_clusters(np.reshape(data,\n", - " (n_samples, 1, dimensions)),\n", - " np.reshape(centroids,\n", - " (1, n_clusters, dimensions)))\n", - " cluster_labels = np_assign_points_to_clusters(distances)\n", - "\n", - " centroid_difference = np.sum(np.abs(centroids - prev_centroids))\n", - " if centroid_difference < tolerance:\n", - " print(f'Converged at iteration: {iteration}')\n", - " print(f'Runtime: {time.time() - start_time} seconds')\n", - "\n", - " return cluster_labels, centroids\n", - "\n", - " print(f'Did not converge in {max_iterations} iterations')\n", - " print(f'Runtime: {time.time() - start_time} seconds')\n", - "\n", - " return cluster_labels, centroids" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "When working towards becoming a data scientist using Python this last step is\n", - "arguably one of the most important. Thinking of ways to avoid explicitly looping\n", - "by adding dimensions to our arrays in such a way that they become broadcastable\n", - "using numpy (also tensorflow and many others). Let us take a look at our the\n", - "fruits of our labor." - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Converged at iteration: 5\n", - "Runtime: 0.003773212432861328 seconds\n" - ] - } - ], - "source": [ - "cluster_labels, centroids = np_k_means(data)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Python 3", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/Clustering.py b/doc/LectureNotes/_build/jupyter_execute/Clustering.py deleted file mode 100644 index 36885f47b..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/Clustering.py +++ /dev/null @@ -1,600 +0,0 @@ - - -# Clustering Analysis -In this chapter we will concern ourselves with the study of **cluster analysis**. -In general terms cluster analysis, or clustering, is the task of grouping a -data-set into different distinct categories based on some measure of equality of -the data. This measure is often referred to as a **metric** or **similarity -measure** in the literature (note: sometimes we deal with a **dissimilarity -measure** instead). Usually, these metrics are formulated as some kind of -distance function between points in a high-dimensional space. - -There exists a lot of such distance measures. The simplest, and also the most -common is the **Euclidean distance** (i.e. Pythagoras). A good source for those of -you wanting a thorough overview is the article (DOI:10.5120/ijca2016907841 -Irani, Pise, Phatak). A few other metrics mentioned there are: *cosine -similarity*, *Manhattan distance*, *Chebychev distance* and the *Minkowski -distance*. The Minkowski distance is a general formulation which encapsulates a -range of metrics. All of these, and many more, can be used in clustering. There -exists different categories of clustering algorithms. A few of the most -common are: *centroid-*, *distribution-*, *density-* and *hierarchical- -clustering*. We will concern ourselves primarily with the first one. - -## Basic Idea of the K-means Clustering Algorithm -The simplest of all clustering algorithms is the aptly named **k-means algorithm** -, sometimes also referred to as *Lloyds algorithm*. It is the simplest and also -the most common. From its simplicity it obtains both strengths and weaknesses. -These will be discussed in more detail later. The k-means algorithm is a -**centroid based** clustering algorithm. - -Assume, we are given $n$ data points and we wish to split the data into $K < n$ -different categories, or clusters. We label each cluster by an integer $k\in\{ -1, \cdots, K \}$. In the basic k-means algorithm each point is assigned to only -one cluster $k$, and these assignments are *non-injective* i.e. many-to-one. We -can think of these mappings as an encoder $k = C(i)$, which assigns the $i$-th -data-point $\bf x_i$ to the $k$-th cluster. Before we jump into the mathematics -let us describe the k-means algorithm in words: -1. We start with guesses / random initializations of our $k$ cluster centers / centroids - -2. For each centroid the points that are most similar are identified - -3. Then we move / replace each centroid with a coordinate average of all the points that were assigned to that centroid. - -4. Iterate this points 2, 3) until the centroids no longer move (to some tolerance) - -Now we consider the method formally. Again, we assume we have $n$ data-points -(vectors) - - -
- -$$ -\begin{equation}\label{eq:kmeanspoints} \tag{1} - \boldsymbol{x_i} = \{x_{i, 1}, \cdots, x_{i, p}\}\in\mathbb{R}^p. -\end{equation} -$$ - -which we wish to group into $K < n$ clusters. For our dissimilarity measure we -will use the *squared Euclidean distance* - - -
- -$$ -\begin{equation}\label{eq:squaredeuclidean} \tag{2} - d(\boldsymbol{x_i}, \boldsymbol{x_i'}) = \sum_{j=1}^p(x_{ij} - x_{i'j})^2 - = ||\boldsymbol{x_i} - \boldsymbol{x_{i'}}||^2 -\end{equation} -$$ - -Next we define the so called *within-cluster point scatter* which gives us a -measure of how close each data point assigned to the same cluster tends to be to -the all the others. - - -
- -$$ -\begin{equation}\label{eq:withincluster} \tag{3} - W(C) = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k} - \sum_{C(i')=k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) = - \sum_{k=1}^KN_k\sum_{C(i)=k}||\boldsymbol{x_i} - \boldsymbol{\overline{x_k}}||^2 -\end{equation} -$$ - -where $\boldsymbol{\overline{x_k}}$ is the mean vector associated with the $k$-th -cluster, and $N_k = \sum_{i=1}^nI(C(i) = k)$, where the $I()$ notation is -similar to the Kronecker delta (*Commonly used in statistics, it just means that -when $i = k$ we have the encoder $C(i)$*). In other words, the within-cluster -scatter measures the compactness of each cluster with respect to the data points -assigned to each cluster. This is the quantity that the $k$-means algorithm aims -to minimize. We refer to this quantity $W(C)$ as the within cluster scatter -because of its relation to the *total scatter*. - - -
- -$$ -\begin{equation}\label{eq:totalscatter} \tag{4} - T = W(C) + B(C) = \frac{1}{2}\sum_{i=1}^n - \sum_{i'=1}^nd(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) - = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k} - \Big(\sum_{C(i') = k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) - + \sum_{C(i')\neq k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})\Big) -\end{equation} -$$ - -Which is a quantity that is conserved throughout the $k$-means algorithm. It can -be thought of as the total amount of information in the data, and it is composed -of the aforementioned within-cluster scatter and the *between-cluster scatter* -$B(C)$. In methods such as principle component analysis the total scatter is not -conserved. - -Given a cluster mean $\boldsymbol{m_k}$ we define the **total cluster variance** - - -
- -$$ -\begin{equation}\label{eq:totalclustervariance} \tag{5} - \min_{C, \{\boldsymbol{m_k}\}_1^K}\sum_{k=1}^KN_k\sum||\boldsymbol{x_i} - \boldsymbol{m_k}||^2 -\end{equation} -$$ - -Now we have all the pieces necessary to formally revisit the k-means algorithm. -If you at this point feel like some of the above definitions came a bit out of -no-where, don't fret, the method does get a whole lot simpler once we start -programming. - -## The K-means Clustering Algorithm -The k-means clustering algorithm goes as follows (note in my opinion this -description is a bit complicated and is lifted directly out of ESL HASTIE for -deeper understanding purposes) - -1. For a given cluster assignment $C$, and $k$ cluster means $\{m_1, \cdots, m_k\}$. We minimize the total cluster variance with respect to the cluster means $\{m_k\}$ yielding the means of the currently assigned clusters. - -2. Given a current set of $k$ means $\{m_k\}$ the total cluster variance is minimized by assigning each observation to the closest (current) cluster mean. That is $$C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2$$ - -3. Steps 1 and 2 are repeated until the assignments do not change. - -As previously stated the above formulation can be a bit difficult to understand, -*at least the first time*, due to the dense notation used. But all in all the -concept is fairly simple when explained in words. The math needs to be -understood but to help you along the way we summarize the algorithm as follows -(try to look at the terms above to match with the summary). - -1. Before we start we specify a number $k$ which is the number of clusters we want to try to separate our data into. - -2. We initially choose $k$ random data points in our data as our initial centroids, *or means* (this is where the name comes from). - -3. Assign each data point to their closest centroid, based on the squared Euclidean distance. - -4. For each of the $k$ cluster we update the centroid by calculating new mean values for all the data points in the cluster. - -5. Iteratively minimize the within cluster scatter by performing steps (3, 4) until the new assignments stop changing (can be to some tolerance) or until a maximum number of iterations have passed. - -That's it, nothing magical happening. - -## Writing Our Own Code -In the following section we will work to develop a deeper understanding of the -previously discussed mathematics through developing codes to do k-means cluster -analysis. - -### Basic Python - -Let us now program the most basic version of the algorithm using nothing but -Python with numpy arrays. This code is kept intentionally simple to gradually -progress our understanding. There is no vectorization of any kind, and even most -helper functions are not utilized. Throughout our implementation process it will -be helpful to keep in mind both the mathematical description of the algorithm -*and* our summary from above. In addition, try to think of ways to optimize this -while reading the next section. We will get to it, take it as a challenge to see -if your optimizations are better. - -First of all we need a dataset to do our cluster analysis on, for clarity (and -lack of googling beforehand) we generate it ourselves using Gaussians. First we -import - -%matplotlib inline - -%matplotlib inline -import matplotlib.pyplot as plt -import numpy as np -import time -from IPython.display import display - -np.random.seed(2021) - -Next we define functions, for ease of use later, to generate Gaussians and to -set up our toy data set. - -def gaussian_points(dim=2, n_points=1000, mean_vector=np.array([0, 0]), - sample_variance=1): - """ - Very simple custom function to generate gaussian distributed point clusters - with variable dimension, number of points, means in each direction - (must match dim) and sample variance. - - Inputs: - dim (int) - n_points (int) - mean_vector (np.array) (where index 0 is x, index 1 is y etc.) - sample_variance (float) - - Returns: - data (np.array): with dimensions (dim x n_points) - """ - - mean_matrix = np.zeros(dim) + mean_vector - covariance_matrix = np.eye(dim) * sample_variance - data = np.random.multivariate_normal(mean_matrix, covariance_matrix, - n_points) - return data - - - -def generate_simple_clustering_dataset(dim=2, n_points=1000, plotting=True, - return_data=True): - """ - Toy model to illustrate k-means clustering - """ - - data1 = gaussian_points(mean_vector=np.array([5, 5])) - data2 = gaussian_points() - data3 = gaussian_points(mean_vector=np.array([1, 4.5])) - data4 = gaussian_points(mean_vector=np.array([5, 1])) - data = np.concatenate((data1, data2, data3, data4), axis=0) - - if plotting: - fig, ax = plt.subplots() - ax.scatter(data[:, 0], data[:, 1], alpha=0.2) - ax.set_title('Toy Model Dataset') - plt.show() - - - if return_data: - return data - - -data = generate_simple_clustering_dataset() - -Now that we are our, albeit very simple, dataset we are ready to start -implementing the k-means algorithm. - -n_samples, dimensions = data.shape -n_clusters = 4 - -# we randomly initialize our centroids -np.random.seed(2021) -centroids = data[np.random.choice(n_samples, n_clusters, replace=False), :] -distances = np.zeros((n_samples, n_clusters)) - -# first we need to calculate the distance to each centroid from our data -for k in range(n_clusters): - for n in range(n_samples): - dist = 0 - for d in range(dimensions): - dist += np.abs(data[n, d] - centroids[k, d])**2 - distances[n, k] = dist - -# we initialize an array to keep track of to which cluster each point belongs -# the way we set it up here the index tracks which point and the value which -# cluster the point belongs to -cluster_labels = np.zeros(n_samples, dtype='int') - -# next we loop through our samples and for every point assign it to the cluster -# to which it has the smallest distance to -for n in range(n_samples): - # tracking variables (all of this is basically just an argmin) - smallest = 1e10 - smallest_row_index = 1e10 - for k in range(n_clusters): - if distances[n, k] < smallest: - smallest = distances[n, k] - smallest_row_index = k - - cluster_labels[n] = smallest_row_index - -Let's plot and see - -fig = plt.figure() -ax = fig.add_subplot() -unique_cluster_labels = np.unique(cluster_labels) -for i in unique_cluster_labels: - ax.scatter(data[cluster_labels == i, 0], - data[cluster_labels == i, 1], - label = i, - alpha = 0.2) - ax.scatter(centroids[:, 0], centroids[:, 1], c='black') - -ax.set_title("First Grouping of Points to Centroids") - -plt.show() - -So what do we have so far? We have 'picked' $k$ centroids at random from our -data points. There are other ways of more intelligently choosing their -initializations, however for our purposes randomly is fine. Then we have -initialized an array 'distances' which holds the information of the distance, -*or dissimilarity*, of every point to of our centroids. Finally, we have -initialized an array 'cluster_labels' which according to our distances array -holds the information of to which centroid every point is assigned. This was the -first pass of our algorithm. Essentially, all we need to do now is repeat the -distance and assignment steps above until we have reached a desired convergence -or a maximum amount of iterations. - - -max_iterations = 100 -tolerance = 1e-8 -start_time = time.time() - -for iteration in range(max_iterations): - prev_centroids = centroids.copy() - for k in range(n_clusters): - # this array will be used to update our centroid positions - vector_mean = np.zeros(dimensions) - mean_divisor = 0 - for n in range(n_samples): - if cluster_labels[n] == k: - vector_mean += data[n, :] - mean_divisor += 1 - - # update according to the k means - centroids[k, :] = vector_mean / mean_divisor - - # we find the dissimilarity - for k in range(n_clusters): - for n in range(n_samples): - dist = 0 - for d in range(dimensions): - dist += np.abs(data[n, d] - centroids[k, d])**2 - distances[n, k] = dist - - # assign each point - for n in range(n_samples): - smallest = 1e10 - smallest_row_index = 1e10 - for k in range(n_clusters): - if distances[n, k] < smallest: - smallest = distances[n, k] - smallest_row_index = k - - cluster_labels[n] = smallest_row_index - - # convergence criteria - centroid_difference = np.sum(np.abs(centroids - prev_centroids)) - if centroid_difference < tolerance: - print(f'Converged at iteration {iteration}') - print(f'Runtime: {time.time() - start_time} seconds') - break - - elif iteration == max_iterations: - print(f'Did not converge in {max_iterations} iterations') - print(f'Runtime: {time.time() - start_time} seconds') - -And thats it! We now have an extremely barebones, un-optimized k-means -clustering implementation. Lets plot the final result - -fig = plt.figure() -ax = fig.add_subplot() -unique_cluster_labels = np.unique(cluster_labels) -for i in unique_cluster_labels: - ax.scatter(data[cluster_labels == i, 0], - data[cluster_labels == i, 1], - label = i, - alpha = 0.2) - ax.scatter(centroids[:, 0], centroids[:, 1], c='black') - -ax.set_title("Final Result of K-means Clustering") - -plt.show() - -Now there are a few glaring improvements to be done here. First of all is -organizing things into functions for better readability. Second is getting rid -of the small inefficiencies like manually calculating distances and argmin. And -finally, we need to optimize for better run-time. It's like we always say: the -best way of looping in Python is to not loop in Python. Let us tackle the first -two improvements. - -## Towards a More Numpythonic Code - - -def get_distances_to_clusters(data, centroids): - """ - Function that for each cluster finds the squared Euclidean distance - from every data point to the cluster centroid and returns a numpy array - containing the distances such that distance[i, j] means the distance between - the i-th point and the j-th centroid. - Inputs: - data (np.array): with dimensions (n_samples x dim) - centroids (np.array): with dimensions (n_clusters x dim) - - Returns: - distances (np.array): with dimensions (n_samples x n_clusters) - """ - - n_samples, dimensions = data.shape - n_clusters = centroids.shape[0] - distances = np.zeros((n_samples, n_clusters)) - for k in range(n_clusters): - for i in range(n_samples): - dist = 0 - for j in range(dimensions): - dist += np.abs(data[i, j] - centroids[k, j])**2 - distances[i, k] = dist - - return distances - - - -def assign_points_to_clusters(distances): - """ - Function to assign each data point to the cluster to which it is the closest - based on the squared Euclidean distance from the get_distances_to_clusters - method. - Inputs: - distances (np.array): with dimensions (n_samples x n_clusters) - - Returns: - cluster_labels (np.array): with dimensions (n_samples) - """ - cluster_labels = np.argmin(distances, axis=1) - - return cluster_labels - - - -def k_means(data, n_clusters=4, max_iterations=100, tolerance=1e-8): - """ - Naive implementation of the k-means clustering algorithm. A short summary of - the algorithm is as follows: we randomly initialize k centroids / means. - Then we assign, using the squared Euclidean distance, every data-point to a - cluster. We then update the position of the k centroids / means, and repeat - until convergence or we reach our desired maximum iterations. The method - returns the cluster assignments of our data-points and a sequence of - centroids. - Inputs: - data (np.array): with dimesions (n_samples x dim) - n_clusters (int): hyperparameter which depends on dataset - max_iterations (int): hyperparameter which depends on dataset - tolerance (float): convergence measure - - Returns: - cluster_labels (np.array): with dimension (n_samples) - centroid_list (list): list of centroids (np.array) - with dimensions (n_clusters x dim) - """ - - samples, dimensions = data.shape - np.random.seed(2021) - centroids = data[np.random.choice(len(data), n_clusters, replace=False), :] - distances = get_distances_to_clusters(data, centroids) - cluster_labels = assign_points_to_clusters(distances) - - start_time = time.time() - - for iteration in range(max_iterations): - prev_centroids = centroids.copy() - for k in range(n_clusters): - vector_mean = np.zeros(dimensions) - mean_divisor = 0 - for n in range(n_samples): - if cluster_labels[n] == k: - vector_mean += data[n, :] - mean_divisor += 1 - # And update according to the new means - centroids[k, :] = vector_mean / mean_divisor - - distances = get_distances_to_clusters(data, centroids) - cluster_labels = assign_points_to_clusters(distances) - - centroid_difference = np.sum(np.abs(centroids - prev_centroids)) - if centroid_difference < tolerance: - print(f'Converged at iteration: {iteration}') - print(f'Runtime: {time.time() - start_time} seconds') - - return cluster_labels, centroids - - print(f'Did not converge in {max_iterations} iterations') - print(f'Runtime: {time.time() - start_time} seconds') - - return cluster_labels, centroids - - -# quirk of numpy / Jupyter need to set seed again -cluster_labels, centroids = k_means(data) - -**Note**: the start of the timing is after the random initialization, and first -'cycle' of our algorithm. This is technically not the correct way to time it but -due to this being in a Jupyter notebook and the way it is structured this way of -comparing our algorithms will produce a more equal result. When timing code we -should always encapsulate our whole computation block. - -So we see an improvement from just switching to numpy's argmin function. There -is a very nice tool (or category of tools) called profilers. These can be -utilized to make clearer which improvements to our code we should care most -about here is an [excellent source](https://ipython-books.github.io/42-profiling-your-code-easily-with-cprofile-and-ipython/) -on the topic. Even before optimizing we can understand which parts of our code -will be taking the most of the run-time. It will be the longest Python loop, -i.e. the loop over all the samples. Nonetheless, let us do some profiling! - -test_data = generate_simple_clustering_dataset(n_points=10000, plotting=False) -%prun -l 10 cluster_labels, centroids = k_means(test_data) - -Here we can see the reason for profiling. We now know for certain a lot can be -gained just by vectorizing our distance function. Ideally we wish to perform -most of our loops in numpy, i.e. C. To do this we need our array shapes to match -and clever reshaping will let us do so. - - -def np_get_distances_to_clusters(data, centroids): - """ - Squared Euclidean distance between all data-points and every centroid. For - the function to work properly it needs data and centroids to be numpy - broadcastable. We sum along the dimension axis. - Inputs: - data (np.array): with dimensions (samples x 1 x dim) - centroids (np.array): with dimensions (1 x n_clusters x dim) - - Returns: - distances (np.array): with dimensions (samples x n_clusters) - """ - - distances = np.sum(np.abs((data - centroids))**2, axis=2) - return distances - -def np_assign_points_to_clusters(distances): - """ - Assigning each data-point to a cluster given an array distances containing - the squared Euclidean distance from every point to each centroid. We do - np.argmin along the cluster axis to find the closest cluster. Returns a - numpy array with corresponding labels. - Inputs: - distances (np.array): with dimensions (samples x n_clusters) - - Returns: - cluster_labels (np.array): with dimensions (samples x None) - """ - cluster_labels = np.argmin(distances, axis=1) - return cluster_labels - - -def np_k_means(data, n_clusters=4, max_iterations=100, tolerance=1e-8): - """ - Numpythonic implementation of the k-means clusting algorithm. - Inputs: - data (np.array): with dimesions (samples x dim) - n_clusters (int): hyperparameter which depends on dataset - max_iterations (int): hyperparameter which depends on dataset - tolerance (float): convergence measure - progression_plot (bool): activation flag for plotting - Returns: - cluster_labels (np.array): with dimension (samples) - centroid_list (list): list of centroids (np.array) - with dimensions (n_clusters x dim) - """ - n_samples, dimensions = data.shape - np.random.seed(2021) - centroids = data[np.random.choice(len(data), n_clusters, replace=False), :] - - distances = np_get_distances_to_clusters(np.reshape(data, - (n_samples, 1, dimensions)), - np.reshape(centroids, - (1, n_clusters, dimensions))) - cluster_labels = np_assign_points_to_clusters(distances) - - start_time = time.time() - - for iteration in range(max_iterations): - prev_centroids = centroids.copy() - for k in range(n_clusters): - points_in_cluster = data[cluster_labels == k] - mean_vector = np.mean(points_in_cluster, axis=0) - centroids[k] = mean_vector - - distances = np_get_distances_to_clusters(np.reshape(data, - (n_samples, 1, dimensions)), - np.reshape(centroids, - (1, n_clusters, dimensions))) - cluster_labels = np_assign_points_to_clusters(distances) - - centroid_difference = np.sum(np.abs(centroids - prev_centroids)) - if centroid_difference < tolerance: - print(f'Converged at iteration: {iteration}') - print(f'Runtime: {time.time() - start_time} seconds') - - return cluster_labels, centroids - - print(f'Did not converge in {max_iterations} iterations') - print(f'Runtime: {time.time() - start_time} seconds') - - return cluster_labels, centroids - -When working towards becoming a data scientist using Python this last step is -arguably one of the most important. Thinking of ways to avoid explicitly looping -by adding dimensions to our arrays in such a way that they become broadcastable -using numpy (also tensorflow and many others). 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basic Elements\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureAug21.mp4?vrtx=view-as-webpage)\n", - "\n", - "\n", - "## Introduction\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "Our emphasis throughout this series of lectures \n", - "is on understanding the mathematical aspects of\n", - "different algorithms used in the fields of data analysis and machine learning. \n", - "\n", - "However, where possible we will emphasize the\n", - "importance of using available software. We start thus with a hands-on\n", - "and top-down approach to machine learning. The aim is thus to start with\n", - "relevant data or data we have produced \n", - "and use these to introduce statistical data analysis\n", - "concepts and machine learning algorithms before we delve into the\n", - "algorithms themselves. The examples we will use in the beginning, start with simple\n", - "polynomials with random noise added. We will use the Python\n", - "software package [Scikit-Learn](http://scikit-learn.org/stable/) and\n", - "introduce various machine learning algorithms to make fits of\n", - "the data and predictions. We move thereafter to more interesting\n", - "cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example).\n", - "These are examples where we can easily set up the data and\n", - "then use machine learning algorithms included in for example\n", - "**Scikit-Learn**. \n", - "\n", - "These examples will serve us the purpose of getting\n", - "started. Furthermore, they allow us to catch more than two birds with\n", - "a stone. They will allow us to bring in some programming specific\n", - "topics and tools as well as showing the power of various Python \n", - "libraries for machine learning and statistical data analysis. \n", - "\n", - "Here, we will mainly focus on two\n", - "specific Python packages for Machine Learning, Scikit-Learn and\n", - "Tensorflow (see below for links etc). Moreover, the examples we\n", - "introduce will serve as inputs to many of our discussions later, as\n", - "well as allowing you to set up models and produce your own data and\n", - "get started with programming.\n", - "\n", - "\n", - "\n", - "## What is Machine Learning?\n", - "\n", - "Statistics, data science and machine learning form important fields of\n", - "research in modern science. They describe how to learn and make\n", - "predictions from data, as well as allowing us to extract important\n", - "correlations about physical process and the underlying laws of motion\n", - "in large data sets. The latter, big data sets, appear frequently in\n", - "essentially all disciplines, from the traditional Science, Technology,\n", - "Mathematics and Engineering fields to Life Science, Law, education\n", - "research, the Humanities and the Social Sciences. \n", - "\n", - "It has become more\n", - "and more common to see research projects on big data in for example\n", - "the Social Sciences where extracting patterns from complicated survey\n", - "data is one of many research directions. Having a solid grasp of data\n", - "analysis and machine learning is thus becoming central to scientific\n", - "computing in many fields, and competences and skills within the fields\n", - "of machine learning and scientific computing are nowadays strongly\n", - "requested by many potential employers. The latter cannot be\n", - "overstated, familiarity with machine learning has almost become a\n", - "prerequisite for many of the most exciting employment opportunities,\n", - "whether they are in bioinformatics, life science, physics or finance,\n", - "in the private or the public sector. This author has had several\n", - "students or met students who have been hired recently based on their\n", - "skills and competences in scientific computing and data science, often\n", - "with marginal knowledge of machine learning.\n", - "\n", - "Machine learning is a subfield of computer science, and is closely\n", - "related to computational statistics. It evolved from the study of\n", - "pattern recognition in artificial intelligence (AI) research, and has\n", - "made contributions to AI tasks like computer vision, natural language\n", - "processing and speech recognition. Many of the methods we will study are also \n", - "strongly rooted in basic mathematics and physics research. \n", - "\n", - "Ideally, machine learning represents the science of giving computers\n", - "the ability to learn without being explicitly programmed. The idea is\n", - "that there exist generic algorithms which can be used to find patterns\n", - "in a broad class of data sets without having to write code\n", - "specifically for each problem. The algorithm will build its own logic\n", - "based on the data. You should however always keep in mind that\n", - "machines and algorithms are to a large extent developed by humans. The\n", - "insights and knowledge we have about a specific system, play a central\n", - "role when we develop a specific machine learning algorithm. \n", - "\n", - "Machine learning is an extremely rich field, in spite of its young\n", - "age. The increases we have seen during the last three decades in\n", - "computational capabilities have been followed by developments of\n", - "methods and techniques for analyzing and handling large date sets,\n", - "relying heavily on statistics, computer science and mathematics. The\n", - "field is rather new and developing rapidly. Popular software packages\n", - "written in Python for machine learning like\n", - "[Scikit-learn](http://scikit-learn.org/stable/),\n", - "[Tensorflow](https://www.tensorflow.org/),\n", - "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all\n", - "freely available at their respective GitHub sites, encompass\n", - "communities of developers in the thousands or more. And the number of\n", - "code developers and contributors keeps increasing. Not all the\n", - "algorithms and methods can be given a rigorous mathematical\n", - "justification, opening up thereby large rooms for experimenting and\n", - "trial and error and thereby exciting new developments. However, a\n", - "solid command of linear algebra, multivariate theory, probability\n", - "theory, statistical data analysis, understanding errors and Monte\n", - "Carlo methods are central elements in a proper understanding of many\n", - "of algorithms and methods we will discuss.\n", - "\n", - "\n", - "\n", - "The approaches to machine learning are many, but are often split into\n", - "two main categories. In *supervised learning* we know the answer to a\n", - "problem, and let the computer deduce the logic behind it. On the other\n", - "hand, *unsupervised learning* is a method for finding patterns and\n", - "relationship in data sets without any prior knowledge of the system.\n", - "Some authours also operate with a third category, namely\n", - "*reinforcement learning*. This is a paradigm of learning inspired by\n", - "behavioral psychology, where learning is achieved by trial-and-error,\n", - "solely from rewards and punishment.\n", - "\n", - "Another way to categorize machine learning tasks is to consider the\n", - "desired output of a system. Some of the most common tasks are:\n", - "\n", - " * Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.\n", - "\n", - " * Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.\n", - "\n", - " * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.\n", - "\n", - "The methods we cover have three main topics in common, irrespective of\n", - "whether we deal with supervised or unsupervised learning. The first\n", - "ingredient is normally our data set (which can be subdivided into\n", - "training and test data), the second item is a model which is normally a\n", - "function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. \n", - "\n", - "The last ingredient is a so-called **cost**\n", - "function which allows us to present an estimate on how good our model\n", - "is in reproducing the data it is supposed to train. \n", - "At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of **gradient** methods.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Software and needed installations\n", - "\n", - "We will make extensive use of Python as programming language and its\n", - "myriad of available libraries. You will find\n", - "Jupyter notebooks invaluable in your work. You can run **R**\n", - "codes in the Jupyter/IPython notebooks, with the immediate benefit of\n", - "visualizing your data. You can also use compiled languages like C++,\n", - "Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be\n", - "on Python.\n", - "\n", - "\n", - "If you have Python installed (we strongly recommend Python3) and you feel\n", - "pretty familiar with installing different packages, we recommend that\n", - "you install the following Python packages via **pip** as \n", - "\n", - "1. pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow \n", - "\n", - "For Python3, replace **pip** with **pip3**.\n", - "\n", - "For OSX users we recommend, after having installed Xcode, to\n", - "install **brew**. Brew allows for a seamless installation of additional\n", - "software via for example \n", - "\n", - "1. brew install python3\n", - "\n", - "For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution,\n", - "you can use **pip** as well and simply install Python as \n", - "\n", - "1. sudo apt-get install python3 (or python for pyhton2.7)\n", - "\n", - "etc etc. \n", - "\n", - "\n", - "\n", - "## Python installers\n", - "\n", - "If you don't want to perform these operations separately and venture\n", - "into the hassle of exploring how to set up dependencies and paths, we\n", - "recommend two widely used distrubutions which set up all relevant\n", - "dependencies for Python, namely \n", - "\n", - "* [Anaconda](https://docs.anaconda.com/), \n", - "\n", - "which is an open source\n", - "distribution of the Python and R programming languages for large-scale\n", - "data processing, predictive analytics, and scientific computing, that\n", - "aims to simplify package management and deployment. Package versions\n", - "are managed by the package management system **conda**. \n", - "\n", - "* [Enthought canopy](https://www.enthought.com/product/canopy/) \n", - "\n", - "is a Python\n", - "distribution for scientific and analytic computing distribution and\n", - "analysis environment, available for free and under a commercial\n", - "license.\n", - "\n", - "Furthermore, [Google's Colab](https://colab.research.google.com/notebooks/welcome.ipynb) is a free Jupyter notebook environment that requires \n", - "no setup and runs entirely in the cloud. Try it out!\n", - "\n", - "\n", - "## Useful Python libraries\n", - "Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)\n", - "\n", - "* [NumPy](https://www.numpy.org/) is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays\n", - "\n", - "* [The pandas](https://pandas.pydata.org/) library provides high-performance, easy-to-use data structures and data analysis tools \n", - "\n", - "* [Xarray](http://xarray.pydata.org/en/stable/) is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!\n", - "\n", - "* [Scipy](https://www.scipy.org/) (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. \n", - "\n", - "* [Matplotlib](https://matplotlib.org/) is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.\n", - "\n", - "* [Autograd](https://github.com/HIPS/autograd) can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives\n", - "\n", - "* [SymPy](https://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. \n", - "\n", - "* [scikit-learn](https://scikit-learn.org/stable/) has simple and efficient tools for machine learning, data mining and data analysis\n", - "\n", - "* [TensorFlow](https://www.tensorflow.org/) is a Python library for fast numerical computing created and released by Google\n", - "\n", - "* [Keras](https://keras.io/) is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano\n", - "\n", - "* And many more such as [pytorch](https://pytorch.org/), [Theano](https://pypi.org/project/Theano/) etc \n", - "\n", - "## Installing R, C++, cython or Julia\n", - "\n", - "You will also find it convenient to utilize **R**. We will mainly\n", - "use Python during our lectures and in various projects and exercises.\n", - "Those of you\n", - "already familiar with **R** should feel free to continue using **R**, keeping\n", - "however an eye on the parallel Python set ups. Similarly, if you are a\n", - "Python afecionado, feel free to explore **R** as well. Jupyter/Ipython\n", - "notebook allows you to run **R** codes interactively in your\n", - "browser. The software library **R** is really tailored for statistical data analysis\n", - "and allows for an easy usage of the tools and algorithms we will discuss in these\n", - "lectures.\n", - "\n", - "To install **R** with Jupyter notebook \n", - "[follow the link here](https://mpacer.org/maths/r-kernel-for-ipython-notebook)\n", - "\n", - "\n", - "\n", - "\n", - "## Installing R, C++, cython, Numba etc\n", - "\n", - "\n", - "For the C++ aficionados, Jupyter/IPython notebook allows you also to\n", - "install C++ and run codes written in this language interactively in\n", - "the browser. Since we will emphasize writing many of the algorithms\n", - "yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming\n", - "languages.\n", - "\n", - "To add more entropy, **cython** can also be used when running your\n", - "notebooks. It means that Python with the jupyter notebook\n", - "setup allows you to integrate widely popular softwares and tools for\n", - "scientific computing. Similarly, the \n", - "[Numba Python package](https://numba.pydata.org/) delivers increased performance\n", - "capabilities with minimal rewrites of your codes. With its\n", - "versatility, including symbolic operations, Python offers a unique\n", - "computational environment. Your jupyter notebook can easily be\n", - "converted into a nicely rendered **PDF** file or a Latex file for\n", - "further processing. For example, convert to latex as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " pycod jupyter nbconvert filename.ipynb --to latex \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "And to add more versatility, the Python package [SymPy](http://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python. \n", - "\n", - "Finally, if you wish to use the light mark-up language \n", - "[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML \n", - "formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using **doconce**.\n", - "\n", - "\n", - "\n", - "## Numpy examples and Important Matrix and vector handling packages\n", - "\n", - "There are several central software libraries for linear algebra and eigenvalue problems. Several of the more\n", - "popular ones have been wrapped into ofter software packages like those from the widely used text **Numerical Recipes**. The original source codes in many of the available packages are often taken from the widely used\n", - "software package LAPACK, which follows two other popular packages\n", - "developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here.\n", - "\n", - " * LINPACK: package for linear equations and least square problems.\n", - "\n", - " * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.\n", - "\n", - " * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from .\n", - "\n", - "## Basic Matrix Features\n", - "\n", - "Matrix properties reminder" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A} =\n", - " \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", - " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", - " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", - " a_{41} & a_{42} & a_{43} & a_{44}\n", - " \\end{bmatrix}\\qquad\n", - "\\mathbf{I} =\n", - " \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n", - " 0 & 1 & 0 & 0 \\\\\n", - " 0 & 0 & 1 & 0 \\\\\n", - " 0 & 0 & 0 & 1\n", - " \\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The inverse of a matrix is defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \\left (A^{T} \\right )^{-1}$ real orthogonal $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \\left (A^{\\dagger} \\right )^{-1}$ unitary $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$
\n", - "\n", - "\n", - "### Some famous Matrices\n", - "\n", - " * Diagonal if $a_{ij}=0$ for $i\\ne j$\n", - "\n", - " * Upper triangular if $a_{ij}=0$ for $i > j$\n", - "\n", - " * Lower triangular if $a_{ij}=0$ for $i < j$\n", - "\n", - " * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$\n", - "\n", - " * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$\n", - "\n", - " * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$\n", - "\n", - " * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$\n", - "\n", - " * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$\n", - "\n", - " * Banded, block upper triangular, block lower triangular....\n", - "\n", - "### More Basic Matrix Features\n", - "\n", - "Some Equivalent Statements\n", - "For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", - "\n", - " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", - "\n", - " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", - "\n", - " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * $\\mathbf{A}$ is a product of elementary matrices.\n", - "\n", - " * $0$ is not eigenvalue of $\\mathbf{A}$.\n", - "\n", - "## Numpy and arrays\n", - "[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution," - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[ 0.52767404 -0.32617572 -0.84692682 0.35650759 -2.05926228 -0.91335551\n", - " 0.68132045 -0.31279832 -0.35453393 -0.0157447 ]\n" - ] - } - ], - "source": [ - "n = 10\n", - "x = np.random.normal(size=n)\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.\n", - "Another alternative is to declare a vector as follows" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1 2 3]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "x = np.array([1, 2, 3])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++\n", - "start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \\dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1.38629436 1.94591015 2.07944154]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4, 7, 8]))\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the last example we used Numpy's unary function $np.log$. This function is\n", - "highly tuned to compute array elements since the code is vectorized\n", - "and does not require looping. We normaly recommend that you use the\n", - "Numpy intrinsic functions instead of the corresponding **log** function\n", - "from Python's **math** module. The looping is done explicitely by the\n", - "**np.log** function. The alternative, and slower way to compute the\n", - "logarithms of a vector would be to write" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1 1 2]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "from math import log\n", - "x = np.array([4, 7, 8])\n", - "for i in range(0, len(x)):\n", - " x[i] = log(x[i])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note that our code is much longer already and we need to import the **log** function from the **math** module. \n", - "The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1.38629436 1.94591015 2.07944154]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4, 7, 8], dtype = np.float64))\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "ename": "SyntaxError", - "evalue": "invalid syntax (, line 3)", - "output_type": "error", - "traceback": [ - "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m3\u001b[0m\n\u001b[0;31m print(x)\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", - "print(x.itemsize)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Matrices in Python\n", - "\n", - "Having defined vectors, we are now ready to try out matrices. We can\n", - "define a $3 \\times 3 $ real matrix $\\hat{A}$ as (recall that we user\n", - "lowercase letters for vectors and uppercase letters for matrices)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "# print the first column, row-major order and elements start with 0\n", - "print(A[:,0])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can continue this was by printing out other columns or rows. The example here prints out the second column" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "# print the first column, row-major order and elements start with 0\n", - "print(A[1,:])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to zero\n", - "A = np.zeros( (n, n) )\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or initializing all elements to" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to one\n", - "A = np.ones( (n, n) )\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or as unitarily distributed random numbers (see the material on random number generators in the statistics part)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \\in [0, 1]\n", - "A = np.random.rand(n, n)\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n", - "As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n", - "$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", - " \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n", - " \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where for example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma_{xy} =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. \n", - "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\hat{W}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n", - " x_1 & y_1 & z_1 \\\\\n", - " x_2 & y_2 & z_2 \\\\\n", - " \\dots & \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2} & z_{n-2} \\\\\n", - " x_{n-1} & y_{n-1} & z_{n-1}\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in turn is converted into into the $3\\times 3$ covariance matrix\n", - "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", - "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", - "function **np.mean(x)**. We can also extract the eigenvalues of the\n", - "covariance matrix through the **np.linalg.eig()** function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "z = x**3+np.random.normal(size=n)\n", - "print(np.mean(z))\n", - "W = np.vstack((x, y, z))\n", - "Sigma = np.cov(W)\n", - "print(Sigma)\n", - "Eigvals, Eigvecs = np.linalg.eig(Sigma)\n", - "print(Eigvals)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from scipy import sparse\n", - "eye = np.eye(4)\n", - "print(eye)\n", - "sparse_mtx = sparse.csr_matrix(eye)\n", - "print(sparse_mtx)\n", - "x = np.linspace(-10,10,100)\n", - "y = np.sin(x)\n", - "plt.plot(x,y,marker='x')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Meet the Pandas\n", - "\n", - "\n", - "\n", - "\n", - "Another useful Python package is\n", - "[pandas](https://pandas.pydata.org/), which is an open source library\n", - "providing high-performance, easy-to-use data structures and data\n", - "analysis tools for Python. **pandas** stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data.\n", - "**pandas** has two major classes, the **DataFrame** class with two-dimensional data objects and tabular data organized in columns and the class **Series** with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. \n", - "**pandas** allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. \n", - "\n", - "The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of **pandas**, in particular in connection with classification of data." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import pandas as pd\n", - "from IPython.display import display\n", - "data = {'First Name': [\"Frodo\", \"Bilbo\", \"Aragorn II\", \"Samwise\"],\n", - " 'Last Name': [\"Baggins\", \"Baggins\",\"Elessar\",\"Gamgee\"],\n", - " 'Place of birth': [\"Shire\", \"Shire\", \"Eriador\", \"Shire\"],\n", - " 'Date of Birth T.A.': [2968, 2890, 2931, 2980]\n", - " }\n", - "data_pandas = pd.DataFrame(data)\n", - "display(data_pandas)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the above we have imported **pandas** with the shorthand **pd**, the latter has become the standard way we import **pandas**. We make then a list of various variables\n", - "and reorganize the aboves lists into a **DataFrame** and then print out a neat table with specific column labels as *Name*, *place of birth* and *date of birth*.\n", - "Displaying these results, we see that the indices are given by the default numbers from zero to three.\n", - "**pandas** is extremely flexible and we can easily change the above indices by defining a new type of indexing as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam'])\n", - "display(data_pandas)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Thereafter we display the content of the row which begins with the index **Aragorn**" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "display(data_pandas.loc['Aragorn'])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can easily append data to this, for example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "new_hobbit = {'First Name': [\"Peregrin\"],\n", - " 'Last Name': [\"Took\"],\n", - " 'Place of birth': [\"Shire\"],\n", - " 'Date of Birth T.A.': [2990]\n", - " }\n", - "data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin']))\n", - "display(data_pandas)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here are other examples where we use the **DataFrame** functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix \n", - "of dimensionality $10\\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import pandas as pd\n", - "from IPython.display import display\n", - "np.random.seed(100)\n", - "# setting up a 10 x 5 matrix\n", - "rows = 10\n", - "cols = 5\n", - "a = np.random.randn(rows,cols)\n", - "df = pd.DataFrame(a)\n", - "display(df)\n", - "print(df.mean())\n", - "print(df.std())\n", - "display(df**2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Thereafter we can select specific columns only and plot final results" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth']\n", - "df.index = np.arange(10)\n", - "\n", - "display(df)\n", - "print(df['Second'].mean() )\n", - "print(df.info())\n", - "print(df.describe())\n", - "\n", - "from pylab import plt, mpl\n", - "plt.style.use('seaborn')\n", - "mpl.rcParams['font.family'] = 'serif'\n", - "\n", - "df.cumsum().plot(lw=2.0, figsize=(10,6))\n", - "plt.show()\n", - "\n", - "\n", - "df.plot.bar(figsize=(10,6), rot=15)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can produce a $4\\times 4$ matrix" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "b = np.arange(16).reshape((4,4))\n", - "print(b)\n", - "df1 = pd.DataFrame(b)\n", - "print(df1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and many other operations. \n", - "\n", - "The **Series** class is another important class included in\n", - "**pandas**. You can view it as a specialization of **DataFrame** but where\n", - "we have just a single column of data. It shares many of the same features as _DataFrame. As with **DataFrame**,\n", - "most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays.\n", - "As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in.\n", - "For multidimensional arrays, we recommend strongly [xarray](http://xarray.pydata.org/en/stable/). **xarray** has much of the same flexibility as **pandas**, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both **pandas** and **xarray**. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "In order to study various Machine Learning algorithms, we need to\n", - "access data. Acccessing data is an essential step in all machine\n", - "learning algorithms. In particular, setting up the so-called **design\n", - "matrix** (to be defined below) is often the first element we need in\n", - "order to perform our calculations. To set up the design matrix means\n", - "reading (and later, when the calculations are done, writing) data\n", - "in various formats, The formats span from reading files from disk,\n", - "loading data from databases and interacting with online sources\n", - "like web application programming interfaces (APIs).\n", - "\n", - "In handling various input formats, as discussed above, we will mainly stay with **pandas**,\n", - "a Python package which allows us, in a seamless and painless way, to\n", - "deal with a multitude of formats, from standard **csv** (comma separated\n", - "values) files, via **excel**, **html** to **hdf5** formats. With **pandas**\n", - "and the **DataFrame** and **Series** functionalities we are able to convert text data\n", - "into the calculational formats we need for a specific algorithm. And our code is going to be \n", - "pretty close the basic mathematical expressions.\n", - "\n", - "Our first data set is going to be a classic from nuclear physics, namely all\n", - "available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set. \n", - "\n", - "We will show some of the\n", - "strengths of packages like **Scikit-Learn** in fitting nuclear binding energies to\n", - "specific functions using linear regression first. Then, as a teaser, we will show you how \n", - "you can easily implement other algorithms like decision trees and random forests and neural networks.\n", - "\n", - "But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as,\n", - "(don't be offended) fitting straight lines!\n", - "\n", - "\n", - "\n", - "\n", - "## Simple linear regression model using **scikit-learn**\n", - "\n", - "We start with perhaps our simplest possible example, using **Scikit-Learn** to perform linear regression analysis on a data set produced by us. \n", - "\n", - "What follows is a simple Python code where we have defined a function\n", - "$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. \n", - "The numbers in the vector $\\hat{x}$ are given\n", - "by random numbers generated with a uniform distribution with entries\n", - "$x_i \\in [0,1]$ (more about probability distribution functions\n", - "later). These values are then used to define a function $y(x)$\n", - "(tabulated again as a vector) with a linear dependence on $x$ plus a\n", - "random noise added via the normal distribution.\n", - "\n", - "\n", - "The Numpy functions are imported used the **import numpy as np**\n", - "statement and the random number generator for the uniform distribution\n", - "is called using the function **np.random.rand()**, where we specificy\n", - "that we want $100$ random variables. Using Numpy we define\n", - "automatically an array with the specified number of elements, $100$ in\n", - "our case. With the Numpy function **randn()** we can compute random\n", - "numbers with the normal distribution (mean value $\\mu$ equal to zero and\n", - "variance $\\sigma^2$ set to one) and produce the values of $y$ assuming a linear\n", - "dependence as function of $x$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y = 2x+N(0,1),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $N(0,1)$ represents random numbers generated by the normal\n", - "distribution. From **Scikit-Learn** we import then the\n", - "**LinearRegression** functionality and make a prediction $\\tilde{y} =\n", - "\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n", - "data $(\\hat{x},\\hat{y})$ for our training data. The Python package\n", - "**scikit-learn** has also a functionality which extracts the above\n", - "fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n", - "distinguish between training data and test data.\n", - "\n", - "For plotting we use the Python package\n", - "[matplotlib](https://matplotlib.org/) which produces publication\n", - "quality figures. Feel free to explore the extensive\n", - "[gallery](https://matplotlib.org/gallery/index.html) of examples. In\n", - "this example we plot our original values of $x$ and $y$ as well as the\n", - "prediction **ypredict** ($\\tilde{y}$), which attempts at fitting our\n", - "data with a straight line.\n", - "\n", - "The Python code follows here." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression\n", - "\n", - "x = np.random.rand(100,1)\n", - "y = 2*x+np.random.randn(100,1)\n", - "linreg = LinearRegression()\n", - "linreg.fit(x,y)\n", - "xnew = np.array([[0],[1]])\n", - "ypredict = linreg.predict(xnew)\n", - "\n", - "plt.plot(xnew, ypredict, \"r-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0,1.0,0, 5.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Simple Linear Regression')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This example serves several aims. It allows us to demonstrate several\n", - "aspects of data analysis and later machine learning algorithms. The\n", - "immediate visualization shows that our linear fit is not\n", - "impressive. It goes through the data points, but there are many\n", - "outliers which are not reproduced by our linear regression. We could\n", - "now play around with this small program and change for example the\n", - "factor in front of $x$ and the normal distribution. Try to change the\n", - "function $y$ to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y = 10x+0.01 \\times N(0,1),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $x$ is defined as before. Does the fit look better? Indeed, by\n", - "reducing the role of the noise given by the normal distribution we see immediately that\n", - "our linear prediction seemingly reproduces better the training\n", - "set. However, this testing 'by the eye' is obviouly not satisfactory in the\n", - "long run. Here we have only defined the training data and our model, and \n", - "have not discussed a more rigorous approach to the **cost** function.\n", - "\n", - "We need more rigorous criteria in defining whether we have succeeded or\n", - "not in modeling our training data. You will be surprised to see that\n", - "many scientists seldomly venture beyond this 'by the eye' approach. A\n", - "standard approach for the *cost* function is the so-called $\\chi^2$\n", - "function (a variant of the mean-squared error (MSE))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\chi^2 = \\frac{1}{n}\n", - "\\sum_{i=0}^{n-1}\\frac{(y_i-\\tilde{y}_i)^2}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\sigma_i^2$ is the variance (to be defined later) of the entry\n", - "$y_i$. We may not know the explicit value of $\\sigma_i^2$, it serves\n", - "however the aim of scaling the equations and make the cost function\n", - "dimensionless. \n", - "\n", - "Minimizing the cost function is a central aspect of\n", - "our discussions to come. Finding its minima as function of the model\n", - "parameters ($\\alpha$ and $\\beta$ in our case) will be a recurring\n", - "theme in these series of lectures. Essentially all machine learning\n", - "algorithms we will discuss center around the minimization of the\n", - "chosen cost function. This depends in turn on our specific\n", - "model for describing the data, a typical situation in supervised\n", - "learning. Automatizing the search for the minima of the cost function is a\n", - "central ingredient in all algorithms. Typical methods which are\n", - "employed are various variants of **gradient** methods. These will be\n", - "discussed in more detail later. Again, you'll be surprised to hear that\n", - "many practitioners minimize the above function ''by the eye', popularly dubbed as \n", - "'chi by the eye'. That is, change a parameter and see (visually and numerically) that \n", - "the $\\chi^2$ function becomes smaller. \n", - "\n", - "There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define \n", - "the relative error (why would we prefer the MSE instead of the relative error?) as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\hat{y} -\\hat{\\tilde{y}}\\vert}{\\vert \\hat{y}\\vert}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The squared cost function results in an arithmetic mean-unbiased\n", - "estimator, and the absolute-value cost function results in a\n", - "median-unbiased estimator (in the one-dimensional case, and a\n", - "geometric median-unbiased estimator for the multi-dimensional\n", - "case). The squared cost function has the disadvantage that it has the tendency\n", - "to be dominated by outliers.\n", - "\n", - "We can modify easily the above Python code and plot the relative error instead" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression\n", - "\n", - "x = np.random.rand(100,1)\n", - "y = 5*x+0.01*np.random.randn(100,1)\n", - "linreg = LinearRegression()\n", - "linreg.fit(x,y)\n", - "ypredict = linreg.predict(x)\n", - "\n", - "plt.plot(x, np.abs(ypredict-y)/abs(y), \"ro\")\n", - "plt.axis([0,1.0,0.0, 0.5])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$\\epsilon_{\\mathrm{relative}}$')\n", - "plt.title(r'Relative error')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Depending on the parameter in front of the normal distribution, we may\n", - "have a small or larger relative error. Try to play around with\n", - "different training data sets and study (graphically) the value of the\n", - "relative error.\n", - "\n", - "As mentioned above, **Scikit-Learn** has an impressive functionality.\n", - "We can for example extract the values of $\\alpha$ and $\\beta$ and\n", - "their error estimates, or the variance and standard deviation and many\n", - "other properties from the statistical data analysis. \n", - "\n", - "Here we show an\n", - "example of the functionality of **Scikit-Learn**." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np \n", - "import matplotlib.pyplot as plt \n", - "from sklearn.linear_model import LinearRegression \n", - "from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error\n", - "\n", - "x = np.random.rand(100,1)\n", - "y = 2.0+ 5*x+0.5*np.random.randn(100,1)\n", - "linreg = LinearRegression()\n", - "linreg.fit(x,y)\n", - "ypredict = linreg.predict(x)\n", - "print('The intercept alpha: \\n', linreg.intercept_)\n", - "print('Coefficient beta : \\n', linreg.coef_)\n", - "# The mean squared error \n", - "print(\"Mean squared error: %.2f\" % mean_squared_error(y, ypredict))\n", - "# Explained variance score: 1 is perfect prediction \n", - "print('Variance score: %.2f' % r2_score(y, ypredict))\n", - "# Mean squared log error \n", - "print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) )\n", - "# Mean absolute error \n", - "print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict))\n", - "plt.plot(x, ypredict, \"r-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0.0,1.0,1.5, 7.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Linear Regression fit ')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The function **coef** gives us the parameter $\\beta$ of our fit while **intercept** yields \n", - "$\\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n", - "\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The smaller the value, the better the fit. Ideally we would like to\n", - "have an MSE equal zero. The attentive reader has probably recognized\n", - "this function as being similar to the $\\chi^2$ function defined above.\n", - "\n", - "The **r2score** function computes $R^2$, the coefficient of\n", - "determination. It provides a measure of how well future samples are\n", - "likely to be predicted by the model. Best possible score is 1.0 and it\n", - "can be negative (because the model can be arbitrarily worse). A\n", - "constant model that always predicts the expected value of $\\hat{y}$,\n", - "disregarding the input features, would get a $R^2$ score of $0.0$.\n", - "\n", - "If $\\tilde{\\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined the mean value of $\\hat{y}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\bar{y} = \\frac{1}{n} \\sum_{i=0}^{n - 1} y_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Another quantity taht we will meet again in our discussions of regression analysis is \n", - " the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error.\n", - "The MAE is defined as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\text{MAE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We present the \n", - "squared logarithmic (quadratic) error" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\text{MSLE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\log_e (x)$ stands for the natural logarithm of $x$. This error\n", - "estimate is best to use when targets having exponential growth, such\n", - "as population counts, average sales of a commodity over a span of\n", - "years etc. \n", - "\n", - "\n", - "Finally, another cost function is the Huber cost function used in robust regression.\n", - "\n", - "The rationale behind this possible cost function is its reduced\n", - "sensitivity to outliers in the data set. In our discussions on\n", - "dimensionality reduction and normalization of data we will meet other\n", - "ways of dealing with outliers.\n", - "\n", - "The Huber cost function is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "H_{\\delta}(a)={\\begin{cases}{\\frac {1}{2}}{a^{2}}&{\\text{for }}|a|\\leq \\delta ,\\\\\\delta (|a|-{\\frac {1}{2}}\\delta ),&{\\text{otherwise.}}\\end{cases}}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here $a=\\boldsymbol{y} - \\boldsymbol{\\tilde{y}}$.\n", - "We will discuss in more\n", - "detail these and other functions in the various lectures. We conclude this part with another example. Instead of \n", - "a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "import random\n", - "from sklearn.linear_model import Ridge\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.linear_model import LinearRegression\n", - "\n", - "x=np.linspace(0.02,0.98,200)\n", - "noise = np.asarray(random.sample((range(200)),200))\n", - "y=x**3*noise\n", - "yn=x**3*100\n", - "poly3 = PolynomialFeatures(degree=3)\n", - "X = poly3.fit_transform(x[:,np.newaxis])\n", - "clf3 = LinearRegression()\n", - "clf3.fit(X,y)\n", - "\n", - "Xplot=poly3.fit_transform(x[:,np.newaxis])\n", - "poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit')\n", - "plt.plot(x,yn, color='red', label=\"True Cubic\")\n", - "plt.scatter(x, y, label='Data', color='orange', s=15)\n", - "plt.legend()\n", - "plt.show()\n", - "\n", - "def error(a):\n", - " for i in y:\n", - " err=(y-yn)/yn\n", - " return abs(np.sum(err))/len(err)\n", - "\n", - "print (error(y))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding\n", - "energies. A basic quantity which can be measured for the ground\n", - "states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with\n", - "atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). \n", - "\n", - "Atomic masses are usually tabulated in terms of the mass excess defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\Delta M(N, Z) = M(N, Z) - uA,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $u$ is the Atomic Mass Unit" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u = M(^{12}\\mathrm{C})/12 = 931.4940954(57) \\hspace{0.1cm} \\mathrm{MeV}/c^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The nucleon masses are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "m_p = 1.00727646693(9)u,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "m_n = 939.56536(8)\\hspace{0.1cm} \\mathrm{MeV}/c^2 = 1.0086649156(6)u.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the [2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu](http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf)\n", - "there are data on masses and decays of 3437 nuclei.\n", - "\n", - "The nuclear binding energy is defined as the energy required to break\n", - "up a given nucleus into its constituent parts of $N$ neutrons and $Z$\n", - "protons. In terms of the atomic masses $M(N, Z)$ the binding energy is\n", - "defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron.\n", - "In terms of the mass excess the binding energy is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "BE(N, Z) = Z\\Delta_H c^2 + N\\Delta_n c^2 -\\Delta(N, Z)c^2 ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\Delta_H c^2 = 7.2890$ MeV and $\\Delta_n c^2 = 8.0713$ MeV.\n", - "\n", - "\n", - "A popular and physically intuitive model which can be used to parametrize \n", - "the experimental binding energies as function of $A$, is the so-called \n", - "**liquid drop model**. The ansatz is based on the following expression" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "BE(N,Z) = a_1A-a_2A^{2/3}-a_3\\frac{Z^2}{A^{1/3}}-a_4\\frac{(N-Z)^2}{A},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit \n", - "to the experimental data. \n", - "\n", - "\n", - "\n", - "\n", - "To arrive at the above expression we have assumed that we can make the following assumptions:\n", - "\n", - " * There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume.\n", - "\n", - " * There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area.\n", - "\n", - " * There is a Coulomb energy term $a_3\\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding. \n", - "\n", - " * There is an asymmetry term $a_4\\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions.\n", - "\n", - "We could also add a so-called pairing term, which is a correction term that\n", - "arises from the tendency of proton pairs and neutron pairs to\n", - "occur. An even number of particles is more stable than an odd number. \n", - "\n", - "\n", - "### Organizing our data\n", - "\n", - "Let us start with reading and organizing our data. \n", - "We start with the compilation of masses and binding energies from 2016.\n", - "After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data.\n", - "\n", - "\n", - "We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of **scikit-learn**." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "import sklearn.linear_model as skl\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error\n", - "import os\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"MassEval2016.dat\"),'r')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various **matplotlib** commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from pylab import plt, mpl\n", - "plt.style.use('seaborn')\n", - "mpl.rcParams['font.family'] = 'serif'\n", - "\n", - "def MakePlot(x,y, styles, labels, axlabels):\n", - " plt.figure(figsize=(10,6))\n", - " for i in range(len(x)):\n", - " plt.plot(x[i], y[i], styles[i], label = labels[i])\n", - " plt.xlabel(axlabels[0])\n", - " plt.ylabel(axlabels[1])\n", - " plt.legend(loc=0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Our next step is to read the data on experimental binding energies and\n", - "reorganize them as functions of the mass number $A$, the number of\n", - "protons $Z$ and neutrons $N$ using **pandas**. Before we do this it is\n", - "always useful (unless you have a binary file or other types of compressed\n", - "data) to actually open the file and simply take a look at it!\n", - "\n", - "\n", - "In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with **pandas**. The file begins with some basic format information." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\"\"\" \n", - "This is taken from the data file of the mass 2016 evaluation. \n", - "All files are 3436 lines long with 124 character per line. \n", - " Headers are 39 lines long. \n", - " col 1 : Fortran character control: 1 = page feed 0 = line feed \n", - " format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 \n", - " These formats are reflected in the pandas widths variable below, see the statement \n", - " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), \n", - " Pandas has also a variable header, with length 39 in this case. \n", - "\"\"\"" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The data we are interested in are in columns 2, 3, 4 and 11, giving us\n", - "the number of neutrons, protons, mass numbers and binding energies,\n", - "respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will\n", - "covert them into the **pandas** DataFrame structure." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Read the experimental data with Pandas\n", - "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", - " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", - " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", - " header=39,\n", - " index_col=False)\n", - "\n", - "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", - "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", - "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", - "Masses = Masses.dropna()\n", - "# Convert from keV to MeV.\n", - "Masses['Ebinding'] /= 1000\n", - "\n", - "# Group the DataFrame by nucleon number, A.\n", - "Masses = Masses.groupby('A')\n", - "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", - "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We have now read in the data, grouped them according to the variables we are interested in. \n", - "We see how easy it is to reorganize the data using **pandas**. If we\n", - "were to do these operations in C/C++ or Fortran, we would have had to\n", - "write various functions/subroutines which perform the above\n", - "reorganizations for us. Having reorganized the data, we can now start\n", - "to make some simple fits using both the functionalities in **numpy** and\n", - "**Scikit-Learn** afterwards. \n", - "\n", - "Now we define five variables which contain\n", - "the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "A = Masses['A']\n", - "Z = Masses['Z']\n", - "N = Masses['N']\n", - "Element = Masses['Element']\n", - "Energies = Masses['Ebinding']\n", - "print(Masses)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The next step, and we will define this mathematically later, is to set up the so-called **design matrix**. We will throughout call this matrix $\\boldsymbol{X}$.\n", - "It has dimensionality $p\\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Now we set up the design matrix X\n", - "X = np.zeros((len(A),5))\n", - "X[:,0] = 1\n", - "X[:,1] = A\n", - "X[:,2] = A**(2.0/3.0)\n", - "X[:,3] = A**(-1.0/3.0)\n", - "X[:,4] = A**(-1.0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With **scikitlearn** we are now ready to use linear regression and fit our data." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "clf = skl.LinearRegression().fit(X, Energies)\n", - "fity = clf.predict(X)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Pretty simple! \n", - "Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# The mean squared error \n", - "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, fity))\n", - "# Explained variance score: 1 is perfect prediction \n", - "print('Variance score: %.2f' % r2_score(Energies, fity))\n", - "# Mean absolute error \n", - "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity))\n", - "print(clf.coef_, clf.intercept_)\n", - "\n", - "Masses['Eapprox'] = fity\n", - "# Generate a plot comparing the experimental with the fitted values values.\n", - "fig, ax = plt.subplots()\n", - "ax.set_xlabel(r'$A = N + Z$')\n", - "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", - "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", - " label='Ame2016')\n", - "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", - " label='Fit')\n", - "ax.legend()\n", - "save_fig(\"Masses2016\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As a teaser, let us now see how we can do this with decision trees using **scikit-learn**. Later we will switch to so-called **random forests**!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "#Decision Tree Regression\n", - "from sklearn.tree import DecisionTreeRegressor\n", - "regr_1=DecisionTreeRegressor(max_depth=5)\n", - "regr_2=DecisionTreeRegressor(max_depth=7)\n", - "regr_3=DecisionTreeRegressor(max_depth=9)\n", - "regr_1.fit(X, Energies)\n", - "regr_2.fit(X, Energies)\n", - "regr_3.fit(X, Energies)\n", - "\n", - "\n", - "y_1 = regr_1.predict(X)\n", - "y_2 = regr_2.predict(X)\n", - "y_3=regr_3.predict(X)\n", - "Masses['Eapprox'] = y_3\n", - "# Plot the results\n", - "plt.figure()\n", - "plt.plot(A, Energies, color=\"blue\", label=\"Data\", linewidth=2)\n", - "plt.plot(A, y_1, color=\"red\", label=\"max_depth=5\", linewidth=2)\n", - "plt.plot(A, y_2, color=\"green\", label=\"max_depth=7\", linewidth=2)\n", - "plt.plot(A, y_3, color=\"m\", label=\"max_depth=9\", linewidth=2)\n", - "\n", - "plt.xlabel(\"$A$\")\n", - "plt.ylabel(\"$E$[MeV]\")\n", - "plt.title(\"Decision Tree Regression\")\n", - "plt.legend()\n", - "save_fig(\"Masses2016Trees\")\n", - "plt.show()\n", - "print(Masses)\n", - "print(np.mean( (Energies-y_1)**2))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The **seaborn** package allows us to visualize data in an efficient way. Note that we use **scikit-learn**'s multi-layer perceptron (or feed forward neural network) \n", - "functionality." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.neural_network import MLPRegressor\n", - "from sklearn.metrics import accuracy_score\n", - "import seaborn as sns\n", - "\n", - "X_train = X\n", - "Y_train = Energies\n", - "n_hidden_neurons = 100\n", - "epochs = 100\n", - "# store models for later use\n", - "eta_vals = np.logspace(-5, 1, 7)\n", - "lmbd_vals = np.logspace(-5, 1, 7)\n", - "# store the models for later use\n", - "DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "sns.set()\n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',\n", - " alpha=lmbd, learning_rate_init=eta, max_iter=epochs)\n", - " dnn.fit(X_train, Y_train)\n", - " DNN_scikit[i][j] = dnn\n", - " train_accuracy[i][j] = dnn.score(X_train, Y_train)\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Linear Regression, basic elements\n", - "\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureAug27.mp4?vrtx=view-as-webpage).\n", - "\n", - "\n", - "Fitting a continuous function with linear parameterization in terms of the parameters $\\boldsymbol{\\beta}$.\n", - "* Method of choice for fitting a continuous function!\n", - "\n", - "* Gives an excellent introduction to central Machine Learning features with **understandable pedagogical** links to other methods like **Neural Networks**, **Support Vector Machines** etc\n", - "\n", - "* Analytical expression for the fitting parameters $\\boldsymbol{\\beta}$\n", - "\n", - "* Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more\n", - "\n", - "* Analytical relation with probabilistic interpretations \n", - "\n", - "* Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics\n", - "\n", - "* Easy to code! And links well with classification problems and logistic regression and neural networks\n", - "\n", - "* Allows for **easy** hands-on understanding of gradient descent methods\n", - "\n", - "* and many more features\n", - "\n", - "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", - "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended.\n", - "\n", - "\n", - "\n", - "Regression modeling deals with the description of the sampling distribution of a given random variable $y$ and how it varies as function of another variable or a set of such variables $\\boldsymbol{x} =[x_0, x_1,\\dots, x_{n-1}]^T$. \n", - "The first variable is called the **dependent**, the **outcome** or the **response** variable while the set of variables $\\boldsymbol{x}$ is called the independent variable, or the predictor variable or the explanatory variable. \n", - "\n", - "A regression model aims at finding a likelihood function $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$, that is the conditional distribution for $\\boldsymbol{y}$ with a given $\\boldsymbol{x}$. The estimation of $p(\\boldsymbol{y}\\vert \\boldsymbol{x})$ is made using a data set with \n", - "* $n$ cases $i = 0, 1, 2, \\dots, n-1$ \n", - "\n", - "* Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \\dots, n-1$ \n", - "\n", - "* $p$ so-called explanatory (independent or predictor) variables $\\boldsymbol{x}_i=[x_{i0}, x_{i1}, \\dots, x_{ip-1}]$ with $i = 0, 1, 2, \\dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples. \n", - "\n", - " The goal of the regression analysis is to extract/exploit relationship between $\\boldsymbol{y}$ and $\\boldsymbol{x}$ in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things.\n", - "\n", - "\n", - "Consider an experiment in which $p$ characteristics of $n$ samples are\n", - "measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix \n", - "$\\mathbf{X}$.\n", - "\n", - "The matrix $\\mathbf{X}$ is called the *design\n", - "matrix*. Additional information of the samples is available in the\n", - "form of $\\boldsymbol{y}$ (also as above). The variable $\\boldsymbol{y}$ is\n", - "generally referred to as the *response variable*. The aim of\n", - "regression analysis is to explain $\\boldsymbol{y}$ in terms of\n", - "$\\boldsymbol{X}$ through a functional relationship like $y_i =\n", - "f(\\mathbf{X}_{i,\\ast})$. When no prior knowledge on the form of\n", - "$f(\\cdot)$ is available, it is common to assume a linear relationship\n", - "between $\\boldsymbol{X}$ and $\\boldsymbol{y}$. This assumption gives rise to\n", - "the *linear regression model* where $\\boldsymbol{\\beta} = [\\beta_0, \\ldots,\n", - "\\beta_{p-1}]^{T}$ are the *regression parameters*. \n", - "\n", - "Linear regression gives us a set of analytical equations for the parameters $\\beta_j$.\n", - "\n", - "\n", - "In order to understand the relation among the predictors $p$, the set of data $n$ and the target (outcome, output etc) $\\boldsymbol{y}$,\n", - "consider the model we discussed for describing nuclear binding energies. \n", - "\n", - "There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model.\n", - "Assuming" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we have five predictors, that is the intercept, the $A$ dependent term, the $A^{2/3}$ term and the $A^{-1/3}$ and $A^{-1}$ terms.\n", - "This gives $p=0,1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n", - "$p\\times n$ matrix $\\boldsymbol{X}$.\n", - "\n", - "Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the\n", - "so-called [credit card default data from Taiwan](https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub). The data set contains data on $n=30000$ credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are $24$ such predictors or attributes leading to a design matrix of dimensionality $24 \\times 30000$. This is however a classification problem and we will come back to it when we discuss Logistic Regression. \n", - "\n", - "\n", - "Before we proceed let us study a case from linear algebra where we aim at fitting a set of data $\\boldsymbol{y}=[y_0,y_1,\\dots,y_{n-1}]$. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables $\\boldsymbol{x}=[x_0,x_1,\\dots,x_{n-1}]$, that is $y_i = y(x_i)$ with $i=0,1,2,\\dots,n-1$. The variables $x_i$ could represent physical quantities like time, temperature, position etc. We assume that $y(x)$ is a smooth function. \n", - "\n", - "Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of $y$ which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree $n-1$ with $n$ points, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y=y(x) \\rightarrow y(x_i)=\\tilde{y}_i+\\epsilon_i=\\sum_{j=0}^{n-1} \\beta_j x_i^j+\\epsilon_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\epsilon_i$ is the error in our approximation. \n", - "\n", - "\n", - "For every set of values $y_i,x_i$ we have thus the corresponding set of equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "y_0&=\\beta_0+\\beta_1x_0^1+\\beta_2x_0^2+\\dots+\\beta_{n-1}x_0^{n-1}+\\epsilon_0\\\\\n", - "y_1&=\\beta_0+\\beta_1x_1^1+\\beta_2x_1^2+\\dots+\\beta_{n-1}x_1^{n-1}+\\epsilon_1\\\\\n", - "y_2&=\\beta_0+\\beta_1x_2^1+\\beta_2x_2^2+\\dots+\\beta_{n-1}x_2^{n-1}+\\epsilon_2\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{n-1}&=\\beta_0+\\beta_1x_{n-1}^1+\\beta_2x_{n-1}^2+\\dots+\\beta_{n-1}x_{n-1}^{n-1}+\\epsilon_{n-1}.\\\\\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Defining the vectors" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y} = [y_0,y_1, y_2,\\dots, y_{n-1}]^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta} = [\\beta_0,\\beta_1, \\beta_2,\\dots, \\beta_{n-1}]^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\epsilon} = [\\epsilon_0,\\epsilon_1, \\epsilon_2,\\dots, \\epsilon_{n-1}]^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the design matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\n", - "\\begin{bmatrix} \n", - "1& x_{0}^1 &x_{0}^2& \\dots & \\dots &x_{0}^{n-1}\\\\\n", - "1& x_{1}^1 &x_{1}^2& \\dots & \\dots &x_{1}^{n-1}\\\\\n", - "1& x_{2}^1 &x_{2}^2& \\dots & \\dots &x_{2}^{n-1}\\\\ \n", - "\\dots& \\dots &\\dots& \\dots & \\dots &\\dots\\\\\n", - "1& x_{n-1}^1 &x_{n-1}^2& \\dots & \\dots &x_{n-1}^{n-1}\\\\\n", - "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we can rewrite our equations as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta}+\\boldsymbol{\\epsilon}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The above design matrix is called a [Vandermonde matrix](https://en.wikipedia.org/wiki/Vandermonde_matrix).\n", - "\n", - "We are obviously not limited to the above polynomial expansions. We\n", - "could replace the various powers of $x$ with elements of Fourier\n", - "series or instead of $x_i^j$ we could have $\\cos{(j x_i)}$ or $\\sin{(j\n", - "x_i)}$, or time series or other orthogonal functions. For every set\n", - "of values $y_i,x_i$ we can then generalize the equations to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "y_0&=\\beta_0x_{00}+\\beta_1x_{01}+\\beta_2x_{02}+\\dots+\\beta_{n-1}x_{0n-1}+\\epsilon_0\\\\\n", - "y_1&=\\beta_0x_{10}+\\beta_1x_{11}+\\beta_2x_{12}+\\dots+\\beta_{n-1}x_{1n-1}+\\epsilon_1\\\\\n", - "y_2&=\\beta_0x_{20}+\\beta_1x_{21}+\\beta_2x_{22}+\\dots+\\beta_{n-1}x_{2n-1}+\\epsilon_2\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{i}&=\\beta_0x_{i0}+\\beta_1x_{i1}+\\beta_2x_{i2}+\\dots+\\beta_{n-1}x_{in-1}+\\epsilon_i\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{n-1}&=\\beta_0x_{n-1,0}+\\beta_1x_{n-1,2}+\\beta_2x_{n-1,2}+\\dots+\\beta_{n-1}x_{n-1,n-1}+\\epsilon_{n-1}.\\\\\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!**\n", - "\n", - "We redefine in turn the matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\n", - "\\begin{bmatrix} \n", - "x_{00}& x_{01} &x_{02}& \\dots & \\dots &x_{0,n-1}\\\\\n", - "x_{10}& x_{11} &x_{12}& \\dots & \\dots &x_{1,n-1}\\\\\n", - "x_{20}& x_{21} &x_{22}& \\dots & \\dots &x_{2,n-1}\\\\ \n", - "\\dots& \\dots &\\dots& \\dots & \\dots &\\dots\\\\\n", - "x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \\dots & \\dots &x_{n-1,n-1}\\\\\n", - "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and without loss of generality we rewrite again our equations as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta}+\\boldsymbol{\\epsilon}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The left-hand side of this equation is kwown. Our error vector $\\boldsymbol{\\epsilon}$ and the parameter vector $\\boldsymbol{\\beta}$ are our unknow quantities. How can we obtain the optimal set of $\\beta_i$ values? \n", - "\n", - "We have defined the matrix $\\boldsymbol{X}$ via the equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "y_0&=\\beta_0x_{00}+\\beta_1x_{01}+\\beta_2x_{02}+\\dots+\\beta_{n-1}x_{0n-1}+\\epsilon_0\\\\\n", - "y_1&=\\beta_0x_{10}+\\beta_1x_{11}+\\beta_2x_{12}+\\dots+\\beta_{n-1}x_{1n-1}+\\epsilon_1\\\\\n", - "y_2&=\\beta_0x_{20}+\\beta_1x_{21}+\\beta_2x_{22}+\\dots+\\beta_{n-1}x_{2n-1}+\\epsilon_1\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{i}&=\\beta_0x_{i0}+\\beta_1x_{i1}+\\beta_2x_{i2}+\\dots+\\beta_{n-1}x_{in-1}+\\epsilon_1\\\\\n", - "\\dots & \\dots \\\\\n", - "y_{n-1}&=\\beta_0x_{n-1,0}+\\beta_1x_{n-1,2}+\\beta_2x_{n-1,2}+\\dots+\\beta_{n-1}x_{n-1,n-1}+\\epsilon_{n-1}.\\\\\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As we noted above, we stayed with a system with the design matrix \n", - " $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times n}$, that is we have $p=n$. For reasons to come later (algorithmic arguments) we will hereafter define \n", - "our matrix as $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors refering to the column numbers and the entries $n$ being the row elements.\n", - "\n", - "In our [introductory notes](https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html) we looked at the so-called [liquid drop model](https://en.wikipedia.org/wiki/Semi-empirical_mass_formula). Let us remind ourselves about what we did by looking at the code.\n", - "\n", - "We restate the parts of the code we are most interested in." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from IPython.display import display\n", - "import os\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"MassEval2016.dat\"),'r')\n", - "\n", - "\n", - "# Read the experimental data with Pandas\n", - "Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),\n", - " names=('N', 'Z', 'A', 'Element', 'Ebinding'),\n", - " widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),\n", - " header=39,\n", - " index_col=False)\n", - "\n", - "# Extrapolated values are indicated by '#' in place of the decimal place, so\n", - "# the Ebinding column won't be numeric. Coerce to float and drop these entries.\n", - "Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')\n", - "Masses = Masses.dropna()\n", - "# Convert from keV to MeV.\n", - "Masses['Ebinding'] /= 1000\n", - "\n", - "# Group the DataFrame by nucleon number, A.\n", - "Masses = Masses.groupby('A')\n", - "# Find the rows of the grouped DataFrame with the maximum binding energy.\n", - "Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])\n", - "A = Masses['A']\n", - "Z = Masses['Z']\n", - "N = Masses['N']\n", - "Element = Masses['Element']\n", - "Energies = Masses['Ebinding']\n", - "\n", - "# Now we set up the design matrix X\n", - "X = np.zeros((len(A),5))\n", - "X[:,0] = 1\n", - "X[:,1] = A\n", - "X[:,2] = A**(2.0/3.0)\n", - "X[:,3] = A**(-1.0/3.0)\n", - "X[:,4] = A**(-1.0)\n", - "# Then nice printout using pandas\n", - "DesignMatrix = pd.DataFrame(X)\n", - "DesignMatrix.index = A\n", - "DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']\n", - "display(DesignMatrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With $\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p\\times 1}$, it means that we will hereafter write our equations for the approximation as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "throughout these lectures. \n", - "\n", - "With the above we use the design matrix to define the approximation $\\boldsymbol{\\tilde{y}}$ via the unknown quantity $\\boldsymbol{\\beta}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}}= \\boldsymbol{X}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and in order to find the optimal parameters $\\beta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parameterized values $\\tilde{y}_i$, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or using the matrix $\\boldsymbol{X}$ and in a more compact matrix-vector notation as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This function is one possible way to define the so-called cost function.\n", - "\n", - "\n", - "\n", - "It is also common to define\n", - "the function $C$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta})=\\frac{1}{2n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "since when taking the first derivative with respect to the unknown parameters $\\beta$, the factor of $2$ cancels out. \n", - "\n", - "The function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "can be linked to the variance of the quantity $y_i$ if we interpret the latter as the mean value. \n", - "When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret $y_i$ as a mean value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_{i}=\\langle y_i \\rangle = \\beta_0x_{i,0}+\\beta_1x_{i,1}+\\beta_2x_{i,2}+\\dots+\\beta_{n-1}x_{i,n-1}+\\epsilon_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\langle y_i \\rangle$ is the mean value. Keep in mind also that\n", - "till now we have treated $y_i$ as the exact value. Normally, the\n", - "response (dependent or outcome) variable $y_i$ the outcome of a\n", - "numerical experiment or another type of experiment and is thus only an\n", - "approximation to the true value. It is then always accompanied by an\n", - "error estimate, often limited to a statistical error estimate given by\n", - "the standard deviation discussed earlier. In the discussion here we\n", - "will treat $y_i$ as our exact value for the response variable.\n", - "\n", - "In order to find the parameters $\\beta_i$ we will then minimize the spread of $C(\\boldsymbol{\\beta})$, that is we are going to solve the problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", - "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In practical terms it means we will require" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_j} = \\frac{\\partial }{\\partial \\beta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}\\right)^2\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which results in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\beta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_{ij}\\left(y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}\\right)\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or in a matrix-vector form as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can rewrite" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{y} = \\boldsymbol{X}^T\\boldsymbol{X}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and if the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$ is invertible we have the solution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta} =\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n", - "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n", - "{\\mathbb{R}}^{p\\times p}$. In the above case we have that $p \\ll n$,\n", - "in our case $p=5$ meaning that we end up with inverting a small\n", - "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n", - "matrices to invert. The methods discussed here and for many other\n", - "supervised learning algorithms like classification with logistic\n", - "regression or support vector machines, exhibit dimensionalities which\n", - "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n", - "$\\boldsymbol{X}^T\\boldsymbol{X}$. \n", - "\n", - "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect? \n", - "\n", - "\n", - "The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and \n", - "matrices as upper case boldfaced letters." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "4\n", - "8\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "4\n", - "9\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "5\n", - "0\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\log{\\vert\\boldsymbol{A}\\vert}}{\\partial \\boldsymbol{A}} = (\\boldsymbol{A}^{-1})^T.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The residuals $\\boldsymbol{\\epsilon}$ are in turn given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\epsilon} = \\boldsymbol{y}-\\boldsymbol{\\tilde{y}} = \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{\\epsilon}=\\boldsymbol{X}^T\\left( \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)= 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "meaning that the solution for $\\boldsymbol{\\beta}$ is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.\n", - "\n", - "\n", - "Let us now return to our nuclear binding energies and simply code the above equations. \n", - "\n", - "\n", - "It is rather straightforward to implement the matrix inversion and obtain the parameters $\\boldsymbol{\\beta}$. After having defined the matrix $\\boldsymbol{X}$ we simply need to \n", - "write" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# matrix inversion to find beta\n", - "beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)\n", - "# and then make the prediction\n", - "ytilde = X @ beta" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Alternatively, you can use the least squares functionality in **Numpy** as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "fit = np.linalg.lstsq(X, Energies, rcond =None)[0]\n", - "ytildenp = np.dot(fit,X.T)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "And finally we plot our fit with and compare with data" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "Masses['Eapprox'] = ytilde\n", - "# Generate a plot comparing the experimental with the fitted values values.\n", - "fig, ax = plt.subplots()\n", - "ax.set_xlabel(r'$A = N + Z$')\n", - "ax.set_ylabel(r'$E_\\mathrm{bind}\\,/\\mathrm{MeV}$')\n", - "ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,\n", - " label='Ame2016')\n", - "ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',\n", - " label='Fit')\n", - "ax.legend()\n", - "save_fig(\"Masses2016OLS\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can easily test our fit by computing the $R2$ score that we discussed in connection with the functionality of **Scikit-Learn** in the introductory slides.\n", - "Since we are not using **Scikit-Learn** here we can define our own $R2$ function as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def R2(y_data, y_model):\n", - " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we would be using it as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "print(R2(Energies,ytilde))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can easily add our **MSE** score as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def MSE(y_data,y_model):\n", - " n = np.size(y_model)\n", - " return np.sum((y_data-y_model)**2)/n\n", - "\n", - "print(MSE(Energies,ytilde))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and finally the relative error as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def RelativeError(y_data,y_model):\n", - " return abs((y_data-y_model)/y_data)\n", - "print(RelativeError(Energies, ytilde))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### The $\\chi^2$ function\n", - "\n", - "Normally, the response (dependent or outcome) variable $y_i$ is the\n", - "outcome of a numerical experiment or another type of experiment and is\n", - "thus only an approximation to the true value. It is then always\n", - "accompanied by an error estimate, often limited to a statistical error\n", - "estimate given by the standard deviation discussed earlier. In the\n", - "discussion here we will treat $y_i$ as our exact value for the\n", - "response variable.\n", - "\n", - "Introducing the standard deviation $\\sigma_i$ for each measurement\n", - "$y_i$, we define now the $\\chi^2$ function (omitting the $1/n$ term)\n", - "as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\chi^2(\\boldsymbol{\\beta})=\\frac{1}{n}\\sum_{i=0}^{n-1}\\frac{\\left(y_i-\\tilde{y}_i\\right)^2}{\\sigma_i^2}=\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)^T\\frac{1}{\\boldsymbol{\\Sigma^2}}\\left(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}}\\right)\\right\\},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the matrix $\\boldsymbol{\\Sigma}$ is a diagonal matrix with $\\sigma_i$ as matrix elements. \n", - "\n", - "\n", - "In order to find the parameters $\\beta_i$ we will then minimize the spread of $\\chi^2(\\boldsymbol{\\beta})$ by requiring" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_j} = \\frac{\\partial }{\\partial \\beta_j}\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)^2\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which results in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_j} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}\\frac{x_{ij}}{\\sigma_i}\\left(\\frac{y_i-\\beta_0x_{i,0}-\\beta_1x_{i,1}-\\beta_2x_{i,2}-\\dots-\\beta_{n-1}x_{i,n-1}}{\\sigma_i}\\right)\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or in a matrix-vector form as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\beta}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined the matrix $\\boldsymbol{A} =\\boldsymbol{X}/\\boldsymbol{\\Sigma}$ with matrix elements $a_{ij} = x_{ij}/\\sigma_i$ and the vector $\\boldsymbol{b}$ with elements $b_i = y_i/\\sigma_i$. \n", - "\n", - "We can rewrite" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = 0 = \\boldsymbol{A}^T\\left( \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{\\beta}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}^T\\boldsymbol{b} = \\boldsymbol{A}^T\\boldsymbol{A}\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and if the matrix $\\boldsymbol{A}^T\\boldsymbol{A}$ is invertible we have the solution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta} =\\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1}\\boldsymbol{A}^T\\boldsymbol{b}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we then introduce the matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{H} = \\left(\\boldsymbol{A}^T\\boldsymbol{A}\\right)^{-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we have then the following expression for the parameters $\\beta_j$ (the matrix elements of $\\boldsymbol{H}$ are $h_{ij}$)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_j = \\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}\\frac{y_i}{\\sigma_i}\\frac{x_{ik}}{\\sigma_i} = \\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}b_ia_{ik}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We state without proof the expression for the uncertainty in the parameters $\\beta_j$ as (we leave this as an exercise)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2(\\beta_j) = \\sum_{i=0}^{n-1}\\sigma_i^2\\left( \\frac{\\partial \\beta_j}{\\partial y_i}\\right)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2(\\beta_j) = \\left(\\sum_{k=0}^{p-1}h_{jk}\\sum_{i=0}^{n-1}a_{ik}\\right)\\left(\\sum_{l=0}^{p-1}h_{jl}\\sum_{m=0}^{n-1}a_{ml}\\right) = h_{jj}!\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The first step here is to approximate the function $y$ with a first-order polynomial, that is we write" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y=y(x) \\rightarrow y(x_i) \\approx \\beta_0+\\beta_1 x_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "By computing the derivatives of $\\chi^2$ with respect to $\\beta_0$ and $\\beta_1$ show that these are given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_0} = -2\\left[ \\frac{1}{n}\\sum_{i=0}^{n-1}\\left(\\frac{y_i-\\beta_0-\\beta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\chi^2(\\boldsymbol{\\beta})}{\\partial \\beta_1} = -\\frac{2}{n}\\left[ \\sum_{i=0}^{n-1}x_i\\left(\\frac{y_i-\\beta_0-\\beta_1x_{i}}{\\sigma_i^2}\\right)\\right]=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For a linear fit (a first-order polynomial) we don't need to invert a matrix!! \n", - "Defining" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma = \\sum_{i=0}^{n-1}\\frac{1}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_x = \\sum_{i=0}^{n-1}\\frac{x_{i}}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_y = \\sum_{i=0}^{n-1}\\left(\\frac{y_i}{\\sigma_i^2}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_{xx} = \\sum_{i=0}^{n-1}\\frac{x_ix_{i}}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_{xy} = \\sum_{i=0}^{n-1}\\frac{y_ix_{i}}{\\sigma_i^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_0 = \\frac{\\gamma_{xx}\\gamma_y-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_1 = \\frac{\\gamma_{xy}\\gamma-\\gamma_x\\gamma_y}{\\gamma\\gamma_{xx}-\\gamma_x^2}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This approach (different linear and non-linear regression) suffers\n", - "often from both being underdetermined and overdetermined in the\n", - "unknown coefficients $\\beta_i$. A better approach is to use the\n", - "Singular Value Decomposition (SVD) method discussed below. Or using\n", - "Lasso and Ridge regression. See below.\n", - "\n", - "\n", - "### Fitting an Equation of State for Dense Nuclear Matter\n", - "\n", - "Before we continue, let us introduce yet another example. We are going to fit the\n", - "nuclear equation of state using results from many-body calculations.\n", - "The equation of state we have made available here, as function of\n", - "density, has been derived using modern nucleon-nucleon potentials with\n", - "[the addition of three-body\n", - "forces](https://www.sciencedirect.com/science/article/pii/S0370157399001106). This\n", - "time the file is presented as a standard **csv** file.\n", - "\n", - "The beginning of the Python code here is similar to what you have seen\n", - "before, with the same initializations and declarations. We use also\n", - "**pandas** again, rather extensively in order to organize our data.\n", - "\n", - "The difference now is that we use **Scikit-Learn's** regression tools\n", - "instead of our own matrix inversion implementation. Furthermore, we\n", - "sneak in **Ridge** regression (to be discussed below) which includes a\n", - "hyperparameter $\\lambda$, also to be explained below." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "import matplotlib.pyplot as plt\n", - "import sklearn.linear_model as skl\n", - "from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"EoS.csv\"),'r')\n", - "\n", - "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", - "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", - "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", - "EoS = EoS.dropna()\n", - "Energies = EoS['Energy']\n", - "Density = EoS['Density']\n", - "# The design matrix now as function of various polytrops\n", - "X = np.zeros((len(Density),4))\n", - "X[:,3] = Density**(4.0/3.0)\n", - "X[:,2] = Density\n", - "X[:,1] = Density**(2.0/3.0)\n", - "X[:,0] = 1\n", - "\n", - "# We use now Scikit-Learn's linear regressor and ridge regressor\n", - "# OLS part\n", - "clf = skl.LinearRegression().fit(X, Energies)\n", - "ytilde = clf.predict(X)\n", - "EoS['Eols'] = ytilde\n", - "# The mean squared error \n", - "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, ytilde))\n", - "# Explained variance score: 1 is perfect prediction \n", - "print('Variance score: %.2f' % r2_score(Energies, ytilde))\n", - "# Mean absolute error \n", - "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde))\n", - "print(clf.coef_, clf.intercept_)\n", - "\n", - "# The Ridge regression with a hyperparameter lambda = 0.1\n", - "_lambda = 0.1\n", - "clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies)\n", - "yridge = clf_ridge.predict(X)\n", - "EoS['Eridge'] = yridge\n", - "# The mean squared error \n", - "print(\"Mean squared error: %.2f\" % mean_squared_error(Energies, yridge))\n", - "# Explained variance score: 1 is perfect prediction \n", - "print('Variance score: %.2f' % r2_score(Energies, yridge))\n", - "# Mean absolute error \n", - "print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge))\n", - "print(clf_ridge.coef_, clf_ridge.intercept_)\n", - "\n", - "fig, ax = plt.subplots()\n", - "ax.set_xlabel(r'$\\rho[\\mathrm{fm}^{-3}]$')\n", - "ax.set_ylabel(r'Energy per particle')\n", - "ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2,\n", - " label='Theoretical data')\n", - "ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m',\n", - " label='OLS')\n", - "ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g',\n", - " label='Ridge $\\lambda = 0.1$')\n", - "ax.legend()\n", - "save_fig(\"EoSfitting\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The above simple polynomial in density $\\rho$ gives an excellent fit\n", - "to the data. \n", - "\n", - "We note also that there is a small deviation between the\n", - "standard OLS and the Ridge regression at higher densities. We discuss this in more detail\n", - "below.\n", - "\n", - "\n", - "## Splitting our Data in Training and Test data\n", - "\n", - "It is normal in essentially all Machine Learning studies to split the\n", - "data in a training set and a test set (sometimes also an additional\n", - "validation set). **Scikit-Learn** has an own function for this. There\n", - "is no explicit recipe for how much data should be included as training\n", - "data and say test data. An accepted rule of thumb is to use\n", - "approximately $2/3$ to $4/5$ of the data as training data. We will\n", - "postpone a discussion of this splitting to the end of these notes and\n", - "our discussion of the so-called **bias-variance** tradeoff. Here we\n", - "limit ourselves to repeat the above equation of state fitting example\n", - "but now splitting the data into a training set and a test set." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.model_selection import train_test_split\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "def R2(y_data, y_model):\n", - " return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)\n", - "def MSE(y_data,y_model):\n", - " n = np.size(y_model)\n", - " return np.sum((y_data-y_model)**2)/n\n", - "\n", - "infile = open(data_path(\"EoS.csv\"),'r')\n", - "\n", - "# Read the EoS data as csv file and organized into two arrays with density and energies\n", - "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", - "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", - "EoS = EoS.dropna()\n", - "Energies = EoS['Energy']\n", - "Density = EoS['Density']\n", - "# The design matrix now as function of various polytrops\n", - "X = np.zeros((len(Density),5))\n", - "X[:,0] = 1\n", - "X[:,1] = Density**(2.0/3.0)\n", - "X[:,2] = Density\n", - "X[:,3] = Density**(4.0/3.0)\n", - "X[:,4] = Density**(5.0/3.0)\n", - "# We split the data in test and training data\n", - "X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)\n", - "# matrix inversion to find beta\n", - "beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)\n", - "# and then make the prediction\n", - "ytilde = X_train @ beta\n", - "print(\"Training R2\")\n", - "print(R2(y_train,ytilde))\n", - "print(\"Training MSE\")\n", - "print(MSE(y_train,ytilde))\n", - "ypredict = X_test @ beta\n", - "print(\"Test R2\")\n", - "print(R2(y_test,ypredict))\n", - "print(\"Test MSE\")\n", - "print(MSE(y_test,ypredict))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The Boston housing data example\n", - "\n", - "The Boston housing \n", - "data set was originally a part of UCI Machine Learning Repository\n", - "and has been removed now. The data set is now included in **Scikit-Learn**'s \n", - "library. There are 506 samples and 13 feature (predictor) variables\n", - "in this data set. The objective is to predict the value of prices of\n", - "the house using the features (predictors) listed here.\n", - "\n", - "The features/predictors are\n", - "1. CRIM: Per capita crime rate by town\n", - "\n", - "2. ZN: Proportion of residential land zoned for lots over 25000 square feet\n", - "\n", - "3. INDUS: Proportion of non-retail business acres per town\n", - "\n", - "4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)\n", - "\n", - "5. NOX: Nitric oxide concentration (parts per 10 million)\n", - "\n", - "6. RM: Average number of rooms per dwelling\n", - "\n", - "7. AGE: Proportion of owner-occupied units built prior to 1940\n", - "\n", - "8. DIS: Weighted distances to five Boston employment centers\n", - "\n", - "9. RAD: Index of accessibility to radial highways\n", - "\n", - "10. TAX: Full-value property tax rate per USD10000\n", - "\n", - "11. B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town\n", - "\n", - "12. LSTAT: Percentage of lower status of the population\n", - "\n", - "13. MEDV: Median value of owner-occupied homes in USD 1000s\n", - "\n", - "## Housing data, the code\n", - "We start by importing the libraries" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt \n", - "\n", - "import pandas as pd \n", - "import seaborn as sns" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and load the Boston Housing DataSet from **Scikit-Learn**" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.datasets import load_boston\n", - "\n", - "boston_dataset = load_boston()\n", - "\n", - "# boston_dataset is a dictionary\n", - "# let's check what it contains\n", - "boston_dataset.keys()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Then we invoke Pandas" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)\n", - "boston.head()\n", - "boston['MEDV'] = boston_dataset.target" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and preprocess the data" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# check for missing values in all the columns\n", - "boston.isnull().sum()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can then visualize the data" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# set the size of the figure\n", - "sns.set(rc={'figure.figsize':(11.7,8.27)})\n", - "\n", - "# plot a histogram showing the distribution of the target values\n", - "sns.distplot(boston['MEDV'], bins=30)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is now useful to look at the correlation matrix" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# compute the pair wise correlation for all columns \n", - "correlation_matrix = boston.corr().round(2)\n", - "# use the heatmap function from seaborn to plot the correlation matrix\n", - "# annot = True to print the values inside the square\n", - "sns.heatmap(data=correlation_matrix, annot=True)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "From the above coorelation plot we can see that **MEDV** is strongly correlated to **LSTAT** and **RM**. We see also that **RAD** and **TAX** are stronly correlated, but we don't include this in our features together to avoid multi-colinearity" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "plt.figure(figsize=(20, 5))\n", - "\n", - "features = ['LSTAT', 'RM']\n", - "target = boston['MEDV']\n", - "\n", - "for i, col in enumerate(features):\n", - " plt.subplot(1, len(features) , i+1)\n", - " x = boston[col]\n", - " y = target\n", - " plt.scatter(x, y, marker='o')\n", - " plt.title(col)\n", - " plt.xlabel(col)\n", - " plt.ylabel('MEDV')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now we start training our model" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])\n", - "Y = boston['MEDV']" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We split the data into training and test sets" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.model_selection import train_test_split\n", - "\n", - "# splits the training and test data set in 80% : 20%\n", - "# assign random_state to any value.This ensures consistency.\n", - "X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "print(Y_train.shape)\n", - "print(Y_test.shape)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Then we use the linear regression functionality from **Scikit-Learn**" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.linear_model import LinearRegression\n", - "from sklearn.metrics import mean_squared_error, r2_score\n", - "\n", - "lin_model = LinearRegression()\n", - "lin_model.fit(X_train, Y_train)\n", - "\n", - "# model evaluation for training set\n", - "\n", - "y_train_predict = lin_model.predict(X_train)\n", - "rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))\n", - "r2 = r2_score(Y_train, y_train_predict)\n", - "\n", - "print(\"The model performance for training set\")\n", - "print(\"--------------------------------------\")\n", - "print('RMSE is {}'.format(rmse))\n", - "print('R2 score is {}'.format(r2))\n", - "print(\"\\n\")\n", - "\n", - "# model evaluation for testing set\n", - "\n", - "y_test_predict = lin_model.predict(X_test)\n", - "# root mean square error of the model\n", - "rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))\n", - "\n", - "# r-squared score of the model\n", - "r2 = r2_score(Y_test, y_test_predict)\n", - "\n", - "print(\"The model performance for testing set\")\n", - "print(\"--------------------------------------\")\n", - "print('RMSE is {}'.format(rmse))\n", - "print('R2 score is {}'.format(r2))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# plotting the y_test vs y_pred\n", - "# ideally should have been a straight line\n", - "plt.scatter(Y_test, y_test_predict)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Reducing the number of degrees of freedom, overarching view\n", - "\n", - "Many Machine Learning problems involve thousands or even millions of\n", - "features for each training instance. Not only does this make training\n", - "extremely slow, it can also make it much harder to find a good\n", - "solution, as we will see. This problem is often referred to as the\n", - "curse of dimensionality. Fortunately, in real-world problems, it is\n", - "often possible to reduce the number of features considerably, turning\n", - "an intractable problem into a tractable one.\n", - "\n", - "Later we will discuss some of the most popular dimensionality reduction\n", - "techniques: the principal component analysis (PCA), Kernel PCA, and\n", - "Locally Linear Embedding (LLE). \n", - "\n", - "\n", - "Principal component analysis and its various variants deal with the\n", - "problem of fitting a low-dimensional [affine\n", - "subspace](https://en.wikipedia.org/wiki/Affine_space) to a set of of\n", - "data points in a high-dimensional space. With its family of methods it\n", - "is one of the most used tools in data modeling, compression and\n", - "visualization.\n", - "\n", - "\n", - "Before we proceed however, we will discuss how to preprocess our\n", - "data. Till now and in connection with our previous examples we have\n", - "not met so many cases where we are too sensitive to the scaling of our\n", - "data. Normally the data may need a rescaling and/or may be sensitive\n", - "to extreme values. Scaling the data renders our inputs much more\n", - "suitable for the algorithms we want to employ.\n", - "\n", - "**Scikit-Learn** has several functions which allow us to rescale the\n", - "data, normally resulting in much better results in terms of various\n", - "accuracy scores. The **StandardScaler** function in **Scikit-Learn**\n", - "ensures that for each feature/predictor we study the mean value is\n", - "zero and the variance is one (every column in the design/feature\n", - "matrix). This scaling has the drawback that it does not ensure that\n", - "we have a particular maximum or minimum in our data set. Another\n", - "function included in **Scikit-Learn** is the **MinMaxScaler** which\n", - "ensures that all features are exactly between $0$ and $1$. The\n", - "\n", - "\n", - "The **Normalizer** scales each data\n", - "point such that the feature vector has a euclidean length of one. In other words, it\n", - "projects a data point on the circle (or sphere in the case of higher dimensions) with a\n", - "radius of 1. This means every data point is scaled by a different number (by the\n", - "inverse of it’s length).\n", - "This normalization is often used when only the direction (or angle) of the data matters,\n", - "not the length of the feature vector.\n", - "\n", - "The **RobustScaler** works similarly to the StandardScaler in that it\n", - "ensures statistical properties for each feature that guarantee that\n", - "they are on the same scale. However, the RobustScaler uses the median\n", - "and quartiles, instead of mean and variance. This makes the\n", - "RobustScaler ignore data points that are very different from the rest\n", - "(like measurement errors). These odd data points are also called\n", - "outliers, and might often lead to trouble for other scaling\n", - "techniques.\n", - "\n", - "\n", - "### Simple preprocessing examples, Franke function and regression" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "import sklearn.linear_model as skl\n", - "from sklearn.metrics import mean_squared_error\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "\n", - "def FrankeFunction(x,y):\n", - "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", - "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", - "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", - "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", - "\treturn term1 + term2 + term3 + term4\n", - "\n", - "\n", - "def create_X(x, y, n ):\n", - "\tif len(x.shape) > 1:\n", - "\t\tx = np.ravel(x)\n", - "\t\ty = np.ravel(y)\n", - "\n", - "\tN = len(x)\n", - "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", - "\tX = np.ones((N,l))\n", - "\n", - "\tfor i in range(1,n+1):\n", - "\t\tq = int((i)*(i+1)/2)\n", - "\t\tfor k in range(i+1):\n", - "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", - "\n", - "\treturn X\n", - "\n", - "\n", - "# Making meshgrid of datapoints and compute Franke's function\n", - "n = 5\n", - "N = 1000\n", - "x = np.sort(np.random.uniform(0, 1, N))\n", - "y = np.sort(np.random.uniform(0, 1, N))\n", - "z = FrankeFunction(x, y)\n", - "X = create_X(x, y, n=n) \n", - "# split in training and test data\n", - "X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=0.2)\n", - "\n", - "\n", - "clf = skl.LinearRegression().fit(X_train, y_train)\n", - "\n", - "# The mean squared error and R2 score\n", - "print(\"MSE before scaling: {:.2f}\".format(mean_squared_error(clf.predict(X_test), y_test)))\n", - "print(\"R2 score before scaling {:.2f}\".format(clf.score(X_test,y_test)))\n", - "\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "print(\"Feature min values before scaling:\\n {}\".format(X_train.min(axis=0)))\n", - "print(\"Feature max values before scaling:\\n {}\".format(X_train.max(axis=0)))\n", - "\n", - "print(\"Feature min values after scaling:\\n {}\".format(X_train_scaled.min(axis=0)))\n", - "print(\"Feature max values after scaling:\\n {}\".format(X_train_scaled.max(axis=0)))\n", - "\n", - "clf = skl.LinearRegression().fit(X_train_scaled, y_train)\n", - "\n", - "\n", - "print(\"MSE after scaling: {:.2f}\".format(mean_squared_error(clf.predict(X_test_scaled), y_test)))\n", - "print(\"R2 score for scaled data: {:.2f}\".format(clf.score(X_test_scaled,y_test)))" - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter1.py b/doc/LectureNotes/_build/jupyter_execute/chapter1.py deleted file mode 100644 index 310ea62e5..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter1.py +++ /dev/null @@ -1,2454 +0,0 @@ -# Linear Regression, basic Elements - -[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureAug21.mp4?vrtx=view-as-webpage) - - -## Introduction - - - - - -Our emphasis throughout this series of lectures -is on understanding the mathematical aspects of -different algorithms used in the fields of data analysis and machine learning. - -However, where possible we will emphasize the -importance of using available software. We start thus with a hands-on -and top-down approach to machine learning. The aim is thus to start with -relevant data or data we have produced -and use these to introduce statistical data analysis -concepts and machine learning algorithms before we delve into the -algorithms themselves. The examples we will use in the beginning, start with simple -polynomials with random noise added. We will use the Python -software package [Scikit-Learn](http://scikit-learn.org/stable/) and -introduce various machine learning algorithms to make fits of -the data and predictions. We move thereafter to more interesting -cases such as data from say experiments (below we will look at experimental nuclear binding energies as an example). -These are examples where we can easily set up the data and -then use machine learning algorithms included in for example -**Scikit-Learn**. - -These examples will serve us the purpose of getting -started. Furthermore, they allow us to catch more than two birds with -a stone. They will allow us to bring in some programming specific -topics and tools as well as showing the power of various Python -libraries for machine learning and statistical data analysis. - -Here, we will mainly focus on two -specific Python packages for Machine Learning, Scikit-Learn and -Tensorflow (see below for links etc). Moreover, the examples we -introduce will serve as inputs to many of our discussions later, as -well as allowing you to set up models and produce your own data and -get started with programming. - - - -## What is Machine Learning? - -Statistics, data science and machine learning form important fields of -research in modern science. They describe how to learn and make -predictions from data, as well as allowing us to extract important -correlations about physical process and the underlying laws of motion -in large data sets. The latter, big data sets, appear frequently in -essentially all disciplines, from the traditional Science, Technology, -Mathematics and Engineering fields to Life Science, Law, education -research, the Humanities and the Social Sciences. - -It has become more -and more common to see research projects on big data in for example -the Social Sciences where extracting patterns from complicated survey -data is one of many research directions. Having a solid grasp of data -analysis and machine learning is thus becoming central to scientific -computing in many fields, and competences and skills within the fields -of machine learning and scientific computing are nowadays strongly -requested by many potential employers. The latter cannot be -overstated, familiarity with machine learning has almost become a -prerequisite for many of the most exciting employment opportunities, -whether they are in bioinformatics, life science, physics or finance, -in the private or the public sector. This author has had several -students or met students who have been hired recently based on their -skills and competences in scientific computing and data science, often -with marginal knowledge of machine learning. - -Machine learning is a subfield of computer science, and is closely -related to computational statistics. It evolved from the study of -pattern recognition in artificial intelligence (AI) research, and has -made contributions to AI tasks like computer vision, natural language -processing and speech recognition. Many of the methods we will study are also -strongly rooted in basic mathematics and physics research. - -Ideally, machine learning represents the science of giving computers -the ability to learn without being explicitly programmed. The idea is -that there exist generic algorithms which can be used to find patterns -in a broad class of data sets without having to write code -specifically for each problem. The algorithm will build its own logic -based on the data. You should however always keep in mind that -machines and algorithms are to a large extent developed by humans. The -insights and knowledge we have about a specific system, play a central -role when we develop a specific machine learning algorithm. - -Machine learning is an extremely rich field, in spite of its young -age. The increases we have seen during the last three decades in -computational capabilities have been followed by developments of -methods and techniques for analyzing and handling large date sets, -relying heavily on statistics, computer science and mathematics. The -field is rather new and developing rapidly. Popular software packages -written in Python for machine learning like -[Scikit-learn](http://scikit-learn.org/stable/), -[Tensorflow](https://www.tensorflow.org/), -[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), all -freely available at their respective GitHub sites, encompass -communities of developers in the thousands or more. And the number of -code developers and contributors keeps increasing. Not all the -algorithms and methods can be given a rigorous mathematical -justification, opening up thereby large rooms for experimenting and -trial and error and thereby exciting new developments. However, a -solid command of linear algebra, multivariate theory, probability -theory, statistical data analysis, understanding errors and Monte -Carlo methods are central elements in a proper understanding of many -of algorithms and methods we will discuss. - - - -The approaches to machine learning are many, but are often split into -two main categories. In *supervised learning* we know the answer to a -problem, and let the computer deduce the logic behind it. On the other -hand, *unsupervised learning* is a method for finding patterns and -relationship in data sets without any prior knowledge of the system. -Some authours also operate with a third category, namely -*reinforcement learning*. This is a paradigm of learning inspired by -behavioral psychology, where learning is achieved by trial-and-error, -solely from rewards and punishment. - -Another way to categorize machine learning tasks is to consider the -desired output of a system. Some of the most common tasks are: - - * Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning. - - * Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values. - - * Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning. - -The methods we cover have three main topics in common, irrespective of -whether we deal with supervised or unsupervised learning. The first -ingredient is normally our data set (which can be subdivided into -training and test data), the second item is a model which is normally a -function of some parameters. The model reflects our knowledge of the system (or lack thereof). As an example, if we know that our data show a behavior similar to what would be predicted by a polynomial, fitting our data to a polynomial of some degree would then determin our model. - -The last ingredient is a so-called **cost** -function which allows us to present an estimate on how good our model -is in reproducing the data it is supposed to train. -At the heart of basically all ML algorithms there are so-called minimization algorithms, often we end up with various variants of **gradient** methods. - - - - - - - -## Software and needed installations - -We will make extensive use of Python as programming language and its -myriad of available libraries. You will find -Jupyter notebooks invaluable in your work. You can run **R** -codes in the Jupyter/IPython notebooks, with the immediate benefit of -visualizing your data. You can also use compiled languages like C++, -Rust, Julia, Fortran etc if you prefer. The focus in these lectures will be -on Python. - - -If you have Python installed (we strongly recommend Python3) and you feel -pretty familiar with installing different packages, we recommend that -you install the following Python packages via **pip** as - -1. pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow - -For Python3, replace **pip** with **pip3**. - -For OSX users we recommend, after having installed Xcode, to -install **brew**. Brew allows for a seamless installation of additional -software via for example - -1. brew install python3 - -For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, -you can use **pip** as well and simply install Python as - -1. sudo apt-get install python3 (or python for pyhton2.7) - -etc etc. - - - -## Python installers - -If you don't want to perform these operations separately and venture -into the hassle of exploring how to set up dependencies and paths, we -recommend two widely used distrubutions which set up all relevant -dependencies for Python, namely - -* [Anaconda](https://docs.anaconda.com/), - -which is an open source -distribution of the Python and R programming languages for large-scale -data processing, predictive analytics, and scientific computing, that -aims to simplify package management and deployment. Package versions -are managed by the package management system **conda**. - -* [Enthought canopy](https://www.enthought.com/product/canopy/) - -is a Python -distribution for scientific and analytic computing distribution and -analysis environment, available for free and under a commercial -license. - -Furthermore, [Google's Colab](https://colab.research.google.com/notebooks/welcome.ipynb) is a free Jupyter notebook environment that requires -no setup and runs entirely in the cloud. Try it out! - - -## Useful Python libraries -Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there) - -* [NumPy](https://www.numpy.org/) is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays - -* [The pandas](https://pandas.pydata.org/) library provides high-performance, easy-to-use data structures and data analysis tools - -* [Xarray](http://xarray.pydata.org/en/stable/) is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun! - -* [Scipy](https://www.scipy.org/) (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering. - -* [Matplotlib](https://matplotlib.org/) is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms. - -* [Autograd](https://github.com/HIPS/autograd) can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives - -* [SymPy](https://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. - -* [scikit-learn](https://scikit-learn.org/stable/) has simple and efficient tools for machine learning, data mining and data analysis - -* [TensorFlow](https://www.tensorflow.org/) is a Python library for fast numerical computing created and released by Google - -* [Keras](https://keras.io/) is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano - -* And many more such as [pytorch](https://pytorch.org/), [Theano](https://pypi.org/project/Theano/) etc - -## Installing R, C++, cython or Julia - -You will also find it convenient to utilize **R**. We will mainly -use Python during our lectures and in various projects and exercises. -Those of you -already familiar with **R** should feel free to continue using **R**, keeping -however an eye on the parallel Python set ups. Similarly, if you are a -Python afecionado, feel free to explore **R** as well. Jupyter/Ipython -notebook allows you to run **R** codes interactively in your -browser. The software library **R** is really tailored for statistical data analysis -and allows for an easy usage of the tools and algorithms we will discuss in these -lectures. - -To install **R** with Jupyter notebook -[follow the link here](https://mpacer.org/maths/r-kernel-for-ipython-notebook) - - - - -## Installing R, C++, cython, Numba etc - - -For the C++ aficionados, Jupyter/IPython notebook allows you also to -install C++ and run codes written in this language interactively in -the browser. Since we will emphasize writing many of the algorithms -yourself, you can thus opt for either Python or C++ (or Fortran or other compiled languages) as programming -languages. - -To add more entropy, **cython** can also be used when running your -notebooks. It means that Python with the jupyter notebook -setup allows you to integrate widely popular softwares and tools for -scientific computing. Similarly, the -[Numba Python package](https://numba.pydata.org/) delivers increased performance -capabilities with minimal rewrites of your codes. With its -versatility, including symbolic operations, Python offers a unique -computational environment. Your jupyter notebook can easily be -converted into a nicely rendered **PDF** file or a Latex file for -further processing. For example, convert to latex as - - pycod jupyter nbconvert filename.ipynb --to latex - - -And to add more versatility, the Python package [SymPy](http://www.sympy.org/en/index.html) is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) and is entirely written in Python. - -Finally, if you wish to use the light mark-up language -[doconce](https://github.com/hplgit/doconce) you can convert a standard ascii text file into various HTML -formats, ipython notebooks, latex files, pdf files etc with minimal edits. These lectures were generated using **doconce**. - - - -## Numpy examples and Important Matrix and vector handling packages - -There are several central software libraries for linear algebra and eigenvalue problems. Several of the more -popular ones have been wrapped into ofter software packages like those from the widely used text **Numerical Recipes**. The original source codes in many of the available packages are often taken from the widely used -software package LAPACK, which follows two other popular packages -developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here. - - * LINPACK: package for linear equations and least square problems. - - * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available. - - * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from . - -## Basic Matrix Features - -Matrix properties reminder - -$$ -\mathbf{A} = - \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix}\qquad -\mathbf{I} = - \begin{bmatrix} 1 & 0 & 0 & 0 \\ - 0 & 1 & 0 & 0 \\ - 0 & 0 & 1 & 0 \\ - 0 & 0 & 0 & 1 - \end{bmatrix} -$$ - -The inverse of a matrix is defined by - -$$ -\mathbf{A}^{-1} \cdot \mathbf{A} = I -$$ - - - - - - - - - - - - -
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \left (A^{T} \right )^{-1}$ real orthogonal $\sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \left (A^{\dagger} \right )^{-1}$ unitary $\sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij}$
- - -### Some famous Matrices - - * Diagonal if $a_{ij}=0$ for $i\ne j$ - - * Upper triangular if $a_{ij}=0$ for $i > j$ - - * Lower triangular if $a_{ij}=0$ for $i < j$ - - * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$ - - * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$ - - * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$ - - * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$ - - * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$ - - * Banded, block upper triangular, block lower triangular.... - -### More Basic Matrix Features - -Some Equivalent Statements -For an $N\times N$ matrix $\mathbf{A}$ the following properties are all equivalent - - * If the inverse of $\mathbf{A}$ exists, $\mathbf{A}$ is nonsingular. - - * The equation $\mathbf{Ax}=0$ implies $\mathbf{x}=0$. - - * The rows of $\mathbf{A}$ form a basis of $R^N$. - - * The columns of $\mathbf{A}$ form a basis of $R^N$. - - * $\mathbf{A}$ is a product of elementary matrices. - - * $0$ is not eigenvalue of $\mathbf{A}$. - -## Numpy and arrays -[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as - -import numpy as np - -Here follows a simple example where we set up an array of ten elements, all determined by random numbers drawn according to the normal distribution, - -n = 10 -x = np.random.normal(size=n) -print(x) - -We defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$. -Another alternative is to declare a vector as follows - -import numpy as np -x = np.array([1, 2, 3]) -print(x) - -Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++ -start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as - -import numpy as np -x = np.log(np.array([4, 7, 8])) -print(x) - -In the last example we used Numpy's unary function $np.log$. This function is -highly tuned to compute array elements since the code is vectorized -and does not require looping. We normaly recommend that you use the -Numpy intrinsic functions instead of the corresponding **log** function -from Python's **math** module. The looping is done explicitely by the -**np.log** function. The alternative, and slower way to compute the -logarithms of a vector would be to write - -import numpy as np -from math import log -x = np.array([4, 7, 8]) -for i in range(0, len(x)): - x[i] = log(x[i]) -print(x) - -We note that our code is much longer already and we need to import the **log** function from the **math** module. -The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automagically our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as - -import numpy as np -x = np.log(np.array([4, 7, 8], dtype = np.float64)) -print(x) - -or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is - -import numpy as np -x = np.log(np.array([4.0, 7.0, 8.0]) -print(x) - -To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as - -import numpy as np -x = np.log(np.array([4.0, 7.0, 8.0]) -print(x.itemsize) - -## Matrices in Python - -Having defined vectors, we are now ready to try out matrices. We can -define a $3 \times 3 $ real matrix $\hat{A}$ as (recall that we user -lowercase letters for vectors and uppercase letters for matrices) - -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -print(A) - -If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as - -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -# print the first column, row-major order and elements start with 0 -print(A[:,0]) - -We can continue this was by printing out other columns or rows. The example here prints out the second column - -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -# print the first column, row-major order and elements start with 0 -print(A[1,:]) - -Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero - -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to zero -A = np.zeros( (n, n) ) -print(A) - -or initializing all elements to - -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to one -A = np.ones( (n, n) ) -print(A) - -or as unitarily distributed random numbers (see the material on random number generators in the statistics part) - -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1] -A = np.random.rand(n, n) -print(A) - -As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. -As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors -$\hat{x}, \hat{y}, \hat{z}$ with $n$ elements each. The covariance matrix is defined as - -$$ -\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ - \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ - \sigma_{zx} & \sigma_{zy} & \sigma_{zz} - \end{bmatrix}, -$$ - -where for example - -$$ -\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -$$ - -The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. -The following simple function uses the **np.vstack** function which takes each vector of dimension $1\times n$ and produces a $3\times n$ matrix $\hat{W}$ - -$$ -\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ - x_1 & y_1 & z_1 \\ - x_2 & y_2 & z_2 \\ - \dots & \dots & \dots \\ - x_{n-2} & y_{n-2} & z_{n-2} \\ - x_{n-1} & y_{n-1} & z_{n-1} - \end{bmatrix}, -$$ - -which in turn is converted into into the $3\times 3$ covariance matrix -$\hat{\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate -the mean value of each set of samples $\hat{x}$ etc using the Numpy -function **np.mean(x)**. We can also extract the eigenvalues of the -covariance matrix through the **np.linalg.eig()** function. - -# Importing various packages -import numpy as np - -n = 100 -x = np.random.normal(size=n) -print(np.mean(x)) -y = 4+3*x+np.random.normal(size=n) -print(np.mean(y)) -z = x**3+np.random.normal(size=n) -print(np.mean(z)) -W = np.vstack((x, y, z)) -Sigma = np.cov(W) -print(Sigma) -Eigvals, Eigvecs = np.linalg.eig(Sigma) -print(Eigvals) - -%matplotlib inline - -import numpy as np -import matplotlib.pyplot as plt -from scipy import sparse -eye = np.eye(4) -print(eye) -sparse_mtx = sparse.csr_matrix(eye) -print(sparse_mtx) -x = np.linspace(-10,10,100) -y = np.sin(x) -plt.plot(x,y,marker='x') -plt.show() - -## Meet the Pandas - - - - -Another useful Python package is -[pandas](https://pandas.pydata.org/), which is an open source library -providing high-performance, easy-to-use data structures and data -analysis tools for Python. **pandas** stands for panel data, a term borrowed from econometrics and is an efficient library for data analysis with an emphasis on tabular data. -**pandas** has two major classes, the **DataFrame** class with two-dimensional data objects and tabular data organized in columns and the class **Series** with a focus on one-dimensional data objects. Both classes allow you to index data easily as we will see in the examples below. -**pandas** allows you also to perform mathematical operations on the data, spanning from simple reshapings of vectors and matrices to statistical operations. - -The following simple example shows how we can, in an easy way make tables of our data. Here we define a data set which includes names, place of birth and date of birth, and displays the data in an easy to read way. We will see repeated use of **pandas**, in particular in connection with classification of data. - -import pandas as pd -from IPython.display import display -data = {'First Name': ["Frodo", "Bilbo", "Aragorn II", "Samwise"], - 'Last Name': ["Baggins", "Baggins","Elessar","Gamgee"], - 'Place of birth': ["Shire", "Shire", "Eriador", "Shire"], - 'Date of Birth T.A.': [2968, 2890, 2931, 2980] - } -data_pandas = pd.DataFrame(data) -display(data_pandas) - -In the above we have imported **pandas** with the shorthand **pd**, the latter has become the standard way we import **pandas**. We make then a list of various variables -and reorganize the aboves lists into a **DataFrame** and then print out a neat table with specific column labels as *Name*, *place of birth* and *date of birth*. -Displaying these results, we see that the indices are given by the default numbers from zero to three. -**pandas** is extremely flexible and we can easily change the above indices by defining a new type of indexing as - -data_pandas = pd.DataFrame(data,index=['Frodo','Bilbo','Aragorn','Sam']) -display(data_pandas) - -Thereafter we display the content of the row which begins with the index **Aragorn** - -display(data_pandas.loc['Aragorn']) - -We can easily append data to this, for example - -new_hobbit = {'First Name': ["Peregrin"], - 'Last Name': ["Took"], - 'Place of birth': ["Shire"], - 'Date of Birth T.A.': [2990] - } -data_pandas=data_pandas.append(pd.DataFrame(new_hobbit, index=['Pippin'])) -display(data_pandas) - -Here are other examples where we use the **DataFrame** functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix -of dimensionality $10\times 5$ and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations. - -import numpy as np -import pandas as pd -from IPython.display import display -np.random.seed(100) -# setting up a 10 x 5 matrix -rows = 10 -cols = 5 -a = np.random.randn(rows,cols) -df = pd.DataFrame(a) -display(df) -print(df.mean()) -print(df.std()) -display(df**2) - -Thereafter we can select specific columns only and plot final results - -df.columns = ['First', 'Second', 'Third', 'Fourth', 'Fifth'] -df.index = np.arange(10) - -display(df) -print(df['Second'].mean() ) -print(df.info()) -print(df.describe()) - -from pylab import plt, mpl -plt.style.use('seaborn') -mpl.rcParams['font.family'] = 'serif' - -df.cumsum().plot(lw=2.0, figsize=(10,6)) -plt.show() - - -df.plot.bar(figsize=(10,6), rot=15) -plt.show() - -We can produce a $4\times 4$ matrix - -b = np.arange(16).reshape((4,4)) -print(b) -df1 = pd.DataFrame(b) -print(df1) - -and many other operations. - -The **Series** class is another important class included in -**pandas**. You can view it as a specialization of **DataFrame** but where -we have just a single column of data. It shares many of the same features as _DataFrame. As with **DataFrame**, -most operations are vectorized, achieving thereby a high performance when dealing with computations of arrays, in particular labeled arrays. -As we will see below it leads also to a very concice code close to the mathematical operations we may be interested in. -For multidimensional arrays, we recommend strongly [xarray](http://xarray.pydata.org/en/stable/). **xarray** has much of the same flexibility as **pandas**, but allows for the extension to higher dimensions than two. We will see examples later of the usage of both **pandas** and **xarray**. - - - - - - -In order to study various Machine Learning algorithms, we need to -access data. Acccessing data is an essential step in all machine -learning algorithms. In particular, setting up the so-called **design -matrix** (to be defined below) is often the first element we need in -order to perform our calculations. To set up the design matrix means -reading (and later, when the calculations are done, writing) data -in various formats, The formats span from reading files from disk, -loading data from databases and interacting with online sources -like web application programming interfaces (APIs). - -In handling various input formats, as discussed above, we will mainly stay with **pandas**, -a Python package which allows us, in a seamless and painless way, to -deal with a multitude of formats, from standard **csv** (comma separated -values) files, via **excel**, **html** to **hdf5** formats. With **pandas** -and the **DataFrame** and **Series** functionalities we are able to convert text data -into the calculational formats we need for a specific algorithm. And our code is going to be -pretty close the basic mathematical expressions. - -Our first data set is going to be a classic from nuclear physics, namely all -available data on binding energies. Don't be intimidated if you are not familiar with nuclear physics. It serves simply as an example here of a data set. - -We will show some of the -strengths of packages like **Scikit-Learn** in fitting nuclear binding energies to -specific functions using linear regression first. Then, as a teaser, we will show you how -you can easily implement other algorithms like decision trees and random forests and neural networks. - -But before we really start with nuclear physics data, let's just look at some simpler polynomial fitting cases, such as, -(don't be offended) fitting straight lines! - - - - -## Simple linear regression model using **scikit-learn** - -We start with perhaps our simplest possible example, using **Scikit-Learn** to perform linear regression analysis on a data set produced by us. - -What follows is a simple Python code where we have defined a function -$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. -The numbers in the vector $\hat{x}$ are given -by random numbers generated with a uniform distribution with entries -$x_i \in [0,1]$ (more about probability distribution functions -later). These values are then used to define a function $y(x)$ -(tabulated again as a vector) with a linear dependence on $x$ plus a -random noise added via the normal distribution. - - -The Numpy functions are imported used the **import numpy as np** -statement and the random number generator for the uniform distribution -is called using the function **np.random.rand()**, where we specificy -that we want $100$ random variables. Using Numpy we define -automatically an array with the specified number of elements, $100$ in -our case. With the Numpy function **randn()** we can compute random -numbers with the normal distribution (mean value $\mu$ equal to zero and -variance $\sigma^2$ set to one) and produce the values of $y$ assuming a linear -dependence as function of $x$ - -$$ -y = 2x+N(0,1), -$$ - -where $N(0,1)$ represents random numbers generated by the normal -distribution. From **Scikit-Learn** we import then the -**LinearRegression** functionality and make a prediction $\tilde{y} = -\alpha + \beta x$ using the function **fit(x,y)**. We call the set of -data $(\hat{x},\hat{y})$ for our training data. The Python package -**scikit-learn** has also a functionality which extracts the above -fitting parameters $\alpha$ and $\beta$ (see below). Later we will -distinguish between training data and test data. - -For plotting we use the Python package -[matplotlib](https://matplotlib.org/) which produces publication -quality figures. Feel free to explore the extensive -[gallery](https://matplotlib.org/gallery/index.html) of examples. In -this example we plot our original values of $x$ and $y$ as well as the -prediction **ypredict** ($\tilde{y}$), which attempts at fitting our -data with a straight line. - -The Python code follows here. - -# Importing various packages -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression - -x = np.random.rand(100,1) -y = 2*x+np.random.randn(100,1) -linreg = LinearRegression() -linreg.fit(x,y) -xnew = np.array([[0],[1]]) -ypredict = linreg.predict(xnew) - -plt.plot(xnew, ypredict, "r-") -plt.plot(x, y ,'ro') -plt.axis([0,1.0,0, 5.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Simple Linear Regression') -plt.show() - -This example serves several aims. It allows us to demonstrate several -aspects of data analysis and later machine learning algorithms. The -immediate visualization shows that our linear fit is not -impressive. It goes through the data points, but there are many -outliers which are not reproduced by our linear regression. We could -now play around with this small program and change for example the -factor in front of $x$ and the normal distribution. Try to change the -function $y$ to - -$$ -y = 10x+0.01 \times N(0,1), -$$ - -where $x$ is defined as before. Does the fit look better? Indeed, by -reducing the role of the noise given by the normal distribution we see immediately that -our linear prediction seemingly reproduces better the training -set. However, this testing 'by the eye' is obviouly not satisfactory in the -long run. Here we have only defined the training data and our model, and -have not discussed a more rigorous approach to the **cost** function. - -We need more rigorous criteria in defining whether we have succeeded or -not in modeling our training data. You will be surprised to see that -many scientists seldomly venture beyond this 'by the eye' approach. A -standard approach for the *cost* function is the so-called $\chi^2$ -function (a variant of the mean-squared error (MSE)) - -$$ -\chi^2 = \frac{1}{n} -\sum_{i=0}^{n-1}\frac{(y_i-\tilde{y}_i)^2}{\sigma_i^2}, -$$ - -where $\sigma_i^2$ is the variance (to be defined later) of the entry -$y_i$. We may not know the explicit value of $\sigma_i^2$, it serves -however the aim of scaling the equations and make the cost function -dimensionless. - -Minimizing the cost function is a central aspect of -our discussions to come. Finding its minima as function of the model -parameters ($\alpha$ and $\beta$ in our case) will be a recurring -theme in these series of lectures. Essentially all machine learning -algorithms we will discuss center around the minimization of the -chosen cost function. This depends in turn on our specific -model for describing the data, a typical situation in supervised -learning. Automatizing the search for the minima of the cost function is a -central ingredient in all algorithms. Typical methods which are -employed are various variants of **gradient** methods. These will be -discussed in more detail later. Again, you'll be surprised to hear that -many practitioners minimize the above function ''by the eye', popularly dubbed as -'chi by the eye'. That is, change a parameter and see (visually and numerically) that -the $\chi^2$ function becomes smaller. - -There are many ways to define the cost function. A simpler approach is to look at the relative difference between the training data and the predicted data, that is we define -the relative error (why would we prefer the MSE instead of the relative error?) as - -$$ -\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}. -$$ - -The squared cost function results in an arithmetic mean-unbiased -estimator, and the absolute-value cost function results in a -median-unbiased estimator (in the one-dimensional case, and a -geometric median-unbiased estimator for the multi-dimensional -case). The squared cost function has the disadvantage that it has the tendency -to be dominated by outliers. - -We can modify easily the above Python code and plot the relative error instead - -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression - -x = np.random.rand(100,1) -y = 5*x+0.01*np.random.randn(100,1) -linreg = LinearRegression() -linreg.fit(x,y) -ypredict = linreg.predict(x) - -plt.plot(x, np.abs(ypredict-y)/abs(y), "ro") -plt.axis([0,1.0,0.0, 0.5]) -plt.xlabel(r'$x$') -plt.ylabel(r'$\epsilon_{\mathrm{relative}}$') -plt.title(r'Relative error') -plt.show() - -Depending on the parameter in front of the normal distribution, we may -have a small or larger relative error. Try to play around with -different training data sets and study (graphically) the value of the -relative error. - -As mentioned above, **Scikit-Learn** has an impressive functionality. -We can for example extract the values of $\alpha$ and $\beta$ and -their error estimates, or the variance and standard deviation and many -other properties from the statistical data analysis. - -Here we show an -example of the functionality of **Scikit-Learn**. - -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression -from sklearn.metrics import mean_squared_error, r2_score, mean_squared_log_error, mean_absolute_error - -x = np.random.rand(100,1) -y = 2.0+ 5*x+0.5*np.random.randn(100,1) -linreg = LinearRegression() -linreg.fit(x,y) -ypredict = linreg.predict(x) -print('The intercept alpha: \n', linreg.intercept_) -print('Coefficient beta : \n', linreg.coef_) -# The mean squared error -print("Mean squared error: %.2f" % mean_squared_error(y, ypredict)) -# Explained variance score: 1 is perfect prediction -print('Variance score: %.2f' % r2_score(y, ypredict)) -# Mean squared log error -print('Mean squared log error: %.2f' % mean_squared_log_error(y, ypredict) ) -# Mean absolute error -print('Mean absolute error: %.2f' % mean_absolute_error(y, ypredict)) -plt.plot(x, ypredict, "r-") -plt.plot(x, y ,'ro') -plt.axis([0.0,1.0,1.5, 7.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Linear Regression fit ') -plt.show() - -The function **coef** gives us the parameter $\beta$ of our fit while **intercept** yields -$\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $alpha =2$ and $\beta =5$. Try to play around with different parameters in front of the normal distribution. The function **meansquarederror** gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as - -$$ -MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n} -\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, -$$ - -The smaller the value, the better the fit. Ideally we would like to -have an MSE equal zero. The attentive reader has probably recognized -this function as being similar to the $\chi^2$ function defined above. - -The **r2score** function computes $R^2$, the coefficient of -determination. It provides a measure of how well future samples are -likely to be predicted by the model. Best possible score is 1.0 and it -can be negative (because the model can be arbitrarily worse). A -constant model that always predicts the expected value of $\hat{y}$, -disregarding the input features, would get a $R^2$ score of $0.0$. - -If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as - -$$ -R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2}, -$$ - -where we have defined the mean value of $\hat{y}$ as - -$$ -\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i. -$$ - -Another quantity taht we will meet again in our discussions of regression analysis is - the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the $l1$-norm loss. In our discussion above we presented the relative error. -The MAE is defined as follows - -$$ -\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|. -$$ - -We present the -squared logarithmic (quadratic) error - -$$ -\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2, -$$ - -where $\log_e (x)$ stands for the natural logarithm of $x$. This error -estimate is best to use when targets having exponential growth, such -as population counts, average sales of a commodity over a span of -years etc. - - -Finally, another cost function is the Huber cost function used in robust regression. - -The rationale behind this possible cost function is its reduced -sensitivity to outliers in the data set. In our discussions on -dimensionality reduction and normalization of data we will meet other -ways of dealing with outliers. - -The Huber cost function is defined as - -$$ -H_{\delta}(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\\delta (|a|-{\frac {1}{2}}\delta ),&{\text{otherwise.}}\end{cases}}}. -$$ - -Here $a=\boldsymbol{y} - \boldsymbol{\tilde{y}}$. -We will discuss in more -detail these and other functions in the various lectures. We conclude this part with another example. Instead of -a linear $x$-dependence we study now a cubic polynomial and use the polynomial regression analysis tools of scikit-learn. - -import matplotlib.pyplot as plt -import numpy as np -import random -from sklearn.linear_model import Ridge -from sklearn.preprocessing import PolynomialFeatures -from sklearn.pipeline import make_pipeline -from sklearn.linear_model import LinearRegression - -x=np.linspace(0.02,0.98,200) -noise = np.asarray(random.sample((range(200)),200)) -y=x**3*noise -yn=x**3*100 -poly3 = PolynomialFeatures(degree=3) -X = poly3.fit_transform(x[:,np.newaxis]) -clf3 = LinearRegression() -clf3.fit(X,y) - -Xplot=poly3.fit_transform(x[:,np.newaxis]) -poly3_plot=plt.plot(x, clf3.predict(Xplot), label='Cubic Fit') -plt.plot(x,yn, color='red', label="True Cubic") -plt.scatter(x, y, label='Data', color='orange', s=15) -plt.legend() -plt.show() - -def error(a): - for i in y: - err=(y-yn)/yn - return abs(np.sum(err))/len(err) - -print (error(y)) - -Let us now dive into nuclear physics and remind ourselves briefly about some basic features about binding -energies. A basic quantity which can be measured for the ground -states of nuclei is the atomic mass $M(N, Z)$ of the neutral atom with -atomic mass number $A$ and charge $Z$. The number of neutrons is $N$. There are indeed several sophisticated experiments worldwide which allow us to measure this quantity to high precision (parts per million even). - -Atomic masses are usually tabulated in terms of the mass excess defined by - -$$ -\Delta M(N, Z) = M(N, Z) - uA, -$$ - -where $u$ is the Atomic Mass Unit - -$$ -u = M(^{12}\mathrm{C})/12 = 931.4940954(57) \hspace{0.1cm} \mathrm{MeV}/c^2. -$$ - -The nucleon masses are - -$$ -m_p = 1.00727646693(9)u, -$$ - -and - -$$ -m_n = 939.56536(8)\hspace{0.1cm} \mathrm{MeV}/c^2 = 1.0086649156(6)u. -$$ - -In the [2016 mass evaluation of by W.J.Huang, G.Audi, M.Wang, F.G.Kondev, S.Naimi and X.Xu](http://nuclearmasses.org/resources_folder/Wang_2017_Chinese_Phys_C_41_030003.pdf) -there are data on masses and decays of 3437 nuclei. - -The nuclear binding energy is defined as the energy required to break -up a given nucleus into its constituent parts of $N$ neutrons and $Z$ -protons. In terms of the atomic masses $M(N, Z)$ the binding energy is -defined by - -$$ -BE(N, Z) = ZM_H c^2 + Nm_n c^2 - M(N, Z)c^2 , -$$ - -where $M_H$ is the mass of the hydrogen atom and $m_n$ is the mass of the neutron. -In terms of the mass excess the binding energy is given by - -$$ -BE(N, Z) = Z\Delta_H c^2 + N\Delta_n c^2 -\Delta(N, Z)c^2 , -$$ - -where $\Delta_H c^2 = 7.2890$ MeV and $\Delta_n c^2 = 8.0713$ MeV. - - -A popular and physically intuitive model which can be used to parametrize -the experimental binding energies as function of $A$, is the so-called -**liquid drop model**. The ansatz is based on the following expression - -$$ -BE(N,Z) = a_1A-a_2A^{2/3}-a_3\frac{Z^2}{A^{1/3}}-a_4\frac{(N-Z)^2}{A}, -$$ - -where $A$ stands for the number of nucleons and the $a_i$s are parameters which are determined by a fit -to the experimental data. - - - - -To arrive at the above expression we have assumed that we can make the following assumptions: - - * There is a volume term $a_1A$ proportional with the number of nucleons (the energy is also an extensive quantity). When an assembly of nucleons of the same size is packed together into the smallest volume, each interior nucleon has a certain number of other nucleons in contact with it. This contribution is proportional to the volume. - - * There is a surface energy term $a_2A^{2/3}$. The assumption here is that a nucleon at the surface of a nucleus interacts with fewer other nucleons than one in the interior of the nucleus and hence its binding energy is less. This surface energy term takes that into account and is therefore negative and is proportional to the surface area. - - * There is a Coulomb energy term $a_3\frac{Z^2}{A^{1/3}}$. The electric repulsion between each pair of protons in a nucleus yields less binding. - - * There is an asymmetry term $a_4\frac{(N-Z)^2}{A}$. This term is associated with the Pauli exclusion principle and reflects the fact that the proton-neutron interaction is more attractive on the average than the neutron-neutron and proton-proton interactions. - -We could also add a so-called pairing term, which is a correction term that -arises from the tendency of proton pairs and neutron pairs to -occur. An even number of particles is more stable than an odd number. - - -### Organizing our data - -Let us start with reading and organizing our data. -We start with the compilation of masses and binding energies from 2016. -After having downloaded this file to our own computer, we are now ready to read the file and start structuring our data. - - -We start with preparing folders for storing our calculations and the data file over masses and binding energies. We import also various modules that we will find useful in order to present various Machine Learning methods. Here we focus mainly on the functionality of **scikit-learn**. - -# Common imports -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -import sklearn.linear_model as skl -from sklearn.model_selection import train_test_split -from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error -import os - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("MassEval2016.dat"),'r') - -Before we proceed, we define also a function for making our plots. You can obviously avoid this and simply set up various **matplotlib** commands every time you need them. You may however find it convenient to collect all such commands in one function and simply call this function. - -from pylab import plt, mpl -plt.style.use('seaborn') -mpl.rcParams['font.family'] = 'serif' - -def MakePlot(x,y, styles, labels, axlabels): - plt.figure(figsize=(10,6)) - for i in range(len(x)): - plt.plot(x[i], y[i], styles[i], label = labels[i]) - plt.xlabel(axlabels[0]) - plt.ylabel(axlabels[1]) - plt.legend(loc=0) - -Our next step is to read the data on experimental binding energies and -reorganize them as functions of the mass number $A$, the number of -protons $Z$ and neutrons $N$ using **pandas**. Before we do this it is -always useful (unless you have a binary file or other types of compressed -data) to actually open the file and simply take a look at it! - - -In particular, the program that outputs the final nuclear masses is written in Fortran with a specific format. It means that we need to figure out the format and which columns contain the data we are interested in. Pandas comes with a function that reads formatted output. After having admired the file, we are now ready to start massaging it with **pandas**. The file begins with some basic format information. - -""" -This is taken from the data file of the mass 2016 evaluation. -All files are 3436 lines long with 124 character per line. - Headers are 39 lines long. - col 1 : Fortran character control: 1 = page feed 0 = line feed - format : a1,i3,i5,i5,i5,1x,a3,a4,1x,f13.5,f11.5,f11.3,f9.3,1x,a2,f11.3,f9.3,1x,i3,1x,f12.5,f11.5 - These formats are reflected in the pandas widths variable below, see the statement - widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), - Pandas has also a variable header, with length 39 in this case. -""" - -The data we are interested in are in columns 2, 3, 4 and 11, giving us -the number of neutrons, protons, mass numbers and binding energies, -respectively. We add also for the sake of completeness the element name. The data are in fixed-width formatted lines and we will -covert them into the **pandas** DataFrame structure. - -# Read the experimental data with Pandas -Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11), - names=('N', 'Z', 'A', 'Element', 'Ebinding'), - widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), - header=39, - index_col=False) - -# Extrapolated values are indicated by '#' in place of the decimal place, so -# the Ebinding column won't be numeric. Coerce to float and drop these entries. -Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce') -Masses = Masses.dropna() -# Convert from keV to MeV. -Masses['Ebinding'] /= 1000 - -# Group the DataFrame by nucleon number, A. -Masses = Masses.groupby('A') -# Find the rows of the grouped DataFrame with the maximum binding energy. -Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()]) - -We have now read in the data, grouped them according to the variables we are interested in. -We see how easy it is to reorganize the data using **pandas**. If we -were to do these operations in C/C++ or Fortran, we would have had to -write various functions/subroutines which perform the above -reorganizations for us. Having reorganized the data, we can now start -to make some simple fits using both the functionalities in **numpy** and -**Scikit-Learn** afterwards. - -Now we define five variables which contain -the number of nucleons $A$, the number of protons $Z$ and the number of neutrons $N$, the element name and finally the energies themselves. - -A = Masses['A'] -Z = Masses['Z'] -N = Masses['N'] -Element = Masses['Element'] -Energies = Masses['Ebinding'] -print(Masses) - -The next step, and we will define this mathematically later, is to set up the so-called **design matrix**. We will throughout call this matrix $\boldsymbol{X}$. -It has dimensionality $p\times n$, where $n$ is the number of data points and $p$ are the so-called predictors. In our case here they are given by the number of polynomials in $A$ we wish to include in the fit. - -# Now we set up the design matrix X -X = np.zeros((len(A),5)) -X[:,0] = 1 -X[:,1] = A -X[:,2] = A**(2.0/3.0) -X[:,3] = A**(-1.0/3.0) -X[:,4] = A**(-1.0) - -With **scikitlearn** we are now ready to use linear regression and fit our data. - -clf = skl.LinearRegression().fit(X, Energies) -fity = clf.predict(X) - -Pretty simple! -Now we can print measures of how our fit is doing, the coefficients from the fits and plot the final fit together with our data. - -# The mean squared error -print("Mean squared error: %.2f" % mean_squared_error(Energies, fity)) -# Explained variance score: 1 is perfect prediction -print('Variance score: %.2f' % r2_score(Energies, fity)) -# Mean absolute error -print('Mean absolute error: %.2f' % mean_absolute_error(Energies, fity)) -print(clf.coef_, clf.intercept_) - -Masses['Eapprox'] = fity -# Generate a plot comparing the experimental with the fitted values values. -fig, ax = plt.subplots() -ax.set_xlabel(r'$A = N + Z$') -ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$') -ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2, - label='Ame2016') -ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m', - label='Fit') -ax.legend() -save_fig("Masses2016") -plt.show() - -As a teaser, let us now see how we can do this with decision trees using **scikit-learn**. Later we will switch to so-called **random forests**! - - -#Decision Tree Regression -from sklearn.tree import DecisionTreeRegressor -regr_1=DecisionTreeRegressor(max_depth=5) -regr_2=DecisionTreeRegressor(max_depth=7) -regr_3=DecisionTreeRegressor(max_depth=9) -regr_1.fit(X, Energies) -regr_2.fit(X, Energies) -regr_3.fit(X, Energies) - - -y_1 = regr_1.predict(X) -y_2 = regr_2.predict(X) -y_3=regr_3.predict(X) -Masses['Eapprox'] = y_3 -# Plot the results -plt.figure() -plt.plot(A, Energies, color="blue", label="Data", linewidth=2) -plt.plot(A, y_1, color="red", label="max_depth=5", linewidth=2) -plt.plot(A, y_2, color="green", label="max_depth=7", linewidth=2) -plt.plot(A, y_3, color="m", label="max_depth=9", linewidth=2) - -plt.xlabel("$A$") -plt.ylabel("$E$[MeV]") -plt.title("Decision Tree Regression") -plt.legend() -save_fig("Masses2016Trees") -plt.show() -print(Masses) -print(np.mean( (Energies-y_1)**2)) - -The **seaborn** package allows us to visualize data in an efficient way. Note that we use **scikit-learn**'s multi-layer perceptron (or feed forward neural network) -functionality. - -from sklearn.neural_network import MLPRegressor -from sklearn.metrics import accuracy_score -import seaborn as sns - -X_train = X -Y_train = Energies -n_hidden_neurons = 100 -epochs = 100 -# store models for later use -eta_vals = np.logspace(-5, 1, 7) -lmbd_vals = np.logspace(-5, 1, 7) -# store the models for later use -DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) -train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) -sns.set() -for i, eta in enumerate(eta_vals): - for j, lmbd in enumerate(lmbd_vals): - dnn = MLPRegressor(hidden_layer_sizes=(n_hidden_neurons), activation='logistic', - alpha=lmbd, learning_rate_init=eta, max_iter=epochs) - dnn.fit(X_train, Y_train) - DNN_scikit[i][j] = dnn - train_accuracy[i][j] = dnn.score(X_train, Y_train) - -fig, ax = plt.subplots(figsize = (10, 10)) -sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") -ax.set_title("Training Accuracy") -ax.set_ylabel("$\eta$") -ax.set_xlabel("$\lambda$") -plt.show() - -## Linear Regression, basic elements - - -[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureAug27.mp4?vrtx=view-as-webpage). - - -Fitting a continuous function with linear parameterization in terms of the parameters $\boldsymbol{\beta}$. -* Method of choice for fitting a continuous function! - -* Gives an excellent introduction to central Machine Learning features with **understandable pedagogical** links to other methods like **Neural Networks**, **Support Vector Machines** etc - -* Analytical expression for the fitting parameters $\boldsymbol{\beta}$ - -* Analytical expressions for statistical propertiers like mean values, variances, confidence intervals and more - -* Analytical relation with probabilistic interpretations - -* Easy to introduce basic concepts like bias-variance tradeoff, cross-validation, resampling and regularization techniques and many other ML topics - -* Easy to code! And links well with classification problems and logistic regression and neural networks - -* Allows for **easy** hands-on understanding of gradient descent methods - -* and many more features - -For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended. -Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended. - - - -Regression modeling deals with the description of the sampling distribution of a given random variable $y$ and how it varies as function of another variable or a set of such variables $\boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T$. -The first variable is called the **dependent**, the **outcome** or the **response** variable while the set of variables $\boldsymbol{x}$ is called the independent variable, or the predictor variable or the explanatory variable. - -A regression model aims at finding a likelihood function $p(\boldsymbol{y}\vert \boldsymbol{x})$, that is the conditional distribution for $\boldsymbol{y}$ with a given $\boldsymbol{x}$. The estimation of $p(\boldsymbol{y}\vert \boldsymbol{x})$ is made using a data set with -* $n$ cases $i = 0, 1, 2, \dots, n-1$ - -* Response (target, dependent or outcome) variable $y_i$ with $i = 0, 1, 2, \dots, n-1$ - -* $p$ so-called explanatory (independent or predictor) variables $\boldsymbol{x}_i=[x_{i0}, x_{i1}, \dots, x_{ip-1}]$ with $i = 0, 1, 2, \dots, n-1$ and explanatory variables running from $0$ to $p-1$. See below for more explicit examples. - - The goal of the regression analysis is to extract/exploit relationship between $\boldsymbol{y}$ and $\boldsymbol{x}$ in or to infer causal dependencies, approximations to the likelihood functions, functional relationships and to make predictions, making fits and many other things. - - -Consider an experiment in which $p$ characteristics of $n$ samples are -measured. The data from this experiment, for various explanatory variables $p$ are normally represented by a matrix -$\mathbf{X}$. - -The matrix $\mathbf{X}$ is called the *design -matrix*. Additional information of the samples is available in the -form of $\boldsymbol{y}$ (also as above). The variable $\boldsymbol{y}$ is -generally referred to as the *response variable*. The aim of -regression analysis is to explain $\boldsymbol{y}$ in terms of -$\boldsymbol{X}$ through a functional relationship like $y_i = -f(\mathbf{X}_{i,\ast})$. When no prior knowledge on the form of -$f(\cdot)$ is available, it is common to assume a linear relationship -between $\boldsymbol{X}$ and $\boldsymbol{y}$. This assumption gives rise to -the *linear regression model* where $\boldsymbol{\beta} = [\beta_0, \ldots, -\beta_{p-1}]^{T}$ are the *regression parameters*. - -Linear regression gives us a set of analytical equations for the parameters $\beta_j$. - - -In order to understand the relation among the predictors $p$, the set of data $n$ and the target (outcome, output etc) $\boldsymbol{y}$, -consider the model we discussed for describing nuclear binding energies. - -There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model. -Assuming - -$$ -BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1}, -$$ - -we have five predictors, that is the intercept, the $A$ dependent term, the $A^{2/3}$ term and the $A^{-1/3}$ and $A^{-1}$ terms. -This gives $p=0,1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a -$p\times n$ matrix $\boldsymbol{X}$. - -Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the -so-called [credit card default data from Taiwan](https://www.sciencedirect.com/science/article/pii/S0957417407006719?via%3Dihub). The data set contains data on $n=30000$ credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are $24$ such predictors or attributes leading to a design matrix of dimensionality $24 \times 30000$. This is however a classification problem and we will come back to it when we discuss Logistic Regression. - - -Before we proceed let us study a case from linear algebra where we aim at fitting a set of data $\boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}]$. We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables $\boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}]$, that is $y_i = y(x_i)$ with $i=0,1,2,\dots,n-1$. The variables $x_i$ could represent physical quantities like time, temperature, position etc. We assume that $y(x)$ is a smooth function. - -Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of $y$ which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree $n-1$ with $n$ points, that is - -$$ -y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i, -$$ - -where $\epsilon_i$ is the error in our approximation. - - -For every set of values $y_i,x_i$ we have thus the corresponding set of equations - -$$ -\begin{align*} -y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ -y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ -y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ -\dots & \dots \\ -y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\ -\end{align*} -$$ - -Defining the vectors - -$$ -\boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T, -$$ - -and - -$$ -\boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T, -$$ - -and - -$$ -\boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T, -$$ - -and the design matrix - -$$ -\boldsymbol{X}= -\begin{bmatrix} -1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\ -1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\ -1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\ -\dots& \dots &\dots& \dots & \dots &\dots\\ -1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\ -\end{bmatrix} -$$ - -we can rewrite our equations as - -$$ -\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. -$$ - -The above design matrix is called a [Vandermonde matrix](https://en.wikipedia.org/wiki/Vandermonde_matrix). - -We are obviously not limited to the above polynomial expansions. We -could replace the various powers of $x$ with elements of Fourier -series or instead of $x_i^j$ we could have $\cos{(j x_i)}$ or $\sin{(j -x_i)}$, or time series or other orthogonal functions. For every set -of values $y_i,x_i$ we can then generalize the equations to - -$$ -\begin{align*} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ -\dots & \dots \\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ -\dots & \dots \\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align*} -$$ - -**Note that we have $p=n$ here. The matrix is symmetric. This is generally not the case!** - -We redefine in turn the matrix $\boldsymbol{X}$ as - -$$ -\boldsymbol{X}= -\begin{bmatrix} -x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\ -x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\ -x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\ -\dots& \dots &\dots& \dots & \dots &\dots\\ -x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\ -\end{bmatrix} -$$ - -and without loss of generality we rewrite again our equations as - -$$ -\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. -$$ - -The left-hand side of this equation is kwown. Our error vector $\boldsymbol{\epsilon}$ and the parameter vector $\boldsymbol{\beta}$ are our unknow quantities. How can we obtain the optimal set of $\beta_i$ values? - -We have defined the matrix $\boldsymbol{X}$ via the equations - -$$ -\begin{align*} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ -\dots & \dots \\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ -\dots & \dots \\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align*} -$$ - -As we noted above, we stayed with a system with the design matrix - $\boldsymbol{X}\in {\mathbb{R}}^{n\times n}$, that is we have $p=n$. For reasons to come later (algorithmic arguments) we will hereafter define -our matrix as $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$, with the predictors refering to the column numbers and the entries $n$ being the row elements. - -In our [introductory notes](https://compphysics.github.io/MachineLearning/doc/pub/How2ReadData/html/How2ReadData.html) we looked at the so-called [liquid drop model](https://en.wikipedia.org/wiki/Semi-empirical_mass_formula). Let us remind ourselves about what we did by looking at the code. - -We restate the parts of the code we are most interested in. - -# Common imports -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from IPython.display import display -import os - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("MassEval2016.dat"),'r') - - -# Read the experimental data with Pandas -Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11), - names=('N', 'Z', 'A', 'Element', 'Ebinding'), - widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1), - header=39, - index_col=False) - -# Extrapolated values are indicated by '#' in place of the decimal place, so -# the Ebinding column won't be numeric. Coerce to float and drop these entries. -Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce') -Masses = Masses.dropna() -# Convert from keV to MeV. -Masses['Ebinding'] /= 1000 - -# Group the DataFrame by nucleon number, A. -Masses = Masses.groupby('A') -# Find the rows of the grouped DataFrame with the maximum binding energy. -Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()]) -A = Masses['A'] -Z = Masses['Z'] -N = Masses['N'] -Element = Masses['Element'] -Energies = Masses['Ebinding'] - -# Now we set up the design matrix X -X = np.zeros((len(A),5)) -X[:,0] = 1 -X[:,1] = A -X[:,2] = A**(2.0/3.0) -X[:,3] = A**(-1.0/3.0) -X[:,4] = A**(-1.0) -# Then nice printout using pandas -DesignMatrix = pd.DataFrame(X) -DesignMatrix.index = A -DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A'] -display(DesignMatrix) - -With $\boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1}$, it means that we will hereafter write our equations for the approximation as - -$$ -\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, -$$ - -throughout these lectures. - -With the above we use the design matrix to define the approximation $\boldsymbol{\tilde{y}}$ via the unknown quantity $\boldsymbol{\beta}$ as - -$$ -\boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, -$$ - -and in order to find the optimal parameters $\beta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parameterized values $\tilde{y}_i$, namely - -$$ -C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, -$$ - -or using the matrix $\boldsymbol{X}$ and in a more compact matrix-vector notation as - -$$ -C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. -$$ - -This function is one possible way to define the so-called cost function. - - - -It is also common to define -the function $C$ as - -$$ -C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2, -$$ - -since when taking the first derivative with respect to the unknown parameters $\beta$, the factor of $2$ cancels out. - -The function - -$$ -C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}, -$$ - -can be linked to the variance of the quantity $y_i$ if we interpret the latter as the mean value. -When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret $y_i$ as a mean value - -$$ -y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, -$$ - -where $\langle y_i \rangle$ is the mean value. Keep in mind also that -till now we have treated $y_i$ as the exact value. Normally, the -response (dependent or outcome) variable $y_i$ the outcome of a -numerical experiment or another type of experiment and is thus only an -approximation to the true value. It is then always accompanied by an -error estimate, often limited to a statistical error estimate given by -the standard deviation discussed earlier. In the discussion here we -will treat $y_i$ as our exact value for the response variable. - -In order to find the parameters $\beta_i$ we will then minimize the spread of $C(\boldsymbol{\beta})$, that is we are going to solve the problem - -$$ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. -$$ - -In practical terms it means we will require - -$$ -\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0, -$$ - -which results in - -$$ -\frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0, -$$ - -or in a matrix-vector form as - -$$ -\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right). -$$ - -We can rewrite - -$$ -\frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right), -$$ - -as - -$$ -\boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}, -$$ - -and if the matrix $\boldsymbol{X}^T\boldsymbol{X}$ is invertible we have the solution - -$$ -\boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. -$$ - -We note also that since our design matrix is defined as $\boldsymbol{X}\in -{\mathbb{R}}^{n\times p}$, the product $\boldsymbol{X}^T\boldsymbol{X} \in -{\mathbb{R}}^{p\times p}$. In the above case we have that $p \ll n$, -in our case $p=5$ meaning that we end up with inverting a small -$5\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional -matrices to invert. The methods discussed here and for many other -supervised learning algorithms like classification with logistic -regression or support vector machines, exhibit dimensionalities which -allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix -$\boldsymbol{X}^T\boldsymbol{X}$. - -**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\boldsymbol{X}^T\boldsymbol{X}$? What kind of problems can we expect? - - -The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and -matrices as upper case boldfaced letters. - -4 -8 - -< -< -< -! -! -M -A -T -H -_ -B -L -O -C -K - -4 -9 - -< -< -< -! -! -M -A -T -H -_ -B -L -O -C -K - -5 -0 - -< -< -< -! -! -M -A -T -H -_ -B -L -O -C -K - -$$ -\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T. -$$ - -The residuals $\boldsymbol{\epsilon}$ are in turn given by - -$$ -\boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}, -$$ - -and with - -$$ -\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, -$$ - -we have - -$$ -\boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, -$$ - -meaning that the solution for $\boldsymbol{\beta}$ is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach. - - -Let us now return to our nuclear binding energies and simply code the above equations. - - -It is rather straightforward to implement the matrix inversion and obtain the parameters $\boldsymbol{\beta}$. After having defined the matrix $\boldsymbol{X}$ we simply need to -write - -# matrix inversion to find beta -beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies) -# and then make the prediction -ytilde = X @ beta - -Alternatively, you can use the least squares functionality in **Numpy** as - -fit = np.linalg.lstsq(X, Energies, rcond =None)[0] -ytildenp = np.dot(fit,X.T) - -And finally we plot our fit with and compare with data - -Masses['Eapprox'] = ytilde -# Generate a plot comparing the experimental with the fitted values values. -fig, ax = plt.subplots() -ax.set_xlabel(r'$A = N + Z$') -ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$') -ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2, - label='Ame2016') -ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m', - label='Fit') -ax.legend() -save_fig("Masses2016OLS") -plt.show() - -We can easily test our fit by computing the $R2$ score that we discussed in connection with the functionality of **Scikit-Learn** in the introductory slides. -Since we are not using **Scikit-Learn** here we can define our own $R2$ function as - -def R2(y_data, y_model): - return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2) - -and we would be using it as - -print(R2(Energies,ytilde)) - -We can easily add our **MSE** score as - -def MSE(y_data,y_model): - n = np.size(y_model) - return np.sum((y_data-y_model)**2)/n - -print(MSE(Energies,ytilde)) - -and finally the relative error as - -def RelativeError(y_data,y_model): - return abs((y_data-y_model)/y_data) -print(RelativeError(Energies, ytilde)) - -### The $\chi^2$ function - -Normally, the response (dependent or outcome) variable $y_i$ is the -outcome of a numerical experiment or another type of experiment and is -thus only an approximation to the true value. It is then always -accompanied by an error estimate, often limited to a statistical error -estimate given by the standard deviation discussed earlier. In the -discussion here we will treat $y_i$ as our exact value for the -response variable. - -Introducing the standard deviation $\sigma_i$ for each measurement -$y_i$, we define now the $\chi^2$ function (omitting the $1/n$ term) -as - -$$ -\chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, -$$ - -where the matrix $\boldsymbol{\Sigma}$ is a diagonal matrix with $\sigma_i$ as matrix elements. - - -In order to find the parameters $\beta_i$ we will then minimize the spread of $\chi^2(\boldsymbol{\beta})$ by requiring - -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0, -$$ - -which results in - -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0, -$$ - -or in a matrix-vector form as - -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right). -$$ - -where we have defined the matrix $\boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma}$ with matrix elements $a_{ij} = x_{ij}/\sigma_i$ and the vector $\boldsymbol{b}$ with elements $b_i = y_i/\sigma_i$. - -We can rewrite - -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right), -$$ - -as - -$$ -\boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta}, -$$ - -and if the matrix $\boldsymbol{A}^T\boldsymbol{A}$ is invertible we have the solution - -$$ -\boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}. -$$ - -If we then introduce the matrix - -$$ -\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}, -$$ - -we have then the following expression for the parameters $\beta_j$ (the matrix elements of $\boldsymbol{H}$ are $h_{ij}$) - -$$ -\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik} -$$ - -We state without proof the expression for the uncertainty in the parameters $\beta_j$ as (we leave this as an exercise) - -$$ -\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2, -$$ - -resulting in - -$$ -\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}! -$$ - -The first step here is to approximate the function $y$ with a first-order polynomial, that is we write - -$$ -y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i. -$$ - -By computing the derivatives of $\chi^2$ with respect to $\beta_0$ and $\beta_1$ show that these are given by - -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0, -$$ - -and - -$$ -\frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0. -$$ - -For a linear fit (a first-order polynomial) we don't need to invert a matrix!! -Defining - -$$ -\gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2}, -$$ - -$$ -\gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2}, -$$ - -$$ -\gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right), -$$ - -$$ -\gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2}, -$$ - -$$ -\gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2}, -$$ - -we obtain - -$$ -\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}, -$$ - -$$ -\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}. -$$ - -This approach (different linear and non-linear regression) suffers -often from both being underdetermined and overdetermined in the -unknown coefficients $\beta_i$. A better approach is to use the -Singular Value Decomposition (SVD) method discussed below. Or using -Lasso and Ridge regression. See below. - - -### Fitting an Equation of State for Dense Nuclear Matter - -Before we continue, let us introduce yet another example. We are going to fit the -nuclear equation of state using results from many-body calculations. -The equation of state we have made available here, as function of -density, has been derived using modern nucleon-nucleon potentials with -[the addition of three-body -forces](https://www.sciencedirect.com/science/article/pii/S0370157399001106). This -time the file is presented as a standard **csv** file. - -The beginning of the Python code here is similar to what you have seen -before, with the same initializations and declarations. We use also -**pandas** again, rather extensively in order to organize our data. - -The difference now is that we use **Scikit-Learn's** regression tools -instead of our own matrix inversion implementation. Furthermore, we -sneak in **Ridge** regression (to be discussed below) which includes a -hyperparameter $\lambda$, also to be explained below. - -# Common imports -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -import matplotlib.pyplot as plt -import sklearn.linear_model as skl -from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("EoS.csv"),'r') - -# Read the EoS data as csv file and organize the data into two arrays with density and energies -EoS = pd.read_csv(infile, names=('Density', 'Energy')) -EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') -EoS = EoS.dropna() -Energies = EoS['Energy'] -Density = EoS['Density'] -# The design matrix now as function of various polytrops -X = np.zeros((len(Density),4)) -X[:,3] = Density**(4.0/3.0) -X[:,2] = Density -X[:,1] = Density**(2.0/3.0) -X[:,0] = 1 - -# We use now Scikit-Learn's linear regressor and ridge regressor -# OLS part -clf = skl.LinearRegression().fit(X, Energies) -ytilde = clf.predict(X) -EoS['Eols'] = ytilde -# The mean squared error -print("Mean squared error: %.2f" % mean_squared_error(Energies, ytilde)) -# Explained variance score: 1 is perfect prediction -print('Variance score: %.2f' % r2_score(Energies, ytilde)) -# Mean absolute error -print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde)) -print(clf.coef_, clf.intercept_) - -# The Ridge regression with a hyperparameter lambda = 0.1 -_lambda = 0.1 -clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies) -yridge = clf_ridge.predict(X) -EoS['Eridge'] = yridge -# The mean squared error -print("Mean squared error: %.2f" % mean_squared_error(Energies, yridge)) -# Explained variance score: 1 is perfect prediction -print('Variance score: %.2f' % r2_score(Energies, yridge)) -# Mean absolute error -print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge)) -print(clf_ridge.coef_, clf_ridge.intercept_) - -fig, ax = plt.subplots() -ax.set_xlabel(r'$\rho[\mathrm{fm}^{-3}]$') -ax.set_ylabel(r'Energy per particle') -ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2, - label='Theoretical data') -ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m', - label='OLS') -ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g', - label='Ridge $\lambda = 0.1$') -ax.legend() -save_fig("EoSfitting") -plt.show() - -The above simple polynomial in density $\rho$ gives an excellent fit -to the data. - -We note also that there is a small deviation between the -standard OLS and the Ridge regression at higher densities. We discuss this in more detail -below. - - -## Splitting our Data in Training and Test data - -It is normal in essentially all Machine Learning studies to split the -data in a training set and a test set (sometimes also an additional -validation set). **Scikit-Learn** has an own function for this. There -is no explicit recipe for how much data should be included as training -data and say test data. An accepted rule of thumb is to use -approximately $2/3$ to $4/5$ of the data as training data. We will -postpone a discussion of this splitting to the end of these notes and -our discussion of the so-called **bias-variance** tradeoff. Here we -limit ourselves to repeat the above equation of state fitting example -but now splitting the data into a training set and a test set. - -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.model_selection import train_test_split -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -def R2(y_data, y_model): - return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2) -def MSE(y_data,y_model): - n = np.size(y_model) - return np.sum((y_data-y_model)**2)/n - -infile = open(data_path("EoS.csv"),'r') - -# Read the EoS data as csv file and organized into two arrays with density and energies -EoS = pd.read_csv(infile, names=('Density', 'Energy')) -EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') -EoS = EoS.dropna() -Energies = EoS['Energy'] -Density = EoS['Density'] -# The design matrix now as function of various polytrops -X = np.zeros((len(Density),5)) -X[:,0] = 1 -X[:,1] = Density**(2.0/3.0) -X[:,2] = Density -X[:,3] = Density**(4.0/3.0) -X[:,4] = Density**(5.0/3.0) -# We split the data in test and training data -X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2) -# matrix inversion to find beta -beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train) -# and then make the prediction -ytilde = X_train @ beta -print("Training R2") -print(R2(y_train,ytilde)) -print("Training MSE") -print(MSE(y_train,ytilde)) -ypredict = X_test @ beta -print("Test R2") -print(R2(y_test,ypredict)) -print("Test MSE") -print(MSE(y_test,ypredict)) - -## The Boston housing data example - -The Boston housing -data set was originally a part of UCI Machine Learning Repository -and has been removed now. The data set is now included in **Scikit-Learn**'s -library. There are 506 samples and 13 feature (predictor) variables -in this data set. The objective is to predict the value of prices of -the house using the features (predictors) listed here. - -The features/predictors are -1. CRIM: Per capita crime rate by town - -2. ZN: Proportion of residential land zoned for lots over 25000 square feet - -3. INDUS: Proportion of non-retail business acres per town - -4. CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise) - -5. NOX: Nitric oxide concentration (parts per 10 million) - -6. RM: Average number of rooms per dwelling - -7. AGE: Proportion of owner-occupied units built prior to 1940 - -8. DIS: Weighted distances to five Boston employment centers - -9. RAD: Index of accessibility to radial highways - -10. TAX: Full-value property tax rate per USD10000 - -11. B: $1000(Bk - 0.63)^2$, where $Bk$ is the proportion of [people of African American descent] by town - -12. LSTAT: Percentage of lower status of the population - -13. MEDV: Median value of owner-occupied homes in USD 1000s - -## Housing data, the code -We start by importing the libraries - -import numpy as np -import matplotlib.pyplot as plt - -import pandas as pd -import seaborn as sns - -and load the Boston Housing DataSet from **Scikit-Learn** - -from sklearn.datasets import load_boston - -boston_dataset = load_boston() - -# boston_dataset is a dictionary -# let's check what it contains -boston_dataset.keys() - -Then we invoke Pandas - -boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names) -boston.head() -boston['MEDV'] = boston_dataset.target - -and preprocess the data - -# check for missing values in all the columns -boston.isnull().sum() - -We can then visualize the data - -# set the size of the figure -sns.set(rc={'figure.figsize':(11.7,8.27)}) - -# plot a histogram showing the distribution of the target values -sns.distplot(boston['MEDV'], bins=30) -plt.show() - -It is now useful to look at the correlation matrix - -# compute the pair wise correlation for all columns -correlation_matrix = boston.corr().round(2) -# use the heatmap function from seaborn to plot the correlation matrix -# annot = True to print the values inside the square -sns.heatmap(data=correlation_matrix, annot=True) - -From the above coorelation plot we can see that **MEDV** is strongly correlated to **LSTAT** and **RM**. We see also that **RAD** and **TAX** are stronly correlated, but we don't include this in our features together to avoid multi-colinearity - -plt.figure(figsize=(20, 5)) - -features = ['LSTAT', 'RM'] -target = boston['MEDV'] - -for i, col in enumerate(features): - plt.subplot(1, len(features) , i+1) - x = boston[col] - y = target - plt.scatter(x, y, marker='o') - plt.title(col) - plt.xlabel(col) - plt.ylabel('MEDV') - -Now we start training our model - -X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM']) -Y = boston['MEDV'] - -We split the data into training and test sets - -from sklearn.model_selection import train_test_split - -# splits the training and test data set in 80% : 20% -# assign random_state to any value.This ensures consistency. -X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5) -print(X_train.shape) -print(X_test.shape) -print(Y_train.shape) -print(Y_test.shape) - -Then we use the linear regression functionality from **Scikit-Learn** - -from sklearn.linear_model import LinearRegression -from sklearn.metrics import mean_squared_error, r2_score - -lin_model = LinearRegression() -lin_model.fit(X_train, Y_train) - -# model evaluation for training set - -y_train_predict = lin_model.predict(X_train) -rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict))) -r2 = r2_score(Y_train, y_train_predict) - -print("The model performance for training set") -print("--------------------------------------") -print('RMSE is {}'.format(rmse)) -print('R2 score is {}'.format(r2)) -print("\n") - -# model evaluation for testing set - -y_test_predict = lin_model.predict(X_test) -# root mean square error of the model -rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict))) - -# r-squared score of the model -r2 = r2_score(Y_test, y_test_predict) - -print("The model performance for testing set") -print("--------------------------------------") -print('RMSE is {}'.format(rmse)) -print('R2 score is {}'.format(r2)) - -# plotting the y_test vs y_pred -# ideally should have been a straight line -plt.scatter(Y_test, y_test_predict) -plt.show() - -## Reducing the number of degrees of freedom, overarching view - -Many Machine Learning problems involve thousands or even millions of -features for each training instance. Not only does this make training -extremely slow, it can also make it much harder to find a good -solution, as we will see. This problem is often referred to as the -curse of dimensionality. Fortunately, in real-world problems, it is -often possible to reduce the number of features considerably, turning -an intractable problem into a tractable one. - -Later we will discuss some of the most popular dimensionality reduction -techniques: the principal component analysis (PCA), Kernel PCA, and -Locally Linear Embedding (LLE). - - -Principal component analysis and its various variants deal with the -problem of fitting a low-dimensional [affine -subspace](https://en.wikipedia.org/wiki/Affine_space) to a set of of -data points in a high-dimensional space. With its family of methods it -is one of the most used tools in data modeling, compression and -visualization. - - -Before we proceed however, we will discuss how to preprocess our -data. Till now and in connection with our previous examples we have -not met so many cases where we are too sensitive to the scaling of our -data. Normally the data may need a rescaling and/or may be sensitive -to extreme values. Scaling the data renders our inputs much more -suitable for the algorithms we want to employ. - -**Scikit-Learn** has several functions which allow us to rescale the -data, normally resulting in much better results in terms of various -accuracy scores. The **StandardScaler** function in **Scikit-Learn** -ensures that for each feature/predictor we study the mean value is -zero and the variance is one (every column in the design/feature -matrix). This scaling has the drawback that it does not ensure that -we have a particular maximum or minimum in our data set. Another -function included in **Scikit-Learn** is the **MinMaxScaler** which -ensures that all features are exactly between $0$ and $1$. The - - -The **Normalizer** scales each data -point such that the feature vector has a euclidean length of one. In other words, it -projects a data point on the circle (or sphere in the case of higher dimensions) with a -radius of 1. This means every data point is scaled by a different number (by the -inverse of it’s length). -This normalization is often used when only the direction (or angle) of the data matters, -not the length of the feature vector. - -The **RobustScaler** works similarly to the StandardScaler in that it -ensures statistical properties for each feature that guarantee that -they are on the same scale. However, the RobustScaler uses the median -and quartiles, instead of mean and variance. This makes the -RobustScaler ignore data points that are very different from the rest -(like measurement errors). These odd data points are also called -outliers, and might often lead to trouble for other scaling -techniques. - - -### Simple preprocessing examples, Franke function and regression - -# Common imports -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -import sklearn.linear_model as skl -from sklearn.metrics import mean_squared_error -from sklearn.model_selection import train_test_split -from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - - -def FrankeFunction(x,y): - term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2)) - term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1)) - term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2)) - term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2) - return term1 + term2 + term3 + term4 - - -def create_X(x, y, n ): - if len(x.shape) > 1: - x = np.ravel(x) - y = np.ravel(y) - - N = len(x) - l = int((n+1)*(n+2)/2) # Number of elements in beta - X = np.ones((N,l)) - - for i in range(1,n+1): - q = int((i)*(i+1)/2) - for k in range(i+1): - X[:,q+k] = (x**(i-k))*(y**k) - - return X - - -# Making meshgrid of datapoints and compute Franke's function -n = 5 -N = 1000 -x = np.sort(np.random.uniform(0, 1, N)) -y = np.sort(np.random.uniform(0, 1, N)) -z = FrankeFunction(x, y) -X = create_X(x, y, n=n) -# split in training and test data -X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=0.2) - - -clf = skl.LinearRegression().fit(X_train, y_train) - -# The mean squared error and R2 score -print("MSE before scaling: {:.2f}".format(mean_squared_error(clf.predict(X_test), y_test))) -print("R2 score before scaling {:.2f}".format(clf.score(X_test,y_test))) - -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) - -print("Feature min values before scaling:\n {}".format(X_train.min(axis=0))) -print("Feature max values before scaling:\n {}".format(X_train.max(axis=0))) - -print("Feature min values after scaling:\n {}".format(X_train_scaled.min(axis=0))) -print("Feature max values after scaling:\n {}".format(X_train_scaled.max(axis=0))) - -clf = skl.LinearRegression().fit(X_train_scaled, y_train) - - -print("MSE after scaling: {:.2f}".format(mean_squared_error(clf.predict(X_test_scaled), y_test))) -print("R2 score for scaled data: {:.2f}".format(clf.score(X_test_scaled,y_test))) \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb deleted file mode 100644 index 7654bacd0..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter10.ipynb +++ /dev/null @@ -1,3315 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Building a Feed Forward Neural Network\n", - "\n", - "We are now gong to develop an example based on the MNIST data\n", - "base. This is a classification problem and we need to use our\n", - "cross-entropy function we discussed in connection with logistic\n", - "regression. The cross-entropy defines our cost function for the\n", - "classificaton problems with neural networks.\n", - "\n", - "In binary classification with two classes $(0, 1)$ we define the\n", - "logistic/sigmoid function as the probability that a particular input\n", - "is in class $0$ or $1$. This is possible because the logistic\n", - "function takes any input from the real numbers and inputs a number\n", - "between 0 and 1, and can therefore be interpreted as a probability. It\n", - "also has other nice properties, such as a derivative that is simple to\n", - "calculate.\n", - "\n", - "For an input $\\boldsymbol{a}$ from the hidden layer, the probability that the input $\\boldsymbol{x}$\n", - "is in class 0 or 1 is just. We let $\\theta$ represent the unknown weights and biases to be adjusted by our equations). The variable $x$\n", - "represents our activation values $z$. We have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(y = 0 \\mid \\hat{x}, \\hat{\\theta}) = \\frac{1}{1 + \\exp{(- \\hat{x}})} ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(y = 1 \\mid \\hat{x}, \\hat{\\theta}) = 1 - P(y = 0 \\mid \\hat{x}, \\hat{\\theta}) ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $y \\in \\{0, 1\\}$ and $\\hat{\\theta}$ represents the weights and biases\n", - "of our network.\n", - "\n", - "\n", - "\n", - "## Defining the cost function\n", - "\n", - "Our cost function is given as (see the Logistic regression lectures)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\theta}) = - \\ln P(\\mathcal{D} \\mid \\hat{\\theta}) = - \\sum_{i=1}^n\n", - "y_i \\ln[P(y_i = 0)] + (1 - y_i) \\ln [1 - P(y_i = 0)] = \\sum_{i=1}^n \\mathcal{L}_i(\\hat{\\theta}) .\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This last equality means that we can interpret our *cost* function as a sum over the *loss* function\n", - "for each point in the dataset $\\mathcal{L}_i(\\hat{\\theta})$. \n", - "The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather\n", - "than maximizing a negative number. \n", - "\n", - "In *multiclass* classification it is common to treat each integer label as a so called *one-hot* vector: \n", - "\n", - "$y = 5 \\quad \\rightarrow \\quad \\hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$ and\n", - "\n", - "\n", - "$y = 1 \\quad \\rightarrow \\quad \\hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$ \n", - "\n", - "\n", - "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset (numbers from $0$ to $9$).. \n", - "\n", - "If $\\hat{x}_i$ is the $i$-th input (image), $y_{ic}$ refers to the $c$-th component of the $i$-th\n", - "output vector $\\hat{y}_i$. \n", - "The probability of $\\hat{x}_i$ being in class $c$ will be given by the softmax function:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(y_{ic} = 1 \\mid \\hat{x}_i, \\hat{\\theta}) = \\frac{\\exp{((\\hat{a}_i^{hidden})^T \\hat{w}_c)}}\n", - "{\\sum_{c'=0}^{C-1} \\exp{((\\hat{a}_i^{hidden})^T \\hat{w}_{c'})}} ,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which reduces to the logistic function in the binary case. \n", - "The likelihood of this $C$-class classifier\n", - "is now given as:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(\\mathcal{D} \\mid \\hat{\\theta}) = \\prod_{i=1}^n \\prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Again we take the negative log-likelihood to define our cost function:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\theta}) = - \\log{P(\\mathcal{D} \\mid \\hat{\\theta})}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "See the logistic regression lectures for a full definition of the cost function.\n", - "\n", - "The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!\n", - "\n", - "\n", - "### Example: binary classification problem\n", - "\n", - "As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters $\\beta$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\beta}) = - \\sum_{i=1}^n \\left(y_i\\log{p(y_i \\vert x_i,\\hat{\\beta})}+(1-y_i)\\log{1-p(y_i \\vert x_i,\\hat{\\beta})}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we had defined the logistic (sigmoid) function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(y_i =1\\vert x_i,\\hat{\\beta})=\\frac{\\exp{(\\beta_0+\\beta_1 x_i)}}{1+\\exp{(\\beta_0+\\beta_1 x_i)}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(y_i =0\\vert x_i,\\hat{\\beta})=1-p(y_i =1\\vert x_i,\\hat{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The parameters $\\hat{\\beta}$ were defined using a minimization method like gradient descent or Newton-Raphson's method. \n", - "\n", - "Now we replace $x_i$ with the activation $z_i^l$ for a given layer $l$ and the outputs as $y_i=a_i^l=f(z_i^l)$, with $z_i^l$ now being a function of the weights $w_{ij}^l$ and biases $b_i^l$. \n", - "We have then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_i^l = y_i = \\frac{\\exp{(z_i^l)}}{1+\\exp{(z_i^l)}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "z_i^l = \\sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the superscript $l-1$ indicates that these are the outputs from layer $l-1$.\n", - "Our cost function at the final layer $l=L$ is now" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{W}) = - \\sum_{i=1}^n \\left(t_i\\log{a_i^L}+(1-t_i)\\log{(1-a_i^L)}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined the targets $t_i$. The derivatives of the cost function with respect to the output $a_i^L$ are then easily calculated and we get" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\hat{W})}{\\partial a_i^L} = \\frac{a_i^L-t_i}{a_i^L(1-a_i^L)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In case we use another activation function than the logistic one, we need to evaluate other derivatives. \n", - "\n", - "\n", - "\n", - "### The Softmax function\n", - "\n", - "In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f(z_i^l)}{\\partial w_{jk}^l} =\n", - "\\frac{\\partial f(z_i^l)}{\\partial z_j^l} \\frac{\\partial z_j^l}{\\partial w_{jk}^l}= \\frac{\\partial f(z_i^l)}{\\partial z_j^l}a_k^{l-1}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For the Softmax function we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(z_i^l) = \\frac{\\exp{(z_i^l)}}{\\sum_{m=1}^K\\exp{(z_m^l)}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Its derivative with respect to $z_j^l$ gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f(z_i^l)}{\\partial z_j^l}= f(z_i^l)\\left(\\delta_{ij}-f(z_j^l)\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in case of the simply binary model reduces to having $i=j$. \n", - "\n", - "\n", - "## Developing a code for doing neural networks with back propagation\n", - "\n", - "\n", - "One can identify a set of key steps when using neural networks to solve supervised learning problems: \n", - "\n", - "1. Collect and pre-process data \n", - "\n", - "2. Define model and architecture \n", - "\n", - "3. Choose cost function and optimizer \n", - "\n", - "4. Train the model \n", - "\n", - "5. Evaluate model performance on test data \n", - "\n", - "6. Adjust hyperparameters (if necessary, network architecture)\n", - "\n", - "### Collect and pre-process data\n", - "\n", - "Here we will be using the MNIST dataset, which is readily available through the **scikit-learn**\n", - "package. You may also find it for example [here](http://yann.lecun.com/exdb/mnist/). \n", - "The *MNIST* (Modified National Institute of Standards and Technology) database is a large database\n", - "of handwritten digits that is commonly used for training various image processing systems. \n", - "The MNIST dataset consists of 70 000 images of size $28\\times 28$ pixels, each labeled from 0 to 9. \n", - "The scikit-learn dataset we will use consists of a selection of 1797 images of size $8\\times 8$ collected and processed from this database. \n", - "\n", - "To feed data into a feed-forward neural network we need to represent\n", - "the inputs as a design/feature matrix $X = (n_{inputs}, n_{features})$. Each\n", - "row represents an *input*, in this case a handwritten digit, and\n", - "each column represents a *feature*, in this case a pixel. The\n", - "correct answers, also known as *labels* or *targets* are\n", - "represented as a 1D array of integers \n", - "$Y = (n_{inputs}) = (5, 3, 1, 8,...)$.\n", - "\n", - "As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from\n", - "measurements of height (in m) \n", - "and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example: \n", - "\n", - "$$ X = \\begin{bmatrix}\n", - "1.85 & 81\\\\\n", - "1.71 & 65\\\\\n", - "1.95 & 103\\\\\n", - "1.55 & 42\\\\\n", - "1.63 & 56\n", - "\\end{bmatrix} ,$$ \n", - "\n", - "and the targets would be: \n", - "\n", - "$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$ \n", - "\n", - "Since each input image is a 2D matrix, we need to flatten the image\n", - "(i.e. \"unravel\" the 2D matrix into a 1D array) to turn the data into a\n", - "design/feature matrix. This means we lose all spatial information in the\n", - "image, such as locality and translational invariance. More complicated\n", - "architectures such as Convolutional Neural Networks can take advantage\n", - "of such information, and are most commonly applied when analyzing\n", - "images." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "inputs = (n_inputs, pixel_width, pixel_height) = (1797, 8, 8)\n", - "labels = (n_inputs) = (1797,)\n", - "X = (n_inputs, n_features) = (1797, 64)\n" - ] - }, - { - "data": { - "image/png": "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\n", - "text/plain": [ - "

" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_33_1.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "%matplotlib inline\n", - "\n", - "# import necessary packages\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn import datasets\n", - "\n", - "\n", - "# ensure the same random numbers appear every time\n", - "np.random.seed(0)\n", - "\n", - "# display images in notebook\n", - "%matplotlib inline\n", - "plt.rcParams['figure.figsize'] = (12,12)\n", - "\n", - "\n", - "# download MNIST dataset\n", - "digits = datasets.load_digits()\n", - "\n", - "# define inputs and labels\n", - "inputs = digits.images\n", - "labels = digits.target\n", - "\n", - "print(\"inputs = (n_inputs, pixel_width, pixel_height) = \" + str(inputs.shape))\n", - "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", - "\n", - "\n", - "# flatten the image\n", - "# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64\n", - "n_inputs = len(inputs)\n", - "inputs = inputs.reshape(n_inputs, -1)\n", - "print(\"X = (n_inputs, n_features) = \" + str(inputs.shape))\n", - "\n", - "\n", - "# choose some random images to display\n", - "indices = np.arange(n_inputs)\n", - "random_indices = np.random.choice(indices, size=5)\n", - "\n", - "for i, image in enumerate(digits.images[random_indices]):\n", - " plt.subplot(1, 5, i+1)\n", - " plt.axis('off')\n", - " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", - " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Train and test datasets\n", - "\n", - "Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions. \n", - "\n", - "We will reserve $80 \\%$ of our dataset for training and $20 \\%$ for testing. \n", - "\n", - "It is important that the train and test datasets are drawn randomly from our dataset, to ensure\n", - "no bias in the sampling. \n", - "Say you are taking measurements of weather data to predict the weather in the coming 5 days.\n", - "You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data\n", - "collected from 12.00 to 24.00." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Number of training images: 1437\n", - "Number of test images: 360\n" - ] - } - ], - "source": [ - "from sklearn.model_selection import train_test_split\n", - "\n", - "# one-liner from scikit-learn library\n", - "train_size = 0.8\n", - "test_size = 1 - train_size\n", - "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", - " test_size=test_size)\n", - "\n", - "# equivalently in numpy\n", - "def train_test_split_numpy(inputs, labels, train_size, test_size):\n", - " n_inputs = len(inputs)\n", - " inputs_shuffled = inputs.copy()\n", - " labels_shuffled = labels.copy()\n", - " \n", - " np.random.shuffle(inputs_shuffled)\n", - " np.random.shuffle(labels_shuffled)\n", - " \n", - " train_end = int(n_inputs*train_size)\n", - " X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]\n", - " Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]\n", - " \n", - " return X_train, X_test, Y_train, Y_test\n", - "\n", - "#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)\n", - "\n", - "print(\"Number of training images: \" + str(len(X_train)))\n", - "print(\"Number of test images: \" + str(len(X_test)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Define model and architecture\n", - "\n", - "Our simple feed-forward neural network will consist of an *input* layer, a single *hidden* layer and an *output* layer. The activation $y$ of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have \n", - "\n", - "$$ z = \\sum_{i=1}^n w_i a_i ,$$\n", - "\n", - "$$ y = f(z) ,$$\n", - "\n", - "where $f$ is the activation function, $a_i$ represents input from neuron $i$ in the preceding layer\n", - "and $w_i$ is the weight to input $i$. \n", - "The activation of the neurons in the input layer is just the features (e.g. a pixel value). \n", - "\n", - "The simplest activation function for a neuron is the *Heaviside* function:\n", - "\n", - "$$ f(z) = \n", - "\\begin{cases}\n", - "1, & z > 0\\\\\n", - "0, & \\text{otherwise}\n", - "\\end{cases}\n", - "$$\n", - "\n", - "A feed-forward neural network with this activation is known as a *perceptron*. \n", - "For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer. \n", - "This activation can be generalized to $k$ classes (using e.g. the *one-against-all* strategy), \n", - "and we call these architectures *multiclass perceptrons*. \n", - "\n", - "However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and \n", - "Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function. \n", - "\n", - "Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU). \n", - "We will be using the sigmoid function $\\sigma(x)$: \n", - "\n", - "$$ f(x) = \\sigma(x) = \\frac{1}{1 + e^{-x}} ,$$\n", - "\n", - "which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.\n", - "\n", - "### Layers\n", - "\n", - "* Input \n", - "\n", - "Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons. \n", - "\n", - "* Hidden layer\n", - "\n", - "We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer. \n", - "Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer. \n", - "\n", - "* Output\n", - "\n", - "If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,\n", - "which could output 0 or 1 according to the Heaviside function. This would be an example of a *hard* classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a *soft* classifier, which outputs the probability of being in class 0 or 1. \n", - "\n", - "For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class. \n", - "\n", - "Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons $j = 0,1,...,9$. The activation of each output neuron $j$ will be according to the *softmax* function: \n", - "\n", - "$$ P(\\text{class $j$} \\mid \\text{input $\\hat{a}$}) = \\frac{\\exp{(\\hat{a}^T \\hat{w}_j)}}\n", - "{\\sum_{c=0}^{9} \\exp{(\\hat{a}^T \\hat{w}_c)}} ,$$ \n", - "\n", - "i.e. each neuron $j$ outputs the probability of being in class $j$ given an input from the hidden layer $\\hat{a}$, with $\\hat{w}_j$ the weights of neuron $j$ to the inputs. \n", - "The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1. \n", - "The exponent is just the weighted sum of inputs as before: \n", - "\n", - "$$ z_j = \\sum_{i=1}^n w_ {ij} a_i+b_j.$$ \n", - "\n", - "Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500\n", - "weights to the output layer.\n", - "\n", - "\n", - "Typically weights are initialized with small values distributed around zero, drawn from a uniform\n", - "or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. \n", - "\n", - "Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range\n", - "of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron $j$, $b_j$: \n", - "\n", - "$$ z_j = \\sum_{i=1}^n w_ {ij} a_i + b_j.$$ \n", - "\n", - "The bias weights $\\hat{b}$ are often initialized to zero, but a small value like $0.01$ ensures all neurons have some output which can be backpropagated in the first training cycle." - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# building our neural network\n", - "\n", - "n_inputs, n_features = X_train.shape\n", - "n_hidden_neurons = 50\n", - "n_categories = 10\n", - "\n", - "# we make the weights normally distributed using numpy.random.randn\n", - "\n", - "# weights and bias in the hidden layer\n", - "hidden_weights = np.random.randn(n_features, n_hidden_neurons)\n", - "hidden_bias = np.zeros(n_hidden_neurons) + 0.01\n", - "\n", - "# weights and bias in the output layer\n", - "output_weights = np.random.randn(n_hidden_neurons, n_categories)\n", - "output_bias = np.zeros(n_categories) + 0.01" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Feed-forward pass\n", - "\n", - "Denote $F$ the number of features, $H$ the number of hidden neurons and $C$ the number of categories. \n", - "For each input image we calculate a weighted sum of input features (pixel values) to each neuron $j$ in the hidden layer $l$: \n", - "\n", - "$$ z_{j}^{l} = \\sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$\n", - "\n", - "this is then passed through our activation function \n", - "\n", - "$$ a_{j}^{l} = f(z_{j}^{l}) .$$ \n", - "\n", - "We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron $j$ in the output layer: \n", - "\n", - "$$ z_{j}^{L} = \\sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$ \n", - "\n", - "Finally we calculate the output of neuron $j$ in the output layer using the softmax function: \n", - "\n", - "$$ a_{j}^{L} = \\frac{\\exp{(z_j^{L})}}\n", - "{\\sum_{c=0}^{C-1} \\exp{(z_c^{L})}} .$$ \n", - "\n", - "\n", - "Since our data has the dimensions $X = (n_{inputs}, n_{features})$ and our weights to the hidden\n", - "layer have the dimensions \n", - "$W_{hidden} = (n_{features}, n_{hidden})$,\n", - "we can easily feed the network all our training data in one go by taking the matrix product \n", - "\n", - "$$ X W^{h} = (n_{inputs}, n_{hidden}),$$ \n", - "\n", - "and obtain a matrix that holds the weighted sum of inputs to the hidden layer\n", - "for each input image and each hidden neuron. \n", - "We also add the bias to obtain a matrix of weighted sums to the hidden layer $Z^{h}$: \n", - "\n", - "$$ \\hat{z}^{l} = \\hat{X} \\hat{W}^{l} + \\hat{b}^{l} ,$$\n", - "\n", - "meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image. \n", - "This is then passed through the activation: \n", - "\n", - "$$ \\hat{a}^{l} = f(\\hat{z}^l) .$$ \n", - "\n", - "This is fed to the output layer: \n", - "\n", - "$$ \\hat{z}^{L} = \\hat{a}^{L} \\hat{W}^{L} + \\hat{b}^{L} .$$\n", - "\n", - "Finally we receive our output values for each image and each category by passing it through the softmax function: \n", - "\n", - "$$ output = softmax (\\hat{z}^{L}) = (n_{inputs}, n_{categories}) .$$" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "probabilities = (n_inputs, n_categories) = (1437, 10)\n", - "probability that image 0 is in category 0,1,2,...,9 = \n", - "[5.41511965e-04 2.17174962e-03 8.84355903e-03 1.44970586e-03\n", - " 1.10378326e-04 5.08318298e-09 2.03256632e-04 1.92507116e-03\n", - " 9.84443254e-01 3.11507992e-04]\n", - "probabilities sum up to: 1.0\n", - "\n", - "predictions = (n_inputs) = (1437,)\n", - "prediction for image 0: 8\n", - "correct label for image 0: 6\n" - ] - } - ], - "source": [ - "# setup the feed-forward pass, subscript h = hidden layer\n", - "\n", - "def sigmoid(x):\n", - " return 1/(1 + np.exp(-x))\n", - "\n", - "def feed_forward(X):\n", - " # weighted sum of inputs to the hidden layer\n", - " z_h = np.matmul(X, hidden_weights) + hidden_bias\n", - " # activation in the hidden layer\n", - " a_h = sigmoid(z_h)\n", - " \n", - " # weighted sum of inputs to the output layer\n", - " z_o = np.matmul(a_h, output_weights) + output_bias\n", - " # softmax output\n", - " # axis 0 holds each input and axis 1 the probabilities of each category\n", - " exp_term = np.exp(z_o)\n", - " probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n", - " \n", - " return probabilities\n", - "\n", - "probabilities = feed_forward(X_train)\n", - "print(\"probabilities = (n_inputs, n_categories) = \" + str(probabilities.shape))\n", - "print(\"probability that image 0 is in category 0,1,2,...,9 = \\n\" + str(probabilities[0]))\n", - "print(\"probabilities sum up to: \" + str(probabilities[0].sum()))\n", - "print()\n", - "\n", - "# we obtain a prediction by taking the class with the highest likelihood\n", - "def predict(X):\n", - " probabilities = feed_forward(X)\n", - " return np.argmax(probabilities, axis=1)\n", - "\n", - "predictions = predict(X_train)\n", - "print(\"predictions = (n_inputs) = \" + str(predictions.shape))\n", - "print(\"prediction for image 0: \" + str(predictions[0]))\n", - "print(\"correct label for image 0: \" + str(Y_train[0]))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Choose cost function and optimizer\n", - "\n", - "To measure how well our neural network is doing we need to introduce a cost function. \n", - "We will call the function that gives the error of a single sample output the *loss* function, and the function\n", - "that gives the total error of our network across all samples the *cost* function.\n", - "A typical choice for multiclass classification is the *cross-entropy* loss, also known as the negative log likelihood. \n", - "\n", - "In *multiclass* classification it is common to treat each integer label as a so called *one-hot* vector: \n", - "\n", - "$$ y = 5 \\quad \\rightarrow \\quad \\hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$ \n", - "\n", - "\n", - "$$ y = 1 \\quad \\rightarrow \\quad \\hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$ \n", - "\n", - "\n", - "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset. \n", - "\n", - "Let $y_{ic}$ denote the $c$-th component of the $i$-th one-hot vector. \n", - "We define the cost function $\\mathcal{C}$ as a sum over the cross-entropy loss for each point $\\hat{x}_i$ in the dataset.\n", - "\n", - "In the one-hot representation only one of the terms in the loss function is non-zero, namely the\n", - "probability of the correct category $c'$ \n", - "(i.e. the category $c'$ such that $y_{ic'} = 1$). This means that the cross entropy loss only punishes you for how wrong\n", - "you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\\hat{\\theta}$ represents the parameters of our network, i.e. all the weights and biases. \n", - "\n", - "\n", - "\n", - "### Optimizing the cost function\n", - "\n", - "The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is *gradient descent* and its generalizations. The idea behind gradient descent\n", - "is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a *local* minimum of the cost function. \n", - "Each parameter $\\theta$ is iteratively adjusted according to the rule \n", - "\n", - "$$ \\theta_{i+1} = \\theta_i - \\eta \\nabla \\mathcal{C}(\\theta_i) ,$$\n", - "\n", - "where $\\eta$ is known as the *learning rate*, which controls how big a step we take towards the minimum. \n", - "This update can be repeated for any number of iterations, or until we are satisfied with the result. \n", - "\n", - "A simple and effective improvement is a variant called *Batch Gradient Descent*. \n", - "Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient\n", - "on a subset of the data called a *minibatch*. \n", - "If there are $N$ data points and we have a minibatch size of $M$, the total number of batches\n", - "is $N/M$. \n", - "We denote each minibatch $B_k$, with $k = 1, 2,...,N/M$. The gradient then becomes: \n", - "\n", - "$$ \\nabla \\mathcal{C}(\\theta) = \\frac{1}{N} \\sum_{i=1}^N \\nabla \\mathcal{L}_i(\\theta) \\quad \\rightarrow \\quad\n", - "\\frac{1}{M} \\sum_{i \\in B_k} \\nabla \\mathcal{L}_i(\\theta) ,$$\n", - "\n", - "i.e. instead of averaging the loss over the entire dataset, we average over a minibatch. \n", - "\n", - "This has two important benefits: \n", - "1. Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima. \n", - "\n", - "2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient. \n", - "\n", - "The various optmization methods, with codes and algorithms, are discussed in our lectures on [Gradient descent approaches](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", - "\n", - "\n", - "### Regularization\n", - "\n", - "It is common to add an extra term to the cost function, proportional\n", - "to the size of the weights. This is equivalent to constraining the\n", - "size of the weights, so that they do not grow out of control.\n", - "Constraining the size of the weights means that the weights cannot\n", - "grow arbitrarily large to fit the training data, and in this way\n", - "reduces *overfitting*.\n", - "\n", - "We will measure the size of the weights using the so called *L2-norm*, meaning our cost function becomes: \n", - "\n", - "$$ \\mathcal{C}(\\theta) = \\frac{1}{N} \\sum_{i=1}^N \\mathcal{L}_i(\\theta) \\quad \\rightarrow \\quad\n", - "\\frac{1}{N} \\sum_{i=1}^N \\mathcal{L}_i(\\theta) + \\lambda \\lvert \\lvert \\hat{w} \\rvert \\rvert_2^2 \n", - "= \\frac{1}{N} \\sum_{i=1}^N \\mathcal{L}(\\theta) + \\lambda \\sum_{ij} w_{ij}^2,$$ \n", - "\n", - "i.e. we sum up all the weights squared. The factor $\\lambda$ is known as a regularization parameter.\n", - "\n", - "\n", - "In order to train the model, we need to calculate the derivative of\n", - "the cost function with respect to every bias and weight in the\n", - "network. In total our network has $(64 + 1)\\times 50=3250$ weights in\n", - "the hidden layer and $(50 + 1)\\times 10=510$ weights to the output\n", - "layer ($+1$ for the bias), and the gradient must be calculated for\n", - "every parameter. We use the *backpropagation* algorithm discussed\n", - "above. This is a clever use of the chain rule that allows us to\n", - "calculate the gradient efficently. \n", - "\n", - "\n", - "### Matrix multiplication\n", - "\n", - "To more efficently train our network these equations are implemented using matrix operations. \n", - "The error in the output layer is calculated simply as, with $\\hat{t}$ being our targets, \n", - "\n", - "$$ \\delta_L = \\hat{t} - \\hat{y} = (n_{inputs}, n_{categories}) .$$ \n", - "\n", - "The gradient for the output weights is calculated as \n", - "\n", - "$$ \\nabla W_{L} = \\hat{a}^T \\delta_L = (n_{hidden}, n_{categories}) ,$$\n", - "\n", - "where $\\hat{a} = (n_{inputs}, n_{hidden})$. This simply means that we are summing up the gradients for each input. \n", - "Since we are going backwards we have to transpose the activation matrix. \n", - "\n", - "The gradient with respect to the output bias is then \n", - "\n", - "$$ \\nabla \\hat{b}_{L} = \\sum_{i=1}^{n_{inputs}} \\delta_L = (n_{categories}) .$$ \n", - "\n", - "The error in the hidden layer is \n", - "\n", - "$$ \\Delta_h = \\delta_L W_{L}^T \\circ f'(z_{h}) = \\delta_L W_{L}^T \\circ a_{h} \\circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$ \n", - "\n", - "where $f'(a_{h})$ is the derivative of the activation in the hidden layer. The matrix products mean\n", - "that we are summing up the products for each neuron in the output layer. The symbol $\\circ$ denotes\n", - "the *Hadamard product*, meaning element-wise multiplication. \n", - "\n", - "This again gives us the gradients in the hidden layer: \n", - "\n", - "$$ \\nabla W_{h} = X^T \\delta_h = (n_{features}, n_{hidden}) ,$$ \n", - "\n", - "$$ \\nabla b_{h} = \\sum_{i=1}^{n_{inputs}} \\delta_h = (n_{hidden}) .$$" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Old accuracy on training data: 0.1440501043841336\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - ":4: RuntimeWarning: overflow encountered in exp\n", - " return 1/(1 + np.exp(-x))\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "New accuracy on training data: 0.1022964509394572\n" - ] - } - ], - "source": [ - "# to categorical turns our integer vector into a onehot representation\n", - "from sklearn.metrics import accuracy_score\n", - "\n", - "# one-hot in numpy\n", - "def to_categorical_numpy(integer_vector):\n", - " n_inputs = len(integer_vector)\n", - " n_categories = np.max(integer_vector) + 1\n", - " onehot_vector = np.zeros((n_inputs, n_categories))\n", - " onehot_vector[range(n_inputs), integer_vector] = 1\n", - " \n", - " return onehot_vector\n", - "\n", - "#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)\n", - "Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)\n", - "\n", - "def feed_forward_train(X):\n", - " # weighted sum of inputs to the hidden layer\n", - " z_h = np.matmul(X, hidden_weights) + hidden_bias\n", - " # activation in the hidden layer\n", - " a_h = sigmoid(z_h)\n", - " \n", - " # weighted sum of inputs to the output layer\n", - " z_o = np.matmul(a_h, output_weights) + output_bias\n", - " # softmax output\n", - " # axis 0 holds each input and axis 1 the probabilities of each category\n", - " exp_term = np.exp(z_o)\n", - " probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n", - " \n", - " # for backpropagation need activations in hidden and output layers\n", - " return a_h, probabilities\n", - "\n", - "def backpropagation(X, Y):\n", - " a_h, probabilities = feed_forward_train(X)\n", - " \n", - " # error in the output layer\n", - " error_output = probabilities - Y\n", - " # error in the hidden layer\n", - " error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)\n", - " \n", - " # gradients for the output layer\n", - " output_weights_gradient = np.matmul(a_h.T, error_output)\n", - " output_bias_gradient = np.sum(error_output, axis=0)\n", - " \n", - " # gradient for the hidden layer\n", - " hidden_weights_gradient = np.matmul(X.T, error_hidden)\n", - " hidden_bias_gradient = np.sum(error_hidden, axis=0)\n", - "\n", - " return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient\n", - "\n", - "print(\"Old accuracy on training data: \" + str(accuracy_score(predict(X_train), Y_train)))\n", - "\n", - "eta = 0.01\n", - "lmbd = 0.01\n", - "for i in range(1000):\n", - " # calculate gradients\n", - " dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)\n", - " \n", - " # regularization term gradients\n", - " dWo += lmbd * output_weights\n", - " dWh += lmbd * hidden_weights\n", - " \n", - " # update weights and biases\n", - " output_weights -= eta * dWo\n", - " output_bias -= eta * dBo\n", - " hidden_weights -= eta * dWh\n", - " hidden_bias -= eta * dBh\n", - "\n", - "print(\"New accuracy on training data: \" + str(accuracy_score(predict(X_train), Y_train)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Improving performance\n", - "\n", - "As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image. \n", - "In order to obtain a network that does something useful, we will have to do a bit more work. \n", - "\n", - "The choice of *hyperparameters* such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a *grid-search* is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates $\\eta = 10^{-6}, 10^{-5},...,10^{-1}$ with different regularization parameters $\\lambda = 10^{-6},...,10^{-0}$. \n", - "\n", - "Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an *iteration*, and a full training period\n", - "going through the entire dataset ($n/M$ batches) an *epoch*.\n", - "\n", - "If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers. \n", - "Andrew Ng goes through some of these considerations in this [video](https://youtu.be/F1ka6a13S9I). You can find a summary of the video [here](https://kevinzakka.github.io/2016/09/26/applying-deep-learning/). \n", - "\n", - "\n", - "It is very natural to think of the network as an object, with specific instances of the network\n", - "being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below." - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "class NeuralNetwork:\n", - " def __init__(\n", - " self,\n", - " X_data,\n", - " Y_data,\n", - " n_hidden_neurons=50,\n", - " n_categories=10,\n", - " epochs=10,\n", - " batch_size=100,\n", - " eta=0.1,\n", - " lmbd=0.0):\n", - "\n", - " self.X_data_full = X_data\n", - " self.Y_data_full = Y_data\n", - "\n", - " self.n_inputs = X_data.shape[0]\n", - " self.n_features = X_data.shape[1]\n", - " self.n_hidden_neurons = n_hidden_neurons\n", - " self.n_categories = n_categories\n", - "\n", - " self.epochs = epochs\n", - " self.batch_size = batch_size\n", - " self.iterations = self.n_inputs // self.batch_size\n", - " self.eta = eta\n", - " self.lmbd = lmbd\n", - "\n", - " self.create_biases_and_weights()\n", - "\n", - " def create_biases_and_weights(self):\n", - " self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)\n", - " self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01\n", - "\n", - " self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)\n", - " self.output_bias = np.zeros(self.n_categories) + 0.01\n", - "\n", - " def feed_forward(self):\n", - " # feed-forward for training\n", - " self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias\n", - " self.a_h = sigmoid(self.z_h)\n", - "\n", - " self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias\n", - "\n", - " exp_term = np.exp(self.z_o)\n", - " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n", - "\n", - " def feed_forward_out(self, X):\n", - " # feed-forward for output\n", - " z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias\n", - " a_h = sigmoid(z_h)\n", - "\n", - " z_o = np.matmul(a_h, self.output_weights) + self.output_bias\n", - " \n", - " exp_term = np.exp(z_o)\n", - " probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n", - " return probabilities\n", - "\n", - " def backpropagation(self):\n", - " error_output = self.probabilities - self.Y_data\n", - " error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)\n", - "\n", - " self.output_weights_gradient = np.matmul(self.a_h.T, error_output)\n", - " self.output_bias_gradient = np.sum(error_output, axis=0)\n", - "\n", - " self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)\n", - " self.hidden_bias_gradient = np.sum(error_hidden, axis=0)\n", - "\n", - " if self.lmbd > 0.0:\n", - " self.output_weights_gradient += self.lmbd * self.output_weights\n", - " self.hidden_weights_gradient += self.lmbd * self.hidden_weights\n", - "\n", - " self.output_weights -= self.eta * self.output_weights_gradient\n", - " self.output_bias -= self.eta * self.output_bias_gradient\n", - " self.hidden_weights -= self.eta * self.hidden_weights_gradient\n", - " self.hidden_bias -= self.eta * self.hidden_bias_gradient\n", - "\n", - " def predict(self, X):\n", - " probabilities = self.feed_forward_out(X)\n", - " return np.argmax(probabilities, axis=1)\n", - "\n", - " def predict_probabilities(self, X):\n", - " probabilities = self.feed_forward_out(X)\n", - " return probabilities\n", - "\n", - " def train(self):\n", - " data_indices = np.arange(self.n_inputs)\n", - "\n", - " for i in range(self.epochs):\n", - " for j in range(self.iterations):\n", - " # pick datapoints with replacement\n", - " chosen_datapoints = np.random.choice(\n", - " data_indices, size=self.batch_size, replace=False\n", - " )\n", - "\n", - " # minibatch training data\n", - " self.X_data = self.X_data_full[chosen_datapoints]\n", - " self.Y_data = self.Y_data_full[chosen_datapoints]\n", - "\n", - " self.feed_forward()\n", - " self.backpropagation()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Evaluate model performance on test data\n", - "\n", - "To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data. \n", - "We measure the performance of the network using the *accuracy* score. \n", - "The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of $1$. \n", - "\n", - "$$ \\text{Accuracy} = \\frac{\\sum_{i=1}^n I(\\hat{y}_i = y_i)}{n} ,$$ \n", - "\n", - "where $I$ is the indicator function, $1$ if $\\hat{y}_i = y_i$ and $0$ otherwise." - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Accuracy score on test set: 0.9361111111111111\n" - ] - } - ], - "source": [ - "epochs = 100\n", - "batch_size = 100\n", - "\n", - "dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,\n", - " n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)\n", - "dnn.train()\n", - "test_predict = dnn.predict(X_test)\n", - "\n", - "# accuracy score from scikit library\n", - "print(\"Accuracy score on test set: \", accuracy_score(Y_test, test_predict))\n", - "\n", - "# equivalent in numpy\n", - "def accuracy_score_numpy(Y_test, Y_pred):\n", - " return np.sum(Y_test == Y_pred) / len(Y_test)\n", - "\n", - "#print(\"Accuracy score on test set: \", accuracy_score_numpy(Y_test, test_predict))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Adjust hyperparameters\n", - "\n", - "We now perform a grid search to find the optimal hyperparameters for the network. \n", - "Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around $98\\%$ ($2\\%$ error rate)." - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.11666666666666667\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.20833333333333334\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.12222222222222222\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.14722222222222223\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.17777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.16111111111111112\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.20277777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.5305555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.5944444444444444\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.5888888888888889\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.6111111111111112\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.5222222222222223\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.5555555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.8055555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.85\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.85\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.875\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.8666666666666667\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.8638888888888889\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.9555555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.925\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.9583333333333334\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.9277777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.9388888888888889\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.9166666666666666\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.9611111111111111\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.8777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.11388888888888889\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - ":4: RuntimeWarning: overflow encountered in exp\n", - " return 1/(1 + np.exp(-x))\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.09166666666666666\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.10555555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.08888888888888889\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.10555555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.11666666666666667\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.12777777777777777\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.09166666666666666\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - ":43: RuntimeWarning: overflow encountered in exp\n", - " exp_term = np.exp(self.z_o)\n", - ":44: RuntimeWarning: invalid value encountered in true_divide\n", - " self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.08888888888888889\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 10.0\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 10.0\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 10.0\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 10.0\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 10.0\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 10.0\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 10.0\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.07777777777777778\n", - "\n" - ] - } - ], - "source": [ - "eta_vals = np.logspace(-5, 1, 7)\n", - "lmbd_vals = np.logspace(-5, 1, 7)\n", - "# store the models for later use\n", - "DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - "\n", - "# grid search\n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,\n", - " n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)\n", - " dnn.train()\n", - " \n", - " DNN_numpy[i][j] = dnn\n", - " \n", - " test_predict = dnn.predict(X_test)\n", - " \n", - " print(\"Learning rate = \", eta)\n", - " print(\"Lambda = \", lmbd)\n", - " print(\"Accuracy score on test set: \", accuracy_score(Y_test, test_predict))\n", - " print()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Visualization" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - ":4: RuntimeWarning: overflow encountered in exp\n", - " return 1/(1 + np.exp(-x))\n" - ] - }, - { - "data": { - "image/png": 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SklGLXmHN3K38snHvnR+QRdzu03ta8fv4evHGmE4EBuflnYHZN/m5mel28Vvv4fgv/P/jnzWqnelhNt3u+KfvspH1S39y/J2clMCqOdvo2KdRtkmEbvv8T+fx7/lqcy5fvEqPhh/g7ePJqBnP0uW5x1j52Q8Z2c1McTev/bR4eJqp1qAc7/X/jKRrKbw+8SmeG9KaWR+sychuOsXtXwe3OQ++25nA4ADeGZAzzoO3stqcV9Xt3bs3nTt3TrU+ICDAsGyzpT4nmW46bgEBAUyYMIGRI0ditVr5z3/+g5eX1x3375ShMU9PT1JSUpeKExMT09XJzBZ1+gIFgvM6lgsVyUfcxatci09K9zYad6nFhyuHMP/dlXwVui4zuplpIs9dpkCh3I7lQoF5iLucwLVbJvsGBudl6vznsVptvNn/M67eZjJxdhN1Jub+j3/nWnz4vxDmj13JV1Ozz0RZgMiIixQIunHCKVQ4L3GXrnItIX3xN+tSk1IVijiWTSZTtvqEHHn2EgUC8ziWCwUHEHcpnmsJ6ZvsXr/FI2xcsYuUZAvxV67x3dd7qFznwTs/MAuIjLhkPPZ3GXtMZCw7N/5J/JVrpCRb2LJ6DxWqPZBZ3c1U9vPgTc+DwDzEXY5PfR4snJepC17AarHy5ks55zzoSgEBARQvXjzV7dZEKDg42FDhiYyMJCgoyLFssVgoXLgwy5cvZ8WKFTz66KOUKFHijvt3SiLUv39/OnXqxIgRI5g2bRrTpk1jxIgRPPnkk/Tv398ZXfhXu7ceoELNMhQtY/+Htu3TmPBvf0/34x/rUJ2XP+zO211D2bbil0zqZebZ/dPfVHi0OEVL2Oe1tO1ak/DtxtJ+ngA/Js9+jh+2/sWHb/+PpGwyByI9dm89QIUaNx3/5+7y+LevzsvjnuLtJ6dmy+O/5/tDVKhWiqLXJ7q3eaYB4Rv/TPfjSz1UhGdfb4PZbMLbx4v2vRvyfdhvmdXdDLfnxyNUqFKCog/Y5/W1eaoO4VsO3OFRNxw9EEGj63PqPDzN1G1WkYN//JMpfc1oe3YcokLVkhR94Pqx71GP8O/2p/vxP3y7j4atK+PtYx9cqNfiUQ7vO5Upfc1su3/6mwqVbjoPdqtF+PZDhjb282AfftiS886D2UH9+vUJDw8nJiaGhIQENm7cSKNGNybnm0wm+vbty/nz57HZbMyfP582bVJP/r+VU4bG2rdvT+3atQkPDycyMhKbzUbNmjUZOHAgwcHBzujCv7ocHceUAQsY8Vl/PL09OXs8ikkvz6Nc1QcYPK03rzZ+718f32dkFzCZGDytt2PdgZ+P8vGbSzK76xni8sWrfPTeakZO+A+eXh6cPX2RSaNXUa5iUUJGdOCVnp/SrltNAgvnpUGTijRocmOO1LBXFhJ3OcGFvb9/l6PjmDLoM0bMf8l+/E9EMemV+fbjH9qLV5uO/dfH9xnZ2X78Q3s51h345SgfX59TltVdvnCF0DeW8M6nffD08uTsP9FMHryYcpVL8NqE7gxoPelfH784dD2vjO3GzE3D8PD0YMfa3w3DZVnd5ZirhL6zgnem9rQ//0/FMHn4Mso9UozXxnZhQJfp//r42ePX8vI77Zm9NgSr1cbv4X+zfG72GBa8HHOV0GHLeGfGs/bY/7nA5KFfUu7R4rw27kkGdPj3K26+WbyT3Pn8mb56MGaziaP7zzD3wzAn9T5jXb54lY/e/ZqRE5+6fh6MYdKo6+fBkR14pcentOtWy34ebFqBBk1vzBUc9nL2Pw/eKitePh8cHExISAi9evUiOTmZbt26UblyZfr168egQYOoVKkS7733Hi+88AJJSUnUq1eP559//o7bNdnSGnTL4p4o0M/VXXAZW5liru6CS5lO5Lzv67gbJn9/V3fBdXL53blNTpbs3tUHa143fu4DG3a/69T97fmnpNP2Vb2kayuo+kJFERERMbC40dcMuk+kIiIiIrdQRUhEREQMnHn5vKupIiQiIiJuSxUhERERMciKV41lFlWERERExG2pIiQiIiIGFpv71EncJ1IRERGRW6giJCIiIgZWN6qTuE+kIiIiIrdQRUhEREQMdNWYiIiIiBtQRUhEREQMdNWYiIiIiBtQIiQiIiJuS0NjIiIiYmDVZGkRERGRnE8VIRERETGwuFGdxH0iFREREbmFKkIiIiJioMvnRURERNyAKkIiIiJioB9dFREREXEDqgiJiIiIgcWm7xESERERyfFUERIREREDfY+QiIiIiBtQRUhEREQMrPoeIREREZGcTxUhERERMdAcIRERERE3oERIRERE3JaGxkRERMRAX6goIiIi4gayZUXo0HsVXN0F18mf5OoeuJattKt74FIe3hZXd8FlrOf8XN0Fl7L6WV3dBZd6pcEWV3fBrehHV0VERETcQLasCImIiEjmsegLFUVERERyPlWERERExMCKrhoTERERyfFUERIREREDzRESERERcQOqCImIiIiBfnRVRERExA2oIiQiIiIGVv3WmIiIiEjOp4qQiIiIGGiOkIiIiIgbUCIkIiIibktDYyIiImJg1RcqioiIiOR8qgiJiIiIgUU/uioiIiKStYSFhdGmTRtatGjB4sWLU92/f/9+unbtSocOHXjppZeIjY294zaVCImIiIiB1WZ22i29zp8/T2hoKEuWLGH16tV89dVXHD161NDmgw8+YNCgQaxZs4bSpUszb968O25XiZCIiIhkeTt37qRu3brky5cPf39/WrVqxfr16w1trFYrV69eBSAhIQFfX987bldzhERERMTAmXOEYmNj0xzCCggIICAgwLEcGRlJYGCgYzkoKIi9e/caHjN8+HD69OnDuHHj8PPzY9myZXfcvxIhERERcZmFCxcyY8aMVOsHDBjAwIEDHcs2my1VG5PpRsKWmJjIO++8w8KFC6lcuTILFixg2LBhzJ49+1/3r0RIREREDJz5PUK9e/emc+fOqdbfXA0CCA4OZteuXY7lyMhIgoKCHMuHDx/Gx8eHypUrA/DUU08xbdq0O+5fc4RERETEZQICAihevHiq262JUP369QkPDycmJoaEhAQ2btxIo0aNHPc/8MADnDt3jmPHjgGwefNmKlWqdMf9qyIkIiIiBpYs+M3SwcHBhISE0KtXL5KTk+nWrRuVK1emX79+DBo0iEqVKvHhhx8yePBgbDYbBQsWZNy4cXfcrhIhERERyRbat29P+/btDevmzJnj+Ltx48Y0btz4rrapREhEREQMrPpmaREREZGcTxUhERERMciKc4Qyi/tEKiIiInILVYRERETEwGrTHCERERGRHE+JkIiIiLgtDY2JiIiIgcWN6iRKhK5r+kBp3qzXEG8PDw5eiGLY5o1cSU4ytHmnQWNaly3P5cREAI5dusjADd/gZTYzplEzahUtDsD2k8f5cOf3WNP4gbisqmnRB3mzahN7/BcjGfbTOq6k3BJ/9Wa0LlmBy9euxx93gYE/rMbb7MHomi2oF/wA8SlJbD5zlKl7d5B9ooemxdKI/9bjX70ZrR+4Kf7Ym+KvdVP8p7Nf/E2KlOWNSk3xNnty6PJ53vr1m1THv3zeQEZXe4I8Xj5YbFZG7F7H/ovnHPfn8fJhadNeDP/1G/68eNbZIdyzpqVKM7T+9dd+dBTDN2/kSpIx9rcfa0ybcuW59P+v/YsXGbT+G0zAmw0a0bRUaWw2GycuXeKdrZuISUhwQST3pmmJMrxZq5E9/pgohn2/PvVzv04T2pR56MZz/1IMA7aEAfBEqfK8WrUO3h4enLkSy5Bt67h0vV128M+ueHZ9EYM12Ub+B7xpOCAQb39jEnDip6vs+fIiJhP45Dbz2CuBBBTxctx/JTqFsGFn6BxaHN8AD2eHIPdJiRBQwNePic2f4MkVSzlx+RLD6jXkzfoNGbV9s6Fd9SJFGbRhLXvORRjW96pcjQJ+/rRa8hlmk4llXbvTtuxDhB056Mww7lkBHz8m1mvLkxsXcSLuIsOqNuHNak0Z9esGQ7vqhYoz6IfV7Ik+Y1j/yiP1KZYrL0+snUuy1cK4Oq15tnwNPj+825lh3DNH/Buux1+tCW9WTSP+wOIM2pFG/I9ej/+b7Bq/PxNqtec/Wz7j5JWLDK3cjKGVmzF6z3pHG18PTz5r1IO3fv2G7ef+5vGi5ZlSpxOt1n8KQOPCDzKiWkuK+edzURT3poCfHxMef4L/LL/+2q9//bW/LY3X/repX/tPPlKJSkFBdPjyC5IsFoY1aMTbjzXmjU3ryQ4K+PoxqfETdFuzhBOxlxheuxHDajdi5I/fGdrVCC7GwM1h7Ik0xl+pUDDvNWhOl9WLOX0llpF1mzK0VkPe+WGTM8O4ZwmXLeyYHkm7D4uRt6gXv3x+gV8XxdDgpUKONinXrGyfGknn0OIEFPHizzWXCJ8XTasRRQA4sjWOPUsvEh9jcVUYmUKTpd1Mw5IPsDfyHCcuXwLgiz//oGP5ioY23mYPHikURL9qNVnX/Vk+ad2eornzADDv990M3PANNiC/rx8B3j5cvpZ9PhE2LFKGvRfOciLuIgBfHPmNjqUeNrTxNnvwSIFg+lWsw7o2ffmkYWeK+tt/EO/RgoUJO3mAJKsFG7Dx1GFal3zI2WHcs1TxH/6NjqVvE//DdVjXti+fNLop/gKFCTtxU/ynD9P6gewT/2PBZdgbE8HJK/b4lxzdTYeSj6Zq88+VS2w/9zcA30UcZlD4Ssf9vcvV5s1f1hCZGOe8jmeAhiUfYN/5m177+/6g40O3vPY9PHgkMIh+1Wuy9uln+aTNjdf+kQvRfPjD9yRZ7G+C+yLPUeyWH4rMyhoWK8XeqHOciL0EwBcHfqdj2TSe+wWDeLFyLb7t0puZj3ekaC57/J3LPsxXh/Zx+kosAFN3/8inf/zi1Bjux5nf4ylUzoe8Re3VnYpPBPD393HYbqrm26xgs0FSvBWA5EQbnl72JOFqTAonf75Ky5GFnd95yTBKhIAiuQM4e+XGCfzclTgCfHzI7eXtWBeUKxc7T//DpPAdtPlyEb+fO8vstp0c96dYrbxZryHbnn2e6IR4fokwVg2ysiL+eTgbH+tYPhcfS4C3L7k9b4rfLzc7z51k0u/baLNuPr9HRzC7cVcA/oiOoN0DFfH39MLLbKZDqUcI8svt9DjuVZFceTh7NY34vW4T/9rr8Te5Hv+FbB6/fwBnE26KPyGWPLcc/9J5ChKdeIUPa7Zj1eN9Wdi4J56mG6ePvjuW8tuF7POc/39pvfbz+PiQ29v42g8//Q8Td+6g7dJF/HbuLLPadwLgt3Nn2R8VCUCAjw8Da9dj3ZHDTo3hfhTNnYezV2/Ef/ZqHAHet577crMz4h8m/vo9rVcu5LfICOa07AxA6bwF8DSZmdOiE9926c17DVpw9ZZhtazsarSF3AVvDIzkKuhJcryN5IQbiZCXn5kG/QsRNvwMS/ue5MC6WGr1KmhvX8CTx4cXJn8J71Tbzu6smJ12czWn9CAiIuJfb65mvk0F0GKzOv4+HRdL329WceyS/VPz7N92UTJvXornufHpb2L4DqrO/ZjTsZd5v8njmdrnjGQ2pf0PsNz0qej01cv03baMY3ExAMz+62dK5slP8Vx5+fRAOEcuRbOiVS++aP40e6JPk2TNPmVi821+U8divSX+rcs4Fns9/gM/UzL39fj3h3Pk8k3xR512VAiyg/Qcf0+zmcZFyvLlsT10/m4+i478ytyG3fE2Z+/5ELd97Vtveu3HxtJ3zSqOX3/tz9lz/bV/U+WnZN68fNX1KXZHnGHR3t8zs8sZypSe137cZfpsWMGxy9fPfXt/pWRAPornyYun2UzzBx7k7R820WblQqISrjK+YSun9D0j2G4zj/OmHJ+Yk0n8tuwiXf9bgqfnP0DVbvnYPPH8bR8r2Y9T5gi99NJLnDhxgqCgoFRPHpPJxObNm2/zSOeIiIujanARx3Lh3Lm5lJhAQkqKY12FgoWoWCiQVYf+cqwzYSLFaqVG4aLEJCZw/NJFUqxW/ndwP2MaNXNqDPcj4mosVQsWdSwX9s/DpWsJJFiSHesq5AukYv5gVh3/07HOhL0Sltfbjzl//cy437YA0PaBipy8PsyUHUTEx1K10D3Eb4IU203x78mm8V+9TJUCN+IP9gtIFX9kwhWOxUXzR4z9g8t3EYcZV7MtJXLl4++4C07vc0Y5ExdHlcI3XvvBt3ntVwgM5OuDqV/7AHWLl+C/T7Rj9u5fmfvbLud1PgNEXImjWuBN575cea7Hf9Nzv0AgFQsEsuroAcc6e/wWIuOvcOhiNFEJVwFYfngfS9o85bwA7lPuQp5EHb7mWL56IQXv3Ga8fG9kQmd+iye4gq9jcnTF1gH8vOAC1+KsOXpitEVzhDLW0qVLKV26NBMnTmTLli2Gm6uTIIAdp05QrXARSuXNB0CPR6uw6fjfhjZWm43RDZs5KkDPPFqFgxeiOHf1CvWLl2TEY03wMJkwAZ3KVyT89CknR3Hvdpw9TrVCxSiVJz8APcpVY9PpI4Y2VpuN0TUep3iuvAA8U646By9FcS4hjhbFyzGuTmsA/D29eKFCbVYf3+/cIO7Djoh0xl/zpvjLV+fgxSjOxacRf8XarD6RjeI/f4yqBYvxQO7r8T9Yne8ijMM7288dpZh/Ph7Jb58LUatQSWzAqauXnNzbjPXDP8bXfs9KVfjuWBqv/UbNHBWgZypV4WB0FOeuXKF64aJ82rYDb2z6NtslQQA7Tp+galBRSgXkA6BnxSpsOnnU0MZqszGmfnOK57n+3K9YlYMx9nPfuuOHaVqiDPl8fAH7FWR7o8+RXRSr6k/k4WtcjrAnfgc3xPFAbX9Dm4JlfDi3P5GES/bk+OQvV8kd5JmjkyB3Y7I5qb63d+9eli9fztixY+97W6VnfJQBPTJq8kBp3qz3GF5mD07GXuL1TespmTcv45u2pO1XiwB7gtO/Rm08TCbOXbnCsC0biLgSh5fZzMiGTalTtDhWbOyKOMMHP24n8aZPlRkmf+aMvzcp+iBvVm1sj//KJV7fGUbJ3PkYX6cNbb+dD0CnUo/Q/5F69vjj4xj20zoi4mPxMJn4oHZrqhUqiofJzJdHf2fuwUyaMJlJz9YmRR/kzWrX44+7Kf66bWi77nr8pW+JP/ym+OvcEv9fmRO/h3fmDLk1Lvwgb1RuhpfZg3+uXGToL6spkSs/42q2pcOmuYA9+RlWpTn+nl4kWSyM/X0ju6ONCf+2tgMYsHNFplw+bz3nl+HbBPtrf2j9x/Dy8OCfy5d4faP9tf9h85a0W2p/7Xd8qCL9a9547Q//zv7a/7xTN6oEF+ZU7GXH9k7FXubltWsyvJ9WP+udG92DJiVKM6xWI8dzf8i2dZTMk5cJjZ6gzcqFAHQq+zAvV6mNh8nM2atxDPt+PRHX5xY9U7Eqzz5cFbPJxJkrsbz5/Xoi469meD9fabAlw7cJcGq3/fJ5S7KNgMJeNH4tkNjzKfzwcRSdQ+1fiXJg3WUOrIvFw8uET24z9foVIn9J47ygeZ2P0XPhA5mWIL358LeZst3bee23p522r2nVljptX2lxWiKUkTIjEco2MikRyjay3bM1Y2VWIpQdZFYilF1kViKUXWRWIpRdKBHKPPoeIRERETGw2lx/NZezuE+kIiIiIrdQRUhEREQMLLf5WpGcSBUhERERcVuqCImIiIiBfmtMRERExA0oERIRERG3paExERERMdDl8yIiIiJuQBUhERERMbDq8nkRERGRnE8VIRERETGw6PJ5ERERkZxPFSEREREx0FVjIiIiIm5AFSEREREx0E9siIiIiLgBVYRERETEQN8jJCIiIuIGVBESERERA80REhEREXEDqgiJiIiIgb5HSERERMQNKBESERERt6WhMRERETHQZGkRERERN6CKkIiIiBjoCxVFRERE3IAqQiIiImKgOUIiIiIibkAVIRERETFQRUhERETEDagiJCIiIgaqCImIiIi4gWxZEZrYeomruyAuEuQR5+ouuFSiLVu+ZDOExc0/t7X0S3R1F1wq0hLv6i64FVWERERERNyA+368FBERkTRl1W+WDgsLY+bMmSQnJ/Pcc8/Rs2dPx31//fUXw4cPdyzHxMSQN29evvnmm3/dphIhERERyfLOnz9PaGgoK1euxNvbm+7du1OnTh3Kli0LQMWKFVm9ejUACQkJPPnkk4wZM+aO21UiJCIiIgbOnCMUGxtLbGxsqvUBAQEEBAQ4lnfu3EndunXJly8fAK1atWL9+vUMGDAg1WNnzZpFrVq1qFmz5h33r0RIREREXGbhwoXMmDEj1foBAwYwcOBAx3JkZCSBgYGO5aCgIPbu3ZvqcbGxsSxbtoywsLB07V+JkIiIiLhM79696dy5c6r1N1eDAGw2W6o2JlPqylVYWBiPP/44BQsWTNf+lQiJiIiIgTOHxm4dArud4OBgdu3a5ViOjIwkKCgoVbvvvvuOl156Kd371+XzIiIikuXVr1+f8PBwYmJiSEhIYOPGjTRq1MjQxmazsX//fqpVq5bu7SoREhEREQOrzeS0W3oFBwcTEhJCr1696NSpE+3ataNy5cr069ePffv2AfZL5r28vPDx8Un3dk22tAbdsrjlf9dwdRfERfTN0u47mq1vltY3S7uzosUinLq/5luHOG1fm5tOcdq+0uK+Z1URERFJk35iQ0RERMQNqCIkIiIiBjZVhERERERyPlWERERExCCr/uhqZlBFSERERNyWKkIiIiJioKvGRERERNyAKkIiIiJioKvGRERERNyAKkIiIiJioDlCIiIiIm5AiZCIiIi4LQ2NiYiIiIEmS4uIiIi4AVWERERExECTpUVERETcgCpCIiIiYmCzuboHzqOKkIiIiLgtVYRERETEwIrmCImIiIjkeKoIiYiIiIG+R0hERETEDagidN2hX1LY+FkSlmQbwaXNdB7si6+/MSM+sDOFzV8kYTKDX24TnV7zoWARey45rvsVAgrdyCsf6+pF1aZeTo3hfrh7/Ht/hlULICUZipWG3iHgl8vY5rcfYc0iMJnAPzf0CoGgomC1wNKP4fA+e7tHa0G3fvZ22cX+n22ELbCSkgxFS8PTIWb8chkD+ONHG98usl4//vD0YDOBRU3Me99CdMSNdhfOQdlK8OK7Hk6O4t7s/9nK2gWW67Gb6B7ige8tse/90cr6RRZMZvDPbeKpwR4UKmpv80OYhZ/WW0lOghJl7Y/39M4+B397OEydbSYpGcqXgbHDrOS+5bm/eIWJJatM+PhAmZI2RoTYyBdgv2/pKhMr1pq4dg0efsjG2DdteHs7P457Ff6TB3Pn+pCcZKJMGQtDhyaS65b4V6704uuvvfH2sfFASSuvvZZIQABcuQKTJvvyzz9mbDZo1TKFp59Ock0gGUzfI+Rmrl62sTL0Gk+/48vgObkoUNjMxgXXDG2Sr9lYPimRHiN8GTDDnwp1PFj7qb1N1GkrfrlNDJjh77hlpyTA3eOPuwQLP4L+I2HsPAgsDCvnG9skXYN5E+DlkTBqJlSpB1/OtN/302Y4dxpGfwojZ9oTot07nB7GPYu7ZGPxFCt9R5oZMc+DgkVMhC0wXjubdM3GoolWnh9lZtgnHlSqa2LFTCsAz4/wYNgn9lv318z45YYnB2SPU8uVSza+nGKhz0hP3p7nRcEiJr5ZYDW0SbpmY/FEC31GeTL0Ey8eqWti5UwLAHt/sLJjjZWXx3sybJYnyUmwbZU1rV1lSTGXYMR4M1PHWln7hZXiRW1MmWV8A/x5D8xbamLeFCsr51lpVBfGTLYf303fw5KVJuZOsbJ6oZXEayYWLs8+b6CXLpmYONGXd8ck8PnnVylS1MrsOT6GNr/95sHSL7356KN45s6Jp06dFD6a4gvA/AU+BBaysWB+PDM/iWf1Gi/2788ez325QUcMOLInhWLlzRQqZv931G7rxR9bU7Dd9EUK1uvntsSr9nVJCeB5/b3+nwMWzB4wb3gC01+JZ8uSJKyW7PMlDO4e/4E98MBDEFzMvty4Hfy8xfg9GlYr2ICEePvytQTw8rpx37VESE62V5QsyeCVjT4RH9xjo2R5CCpmfwN7rK2JXVtshuNv+//4r9qXryWkjjEl2cYXH1np8pKZ/IHZ483w0B4bJcqbCLwee4O2ZnZvsaYZe+L12JNuiv3XzVaadvEgVx4TZrOJJwd6ULN59jmt7vzVxKMV4IHi9uXuHW2s/c5keO4fOGyibg0bhYPsy483srFtJyQlw5oNZno/Za8Omc0w+nUrHVpmn9f+r7s8eOghK8WL2/vcsUMymzd7GeI/fNhMjRoWAgPtKxs2TCE83JPkZBg44Bovv2z/QBgTYyI5mVTVpOzKZnPezdWcNjT23XffcfbsWRo3bkzJkiUd67/66iueeuopZ3UjTZejbOQtdOPEHVDIxLV4+8ne19++zsfPRIcBPsx+PQH/ABNWK7w42Q+wvxE+WM2TJ573JvkaLBqdgK8/1O+UPd4N3T3+mCgoUOjGcv5ASIw3kRhvcwyP+frBMwNhQgjkymOPedgU+331W8Du72FYT7BY4OHqUKWu8+O4V5eiMCQu+QIhMd5++//4ffxMPDXQROgQK7ny2JODwVOMb/jhG2zkLQBVGmSPJAjgYpSNfDfFnvd67Nfiwfem2J8c6MG0ISmOYz9oiv3UGXXGRtxlG7PeSeHyBRtlHjXT/oXskwidjYTCQTfeiYID4cpVE1fjcQyPVapo44sVZiLO2ShaGFZ9ayI52cTlWDhxCipVgBeHmomKhuqVbbzePwu8s6VTVKSZoKAbFbzAQBtXr5qIj7+R0FSoYGXlKm/OnTNRuLCN9eu9SE42ERtromBBGx4e8ME4X7Zv96ThYymUKJF9KoJi55RX7OTJk/niiy84ceIE3bt3Z/Xq1Y77vvzyS2d04V/dLiM13/TfOXfcwtYlSQya5c+wL3LR5Ckvln6QiM1mo9YTXrTr74Onlwm/3CYadPbmwM4U53Q+A7h9/Lc5b5lvmuJy+jh8sxjGzIZJS6HN0/DpWPv/LuwLyJ0PJn8JExbD1TjY+D+ndD1D3Pb43xR/xHEb6xfbeHuWmfeXeNCyu4l5Y42Vk22rbLR6OvskAXD72E23xL5xsYXhszx5d4kXLbqb+WysvWJqSYHDe2z0ftuDIdM9iY+zse6z7PNGeNvn/k2HsWYVeOU5G4NGmPnPi2bMJsgbYMPLE1JSYOcuE1PGWPlqtpXLsTBtbvZJhK3pOPdVqWKhV68kRo3y46X+/pjMEBBgw9PzxoPfeTuR1V9fITbOxOeLsscHwDux2UxOu7maU85a27dvZ+7cuYwcOZIlS5Ywbdo0vv32WwDDidRV8gWaiLt4ox+x0Tb8coO3740DdHSPhZIPezgmB9dp58X5k1biY+G3zcmcO25xtLUBZk/XH9z0cvf4CwTB5Zgby5eiwT+3DR/fG+sO7IKyj9gnRwM0bQ9nTsKVWPsk6gYt7UOF/rnsFaJDfzg3hvuRPxAux9w4/pej7ZPBfW46/n/ttlHmEROB1ycIN2xv4uxJuBprv//UURtWC5St7NSu37f8gSbi7hD7od1WSj1ickyOfqy92RF73oImKtU34ZvLhKeXiRrNzJz4y/XntPQqEgxRF27EGhkNAXls+PvdaHM1HmpWsfG/uVaWzbbSorE9vrwBEFQIHm9oI3cu8PaC9i1t/LE/+7z2g4OsXLgp/qgoE3ny2PC7Kf74eKhaJYXZs+OZ9Wk8jRraP+QFBMAvv3oQHW1/vJ8fNG+WzJHD2eMiAbnBKYmQzWbDdP0SmlKlSjFr1iw++OADfv75Z8d6Vypb3YNTB61En7F/PPp1XTIV6hpHDYs8aObEPgtXLtrb/BVuIX+wiVx5TUSetLJ5kX1eTPI1Gz+FJVGpUfa5IM/d43+4Bhw7COfP2Je3r4Wq9YxtSpazT4KOvWhf/m0nFAqGPHmhZFn70BjYPyH/8ROUqei8/t+vCjVMnDwIkWfsb3A/rLVRqZ7xdVmirImje23EXk+Y94ZDwWDIndfe7ug+G+WqmLLE6/luPFTDxImDNqKux75zrZVHb4m9eFkTf++1OT4s7Au3OWKv/JiJ33fYSLpmn1P1Z7iVkuWzz/+gfi0bew/AydP25a/WmGjWwJjIRUZDn8FmrlyfI/Xp5ybaNLdhMkHLxjY2bDOReM1eXdu8w8SjFbJPIlizpoW//vLg9Gn7MQsL86JBfWM1OzraxOAQf65ej3/RIm+aNU3GZIJt2zxZ+Lk3NhskJcG2bV5Uq5Z9quFiZ7I5oSQzY8YMdu7cyfDhw6lc2f6Rcffu3QwYMICkpCR27959V9tb/neNDO/joV9T2PRZEpYUGwUKm+n6hi8Xz1pZ9d9rDJhhnyjzU1gSP3+TjIenCb88Jtq97E3wAx4kJdr4ZuY1Th20YLXAI4950qK3d7Z6U8gu8Qd5xGX4NgH2/QKr5tsTmcAi0HcoRJ2Dz0PtV4kBbF1jv3l6gn8e6PEqFC1lrwp9+TGcPGovqVesBt1etLfLaIm2zEkw9/9iv3zekgKFisAzQ81cOAtLp1oZ9on9E+73a6zsCLPhcT3+J18xU6SU/Rgvm2ElbwFo1SPzPltZMulz24Ffrl8+nwKFipjoMdSDC2dtfDXVwtBP7DPif1hjYUeY9fqxN9HlFQ+KlDJhtdjYuNTK799bsVrsSdN/BqW+/D4jtPRLzPBtAnz/E4TONpOSDCWKwbi3rZyOgFGTzKycZ//gs3iliS9XmbDaoHolG+8MtuHrY58TN2uRiW+32OcNVixnY8wbtlSX32eESEt8xm8U+OknD+bM9SElBYoWtfHW8ATOnjUzabIvc+fY97lqlRdfr/bCZjXxaKUUXht0DR8f++XzU0J9OX7cjMkEjzVI4bnnkgxDaxmlaLGIOzfKQJXWjHbavvZ1eNdp+0qLUxIhgPDwcIKCgnjwwQcd686ePcv8+fN555137mpbmZEISfaQWYlQdpFZiVB2kFmJUHaRWYlQdpFZiVB2oUQo8zjtrFqvXr1U64oUKXLXSZCIiIhkLn2hooiIiIgbcN86u4iIiKQpC1zQ7TSqCImIiIjbUkVIREREDLLCFx06iypCIiIi4rZUERIREREDVYRERERE3IAqQiIiImLgRheNqSIkIiIi7ksVIRERETHQHCERERERN6CKkIiIiBi50SQhVYRERETEbSkREhEREbeloTEREREx0GRpERERETegipCIiIgY2DRZWkRERCTnU0VIREREDDRHSERERCSLCQsLo02bNrRo0YLFixenuv/YsWM8++yzdOjQgeeff57Lly/fcZtKhERERMTIZnLeLZ3Onz9PaGgoS5YsYfXq1Xz11VccPXr0RpdtNl5++WX69evHmjVrqFixIrNnz77jdpUIiYiISJa3c+dO6tatS758+fD396dVq1asX7/ecf/+/fvx9/enUaNGAPTv35+ePXvecbuaIyQiIiIGzrxqLDY2ltjY2FTrAwICCAgIcCxHRkYSGBjoWA4KCmLv3r2O5X/++YdChQoxbNgwDhw4QPny5Rk5cuQd96+KkIiIiLjMwoULad68earbwoULDe1saWRnJtONobWUlBR++eUXnnnmGcLCwihRogTjx4+/4/5VERIREREjJ1aEevfuTefOnVOtv7kaBBAcHMyuXbscy5GRkQQFBTmWAwMDeeCBB6hUqRIA7dq1Y9CgQXfcvypCIiIi4jIBAQEUL1481e3WRKh+/fqEh4cTExNDQkICGzdudMwHAqhWrRoxMTEcPHgQgC1btvDII4/ccf+qCImIiIhBVvweoeDgYEJCQujVqxfJycl069aNypUr069fPwYNGkSlSpX4+OOPGTFiBAkJCRQuXJiJEyfecbsmW1qDblnc8r9ruLoL4iJBHnGu7oJLJdrc97OLxc0L2C39El3dBZeKtMS7ugsuVbRYhFP3V3rxh07b1/GebzltX2lx37OqiIiIpC3blUjunXt/xBIRERG3pkRIRERE3JaGxkRERMQgK06WziyqCImIiIjbypYVoa653PfKoUjLFVd3waV8TR6u7oJLXbKmuLoLLlPSM4+ru+BSOxLd+3Prh9XauLoLLrX+opN3qMnSIiIiIjlftqwIiYiISGbSHCERERGRHE8VIRERETHSHCERERGRnE8VIRERETFSRUhEREQk51NFSERERIz0zdIiIiIiOZ8qQiIiImJg0xwhERERkZxPFSERERExUkVIREREJOdTIiQiIiJuS0NjIiIiYqTL50VERERyPlWERERExMCkydIiIiIiOZ8qQiIiImKkipCIiIhIzqeKkIiIiBjpqjERERGRnE8VIRERETHSHCERERGRnE8VIRERETFSRUhEREQk51NFSERERIxUERIRERHJ+VQREhERESN9j5CIiIhIzqdESERERNyWhsZERETEwKTJ0iIiIiI5nypCIiIiYuRGFSElQtdtC4fQ2ZCUDA+VgfeHQe5cxjZfrIDFq8DXB8qUhJEhkC/Aft+SVfC/tXDtGjzyELz/Jnh7Oz+OexX+kwez5/qQnGSiTBkLw4YmkuuW+Fes9GLV1974+NgoWdJKyGuJBATAlSswcbIv//xjxmqDJ1qm0OPpJNcEco9+CDczc643SclQtoyVd4YmpTr+y1Z6svxrT3y8odQDVoa+lkTeAGObYaO8KVTQxtDXkp3X+Qzw80+eLJjrQ3IylC5jJeSNhFTHf/UqL9Z87Y23D5QsaeXVQQkEBNif8zP+68vhQx5YrVChooUBgxLx8XFNLHfL3V/7e3+GVQsgJRmKlYbeIeB3S/y//QhrFoHJBP65oVcIBBUFqwWWfgyH99nbPVoLuvWzt8tOaresRJ9RXfHy9uT4/tOEDvqM+LjENNu+/nEfTvx1hhUzNjrW5QrwY/K6YUwZsIAjv590Vrclg2hoDIi5BO+Mh2lj4dsvoHhR+GiWsc3Pe2DuUlgwBVbNg0Z1YfRk+30bv4fFK2H+FAhbCInXYOFyp4dxzy5dMjF+oi9jxyTwxedXKVrUyqw5xnexPb95sPRLb6Z8FM+8OfHUrZPC5Cm+AMxb4ENgIRufzY9n1ifxrF7jxZ/7s89T6+IleH+iDx++e43lnydSrIiNT2Z7Gdrs+s3M50s9mfHRNb6Ym0j9OhY+/Mj4brdoqSe/7/VwYs8zxqVLJj6a5MvIMQnMW3iVwkWszJ/ra2jz+28eLPvSh/GT45k5+yq16qQwbYofAEsX+2CxwMzZV/l0zlWSrpn4ckn2yILc/bUfdwkWfgT9R8LYeRBYGFbON7ZJugbzJsDLI2HUTKhSD76cab/vp81w7jSM/hRGzrQnRLt3OD2M+5K3YG6GzOjD2F6f8ELtEZw9GUWf0V1TtStRvgjjV79Ow041DetrtajEtM3vULxcYWd1WTKY096tTpw4wfnz5wFYvnw577//PuvWrXPW7v/Vj7/CoxWgVHH78tMd4ZvvwHZTaXD/YahXAwoH2ZdbNIKtO+2fItdsgOeesn9CNJthzOvQoaXz47hXv+7yoMJDVooXtwfcsUMy3232MsR/+LCZGjUsBAXaVzZqmMLOcE+Sk2HQgGu8/PI1AC7EmEhKTv2JOiv7+VcPKj5kpeT1+Lt0TGH9Zk9D/AcPm6ldw0rw9fibNrTwQ7gHydcLP7t+MxP+qwedO6Q4u/v3bc8uDx56yEKx4lYA2nVIYsstx//IEQ+qVU8h8Hr8jz2WzM8/2Y9/pcop9OiZhNkMHh7wYFkLkeezR0nA3V/7B/bAAw9BcDH7cuN28PMWY/xWq32UJCHevnwtAby8btx3LRGSk+0VJUsyeGWjahhA9WaPcPi3E0QciwRg7bxtNHuyTqp27V9oyqYlP7Lj612G9R1fbM5Hr8wn5twlZ3RXMsEdh8bmz5/PjBkzsFgsFC1alIceeshxK1++PMWLF7/jTj777DMWLVqE1Wqlbt26nD17lhYtWrBixQqOHz/Oq6++miHB3KtzkVAk6MZycCBcuWriarzN8YZeqaK9PH7mHBQrDKu+heRkE5dibZw4BZUqQL+hEBkNNSrDG/1dE8u9iIw0ExRkdSwHBtq4etVEfDyO4ZGKFaysWOXNuXMmChe28e16L5KTTcTGmihY0IanB7w/zpft2z157LEUSpSw3mZvWc/5KBPBN8UfdD3+q/E3ErpHKlhZttKTs+dMFCls45v1niQnm7gcCzabidAZ3kybmMiqMK/b7CXriooyUyjwxjtfYKCN+FuOf4UKFlav8ub8eRPBwTY2bLhx/GvUtDgee/68iVUrvXktJO1hhazG3V/7MVFQoNCN5fyBkBhvIjHe5hge8/WDZwbChBDIlcee/AybYr+vfgvY/T0M6wkWCzxcHarUdX4c9yOwWAGizsQ4lqMiLpIrwB//PL6G4bFP3lwCQNVGFQ2PH/HkVKf009l01dhNZs2axcSJE9m0aRPvv/8+derU4fz588yZM4dOnTpRs2ZNnn766X/dxooVK1i3bh1ffPEF69evZ9asWfTs2ZOZM2eyYcOGDAvmXllv855tvum/U6sKvPIcDBwB3V60j4HnDbDh5QnJKbBzF4SOgeWz4XIsTJ3rjJ5nDOttnvA3x1+lioXneiUxYpQfL/b3x2SGgAAbnp43Hjzi7URWf32FuDgTCxdln4+Fttscf4+b4q9WxcoLvZIZNsqH3i/5YDLZ4zeZYMRYb0JeTaJQQef0N6Pd7vl/c/yVKlt45tlrvDfKnwEv58Jsgjx5rHjd9FHqyGEzrw/ORYeOSdStlz0qY+7+2r/dc9980wjv6ePwzWIYMxsmLYU2T8OnY+1Vo7AvIHc+mPwlTFgMV+Ng4/+c0vUMYzKnXb20WLLPhzm5P3esCOXOnZsmTZrg6elJUFAQNWrUMNx/+vRpjhw58q/bsFqteHt7U6xYMfr27YvPTbMoLRbLvzzSOYoEw96/biyfj4a8eWz4+91YdzXefkLs1ta+HB0D/51vL4kHFYLHG96oHrRvCTMXOq//9ys4yMpff914KkRHmciTx4bfTfHHx0OVKim0bWMfC4qJMTF/gQ8BAfDLrx6UKW2lUCH7/6x5s2S2f599KiPBwTb+/OvGO19UlImAW+K/Gg/Vqlrp0Nb+CfFCDMxa4EXEWTMRZ01M/cT7+noTViskJZl4Z2j2mDAeFGTj4MEbbwbR0SZy57Hhe8vxr1QlhSeuH/+LMSYWLvAhT4A9Ed62xZPp//Xj1YEJNGuePZIg0Gu/QBAcP3hj+VI0+Oe24XPTFLEDu6DsI/bJ0QBN28OyWXAl1j6Juvsr4Ollv9VvYZ8j1LKbc+O4H1GnY6hQo7RjuVDRfMRdvMq1+Ozx+s00+omNG1588UWWL7/97L/ixYvTtGnTf91Gy5YteeaZZ7BYLAwcOBCAgwcP0qNHD1q3bn2XXc54DWrBHwfgxGn78ldroFkDY5vIaOg9GK5ctS/P/BzaNrd/OmzVGDZss0+UtNlg8w77vIPsolZNCwf+8uD0afsTf02YFw3qG9/MoqNNDA7x5+r1+D9f5E3zpsmYTLB1myeffe6NzQZJSbB1mxfVq2WfN8M6NS38+ZcH/1yPf2WYJw0bGBP06GgTLw/2cRz/+Yu8aNkshUqPWAlblsgXc+23Lh1SeLxpSrZJggBq1Ezh4AEPzpy2nw7WhnlTr77xqrcLF8y8OSSX4/gv/sKHJs1SMJlgx3ZPPvnYlw8nXM1WSRDotf9wDTh2EM6fsS9vXwtV6xnblCxnnwQde9G+/NtOKBQMefJCybL2oTGAlBT44ycoYxw5yvJ2b9lPhZoPUrSMfYy0bZ8mhK/73aV9Eucy2Wy2fx0JrFatGsnJyTRq1IiGDRtSsWJFHnroIfxu/ricDr/++iu1atVyLB87doxTp07RuHHju+609Vz5u37MnWz/yX4JbXIylCgG49+G0xEwcpL9ShGwXx2yZJV9KKl6JRg52H45rcUCny6Cb7eAxQoPl4N338icCcORlisZv1Hgp/+/fD4FihW18fbwBCLOmpk02Zd5c+yzJFeu8mLVai9sVhOVKqUweNA1fHwg7gpMCfXl+HEzmKBhgxT6PJdkGF7IKL6mzLkq68efzHwyx5uU6/GPfusaEWfNfDDJmy/m2qtAy1d58r+vPbHaoMqjVt54LQnfWy6OmvOZF5cuk2mXz1+yZk6i8cvPnsyf60NKChQpYmXo8ATOnTUT+pEfM2fbM4DVX3sRttobmxUeedTCq9cvke/TKxdXr5goWOjGqeSRRywMeC1j5wmV9MyTodv7f9nltb8jk6Zd7fsFVs23JzKBRaDvUIg6B5+H2q8SA9i6xn7z9AT/PNDjVShayl4V+vJjOHnUPpxYsZp9+NAzE76Y5cNqjTJ+o9fValGJPqO64OnlydkTkUzqP58ipQox+L+9ebXRe4a2aV0+D7Dwj/G833tmpl0+v/6ic8dcy0yd4rR9HRs8xGn7SssdE6FTp05x8OBBDh06xKFDhzh48CAREREUL17cZfN7MiMRyi4yKxHKLjIrEcouMisRyg4yKxHKLjIrEcouMjMRyg6UCGWeO+btJUqUoESJErRo0cKxLj4+nsOHD2dqx0RERMRFdNXYv/P396dq1aoZ3BURERER59JPbIiIiIiBvkdIRERExA0oERIREREjmxNvdyEsLIw2bdrQokULFi9enOr+GTNm0LRpUzp27EjHjh3TbHMrDY2JiIhIlnf+/HlCQ0NZuXIl3t7edO/enTp16lC2bFlHmz///JMpU6ZQrVq1dG9XiZCIiIi4TGxsLLGxsanWBwQEEBAQ4FjeuXMndevWJV++fAC0atWK9evXM2DAAEebP//8kzlz5nDq1Clq1arFsGHDDL9mkRYNjYmIiIiRE4fGFi5cSPPmzVPdFi40/l5NZGQkgYGBjuWgoCDOnz/vWL569SoVK1Zk2LBhrFq1itjYWD755JM7hqqKkIiIiLhM79696dy5c6r1N1eDANL6/meT6cZvouXKlYs5c+Y4lvv27cvbb79NSEjIv+5fiZCIiIgYOPPy+VuHwG4nODiYXbt2OZYjIyMJCgpyLEdERLBz5066dbP/6q/NZsMzHb/3oqExERERyfLq169PeHg4MTExJCQksHHjRho1uvHTK76+vkyaNIlTp05hs9lYvHix4VcxbkcVIRERETGyme7cxsmCg4MJCQmhV69eJCcn061bNypXrky/fv0YNGgQlSpV4r333uPll18mOTmZ6tWr06dPnztuV4mQiIiIZAvt27enffv2hnU3zwtq1aoVrVq1uqttKhESERERI/3EhoiIiEjOp4qQiIiIGOhHV0VERETcgCpCIiIiYqSKkIiIiEjOp4qQiIiIGGiOkIiIiIgbUEVIREREjFQREhEREcn5lAiJiIiI29LQmIiIiBhpaExEREQk51NFSERERAx0+byIiIiIG1AiJCIiIm5LiZCIiIi4Lc0REhERESPNERIRERHJ+VQREhEREQNdNSYiIiLiBrJlRah1qVqu7oLLmLy9Xd0Fl7ImXnN1F1zK7Ovj6i64jme2PF1lGJOXe8dvS4h1dRfciypCIiIiIjmfe3/EEBERkdRUERIRERHJ+VQREhEREQNdNSYiIiLiBpQIiYiIiNvS0JiIiIgYaWhMREREJOdTRUhEREQMNFlaRERExA2oIiQiIiJGqgiJiIiI5HyqCImIiIiRKkIiIiIiOZ8qQiIiImKgq8ZERERE3IAqQiIiImKkipCIiIhIzqeKkIiIiBipIiQiIiKS86kiJCIiIga6akxERETEDSgREhEREbeloTEREREx0tCYiIiISM6nipCIiIgYaLK0iIiIiBtQRUhERESMVBESERERyflUERIREREjVYREREREcj4lQiIiImJgcuLtboSFhdGmTRtatGjB4sWLb9tu27ZtNGvWLF3b1NCYiIiIZHnnz58nNDSUlStX4u3tTffu3alTpw5ly5Y1tIuOjmbChAnp3q4qQiIiImJkc94tNjaW06dPp7rFxsYaurRz507q1q1Lvnz58Pf3p1WrVqxfvz5V10eMGMGAAQPSHapbV4RqP1GVPmOfwsvHk+P7ThHafw7xcQnpapMnfy4G/rcvZaqUJPHqNTZ+/j1rZm4EoHyNMvSf/Cy+/j6YPcws+yiMLUt/dEWId6V2q8r0Gd3NHuufpwkdMJ/4uMQ0274+83lOHDjDiuk3noS58vox+du3mPLqfI78dsJJvb53tVtXpe/73R3HdsqLs1Mf/9u0MZtNvDqtD5UbVQDgl29/Z87wJQBUafwwL07siYenB7EX4vj0jUUc2/uP0+O7G+527G9Wu0Ul+ozqgpe3J8cPnCZ00MLbxz6jDycOnmHFDPtr3dvXi1cn9aB8tVKYzWYO7j7Gx0OXkJSY7MwQ7kutxx+lzzsd8fL24viB00wN+YL4K2nHP2RaL04ejGDFzO8A8M/jS0josxQvVxizycR3y35i+fX/TVamc3/WsnDhQmbMmJFq/YABAxg4cKBjOTIyksDAQMdyUFAQe/fuNTzm888/5+GHH6ZKlSrp3r/bVoTyFsrD67NfZGz3qbxQeSjnjkfS9/2n0t3mpUnPkHA1kRervsngRqOp1aoKdVpXA2Dkl6+xaOz/eKXO24zoOJGXJvSk6IPBTo/xbuQtmIchnzzP2Gc/5oUab3P2RBR93n0yVbsS5YswPuxNGnauZVhfq2Vlpm0dRfHyRZzV5fuSt1Ae3pjzEu89NZXnH32Ds8fP8/wH3dPdpnnPhpQoX4SXqg2jf423qNyoIg271sE/wI9Ry0KYM3wJ/WsMZ/rABbyzZBBe3ln3M4e7Hfub5S2YmyEznmNs75m8UGckZ09E02dUl1TtSpQvzPivX6dhpxqG9U8PaYuHhwevNHyPlx8bg4+vN0+FtHZW9+9b3oK5GTKtF+/3nU2/BmM4dzKaPiM6pWpXolxhPlwxmIYdjPH3Gt6B6LOXeLnxWAY9MZ62vRtRoWZpJ/X+3ujcnz4mm/NuvXv3ZvPmzaluvXv3NvTJZkt9KZvJdGOW0eHDh9m4cSOvvPLKXcXqkkRo/PjxrtitQfXHK3Fo9zEi/j4PwDdzvqNZ9wbpblOuWmk2L/kBq9VGSrKFX779nce61MbLx4svPljJb1v2AxB9JobLF64QWLygE6O7e9WbP8LhPccdsa6dt4VmT9ZN1a79i83ZtHgHO1b9aljfsf/jfNR/LjFnLzmju/etRovKHNp1jIij5wD4ZtZ3NHu6QbrbeHiY8c3lg5ePF14+nnh6e5KcmEyxsoW5ejme37faj/+pQxHExyZQsW45J0Z3d9zt2N+setNHOPzbCSKORQKwdv42mj1ZJ1W79s83ZdOSH9nx9W7D+n3hh1n60VpsNhtWq42je/8hKIu/1m9WvUlFe/zHowD4ZuH3NO1aO1W7dn0as2npTnasMcb/6TvLmDNmBQAFgvLi5eNJfGxCqsdnJTr3Zz0BAQEUL1481S0gIMDQLjg4mOjoaMdyZGQkQUFBjuX169cTFRVF165defHFF4mMjKRHjx533H+mf0x96623Uq3bsmULly9fBuDDDz/M7C6kKbB4QaJPxziWo07HkCuvP/55/Bwl0n9rc/DXv2ne4zH27zyMl48nj3WuRUqyheRryWz4bLvjMa2fb4pfbh/++vmI84K7B4HFChB1c6xnLl6P1dcwTPDJG18AULXxw4bHj+gyxTkdzSCBxQsQdfqCYznt43/7Nhs/307DrnVYcuJjPDzN7P5uHz+t3YN/Hj/8cvtS4/FK7P5uH+VrlOGBh4tToEg+Z4eYbu527G8WWCw/UWcuOpajIi6SKyCN2IctBaBqo4qGx+/ZesDxd1DxAnTu/zjTQhZlcq8zTqGi+YmKuBF/dMQlcgX44Z/b1zA8NvPtrwCo2rBCqm1YLVaGfvwcj7Wrzs5vf+f00fOZ3/H7oHN/OmXB7xGqX78+06dPJyYmBj8/PzZu3MjYsWMd9w8aNIhBgwYBcPr0aXr16sWSJUvuuN1Mrwjly5ePbdu2UaFCBWrXrk3t2rXx9/d3/O0qZnPaF+1ZLNZ0tZk9bDE2m41Pfv6A0ctC2LP5T1KSUgzt/vNGe54d0ZXRXT7K8nMGTOn4f+QkJnPaT33rTfH+W5tnRnblcnQsTxXvT4/SA8iTPxddB7chPi6B0V0/ovuwjszc9SGPP9OQ37fuT/XcyErc7djf7HbH+G5jL1ulJJPXvcmauVv5ZePeOz8gizDfLn7r3cU/6dXPeKriUPLky0WP19tmRNcyjc792VdwcDAhISH06tWLTp060a5dOypXrky/fv3Yt2/fPW830ytCw4YNo1GjRkydOpUhQ4ZQp04dFi5cSOfOnTN71/8q8tQFKtS6ccldoWIFiIu5wrX4a+lqE1CiIPPeXkrcxasA/Of1do4yqpe3J6/PfYkHKhQjpMkYzp+8UcrLqqJOx1Ch5oOO5UJF8xN38QrX4pNc2KvME3Uqmgq1b4q3WAFiY66QeNPx/7c2j3WqxceDF5KSbCElOYFNi3bQsEttVk77lsSriQxt8b7jcXP3TnI8N7Iidzv2N4s6fYEKNW7MaSlUJB9xF6/eVeyNu9RiwKSefPzmErat+CUzuplpIk/H8FD1Uo7lu42/epOKnPgrgpjzl0mMv8a2Vb/yWLtqmdTbjKFzf/bWvn172rdvb1g3Z86cVO2KFy/Oli1b0rVNp8wRqlevHrNmzWLJkiVMmDABi8XijN3+q93f7aNC7bKOiWxt+zUn/Jvd6W7Trl9zeo3qBkC+oABa923K1q92AvDOkkH45/FjcJN3s80LYffmP6lQq8yNWPs2JXztby7uVebZvWkfFWuXo2jZwgC0e7E54WG7093myG8naNTNPo/Gw9ODeu2rc/CXo9hsNt5f/SblqtvfXBt2rUNKsiVLXzXmbsf+Zru3HqBCzTIULWOfZ9C2T2PCv/093Y9/rEN1Xv6wO293Dc12SRDAnu1/UaFGaYqWtl+J06Z3Q8LX/5HuxzfqWIOeb9grQF7enjTqUIPffziUKX3NKDr3p5MTL593NaddypIvXz6mTZvG8uXLOXTI9S+Uy1GxfPTiLEYufQ1Pb0/OHotk0vMzKVe9NCEz+/FKnbdv2wbgy4lreHP+y8zaPR6TycSi91dyePcxHq5XnnrtanDqcAShW0c79jfvnaXs/u7eS3eZ7XJ0HFNemc+Iz1+xx3o8kkkvzaVctVIMnt6HVx8bfeeNZCOXomKZ3G8WI798DS9vTyL+Ps+kvvbjP2RWP16u9fZt2wB8+sYiXp3am3n7JmOxWPl9y598NSkMgA97zSDk0354ensSc/YSY7pl7Tk07nbsb3Y5Oo4pAxYw4rP+12OPYtLL8yhX9QEGT+vNq43f+9fH9xnZBUwmBk+7cXXLgZ+P8vGbd56XkBVcjo4j9LXPeWfei3h6eXD2ZDSTB3xGuSoleW3KMwxoPu5fHz9n9AoGTurBzO0jsdlshH/7B6tnb3VS7++Nzv1yK5MtrevRsrhWvj1d3QWXMXl7u7oLLmVNvHbnRjmY2dfH1V1wHc+s+xUEzmDycu/4rZdj79woB9uQePufk8gMVQeGOm1fv08Pcdq+0uK23yMkIiIi4t4fMURERCS1bDdWdO9UERIRERG3pYqQiIiIGJhUERIRERHJ+VQREhERESNVhERERERyPlWERERExEBzhERERETcgCpCIiIiYqSKkIiIiEjOp4qQiIiIGKkiJCIiIpLzKRESERERt6WhMRERETHQ5fMiIiIibkAVIRERETFSRUhEREQk51NFSERERAxMNvcpCakiJCIiIm5LFSERERExcp+CkCpCIiIi4r5UERIREREDfY+QiIiIiBtQRUhERESMVBESERERyflUERIREREDzRESERERcQOqCImIiIiRKkIiIiIiOZ8SIREREXFbGhoTERERA02WFhEREXED2bIiZE1KcnUXXMedYxcsV5Jd3QVxEZOnl6u74FK2FD33nUoVIREREZGcL1tWhERERCTzaI6QiIiIiBtQRUhERESMbO5TElJFSERERNyWKkIiIiJioDlCIiIiIm5AFSERERExUkVIREREJOdTRUhEREQMTFZX98B5VBESERERt6WKkIiIiBhpjpCIiIhIzqdESERERNyWhsZERETEQF+oKCIiIpLFhIWF0aZNG1q0aMHixYtT3b9p0ybat29P27ZtGT58OElJSXfcphIhERERMbLZnHdLp/PnzxMaGsqSJUtYvXo1X331FUePHnXcHx8fz3vvvceCBQtYu3Yt165dY9WqVXfcrhIhERERyfJ27txJ3bp1yZcvH/7+/rRq1Yr169c77vf392fLli0UKlSI+Ph4Lly4QEBAwB23qzlCIiIiYuDMOUKxsbHExsamWh8QEGBIZCIjIwkMDHQsBwUFsXfvXsNjvLy82L59O2+++SZBQUE89thjd9y/KkIiIiLiMgsXLqR58+apbgsXLjS0s6UxjGYymVKta9y4MT///DNNmzZlzJgxd9y/KkIiIiJi5MSKUO/evencuXOq9bcOawUHB7Nr1y7HcmRkJEFBQY7lS5cu8eeffzqqQO3btyckJOSO+1dFSERERFwmICCA4sWLp7rdmgjVr1+f8PBwYmJiSEhIYOPGjTRq1Mhxv81mY+jQoURERADw7bffUr169TvuXxUhERERMciK3yMUHBxMSEgIvXr1Ijk5mW7dulG5cmX69evHoEGDqFSpEmPHjuWll17CZDJRtmxZ3n333Ttu12RLa9Ati2thftLVXRARcSqTp5eru+BStpRkV3fBpTZZlzt1fw07T3bavnasesNp+0qLKkIiIiJilP1qJPdMc4RERETEbakiJCIiIgZZcY5QZlFFSERERNyWKkIiIiJipIqQiIiISM6nREhERETclobGRERExECTpUVERETcgCpCIiIiYmR1n5KQWydCtdtU5/lxPfDy8eL43pN89MJM4uMS0tVm5LLXKVa2sKNd4dJB7N1+gFGdJlCyYnFCZr2EX25fbDYb895azK6Nfzg7vDtS/O4bvzvHDoq/duuq9H2/O14+nhzfd4opL85OHf9t2pjNJl6d1ofKjSoA8Mu3vzNn+BIAqjR+mBcn9sTD04PYC3F8+sYiju39x+nx3Ym7H38xctvfGstbKIA5f04h5LERnDl6jhfG98Qvjx/TX517V20Aytd8kFHLXyek4UiiTl9g8pYxbFq0nQ0LtvJg1VJ8tPVduhTqg9Vive9+ZxTF777xu3PskH3jz6jfGstbKA9zfp/I4CbvEnH0HM+P645/bj+mD1qQrjYtnm1Ei2caMrz1OExmM1O/H8Oyj75h96a9LDryX8Z2n8rvW/dT4qGijFkxhP7Vh5OclHLf/c6o3xrLrsff2b811rjNRKfta/u6N522r7Q4ZY7Q3r17HX+Hh4czfvx4Jk+ezB9/uC5TrtGyMod//ZszR88BEDZzI817NLzrNp5enrz52QBmhnxG1OkLAJg9zOTJnwsA/zx+JCUmZXY4d03xu2/87hw7KP4aLSpzaNcxIq7H9s2s72j2dIN0t/HwMOObywcvHy+8fDzx9PYkOTGZYmULc/VyPL9v3Q/AqUMRxMcmULFuOSdGd2fufvwlNacMjY0ePZpVq1axePFivvzyS7p27QrAqFGjePLJJ3nmmWec0Q2DwBKFiDod7ViOOn2BXHn98c/j5yiRpqfNE88340JEDD9+/Yuj3fQBc5m0eTRdBrcjX1Bexj0dmqU+EYPid+f43Tl2UPyBxQs43rgBok7HpI7/X9ps/Hw7DbvWYcmJj/HwNLP7u338tHYP/nn88MvtS43HK7H7u32Ur1GGBx4uToEi+Zwd4r9y9+OfXu501ZhT5wgtW7aMzz//nPz58wPQrVs3unXr5pJEyGw2pbn+5idtetp0HdyW0JdmOZa9fLwY8WUIk/p8zM9r91CxTjneWzOcQ7/+bTixuJrid9/43Tl2UPwmc9oDATfH9m9tnhnZlcvRsTxVvD/eft6M+d8Qug5uw4qp6xjd9SP6vPcfXhjfg307DvL71v2kZMCwWEZy9+MvqTllaCwlJQWr1UrBggXx9/d3rPf29sZ8mxdcZov8J5oChfM7lgsVK0BszBUS46+lu82DVUvh4enB3u0HHG1KP1oCH38ffl67B4C/fj7Cyf2nqFAna5WHFb/7xu/OsYPijzoVbajSpBX/v7V5rFMtNny2nZRkC/GxCWxatIMqjR/GZDKReDWRoS3e5+Wab/FJyEKKPhhMxN/nnRjdnbn78U83m815NxdzShaSP39+GjduzNGjRxk9ejRgnyvUvXt3nnjiCWd0IZXdG/+gYt1yjtn/7fq3JHz1r3fVpnLjh/l965+Gx5w5eo5cef15uF55AIqUCaZkxWIc/e14ZoZz1xS/+8bvzrGD4t+9aR8Va5ej6P/H9mJzwsN2p7vNkd9O0KhbXQA8PD2o1746B385is1m4/3Vb1KuemkAGnatQ0qyJctdNebux19Sc+pVY8eOHSM2NpaqVauye/du4uLiaNKkyV1vJyOuGgOo3boafcf1wMvbk4i/zzOx9wyKlAliyJyX6V996G3bxF28AsDAGc9z4exFlnyw0rDdKk0eod+EZ/D29SYlOYUvxv6Pnbe80LICxe++8btz7JA948+oq8YAaj1Rlb7vP+WIbVLfmRQuHcSQWf14udbbt20Td/EqeQrk5tWpvSlXrTQWi5Xft/zJrDcXY0mxUKlhBV7+qBee3p7EnL3E1Ffmcu54ZIb0OaOuGoPsefydfdVY01YTnLavrRuGOW1faXHby+dFRLKTjEyEsqOMTISyIyVCmcetv1BRRERE0pDtSiT3Tr81JiIiIm5LFSERERExMGW/WTP3TBUhERERcVtKhERERMRtaWhMREREjLLnL4PcE1WERERExG2pIiQiIiIGmiwtIiIi4gZUERIREREj9ykIqSIkIiIi7ksVIRERETHSHCERERGRnE8VIRERETEwuU9BSBUhERERcV+qCImIiIiR5giJiIiI5HyqCImIiIiBSb81JiIiIpLzqSIkIiIiRpojJCIiIpLzqSIkIiIiRu5TEFJFSERERNyXEiERERFxWxoaExEREQOTJkuLiIiI5HyqCImIiIiRKkIiIiIiOZ8qQiIiImKkn9gQERERyflUERIREREDXTUmIiIi4gZUERIREREjVYREREREcj5VhEREsgFbSrKruyDuJItWhMLCwpg5cybJyck899xz9OzZ03D/d999x/Tp07HZbBQvXpwPP/yQvHnz/us2VRESERGRLO/8+fOEhoayZMkSVq9ezVdffcXRo0cd91+5coUxY8Ywe/Zs1qxZw0MPPcT06dPvuF0lQiIiImJkdd4tNjaW06dPp7rFxsYaurRz507q1q1Lvnz58Pf3p1WrVqxfv95xf3JyMmPGjCE4OBiAhx56iLNnz94xVA2NiYiIiMssXLiQGTNmpFo/YMAABg4c6FiOjIwkMDDQsRwUFMTevXsdy/nz5+fxxx8HIDExkdmzZ/Pss8/ecf9KhERERMTAmd8j1Lt3bzp37pxqfUBAgGHZlkafTCZTqnVxcXG88sorVKhQIc3t3kqJkIiIiLhMQEBAqqQnLcHBwezatcuxHBkZSVBQkKFNZGQkzz//PHXr1uXtt99O1/41R0hERESyvPr16xMeHk5MTAwJCQls3LiRRo0aOe63WCz079+f1q1b884776RZLUqLKkIiIiJilAUvnw8ODiYkJIRevXqRnJxMt27dqFy5Mv369WPQoEGcO3eOAwcOYLFY2LBhAwCPPvooH3zwwb9u12RLa9Ati2thftLVXRAREXGaTdblTt3fE1VGOm1f6/8Y67R9pUUVIRERETHKfjWSe6Y5QiIiIuK2VBESERERI1WERERERHI+VYRERETEyOrqDjiPKkIiIiLitlQREhEREQNn/sSGq6kiJCIiIm5LFSERERExUkVIREREJOdTRUhERESMrKoIiYiIiOR4qgiJiIiIkeYIiYiIiOR8SoRERETEbWloTERERIw0NCYiIiKS86kiJCIiIkaqCImIiIjkfKoIiYiIiJG+UFFEREQk51NFSERERIxsVlf3wGlUERIRERG3pYqQiIiIGOmqMREREZGcTxUhERERMXKjq8bcOhGq3aY6z4/rgZePF8f3nuSjF2YSH5eQrjYjl71OsbKFHe0Klw5i7/YDjOo0gZIVixMy6yX8cvtis9mY99Zidm38w9nh3ZHid9/43Tl2UPyK373jFyOTzZb9BgJbmJ+8723kLRTAnD+nEPLYCM4cPccL43vil8eP6a/Ovas2AOVrPsio5a8T0nAkUacvMHnLGDYt2s6GBVt5sGopPtr6Ll0K9cFqyTqz8BW/+8bvzrGD4lf82TP+Tdbl972Nu9G6xGtO29e3p6Y5bV9pcdocoR07dhAbGwvA119/zXvvvceKFSuctftUarSszOFf/+bM0XMAhM3cSPMeDe+6jaeXJ29+NoCZIZ8RdfoCAGYPM3ny5wLAP48fSYlJmR3OXVP87hu/O8cOil/xu3f8kppThsY++OAD/vrrL0JDQ5k6dSr79u2jefPmbNq0ib/++osRI0Y4oxsGgSUKEXU62rEcdfoCufL645/Hz1EiTU+bJ55vxoWIGH78+hdHu+kD5jJp82i6DG5HvqC8jHs6NEt9IgLF787xu3PsoPgVv3vHn27Zb7DonjklEfrxxx8JCwvDw8ODbdu2sWzZMry9vXnqqado166dM7qQitlsSnP9zU/a9LTpOrgtoS/Ncix7+Xgx4ssQJvX5mJ/X7qFinXK8t2Y4h3792/GpIStQ/O4bvzvHDopf8bt3/JKaU4bGfH19uXDB/kQoWLAg8fHxACQkJODp6Zr52pH/RFOgcH7HcqFiBYiNuUJi/LV0t3mwaik8PD3Yu/2Ao03pR0vg4+/Dz2v3APDXz0c4uf8UFeqUy+yQ7orid9/43Tl2UPyK373jl9SckggNGDCAbt26MWHCBMqUKcOzzz7LuHHj+M9//kOfPn2c0YVUdm/8g4p1yzlm/7fr35Lw1b/eVZvKjR/m961/Gh5z5ug5cuX15+F65QEoUiaYkhWLcfS345kZzl1T/O4bvzvHDopf8bt3/Olmsznv5mJOu2rs1KlTfPfdd5w8eRKLxUKhQoVo2rQplStXvuttZcRVYwC1W1ej77geeHl7EvH3eSb2nkGRMkEMmfMy/asPvW2buItXABg443kunL3Ikg9WGrZbpckj9JvwDN6+3qQkp/DF2P+x85YXWlag+N03fneOHRS/4s9+8Tv9qrFiA522r2/PTHfavtLitpfPi4iIZBdOT4SKvOq0fX179mOn7Sst+okNERERcVtu/c3SIiIikobsN1h0z1QREhEREbelipCIiIgYqSIkIiIikvOpIiQiIiJGVlWERERERHI8VYRERETEwGbLpj8Wew9UERIRERG3pYqQiIiIGGmOkIiIiEjOp4qQiIiIGOl7hERERERyPiVCIiIi4rY0NCYiIiJGVl0+LyIiIpLjqSIkIiIiRposLSIiIpK1hIWF0aZNG1q0aMHixYtv227YsGGsXLkyXdtURUhEREQMbFlwjtD58+cJDQ1l5cqVeHt70717d+rUqUPZsmUNbUaPHk14eDh16tRJ13ZVERIREZEsb+fOndStW5d8+fLh7+9Pq1atWL9+vaFNWFgYzZs3p3Xr1uneripCIiIiYuTEOUKxsbHExsamWh8QEEBAQIBjOTIyksDAQMdyUFAQe/fuNTzmhRdeAGD37t3p3r8SIREREXGZhQsXMmPGjFTrBwwYwMCBAx3LtjSSM5PJdN/7VyIkIiIiRk780dXevXvTuXPnVOtvrgYBBAcHs2vXLsdyZGQkQUFB971/JUIiIiLiMrcOgd1O/fr1mT59OjExMfj5+bFx40bGjh173/vXZGkRERExslmdd0un4OBgQkJC6NWrF506daJdu3ZUrlyZfv36sW/fvnsO1WRLa9Ati2thftLVXRAREXGaTdblTt1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- "text/plain": [ - "
" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_49_2.png" - } - }, - "output_type": "display_data" - } - ], - "source": [ - "# visual representation of grid search\n", - "# uses seaborn heatmap, you can also do this with matplotlib imshow\n", - "import seaborn as sns\n", - "\n", - "sns.set()\n", - "\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "\n", - "for i in range(len(eta_vals)):\n", - " for j in range(len(lmbd_vals)):\n", - " dnn = DNN_numpy[i][j]\n", - " \n", - " train_pred = dnn.predict(X_train) \n", - " test_pred = dnn.predict(X_test)\n", - "\n", - " train_accuracy[i][j] = accuracy_score(Y_train, train_pred)\n", - " test_accuracy[i][j] = accuracy_score(Y_test, test_pred)\n", - "\n", - " \n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Test Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## scikit-learn implementation\n", - "\n", - "**scikit-learn** focuses more\n", - "on traditional machine learning methods, such as regression,\n", - "clustering, decision trees, etc. As such, it has only two types of\n", - "neural networks: Multi Layer Perceptron outputting continuous values,\n", - "*MPLRegressor*, and Multi Layer Perceptron outputting labels,\n", - "*MLPClassifier*. We will see how simple it is to use these classes.\n", - "\n", - "**scikit-learn** implements a few improvements from our neural network,\n", - "such as early stopping, a varying learning rate, different\n", - "optimization methods, etc. We would therefore expect a better\n", - "performance overall." - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.18333333333333332\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.18611111111111112\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.13055555555555556\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.24444444444444444\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.23333333333333334\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.12777777777777777\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1e-05\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.1527777777777778\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.9111111111111111\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.8888888888888888\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.8722222222222222\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.8305555555555556\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.0001\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.8888888888888888\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - 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warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.975\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.9777777777777777\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.9805555555555555\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.9861111111111112\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.9805555555555555\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.9777777777777777\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.001\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.9444444444444444\n", - "\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/neural_network/_multilayer_perceptron.py:582: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (100) reached and the optimization hasn't converged yet.\n", - " warnings.warn(\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.9861111111111112\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.9888888888888889\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.9888888888888889\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.9861111111111112\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.9888888888888889\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.01\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.9722222222222222\n", - "\n", - "Learning rate = 0.01\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.9527777777777777\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.9111111111111111\n", - "\n", - "Learning rate = 0.1\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.9222222222222223\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.9111111111111111\n", - "\n", - "Learning rate = 0.1\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.9305555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.8388888888888889\n", - "\n", - "Learning rate = 0.1\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.9055555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 0.1\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.8666666666666667\n", - "\n", - "Learning rate = 1.0\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.09166666666666666\n", - "\n", - "Learning rate = 1.0\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.11944444444444445\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 0.001\n", - "Accuracy score on test set: 0.1361111111111111\n", - "\n", - "Learning rate = 1.0\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.1527777777777778\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.16666666666666666\n", - "\n", - "Learning rate = 1.0\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.1111111111111111\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 1.0\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.05\n", - "\n", - "Learning rate = 10.0\n", - "Lambda = 1e-05\n", - "Accuracy score on test set: 0.08888888888888889\n", - "\n", - "Learning rate = 10.0\n", - "Lambda = 0.0001\n", - "Accuracy score on test set: 0.08611111111111111\n", - "\n", - "Learning rate = 10.0\n", - "Lambda = 0.001\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Accuracy score on test set: 0.08888888888888889\n", - "\n", - "Learning rate = 10.0\n", - "Lambda = 0.01\n", - "Accuracy score on test set: 0.08888888888888889\n", - "\n", - "Learning rate = 10.0\n", - "Lambda = 0.1\n", - "Accuracy score on test set: 0.10555555555555556\n", - "\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Learning rate = 10.0\n", - "Lambda = 1.0\n", - "Accuracy score on test set: 0.1111111111111111\n", - "\n", - "Learning rate = 10.0\n", - "Lambda = 10.0\n", - "Accuracy score on test set: 0.1527777777777778\n", - "\n" - ] - } - ], - "source": [ - "from sklearn.neural_network import MLPClassifier\n", - "# store models for later use\n", - "DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - "\n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',\n", - " alpha=lmbd, learning_rate_init=eta, max_iter=epochs)\n", - " dnn.fit(X_train, Y_train)\n", - " \n", - " DNN_scikit[i][j] = dnn\n", - " \n", - " print(\"Learning rate = \", eta)\n", - " print(\"Lambda = \", lmbd)\n", - " print(\"Accuracy score on test set: \", dnn.score(X_test, Y_test))\n", - " print()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Visualization" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "data": { - "image/png": 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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter10_53_1.png" - } - }, - "output_type": "display_data" - } - ], - "source": [ - "# optional\n", - "# visual representation of grid search\n", - "# uses seaborn heatmap, could probably do this in matplotlib\n", - "import seaborn as sns\n", - "\n", - "sns.set()\n", - "\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "\n", - "for i in range(len(eta_vals)):\n", - " for j in range(len(lmbd_vals)):\n", - " dnn = DNN_scikit[i][j]\n", - " \n", - " train_pred = dnn.predict(X_train) \n", - " test_pred = dnn.predict(X_test)\n", - "\n", - " train_accuracy[i][j] = accuracy_score(Y_train, train_pred)\n", - " test_accuracy[i][j] = accuracy_score(Y_test, test_pred)\n", - "\n", - " \n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Test Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Building neural networks in Tensorflow and Keras\n", - "\n", - "Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn\n", - "and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy\n", - "and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer. \n", - "\n", - "In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite\n", - "clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or\n", - "NumPy arrays.\n", - "\n", - "\n", - "Tensorflow is an open source library machine learning library\n", - "developed by the Google Brain team for internal use. It was released\n", - "under the Apache 2.0 open source license in November 9, 2015.\n", - "\n", - "Tensorflow is a computational framework that allows you to construct\n", - "machine learning models at different levels of abstraction, from\n", - "high-level, object-oriented APIs like Keras, down to the C++ kernels\n", - "that Tensorflow is built upon. The higher levels of abstraction are\n", - "simpler to use, but less flexible, and our choice of implementation\n", - "should reflect the problems we are trying to solve.\n", - "\n", - "[Tensorflow uses](https://www.tensorflow.org/guide/graphs) so-called graphs to represent your computation\n", - "in terms of the dependencies between individual operations, such that you first build a Tensorflow *graph*\n", - "to represent your model, and then create a Tensorflow *session* to run the graph.\n", - "\n", - "In this guide we will analyze the same data as we did in our NumPy and\n", - "scikit-learn tutorial, gathered from the MNIST database of images. We\n", - "will give an introduction to the lower level Python Application\n", - "Program Interfaces (APIs), and see how we use them to build our graph.\n", - "Then we will build (effectively) the same graph in Keras, to see just\n", - "how simple solving a machine learning problem can be.\n", - "\n", - "To install tensorflow on Unix/Linux systems, use pip as" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "ename": "SyntaxError", - "evalue": "invalid syntax (, line 1)", - "output_type": "error", - "traceback": [ - "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m1\u001b[0m\n\u001b[0;31m pip3 install tensorflow\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" - ] - } - ], - "source": [ - "pip3 install tensorflow" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and/or if you use **anaconda**, just write (or install from the graphical user interface)\n", - "(current release of CPU-only TensorFlow)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "conda create -n tf tensorflow\n", - "conda activate tf" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To install the current release of GPU TensorFlow" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "conda create -n tf-gpu tensorflow-gpu\n", - "conda activate tf-gpu" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Keras is a high level [neural network](https://en.wikipedia.org/wiki/Application_programming_interface)\n", - "that supports Tensorflow, CTNK and Theano as backends. \n", - "If you have Anaconda installed you may run the following command" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "conda install keras" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "You can look up the [instructions here](https://keras.io/) for more information.\n", - "\n", - "We will to a large extent use **keras** in this course. \n", - "\n", - "\n", - "Let us look again at the MINST data set." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# import necessary packages\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "import tensorflow as tf\n", - "from sklearn import datasets\n", - "\n", - "\n", - "# ensure the same random numbers appear every time\n", - "np.random.seed(0)\n", - "\n", - "# display images in notebook\n", - "%matplotlib inline\n", - "plt.rcParams['figure.figsize'] = (12,12)\n", - "\n", - "\n", - "# download MNIST dataset\n", - "digits = datasets.load_digits()\n", - "\n", - "# define inputs and labels\n", - "inputs = digits.images\n", - "labels = digits.target\n", - "\n", - "print(\"inputs = (n_inputs, pixel_width, pixel_height) = \" + str(inputs.shape))\n", - "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", - "\n", - "\n", - "# flatten the image\n", - "# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64\n", - "n_inputs = len(inputs)\n", - "inputs = inputs.reshape(n_inputs, -1)\n", - "print(\"X = (n_inputs, n_features) = \" + str(inputs.shape))\n", - "\n", - "\n", - "# choose some random images to display\n", - "indices = np.arange(n_inputs)\n", - "random_indices = np.random.choice(indices, size=5)\n", - "\n", - "for i, image in enumerate(digits.images[random_indices]):\n", - " plt.subplot(1, 5, i+1)\n", - " plt.axis('off')\n", - " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", - " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from tensorflow.keras.layers import Input\n", - "from tensorflow.keras.models import Sequential #This allows appending layers to existing models\n", - "from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer\n", - "from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)\n", - "from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)\n", - "from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function\n", - "\n", - "from sklearn.model_selection import train_test_split\n", - "\n", - "# one-hot representation of labels\n", - "labels = to_categorical(labels)\n", - "\n", - "# split into train and test data\n", - "train_size = 0.8\n", - "test_size = 1 - train_size\n", - "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", - " test_size=test_size)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "epochs = 100\n", - "batch_size = 100\n", - "n_neurons_layer1 = 100\n", - "n_neurons_layer2 = 50\n", - "n_categories = 10\n", - "eta_vals = np.logspace(-5, 1, 7)\n", - "lmbd_vals = np.logspace(-5, 1, 7)\n", - "def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd):\n", - " model = Sequential()\n", - " model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))\n", - " model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd)))\n", - " model.add(Dense(n_categories, activation='softmax'))\n", - " \n", - " sgd = optimizers.SGD(lr=eta)\n", - " model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])\n", - " \n", - " return model" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - " \n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories,\n", - " eta=eta, lmbd=lmbd)\n", - " DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)\n", - " scores = DNN.evaluate(X_test, Y_test)\n", - " \n", - " DNN_keras[i][j] = DNN\n", - " \n", - " print(\"Learning rate = \", eta)\n", - " print(\"Lambda = \", lmbd)\n", - " print(\"Test accuracy: %.3f\" % scores[1])\n", - " print()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# optional\n", - "# visual representation of grid search\n", - "# uses seaborn heatmap, could probably do this in matplotlib\n", - "import seaborn as sns\n", - "\n", - "sns.set()\n", - "\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "\n", - "for i in range(len(eta_vals)):\n", - " for j in range(len(lmbd_vals)):\n", - " DNN = DNN_keras[i][j]\n", - "\n", - " train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1]\n", - " test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1]\n", - "\n", - " \n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Test Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The Breast Cancer Data, now with Keras" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "import tensorflow as tf\n", - "from tensorflow.keras.layers import Input\n", - "from tensorflow.keras.models import Sequential #This allows appending layers to existing models\n", - "from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer\n", - "from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)\n", - "from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)\n", - "from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "import seaborn as sns\n", - "from sklearn.model_selection import train_test_split as splitter\n", - "from sklearn.datasets import load_breast_cancer\n", - "import pickle\n", - "import os \n", - "\n", - "\n", - "\"\"\"Load breast cancer dataset\"\"\"\n", - "\n", - "np.random.seed(0) #create same seed for random number every time\n", - "\n", - "cancer=load_breast_cancer() #Download breast cancer dataset\n", - "\n", - "inputs=cancer.data #Feature matrix of 569 rows (samples) and 30 columns (parameters)\n", - "outputs=cancer.target #Label array of 569 rows (0 for benign and 1 for malignant)\n", - "labels=cancer.feature_names[0:30]\n", - "\n", - "print('The content of the breast cancer dataset is:') #Print information about the datasets\n", - "print(labels)\n", - "print('-------------------------')\n", - "print(\"inputs = \" + str(inputs.shape))\n", - "print(\"outputs = \" + str(outputs.shape))\n", - "print(\"labels = \"+ str(labels.shape))\n", - "\n", - "x=inputs #Reassign the Feature and Label matrices to other variables\n", - "y=outputs\n", - "\n", - "#%% \n", - "\n", - "# Visualisation of dataset (for correlation analysis)\n", - "\n", - "plt.figure()\n", - "plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral)\n", - "plt.xlabel('Mean radius',fontweight='bold')\n", - "plt.ylabel('Mean perimeter',fontweight='bold')\n", - "plt.show()\n", - "\n", - "plt.figure()\n", - "plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral)\n", - "plt.xlabel('Mean compactness',fontweight='bold')\n", - "plt.ylabel('Mean concavity',fontweight='bold')\n", - "plt.show()\n", - "\n", - "\n", - "plt.figure()\n", - "plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)\n", - "plt.xlabel('Mean radius',fontweight='bold')\n", - "plt.ylabel('Mean texture',fontweight='bold')\n", - "plt.show()\n", - "\n", - "plt.figure()\n", - "plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral)\n", - "plt.xlabel('Mean perimeter',fontweight='bold')\n", - "plt.ylabel('Mean compactness',fontweight='bold')\n", - "plt.show()\n", - "\n", - "\n", - "# Generate training and testing datasets\n", - "\n", - "#Select features relevant to classification (texture,perimeter,compactness and symmetery) \n", - "#and add to input matrix\n", - "\n", - "temp1=np.reshape(x[:,1],(len(x[:,1]),1))\n", - "temp2=np.reshape(x[:,2],(len(x[:,2]),1))\n", - "X=np.hstack((temp1,temp2)) \n", - "temp=np.reshape(x[:,5],(len(x[:,5]),1))\n", - "X=np.hstack((X,temp)) \n", - "temp=np.reshape(x[:,8],(len(x[:,8]),1))\n", - "X=np.hstack((X,temp)) \n", - "\n", - "X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1) #Split datasets into training and testing\n", - "\n", - "y_train=to_categorical(y_train) #Convert labels to categorical when using categorical cross entropy\n", - "y_test=to_categorical(y_test)\n", - "\n", - "del temp1,temp2,temp\n", - "\n", - "# %%\n", - "\n", - "# Define tunable parameters\"\n", - "\n", - "eta=np.logspace(-3,-1,3) #Define vector of learning rates (parameter to SGD optimiser)\n", - "lamda=0.01 #Define hyperparameter\n", - "n_layers=2 #Define number of hidden layers in the model\n", - "n_neuron=np.logspace(0,3,4,dtype=int) #Define number of neurons per layer\n", - "epochs=100 #Number of reiterations over the input data\n", - "batch_size=100 #Number of samples per gradient update\n", - "\n", - "# %%\n", - "\n", - "\"\"\"Define function to return Deep Neural Network model\"\"\"\n", - "\n", - "def NN_model(inputsize,n_layers,n_neuron,eta,lamda):\n", - " model=Sequential() \n", - " for i in range(n_layers): #Run loop to add hidden layers to the model\n", - " if (i==0): #First layer requires input dimensions\n", - " model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize))\n", - " else: #Subsequent layers are capable of automatic shape inferencing\n", - " model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda)))\n", - " model.add(Dense(2,activation='softmax')) #2 outputs - ordered and disordered (softmax for prob)\n", - " sgd=optimizers.SGD(lr=eta)\n", - " model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy'])\n", - " return model\n", - "\n", - " \n", - "Train_accuracy=np.zeros((len(n_neuron),len(eta))) #Define matrices to store accuracy scores as a function\n", - "Test_accuracy=np.zeros((len(n_neuron),len(eta))) #of learning rate and number of hidden neurons for \n", - "\n", - "for i in range(len(n_neuron)): #run loops over hidden neurons and learning rates to calculate \n", - " for j in range(len(eta)): #accuracy scores \n", - " DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda)\n", - " DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1)\n", - " Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1]\n", - " Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1]\n", - " \n", - "\n", - "def plot_data(x,y,data,title=None):\n", - "\n", - " # plot results\n", - " fontsize=16\n", - "\n", - "\n", - " fig = plt.figure()\n", - " ax = fig.add_subplot(111)\n", - " cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1)\n", - " \n", - " cbar=fig.colorbar(cax)\n", - " cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize)\n", - " cbar.set_ticks([0,.2,.4,0.6,0.8,1.0])\n", - " cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%'])\n", - "\n", - " # put text on matrix elements\n", - " for i, x_val in enumerate(np.arange(len(x))):\n", - " for j, y_val in enumerate(np.arange(len(y))):\n", - " c = \"${0:.1f}\\\\%$\".format( 100*data[j,i]) \n", - " ax.text(x_val, y_val, c, va='center', ha='center')\n", - "\n", - " # convert axis vaues to to string labels\n", - " x=[str(i) for i in x]\n", - " y=[str(i) for i in y]\n", - "\n", - "\n", - " ax.set_xticklabels(['']+x)\n", - " ax.set_yticklabels(['']+y)\n", - "\n", - " ax.set_xlabel('$\\\\mathrm{learning\\\\ rate}$',fontsize=fontsize)\n", - " ax.set_ylabel('$\\\\mathrm{hidden\\\\ neurons}$',fontsize=fontsize)\n", - " if title is not None:\n", - " ax.set_title(title)\n", - "\n", - " plt.tight_layout()\n", - "\n", - " plt.show()\n", - " \n", - "plot_data(eta,n_neuron,Train_accuracy, 'training')\n", - "plot_data(eta,n_neuron,Test_accuracy, 'testing')" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Fine-tuning neural network hyperparameters\n", - "\n", - "The flexibility of neural networks is also one of their main\n", - "drawbacks: there are many hyperparameters to tweak. Not only can you\n", - "use any imaginable network topology (how neurons/nodes are interconnected),\n", - "but even in a simple FFNN you can change the number of layers, the\n", - "number of neurons per layer, the type of activation function to use in\n", - "each layer, the weight initialization logic, the stochastic gradient optmized and much more. How do you\n", - "know what combination of hyperparameters is the best for your task?\n", - "\n", - "* You can use grid search with cross-validation to find the right hyperparameters.\n", - "\n", - "However,since there are many hyperparameters to tune, and since\n", - "training a neural network on a large dataset takes a lot of time, you\n", - "will only be able to explore a tiny part of the hyperparameter space.\n", - "\n", - "\n", - "* You can use randomized search.\n", - "\n", - "* Or use tools like [Oscar](http://oscar.calldesk.ai/), which implements more complex algorithms to help you find a good set of hyperparameters quickly. \n", - "\n", - "For many problems you can start with just one or two hidden layers and it will work just fine.\n", - "For the MNIST data set you ca easily get a high accuracy using just one hidden layer with a\n", - "few hundred neurons.\n", - "You can reach for this data set above 98% accuracy using two hidden layers with the same total amount of\n", - "neurons, in roughly the same amount of training time. \n", - "\n", - "For more complex problems, you can gradually\n", - "ramp up the number of hidden layers, until you start overfitting the training set. Very complex tasks, such\n", - "as large image classification or speech recognition, typically require networks with dozens of layers\n", - "and they need a huge amount\n", - "of training data. However, you will rarely have to train such networks from scratch: it is much more\n", - "common to reuse parts of a pretrained state-of-the-art network that performs a similar task.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Which activation function should I use?\n", - "\n", - "The Back propagation algorithm we derived above works by going from\n", - "the output layer to the input layer, propagating the error gradient on\n", - "the way. Once the algorithm has computed the gradient of the cost\n", - "function with regards to each parameter in the network, it uses these\n", - "gradients to update each parameter with a Gradient Descent (GD) step.\n", - "\n", - "\n", - "Unfortunately for us, the gradients often get smaller and smaller as the\n", - "algorithm progresses down to the first hidden layers. As a result, the\n", - "GD update leaves the lower layer connection weights\n", - "virtually unchanged, and training never converges to a good\n", - "solution. This is known in the literature as \n", - "**the vanishing gradients problem**. \n", - "\n", - "In other cases, the opposite can happen, namely the the gradients can grow bigger and\n", - "bigger. The result is that many of the layers get large updates of the \n", - "weights the\n", - "algorithm diverges. This is the **exploding gradients problem**, which is\n", - "mostly encountered in recurrent neural networks. More generally, deep\n", - "neural networks suffer from unstable gradients, different layers may\n", - "learn at widely different speeds\n", - "\n", - "\n", - "\n", - "\n", - "Although this unfortunate behavior has been empirically observed for\n", - "quite a while (it was one of the reasons why deep neural networks were\n", - "mostly abandoned for a long time), it is only around 2010 that\n", - "significant progress was made in understanding it.\n", - "\n", - "A paper titled [Understanding the Difficulty of Training Deep\n", - "Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio](http://proceedings.mlr.press/v9/glorot10a.html) found that\n", - "the problems with the popular logistic\n", - "sigmoid activation function and the weight initialization technique\n", - "that was most popular at the time, namely random initialization using\n", - "a normal distribution with a mean of 0 and a standard deviation of\n", - "1. \n", - "\n", - "They showed that with this activation function and this\n", - "initialization scheme, the variance of the outputs of each layer is\n", - "much greater than the variance of its inputs. Going forward in the\n", - "network, the variance keeps increasing after each layer until the\n", - "activation function saturates at the top layers. This is actually made\n", - "worse by the fact that the logistic function has a mean of 0.5, not 0\n", - "(the hyperbolic tangent function has a mean of 0 and behaves slightly\n", - "better than the logistic function in deep networks).\n", - "\n", - "\n", - "\n", - "Looking at the logistic activation function, when inputs become large\n", - "(negative or positive), the function saturates at 0 or 1, with a\n", - "derivative extremely close to 0. Thus when backpropagation kicks in,\n", - "it has virtually no gradient to propagate back through the network,\n", - "and what little gradient exists keeps getting diluted as\n", - "backpropagation progresses down through the top layers, so there is\n", - "really nothing left for the lower layers.\n", - "\n", - "In their paper, Glorot and Bengio propose a way to significantly\n", - "alleviate this problem. We need the signal to flow properly in both\n", - "directions: in the forward direction when making predictions, and in\n", - "the reverse direction when backpropagating gradients. We don’t want\n", - "the signal to die out, nor do we want it to explode and saturate. For\n", - "the signal to flow properly, the authors argue that we need the\n", - "variance of the outputs of each layer to be equal to the variance of\n", - "its inputs, and we also need the gradients to have equal variance\n", - "before and after flowing through a layer in the reverse direction.\n", - "\n", - "\n", - "\n", - "One of the insights in the 2010 paper by Glorot and Bengio was that\n", - "the vanishing/exploding gradients problems were in part due to a poor\n", - "choice of activation function. Until then most people had assumed that\n", - "if Nature had chosen to use roughly sigmoid activation functions in\n", - "biological neurons, they must be an excellent choice. But it turns out\n", - "that other activation functions behave much better in deep neural\n", - "networks, in particular the ReLU activation function, mostly because\n", - "it does not saturate for positive values (and also because it is quite\n", - "fast to compute).\n", - "\n", - "\n", - "## The RELU function family\n", - "\n", - "The ReLU activation function suffers from a problem known as the dying\n", - "ReLUs: during training, some neurons effectively die, meaning they\n", - "stop outputting anything other than 0.\n", - "\n", - "In some cases, you may find that half of your network’s neurons are\n", - "dead, especially if you used a large learning rate. During training,\n", - "if a neuron’s weights get updated such that the weighted sum of the\n", - "neuron’s inputs is negative, it will start outputting 0. When this\n", - "happen, the neuron is unlikely to come back to life since the gradient\n", - "of the ReLU function is 0 when its input is negative.\n", - "\n", - "To solve this problem, nowadays practitioners use a variant of the ReLU\n", - "function, such as the leaky ReLU discussed above or the so-called\n", - "exponential linear unit (ELU) function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "ELU(z) = \\left\\{\\begin{array}{cc} \\alpha\\left( \\exp{(z)}-1\\right) & z < 0,\\\\ z & z \\ge 0.\\end{array}\\right.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In general it seems that the ELU activation function is better than\n", - "the leaky ReLU function (and its variants), which is better than\n", - "ReLU. ReLU performs better than $\\tanh$ which in turn performs better\n", - "than the logistic function. \n", - "\n", - "If runtime\n", - "performance is an issue, then you may opt for the leaky ReLU function over the \n", - "ELU function If you don’t\n", - "want to tweak yet another hyperparameter, you may just use the default\n", - "$\\alpha$ of $0.01$ for the leaky ReLU, and $1$ for ELU. If you have\n", - "spare time and computing power, you can use cross-validation or\n", - "bootstrap to evaluate other activation functions.\n", - "\n", - "\n", - "\n", - "In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).\n", - "\n", - "It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.\n", - "\n", - "**For the output layer:**\n", - "\n", - "* For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).\n", - "\n", - "* For regression tasks, you can simply use no activation function at all.\n", - "\n", - "## Batch Normalization\n", - "\n", - "Batch Normalization\n", - "aims to address the vanishing/exploding gradients problems, and more generally the problem that the\n", - "distribution of each layer’s inputs changes during training, as the parameters of the previous layers change.\n", - "\n", - "The technique consists of adding an operation in the model just before the activation function of each\n", - "layer, simply zero-centering and normalizing the inputs, then scaling and shifting the result using two new\n", - "parameters per layer (one for scaling, the other for shifting). In other words, this operation lets the model\n", - "learn the optimal scale and mean of the inputs for each layer.\n", - "In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and\n", - "standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current\n", - "mini-batch, from this the name batch normalization.\n", - "\n", - "## Dropout\n", - "\n", - "It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but\n", - "excluding the output neurons) has a probability $p$ of being temporarily dropped out, meaning it will be\n", - "entirely ignored during this training step, but it may be active during the next step.\n", - "\n", - "The\n", - "hyperparameter $p$ is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore.\n", - " It is viewed as one of the most popular regularization techniques.\n", - "\n", - "## Gradient Clipping\n", - "\n", - "A popular technique to lessen the exploding gradients problem is to simply clip the gradients during\n", - "backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural\n", - "networks).\n", - "\n", - "This technique is called Gradient Clipping.\n", - "\n", - "In general however, Batch\n", - "Normalization is preferred.\n", - "\n", - "\n", - "## A top-down perspective on Neural networks\n", - "\n", - "\n", - "The first thing we would like to do is divide the data into two or three\n", - "parts. A training set, a validation or dev (development) set, and a\n", - "test set. The test set is the data on which we want to make\n", - "predictions. The dev set is a subset of the training data we use to\n", - "check how well we are doing out-of-sample, after training the model on\n", - "the training dataset. We use the validation error as a proxy for the\n", - "test error in order to make tweaks to our model. It is crucial that we\n", - "do not use any of the test data to train the algorithm. This is a\n", - "cardinal sin in ML. Then:\n", - "\n", - "\n", - "* Estimate optimal error rate\n", - "\n", - "* Minimize underfitting (bias) on training data set.\n", - "\n", - "* Make sure you are not overfitting.\n", - "\n", - "If the validation and test sets are drawn from the same distributions,\n", - "then a good performance on the validation set should lead to similarly\n", - "good performance on the test set. \n", - "\n", - "However, sometimes\n", - "the training data and test data differ in subtle ways because, for\n", - "example, they are collected using slightly different methods, or\n", - "because it is cheaper to collect data in one way versus another. In\n", - "this case, there can be a mismatch between the training and test\n", - "data. This can lead to the neural network overfitting these small\n", - "differences between the test and training sets, and a poor performance\n", - "on the test set despite having a good performance on the validation\n", - "set. To rectify this, Andrew Ng suggests making two validation or dev\n", - "sets, one constructed from the training data and one constructed from\n", - "the test data. The difference between the performance of the algorithm\n", - "on these two validation sets quantifies the train-test mismatch. This\n", - "can serve as another important diagnostic when using DNNs for\n", - "supervised learning.\n", - "\n", - "\n", - "## Limitations of supervised learning with deep networks\n", - "\n", - "Like all statistical methods, supervised learning using neural\n", - "networks has important limitations. This is especially important when\n", - "one seeks to apply these methods, especially to physics problems. Like\n", - "all tools, DNNs are not a universal solution. Often, the same or\n", - "better performance on a task can be achieved by using a few\n", - "hand-engineered features (or even a collection of random\n", - "features). \n", - "\n", - "Here we list some of the important limitations of supervised neural network based models. \n", - "\n", - "\n", - "\n", - "* **Need labeled data**. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images).\n", - "\n", - "* **Supervised neural networks are extremely data intensive.** DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs.\n", - "\n", - "* **Homogeneous data.** Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types.\n", - "\n", - "* **Many problems are not about prediction.** In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a *wrong* model. The model might or might not be useful for understanding the underlying science.\n", - "\n", - "Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. This motivates the use of unsupervised methods which in part circumvent these problems." - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter10.py b/doc/LectureNotes/_build/jupyter_execute/chapter10.py deleted file mode 100644 index dfcd72372..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter10.py +++ /dev/null @@ -1,1569 +0,0 @@ -# Building a Feed Forward Neural Network - -We are now gong to develop an example based on the MNIST data -base. This is a classification problem and we need to use our -cross-entropy function we discussed in connection with logistic -regression. The cross-entropy defines our cost function for the -classificaton problems with neural networks. - -In binary classification with two classes $(0, 1)$ we define the -logistic/sigmoid function as the probability that a particular input -is in class $0$ or $1$. This is possible because the logistic -function takes any input from the real numbers and inputs a number -between 0 and 1, and can therefore be interpreted as a probability. It -also has other nice properties, such as a derivative that is simple to -calculate. - -For an input $\boldsymbol{a}$ from the hidden layer, the probability that the input $\boldsymbol{x}$ -is in class 0 or 1 is just. We let $\theta$ represent the unknown weights and biases to be adjusted by our equations). The variable $x$ -represents our activation values $z$. We have - -$$ -P(y = 0 \mid \hat{x}, \hat{\theta}) = \frac{1}{1 + \exp{(- \hat{x}})} , -$$ - -and - -$$ -P(y = 1 \mid \hat{x}, \hat{\theta}) = 1 - P(y = 0 \mid \hat{x}, \hat{\theta}) , -$$ - -where $y \in \{0, 1\}$ and $\hat{\theta}$ represents the weights and biases -of our network. - - - -## Defining the cost function - -Our cost function is given as (see the Logistic regression lectures) - -$$ -\mathcal{C}(\hat{\theta}) = - \ln P(\mathcal{D} \mid \hat{\theta}) = - \sum_{i=1}^n -y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\hat{\theta}) . -$$ - -This last equality means that we can interpret our *cost* function as a sum over the *loss* function -for each point in the dataset $\mathcal{L}_i(\hat{\theta})$. -The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather -than maximizing a negative number. - -In *multiclass* classification it is common to treat each integer label as a so called *one-hot* vector: - -$y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$ and - - -$y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$ - - -i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset (numbers from $0$ to $9$).. - -If $\hat{x}_i$ is the $i$-th input (image), $y_{ic}$ refers to the $c$-th component of the $i$-th -output vector $\hat{y}_i$. -The probability of $\hat{x}_i$ being in class $c$ will be given by the softmax function: - -$$ -P(y_{ic} = 1 \mid \hat{x}_i, \hat{\theta}) = \frac{\exp{((\hat{a}_i^{hidden})^T \hat{w}_c)}} -{\sum_{c'=0}^{C-1} \exp{((\hat{a}_i^{hidden})^T \hat{w}_{c'})}} , -$$ - -which reduces to the logistic function in the binary case. -The likelihood of this $C$-class classifier -is now given as: - -$$ -P(\mathcal{D} \mid \hat{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} . -$$ - -Again we take the negative log-likelihood to define our cost function: - -$$ -\mathcal{C}(\hat{\theta}) = - \log{P(\mathcal{D} \mid \hat{\theta})}. -$$ - -See the logistic regression lectures for a full definition of the cost function. - -The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before! - - -### Example: binary classification problem - -As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters $\beta$ as - -$$ -\mathcal{C}(\hat{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\hat{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\hat{\beta})}\right), -$$ - -where we had defined the logistic (sigmoid) function - -$$ -p(y_i =1\vert x_i,\hat{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}}, -$$ - -and - -$$ -p(y_i =0\vert x_i,\hat{\beta})=1-p(y_i =1\vert x_i,\hat{\beta}). -$$ - -The parameters $\hat{\beta}$ were defined using a minimization method like gradient descent or Newton-Raphson's method. - -Now we replace $x_i$ with the activation $z_i^l$ for a given layer $l$ and the outputs as $y_i=a_i^l=f(z_i^l)$, with $z_i^l$ now being a function of the weights $w_{ij}^l$ and biases $b_i^l$. -We have then - -$$ -a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}}, -$$ - -with - -$$ -z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l, -$$ - -where the superscript $l-1$ indicates that these are the outputs from layer $l-1$. -Our cost function at the final layer $l=L$ is now - -$$ -\mathcal{C}(\hat{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right), -$$ - -where we have defined the targets $t_i$. The derivatives of the cost function with respect to the output $a_i^L$ are then easily calculated and we get - -$$ -\frac{\partial \mathcal{C}(\hat{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}. -$$ - -In case we use another activation function than the logistic one, we need to evaluate other derivatives. - - - -### The Softmax function - -In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need - -$$ -\frac{\partial f(z_i^l)}{\partial w_{jk}^l} = -\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}. -$$ - -For the Softmax function we have - -$$ -f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}. -$$ - -Its derivative with respect to $z_j^l$ gives - -$$ -\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right), -$$ - -which in case of the simply binary model reduces to having $i=j$. - - -## Developing a code for doing neural networks with back propagation - - -One can identify a set of key steps when using neural networks to solve supervised learning problems: - -1. Collect and pre-process data - -2. Define model and architecture - -3. Choose cost function and optimizer - -4. Train the model - -5. Evaluate model performance on test data - -6. Adjust hyperparameters (if necessary, network architecture) - -### Collect and pre-process data - -Here we will be using the MNIST dataset, which is readily available through the **scikit-learn** -package. You may also find it for example [here](http://yann.lecun.com/exdb/mnist/). -The *MNIST* (Modified National Institute of Standards and Technology) database is a large database -of handwritten digits that is commonly used for training various image processing systems. -The MNIST dataset consists of 70 000 images of size $28\times 28$ pixels, each labeled from 0 to 9. -The scikit-learn dataset we will use consists of a selection of 1797 images of size $8\times 8$ collected and processed from this database. - -To feed data into a feed-forward neural network we need to represent -the inputs as a design/feature matrix $X = (n_{inputs}, n_{features})$. Each -row represents an *input*, in this case a handwritten digit, and -each column represents a *feature*, in this case a pixel. The -correct answers, also known as *labels* or *targets* are -represented as a 1D array of integers -$Y = (n_{inputs}) = (5, 3, 1, 8,...)$. - -As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from -measurements of height (in m) -and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example: - -$$ X = \begin{bmatrix} -1.85 & 81\\ -1.71 & 65\\ -1.95 & 103\\ -1.55 & 42\\ -1.63 & 56 -\end{bmatrix} ,$$ - -and the targets would be: - -$$ Y = (23.7, 22.2, 27.1, 17.5, 21.1) $$ - -Since each input image is a 2D matrix, we need to flatten the image -(i.e. "unravel" the 2D matrix into a 1D array) to turn the data into a -design/feature matrix. This means we lose all spatial information in the -image, such as locality and translational invariance. More complicated -architectures such as Convolutional Neural Networks can take advantage -of such information, and are most commonly applied when analyzing -images. - -%matplotlib inline - -# import necessary packages -import numpy as np -import matplotlib.pyplot as plt -from sklearn import datasets - - -# ensure the same random numbers appear every time -np.random.seed(0) - -# display images in notebook -%matplotlib inline -plt.rcParams['figure.figsize'] = (12,12) - - -# download MNIST dataset -digits = datasets.load_digits() - -# define inputs and labels -inputs = digits.images -labels = digits.target - -print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape)) -print("labels = (n_inputs) = " + str(labels.shape)) - - -# flatten the image -# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64 -n_inputs = len(inputs) -inputs = inputs.reshape(n_inputs, -1) -print("X = (n_inputs, n_features) = " + str(inputs.shape)) - - -# choose some random images to display -indices = np.arange(n_inputs) -random_indices = np.random.choice(indices, size=5) - -for i, image in enumerate(digits.images[random_indices]): - plt.subplot(1, 5, i+1) - plt.axis('off') - plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest') - plt.title("Label: %d" % digits.target[random_indices[i]]) -plt.show() - -### Train and test datasets - -Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions. - -We will reserve $80 \%$ of our dataset for training and $20 \%$ for testing. - -It is important that the train and test datasets are drawn randomly from our dataset, to ensure -no bias in the sampling. -Say you are taking measurements of weather data to predict the weather in the coming 5 days. -You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data -collected from 12.00 to 24.00. - -from sklearn.model_selection import train_test_split - -# one-liner from scikit-learn library -train_size = 0.8 -test_size = 1 - train_size -X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size, - test_size=test_size) - -# equivalently in numpy -def train_test_split_numpy(inputs, labels, train_size, test_size): - n_inputs = len(inputs) - inputs_shuffled = inputs.copy() - labels_shuffled = labels.copy() - - np.random.shuffle(inputs_shuffled) - np.random.shuffle(labels_shuffled) - - train_end = int(n_inputs*train_size) - X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:] - Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:] - - return X_train, X_test, Y_train, Y_test - -#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size) - -print("Number of training images: " + str(len(X_train))) -print("Number of test images: " + str(len(X_test))) - -### Define model and architecture - -Our simple feed-forward neural network will consist of an *input* layer, a single *hidden* layer and an *output* layer. The activation $y$ of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have - -$$ z = \sum_{i=1}^n w_i a_i ,$$ - -$$ y = f(z) ,$$ - -where $f$ is the activation function, $a_i$ represents input from neuron $i$ in the preceding layer -and $w_i$ is the weight to input $i$. -The activation of the neurons in the input layer is just the features (e.g. a pixel value). - -The simplest activation function for a neuron is the *Heaviside* function: - -$$ f(z) = -\begin{cases} -1, & z > 0\\ -0, & \text{otherwise} -\end{cases} -$$ - -A feed-forward neural network with this activation is known as a *perceptron*. -For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer. -This activation can be generalized to $k$ classes (using e.g. the *one-against-all* strategy), -and we call these architectures *multiclass perceptrons*. - -However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and -Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function. - -Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU). -We will be using the sigmoid function $\sigma(x)$: - -$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$ - -which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions. - -### Layers - -* Input - -Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons. - -* Hidden layer - -We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer. -Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer. - -* Output - -If we were building a binary classifier, it would be sufficient with a single neuron in the output layer, -which could output 0 or 1 according to the Heaviside function. This would be an example of a *hard* classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a *soft* classifier, which outputs the probability of being in class 0 or 1. - -For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class. - -Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons $j = 0,1,...,9$. The activation of each output neuron $j$ will be according to the *softmax* function: - -$$ P(\text{class $j$} \mid \text{input $\hat{a}$}) = \frac{\exp{(\hat{a}^T \hat{w}_j)}} -{\sum_{c=0}^{9} \exp{(\hat{a}^T \hat{w}_c)}} ,$$ - -i.e. each neuron $j$ outputs the probability of being in class $j$ given an input from the hidden layer $\hat{a}$, with $\hat{w}_j$ the weights of neuron $j$ to the inputs. -The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1. -The exponent is just the weighted sum of inputs as before: - -$$ z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.$$ - -Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500 -weights to the output layer. - - -Typically weights are initialized with small values distributed around zero, drawn from a uniform -or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. - -Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range -of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron $j$, $b_j$: - -$$ z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.$$ - -The bias weights $\hat{b}$ are often initialized to zero, but a small value like $0.01$ ensures all neurons have some output which can be backpropagated in the first training cycle. - -# building our neural network - -n_inputs, n_features = X_train.shape -n_hidden_neurons = 50 -n_categories = 10 - -# we make the weights normally distributed using numpy.random.randn - -# weights and bias in the hidden layer -hidden_weights = np.random.randn(n_features, n_hidden_neurons) -hidden_bias = np.zeros(n_hidden_neurons) + 0.01 - -# weights and bias in the output layer -output_weights = np.random.randn(n_hidden_neurons, n_categories) -output_bias = np.zeros(n_categories) + 0.01 - -### Feed-forward pass - -Denote $F$ the number of features, $H$ the number of hidden neurons and $C$ the number of categories. -For each input image we calculate a weighted sum of input features (pixel values) to each neuron $j$ in the hidden layer $l$: - -$$ z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},$$ - -this is then passed through our activation function - -$$ a_{j}^{l} = f(z_{j}^{l}) .$$ - -We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron $j$ in the output layer: - -$$ z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.$$ - -Finally we calculate the output of neuron $j$ in the output layer using the softmax function: - -$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}} -{\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$ - - -Since our data has the dimensions $X = (n_{inputs}, n_{features})$ and our weights to the hidden -layer have the dimensions -$W_{hidden} = (n_{features}, n_{hidden})$, -we can easily feed the network all our training data in one go by taking the matrix product - -$$ X W^{h} = (n_{inputs}, n_{hidden}),$$ - -and obtain a matrix that holds the weighted sum of inputs to the hidden layer -for each input image and each hidden neuron. -We also add the bias to obtain a matrix of weighted sums to the hidden layer $Z^{h}$: - -$$ \hat{z}^{l} = \hat{X} \hat{W}^{l} + \hat{b}^{l} ,$$ - -meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image. -This is then passed through the activation: - -$$ \hat{a}^{l} = f(\hat{z}^l) .$$ - -This is fed to the output layer: - -$$ \hat{z}^{L} = \hat{a}^{L} \hat{W}^{L} + \hat{b}^{L} .$$ - -Finally we receive our output values for each image and each category by passing it through the softmax function: - -$$ output = softmax (\hat{z}^{L}) = (n_{inputs}, n_{categories}) .$$ - -# setup the feed-forward pass, subscript h = hidden layer - -def sigmoid(x): - return 1/(1 + np.exp(-x)) - -def feed_forward(X): - # weighted sum of inputs to the hidden layer - z_h = np.matmul(X, hidden_weights) + hidden_bias - # activation in the hidden layer - a_h = sigmoid(z_h) - - # weighted sum of inputs to the output layer - z_o = np.matmul(a_h, output_weights) + output_bias - # softmax output - # axis 0 holds each input and axis 1 the probabilities of each category - exp_term = np.exp(z_o) - probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True) - - return probabilities - -probabilities = feed_forward(X_train) -print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape)) -print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0])) -print("probabilities sum up to: " + str(probabilities[0].sum())) -print() - -# we obtain a prediction by taking the class with the highest likelihood -def predict(X): - probabilities = feed_forward(X) - return np.argmax(probabilities, axis=1) - -predictions = predict(X_train) -print("predictions = (n_inputs) = " + str(predictions.shape)) -print("prediction for image 0: " + str(predictions[0])) -print("correct label for image 0: " + str(Y_train[0])) - -### Choose cost function and optimizer - -To measure how well our neural network is doing we need to introduce a cost function. -We will call the function that gives the error of a single sample output the *loss* function, and the function -that gives the total error of our network across all samples the *cost* function. -A typical choice for multiclass classification is the *cross-entropy* loss, also known as the negative log likelihood. - -In *multiclass* classification it is common to treat each integer label as a so called *one-hot* vector: - -$$ y = 5 \quad \rightarrow \quad \hat{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) ,$$ - - -$$ y = 1 \quad \rightarrow \quad \hat{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,$$ - - -i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset. - -Let $y_{ic}$ denote the $c$-th component of the $i$-th one-hot vector. -We define the cost function $\mathcal{C}$ as a sum over the cross-entropy loss for each point $\hat{x}_i$ in the dataset. - -In the one-hot representation only one of the terms in the loss function is non-zero, namely the -probability of the correct category $c'$ -(i.e. the category $c'$ such that $y_{ic'} = 1$). This means that the cross entropy loss only punishes you for how wrong -you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\hat{\theta}$ represents the parameters of our network, i.e. all the weights and biases. - - - -### Optimizing the cost function - -The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is *gradient descent* and its generalizations. The idea behind gradient descent -is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a *local* minimum of the cost function. -Each parameter $\theta$ is iteratively adjusted according to the rule - -$$ \theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,$$ - -where $\eta$ is known as the *learning rate*, which controls how big a step we take towards the minimum. -This update can be repeated for any number of iterations, or until we are satisfied with the result. - -A simple and effective improvement is a variant called *Batch Gradient Descent*. -Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient -on a subset of the data called a *minibatch*. -If there are $N$ data points and we have a minibatch size of $M$, the total number of batches -is $N/M$. -We denote each minibatch $B_k$, with $k = 1, 2,...,N/M$. The gradient then becomes: - -$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}_i(\theta) \quad \rightarrow \quad -\frac{1}{M} \sum_{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$ - -i.e. instead of averaging the loss over the entire dataset, we average over a minibatch. - -This has two important benefits: -1. Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima. - -2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient. - -The various optmization methods, with codes and algorithms, are discussed in our lectures on [Gradient descent approaches](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html). - - -### Regularization - -It is common to add an extra term to the cost function, proportional -to the size of the weights. This is equivalent to constraining the -size of the weights, so that they do not grow out of control. -Constraining the size of the weights means that the weights cannot -grow arbitrarily large to fit the training data, and in this way -reduces *overfitting*. - -We will measure the size of the weights using the so called *L2-norm*, meaning our cost function becomes: - -$$ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) \quad \rightarrow \quad -\frac{1}{N} \sum_{i=1}^N \mathcal{L}_i(\theta) + \lambda \lvert \lvert \hat{w} \rvert \rvert_2^2 -= \frac{1}{N} \sum_{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$ - -i.e. we sum up all the weights squared. The factor $\lambda$ is known as a regularization parameter. - - -In order to train the model, we need to calculate the derivative of -the cost function with respect to every bias and weight in the -network. In total our network has $(64 + 1)\times 50=3250$ weights in -the hidden layer and $(50 + 1)\times 10=510$ weights to the output -layer ($+1$ for the bias), and the gradient must be calculated for -every parameter. We use the *backpropagation* algorithm discussed -above. This is a clever use of the chain rule that allows us to -calculate the gradient efficently. - - -### Matrix multiplication - -To more efficently train our network these equations are implemented using matrix operations. -The error in the output layer is calculated simply as, with $\hat{t}$ being our targets, - -$$ \delta_L = \hat{t} - \hat{y} = (n_{inputs}, n_{categories}) .$$ - -The gradient for the output weights is calculated as - -$$ \nabla W_{L} = \hat{a}^T \delta_L = (n_{hidden}, n_{categories}) ,$$ - -where $\hat{a} = (n_{inputs}, n_{hidden})$. This simply means that we are summing up the gradients for each input. -Since we are going backwards we have to transpose the activation matrix. - -The gradient with respect to the output bias is then - -$$ \nabla \hat{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .$$ - -The error in the hidden layer is - -$$ \Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,$$ - -where $f'(a_{h})$ is the derivative of the activation in the hidden layer. The matrix products mean -that we are summing up the products for each neuron in the output layer. The symbol $\circ$ denotes -the *Hadamard product*, meaning element-wise multiplication. - -This again gives us the gradients in the hidden layer: - -$$ \nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) ,$$ - -$$ \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .$$ - -# to categorical turns our integer vector into a onehot representation -from sklearn.metrics import accuracy_score - -# one-hot in numpy -def to_categorical_numpy(integer_vector): - n_inputs = len(integer_vector) - n_categories = np.max(integer_vector) + 1 - onehot_vector = np.zeros((n_inputs, n_categories)) - onehot_vector[range(n_inputs), integer_vector] = 1 - - return onehot_vector - -#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test) -Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test) - -def feed_forward_train(X): - # weighted sum of inputs to the hidden layer - z_h = np.matmul(X, hidden_weights) + hidden_bias - # activation in the hidden layer - a_h = sigmoid(z_h) - - # weighted sum of inputs to the output layer - z_o = np.matmul(a_h, output_weights) + output_bias - # softmax output - # axis 0 holds each input and axis 1 the probabilities of each category - exp_term = np.exp(z_o) - probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True) - - # for backpropagation need activations in hidden and output layers - return a_h, probabilities - -def backpropagation(X, Y): - a_h, probabilities = feed_forward_train(X) - - # error in the output layer - error_output = probabilities - Y - # error in the hidden layer - error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h) - - # gradients for the output layer - output_weights_gradient = np.matmul(a_h.T, error_output) - output_bias_gradient = np.sum(error_output, axis=0) - - # gradient for the hidden layer - hidden_weights_gradient = np.matmul(X.T, error_hidden) - hidden_bias_gradient = np.sum(error_hidden, axis=0) - - return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient - -print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train))) - -eta = 0.01 -lmbd = 0.01 -for i in range(1000): - # calculate gradients - dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot) - - # regularization term gradients - dWo += lmbd * output_weights - dWh += lmbd * hidden_weights - - # update weights and biases - output_weights -= eta * dWo - output_bias -= eta * dBo - hidden_weights -= eta * dWh - hidden_bias -= eta * dBh - -print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train))) - -## Improving performance - -As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image. -In order to obtain a network that does something useful, we will have to do a bit more work. - -The choice of *hyperparameters* such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a *grid-search* is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates $\eta = 10^{-6}, 10^{-5},...,10^{-1}$ with different regularization parameters $\lambda = 10^{-6},...,10^{-0}$. - -Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an *iteration*, and a full training period -going through the entire dataset ($n/M$ batches) an *epoch*. - -If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers. -Andrew Ng goes through some of these considerations in this [video](https://youtu.be/F1ka6a13S9I). You can find a summary of the video [here](https://kevinzakka.github.io/2016/09/26/applying-deep-learning/). - - -It is very natural to think of the network as an object, with specific instances of the network -being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below. - -class NeuralNetwork: - def __init__( - self, - X_data, - Y_data, - n_hidden_neurons=50, - n_categories=10, - epochs=10, - batch_size=100, - eta=0.1, - lmbd=0.0): - - self.X_data_full = X_data - self.Y_data_full = Y_data - - self.n_inputs = X_data.shape[0] - self.n_features = X_data.shape[1] - self.n_hidden_neurons = n_hidden_neurons - self.n_categories = n_categories - - self.epochs = epochs - self.batch_size = batch_size - self.iterations = self.n_inputs // self.batch_size - self.eta = eta - self.lmbd = lmbd - - self.create_biases_and_weights() - - def create_biases_and_weights(self): - self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons) - self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01 - - self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories) - self.output_bias = np.zeros(self.n_categories) + 0.01 - - def feed_forward(self): - # feed-forward for training - self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias - self.a_h = sigmoid(self.z_h) - - self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias - - exp_term = np.exp(self.z_o) - self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True) - - def feed_forward_out(self, X): - # feed-forward for output - z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias - a_h = sigmoid(z_h) - - z_o = np.matmul(a_h, self.output_weights) + self.output_bias - - exp_term = np.exp(z_o) - probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True) - return probabilities - - def backpropagation(self): - error_output = self.probabilities - self.Y_data - error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h) - - self.output_weights_gradient = np.matmul(self.a_h.T, error_output) - self.output_bias_gradient = np.sum(error_output, axis=0) - - self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden) - self.hidden_bias_gradient = np.sum(error_hidden, axis=0) - - if self.lmbd > 0.0: - self.output_weights_gradient += self.lmbd * self.output_weights - self.hidden_weights_gradient += self.lmbd * self.hidden_weights - - self.output_weights -= self.eta * self.output_weights_gradient - self.output_bias -= self.eta * self.output_bias_gradient - self.hidden_weights -= self.eta * self.hidden_weights_gradient - self.hidden_bias -= self.eta * self.hidden_bias_gradient - - def predict(self, X): - probabilities = self.feed_forward_out(X) - return np.argmax(probabilities, axis=1) - - def predict_probabilities(self, X): - probabilities = self.feed_forward_out(X) - return probabilities - - def train(self): - data_indices = np.arange(self.n_inputs) - - for i in range(self.epochs): - for j in range(self.iterations): - # pick datapoints with replacement - chosen_datapoints = np.random.choice( - data_indices, size=self.batch_size, replace=False - ) - - # minibatch training data - self.X_data = self.X_data_full[chosen_datapoints] - self.Y_data = self.Y_data_full[chosen_datapoints] - - self.feed_forward() - self.backpropagation() - -## Evaluate model performance on test data - -To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data. -We measure the performance of the network using the *accuracy* score. -The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of $1$. - -$$ \text{Accuracy} = \frac{\sum_{i=1}^n I(\hat{y}_i = y_i)}{n} ,$$ - -where $I$ is the indicator function, $1$ if $\hat{y}_i = y_i$ and $0$ otherwise. - -epochs = 100 -batch_size = 100 - -dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size, - n_hidden_neurons=n_hidden_neurons, n_categories=n_categories) -dnn.train() -test_predict = dnn.predict(X_test) - -# accuracy score from scikit library -print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict)) - -# equivalent in numpy -def accuracy_score_numpy(Y_test, Y_pred): - return np.sum(Y_test == Y_pred) / len(Y_test) - -#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict)) - -## Adjust hyperparameters - -We now perform a grid search to find the optimal hyperparameters for the network. -Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around $98\%$ ($2\%$ error rate). - -eta_vals = np.logspace(-5, 1, 7) -lmbd_vals = np.logspace(-5, 1, 7) -# store the models for later use -DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) - -# grid search -for i, eta in enumerate(eta_vals): - for j, lmbd in enumerate(lmbd_vals): - dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size, - n_hidden_neurons=n_hidden_neurons, n_categories=n_categories) - dnn.train() - - DNN_numpy[i][j] = dnn - - test_predict = dnn.predict(X_test) - - print("Learning rate = ", eta) - print("Lambda = ", lmbd) - print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict)) - print() - -## Visualization - -# visual representation of grid search -# uses seaborn heatmap, you can also do this with matplotlib imshow -import seaborn as sns - -sns.set() - -train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) -test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) - -for i in range(len(eta_vals)): - for j in range(len(lmbd_vals)): - dnn = DNN_numpy[i][j] - - train_pred = dnn.predict(X_train) - test_pred = dnn.predict(X_test) - - train_accuracy[i][j] = accuracy_score(Y_train, train_pred) - test_accuracy[i][j] = accuracy_score(Y_test, test_pred) - - -fig, ax = plt.subplots(figsize = (10, 10)) -sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") -ax.set_title("Training Accuracy") -ax.set_ylabel("$\eta$") -ax.set_xlabel("$\lambda$") -plt.show() - -fig, ax = plt.subplots(figsize = (10, 10)) -sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") -ax.set_title("Test Accuracy") -ax.set_ylabel("$\eta$") -ax.set_xlabel("$\lambda$") -plt.show() - -## scikit-learn implementation - -**scikit-learn** focuses more -on traditional machine learning methods, such as regression, -clustering, decision trees, etc. As such, it has only two types of -neural networks: Multi Layer Perceptron outputting continuous values, -*MPLRegressor*, and Multi Layer Perceptron outputting labels, -*MLPClassifier*. We will see how simple it is to use these classes. - -**scikit-learn** implements a few improvements from our neural network, -such as early stopping, a varying learning rate, different -optimization methods, etc. We would therefore expect a better -performance overall. - -from sklearn.neural_network import MLPClassifier -# store models for later use -DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) - -for i, eta in enumerate(eta_vals): - for j, lmbd in enumerate(lmbd_vals): - dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic', - alpha=lmbd, learning_rate_init=eta, max_iter=epochs) - dnn.fit(X_train, Y_train) - - DNN_scikit[i][j] = dnn - - print("Learning rate = ", eta) - print("Lambda = ", lmbd) - print("Accuracy score on test set: ", dnn.score(X_test, Y_test)) - print() - -## Visualization - -# optional -# visual representation of grid search -# uses seaborn heatmap, could probably do this in matplotlib -import seaborn as sns - -sns.set() - -train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) -test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) - -for i in range(len(eta_vals)): - for j in range(len(lmbd_vals)): - dnn = DNN_scikit[i][j] - - train_pred = dnn.predict(X_train) - test_pred = dnn.predict(X_test) - - train_accuracy[i][j] = accuracy_score(Y_train, train_pred) - test_accuracy[i][j] = accuracy_score(Y_test, test_pred) - - -fig, ax = plt.subplots(figsize = (10, 10)) -sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") -ax.set_title("Training Accuracy") -ax.set_ylabel("$\eta$") -ax.set_xlabel("$\lambda$") -plt.show() - -fig, ax = plt.subplots(figsize = (10, 10)) -sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") -ax.set_title("Test Accuracy") -ax.set_ylabel("$\eta$") -ax.set_xlabel("$\lambda$") -plt.show() - -## Building neural networks in Tensorflow and Keras - -Now we want to build on the experience gained from our neural network implementation in NumPy and scikit-learn -and use it to construct a neural network in Tensorflow. Once we have constructed a neural network in NumPy -and Tensorflow, building one in Keras is really quite trivial, though the performance may suffer. - -In our previous example we used only one hidden layer, and in this we will use two. From this it should be quite -clear how to build one using an arbitrary number of hidden layers, using data structures such as Python lists or -NumPy arrays. - - -Tensorflow is an open source library machine learning library -developed by the Google Brain team for internal use. It was released -under the Apache 2.0 open source license in November 9, 2015. - -Tensorflow is a computational framework that allows you to construct -machine learning models at different levels of abstraction, from -high-level, object-oriented APIs like Keras, down to the C++ kernels -that Tensorflow is built upon. The higher levels of abstraction are -simpler to use, but less flexible, and our choice of implementation -should reflect the problems we are trying to solve. - -[Tensorflow uses](https://www.tensorflow.org/guide/graphs) so-called graphs to represent your computation -in terms of the dependencies between individual operations, such that you first build a Tensorflow *graph* -to represent your model, and then create a Tensorflow *session* to run the graph. - -In this guide we will analyze the same data as we did in our NumPy and -scikit-learn tutorial, gathered from the MNIST database of images. We -will give an introduction to the lower level Python Application -Program Interfaces (APIs), and see how we use them to build our graph. -Then we will build (effectively) the same graph in Keras, to see just -how simple solving a machine learning problem can be. - -To install tensorflow on Unix/Linux systems, use pip as - -pip3 install tensorflow - -and/or if you use **anaconda**, just write (or install from the graphical user interface) -(current release of CPU-only TensorFlow) - -conda create -n tf tensorflow -conda activate tf - -To install the current release of GPU TensorFlow - -conda create -n tf-gpu tensorflow-gpu -conda activate tf-gpu - -Keras is a high level [neural network](https://en.wikipedia.org/wiki/Application_programming_interface) -that supports Tensorflow, CTNK and Theano as backends. -If you have Anaconda installed you may run the following command - -conda install keras - -You can look up the [instructions here](https://keras.io/) for more information. - -We will to a large extent use **keras** in this course. - - -Let us look again at the MINST data set. - -# import necessary packages -import numpy as np -import matplotlib.pyplot as plt -import tensorflow as tf -from sklearn import datasets - - -# ensure the same random numbers appear every time -np.random.seed(0) - -# display images in notebook -%matplotlib inline -plt.rcParams['figure.figsize'] = (12,12) - - -# download MNIST dataset -digits = datasets.load_digits() - -# define inputs and labels -inputs = digits.images -labels = digits.target - -print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape)) -print("labels = (n_inputs) = " + str(labels.shape)) - - -# flatten the image -# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64 -n_inputs = len(inputs) -inputs = inputs.reshape(n_inputs, -1) -print("X = (n_inputs, n_features) = " + str(inputs.shape)) - - -# choose some random images to display -indices = np.arange(n_inputs) -random_indices = np.random.choice(indices, size=5) - -for i, image in enumerate(digits.images[random_indices]): - plt.subplot(1, 5, i+1) - plt.axis('off') - plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest') - plt.title("Label: %d" % digits.target[random_indices[i]]) -plt.show() - -from tensorflow.keras.layers import Input -from tensorflow.keras.models import Sequential #This allows appending layers to existing models -from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer -from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop) -from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2) -from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function - -from sklearn.model_selection import train_test_split - -# one-hot representation of labels -labels = to_categorical(labels) - -# split into train and test data -train_size = 0.8 -test_size = 1 - train_size -X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size, - test_size=test_size) - - -epochs = 100 -batch_size = 100 -n_neurons_layer1 = 100 -n_neurons_layer2 = 50 -n_categories = 10 -eta_vals = np.logspace(-5, 1, 7) -lmbd_vals = np.logspace(-5, 1, 7) -def create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, eta, lmbd): - model = Sequential() - model.add(Dense(n_neurons_layer1, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd))) - model.add(Dense(n_neurons_layer2, activation='sigmoid', kernel_regularizer=regularizers.l2(lmbd))) - model.add(Dense(n_categories, activation='softmax')) - - sgd = optimizers.SGD(lr=eta) - model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy']) - - return model - -DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object) - -for i, eta in enumerate(eta_vals): - for j, lmbd in enumerate(lmbd_vals): - DNN = create_neural_network_keras(n_neurons_layer1, n_neurons_layer2, n_categories, - eta=eta, lmbd=lmbd) - DNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0) - scores = DNN.evaluate(X_test, Y_test) - - DNN_keras[i][j] = DNN - - print("Learning rate = ", eta) - print("Lambda = ", lmbd) - print("Test accuracy: %.3f" % scores[1]) - print() - -# optional -# visual representation of grid search -# uses seaborn heatmap, could probably do this in matplotlib -import seaborn as sns - -sns.set() - -train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) -test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals))) - -for i in range(len(eta_vals)): - for j in range(len(lmbd_vals)): - DNN = DNN_keras[i][j] - - train_accuracy[i][j] = DNN.evaluate(X_train, Y_train)[1] - test_accuracy[i][j] = DNN.evaluate(X_test, Y_test)[1] - - -fig, ax = plt.subplots(figsize = (10, 10)) -sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis") -ax.set_title("Training Accuracy") -ax.set_ylabel("$\eta$") -ax.set_xlabel("$\lambda$") -plt.show() - -fig, ax = plt.subplots(figsize = (10, 10)) -sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis") -ax.set_title("Test Accuracy") -ax.set_ylabel("$\eta$") -ax.set_xlabel("$\lambda$") -plt.show() - -## The Breast Cancer Data, now with Keras - - -import tensorflow as tf -from tensorflow.keras.layers import Input -from tensorflow.keras.models import Sequential #This allows appending layers to existing models -from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer -from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop) -from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2) -from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function -import numpy as np -import matplotlib.pyplot as plt -import seaborn as sns -from sklearn.model_selection import train_test_split as splitter -from sklearn.datasets import load_breast_cancer -import pickle -import os - - -"""Load breast cancer dataset""" - -np.random.seed(0) #create same seed for random number every time - -cancer=load_breast_cancer() #Download breast cancer dataset - -inputs=cancer.data #Feature matrix of 569 rows (samples) and 30 columns (parameters) -outputs=cancer.target #Label array of 569 rows (0 for benign and 1 for malignant) -labels=cancer.feature_names[0:30] - -print('The content of the breast cancer dataset is:') #Print information about the datasets -print(labels) -print('-------------------------') -print("inputs = " + str(inputs.shape)) -print("outputs = " + str(outputs.shape)) -print("labels = "+ str(labels.shape)) - -x=inputs #Reassign the Feature and Label matrices to other variables -y=outputs - -#%% - -# Visualisation of dataset (for correlation analysis) - -plt.figure() -plt.scatter(x[:,0],x[:,2],s=40,c=y,cmap=plt.cm.Spectral) -plt.xlabel('Mean radius',fontweight='bold') -plt.ylabel('Mean perimeter',fontweight='bold') -plt.show() - -plt.figure() -plt.scatter(x[:,5],x[:,6],s=40,c=y, cmap=plt.cm.Spectral) -plt.xlabel('Mean compactness',fontweight='bold') -plt.ylabel('Mean concavity',fontweight='bold') -plt.show() - - -plt.figure() -plt.scatter(x[:,0],x[:,1],s=40,c=y,cmap=plt.cm.Spectral) -plt.xlabel('Mean radius',fontweight='bold') -plt.ylabel('Mean texture',fontweight='bold') -plt.show() - -plt.figure() -plt.scatter(x[:,2],x[:,1],s=40,c=y,cmap=plt.cm.Spectral) -plt.xlabel('Mean perimeter',fontweight='bold') -plt.ylabel('Mean compactness',fontweight='bold') -plt.show() - - -# Generate training and testing datasets - -#Select features relevant to classification (texture,perimeter,compactness and symmetery) -#and add to input matrix - -temp1=np.reshape(x[:,1],(len(x[:,1]),1)) -temp2=np.reshape(x[:,2],(len(x[:,2]),1)) -X=np.hstack((temp1,temp2)) -temp=np.reshape(x[:,5],(len(x[:,5]),1)) -X=np.hstack((X,temp)) -temp=np.reshape(x[:,8],(len(x[:,8]),1)) -X=np.hstack((X,temp)) - -X_train,X_test,y_train,y_test=splitter(X,y,test_size=0.1) #Split datasets into training and testing - -y_train=to_categorical(y_train) #Convert labels to categorical when using categorical cross entropy -y_test=to_categorical(y_test) - -del temp1,temp2,temp - -# %% - -# Define tunable parameters" - -eta=np.logspace(-3,-1,3) #Define vector of learning rates (parameter to SGD optimiser) -lamda=0.01 #Define hyperparameter -n_layers=2 #Define number of hidden layers in the model -n_neuron=np.logspace(0,3,4,dtype=int) #Define number of neurons per layer -epochs=100 #Number of reiterations over the input data -batch_size=100 #Number of samples per gradient update - -# %% - -"""Define function to return Deep Neural Network model""" - -def NN_model(inputsize,n_layers,n_neuron,eta,lamda): - model=Sequential() - for i in range(n_layers): #Run loop to add hidden layers to the model - if (i==0): #First layer requires input dimensions - model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda),input_dim=inputsize)) - else: #Subsequent layers are capable of automatic shape inferencing - model.add(Dense(n_neuron,activation='relu',kernel_regularizer=regularizers.l2(lamda))) - model.add(Dense(2,activation='softmax')) #2 outputs - ordered and disordered (softmax for prob) - sgd=optimizers.SGD(lr=eta) - model.compile(loss='categorical_crossentropy',optimizer=sgd,metrics=['accuracy']) - return model - - -Train_accuracy=np.zeros((len(n_neuron),len(eta))) #Define matrices to store accuracy scores as a function -Test_accuracy=np.zeros((len(n_neuron),len(eta))) #of learning rate and number of hidden neurons for - -for i in range(len(n_neuron)): #run loops over hidden neurons and learning rates to calculate - for j in range(len(eta)): #accuracy scores - DNN_model=NN_model(X_train.shape[1],n_layers,n_neuron[i],eta[j],lamda) - DNN_model.fit(X_train,y_train,epochs=epochs,batch_size=batch_size,verbose=1) - Train_accuracy[i,j]=DNN_model.evaluate(X_train,y_train)[1] - Test_accuracy[i,j]=DNN_model.evaluate(X_test,y_test)[1] - - -def plot_data(x,y,data,title=None): - - # plot results - fontsize=16 - - - fig = plt.figure() - ax = fig.add_subplot(111) - cax = ax.matshow(data, interpolation='nearest', vmin=0, vmax=1) - - cbar=fig.colorbar(cax) - cbar.ax.set_ylabel('accuracy (%)',rotation=90,fontsize=fontsize) - cbar.set_ticks([0,.2,.4,0.6,0.8,1.0]) - cbar.set_ticklabels(['0%','20%','40%','60%','80%','100%']) - - # put text on matrix elements - for i, x_val in enumerate(np.arange(len(x))): - for j, y_val in enumerate(np.arange(len(y))): - c = "${0:.1f}\\%$".format( 100*data[j,i]) - ax.text(x_val, y_val, c, va='center', ha='center') - - # convert axis vaues to to string labels - x=[str(i) for i in x] - y=[str(i) for i in y] - - - ax.set_xticklabels(['']+x) - ax.set_yticklabels(['']+y) - - ax.set_xlabel('$\\mathrm{learning\\ rate}$',fontsize=fontsize) - ax.set_ylabel('$\\mathrm{hidden\\ neurons}$',fontsize=fontsize) - if title is not None: - ax.set_title(title) - - plt.tight_layout() - - plt.show() - -plot_data(eta,n_neuron,Train_accuracy, 'training') -plot_data(eta,n_neuron,Test_accuracy, 'testing') - -## Fine-tuning neural network hyperparameters - -The flexibility of neural networks is also one of their main -drawbacks: there are many hyperparameters to tweak. Not only can you -use any imaginable network topology (how neurons/nodes are interconnected), -but even in a simple FFNN you can change the number of layers, the -number of neurons per layer, the type of activation function to use in -each layer, the weight initialization logic, the stochastic gradient optmized and much more. How do you -know what combination of hyperparameters is the best for your task? - -* You can use grid search with cross-validation to find the right hyperparameters. - -However,since there are many hyperparameters to tune, and since -training a neural network on a large dataset takes a lot of time, you -will only be able to explore a tiny part of the hyperparameter space. - - -* You can use randomized search. - -* Or use tools like [Oscar](http://oscar.calldesk.ai/), which implements more complex algorithms to help you find a good set of hyperparameters quickly. - -For many problems you can start with just one or two hidden layers and it will work just fine. -For the MNIST data set you ca easily get a high accuracy using just one hidden layer with a -few hundred neurons. -You can reach for this data set above 98% accuracy using two hidden layers with the same total amount of -neurons, in roughly the same amount of training time. - -For more complex problems, you can gradually -ramp up the number of hidden layers, until you start overfitting the training set. Very complex tasks, such -as large image classification or speech recognition, typically require networks with dozens of layers -and they need a huge amount -of training data. However, you will rarely have to train such networks from scratch: it is much more -common to reuse parts of a pretrained state-of-the-art network that performs a similar task. - - - - - -## Which activation function should I use? - -The Back propagation algorithm we derived above works by going from -the output layer to the input layer, propagating the error gradient on -the way. Once the algorithm has computed the gradient of the cost -function with regards to each parameter in the network, it uses these -gradients to update each parameter with a Gradient Descent (GD) step. - - -Unfortunately for us, the gradients often get smaller and smaller as the -algorithm progresses down to the first hidden layers. As a result, the -GD update leaves the lower layer connection weights -virtually unchanged, and training never converges to a good -solution. This is known in the literature as -**the vanishing gradients problem**. - -In other cases, the opposite can happen, namely the the gradients can grow bigger and -bigger. The result is that many of the layers get large updates of the -weights the -algorithm diverges. This is the **exploding gradients problem**, which is -mostly encountered in recurrent neural networks. More generally, deep -neural networks suffer from unstable gradients, different layers may -learn at widely different speeds - - - - -Although this unfortunate behavior has been empirically observed for -quite a while (it was one of the reasons why deep neural networks were -mostly abandoned for a long time), it is only around 2010 that -significant progress was made in understanding it. - -A paper titled [Understanding the Difficulty of Training Deep -Feedforward Neural Networks by Xavier Glorot and Yoshua Bengio](http://proceedings.mlr.press/v9/glorot10a.html) found that -the problems with the popular logistic -sigmoid activation function and the weight initialization technique -that was most popular at the time, namely random initialization using -a normal distribution with a mean of 0 and a standard deviation of -1. - -They showed that with this activation function and this -initialization scheme, the variance of the outputs of each layer is -much greater than the variance of its inputs. Going forward in the -network, the variance keeps increasing after each layer until the -activation function saturates at the top layers. This is actually made -worse by the fact that the logistic function has a mean of 0.5, not 0 -(the hyperbolic tangent function has a mean of 0 and behaves slightly -better than the logistic function in deep networks). - - - -Looking at the logistic activation function, when inputs become large -(negative or positive), the function saturates at 0 or 1, with a -derivative extremely close to 0. Thus when backpropagation kicks in, -it has virtually no gradient to propagate back through the network, -and what little gradient exists keeps getting diluted as -backpropagation progresses down through the top layers, so there is -really nothing left for the lower layers. - -In their paper, Glorot and Bengio propose a way to significantly -alleviate this problem. We need the signal to flow properly in both -directions: in the forward direction when making predictions, and in -the reverse direction when backpropagating gradients. We don’t want -the signal to die out, nor do we want it to explode and saturate. For -the signal to flow properly, the authors argue that we need the -variance of the outputs of each layer to be equal to the variance of -its inputs, and we also need the gradients to have equal variance -before and after flowing through a layer in the reverse direction. - - - -One of the insights in the 2010 paper by Glorot and Bengio was that -the vanishing/exploding gradients problems were in part due to a poor -choice of activation function. Until then most people had assumed that -if Nature had chosen to use roughly sigmoid activation functions in -biological neurons, they must be an excellent choice. But it turns out -that other activation functions behave much better in deep neural -networks, in particular the ReLU activation function, mostly because -it does not saturate for positive values (and also because it is quite -fast to compute). - - -## The RELU function family - -The ReLU activation function suffers from a problem known as the dying -ReLUs: during training, some neurons effectively die, meaning they -stop outputting anything other than 0. - -In some cases, you may find that half of your network’s neurons are -dead, especially if you used a large learning rate. During training, -if a neuron’s weights get updated such that the weighted sum of the -neuron’s inputs is negative, it will start outputting 0. When this -happen, the neuron is unlikely to come back to life since the gradient -of the ReLU function is 0 when its input is negative. - -To solve this problem, nowadays practitioners use a variant of the ReLU -function, such as the leaky ReLU discussed above or the so-called -exponential linear unit (ELU) function - -$$ -ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\ z & z \ge 0.\end{array}\right. -$$ - -In general it seems that the ELU activation function is better than -the leaky ReLU function (and its variants), which is better than -ReLU. ReLU performs better than $\tanh$ which in turn performs better -than the logistic function. - -If runtime -performance is an issue, then you may opt for the leaky ReLU function over the -ELU function If you don’t -want to tweak yet another hyperparameter, you may just use the default -$\alpha$ of $0.01$ for the leaky ReLU, and $1$ for ELU. If you have -spare time and computing power, you can use cross-validation or -bootstrap to evaluate other activation functions. - - - -In most cases you can use the ReLU activation function in the hidden layers (or one of its variants). - -It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck. - -**For the output layer:** - -* For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive). - -* For regression tasks, you can simply use no activation function at all. - -## Batch Normalization - -Batch Normalization -aims to address the vanishing/exploding gradients problems, and more generally the problem that the -distribution of each layer’s inputs changes during training, as the parameters of the previous layers change. - -The technique consists of adding an operation in the model just before the activation function of each -layer, simply zero-centering and normalizing the inputs, then scaling and shifting the result using two new -parameters per layer (one for scaling, the other for shifting). In other words, this operation lets the model -learn the optimal scale and mean of the inputs for each layer. -In order to zero-center and normalize the inputs, the algorithm needs to estimate the inputs’ mean and -standard deviation. It does so by evaluating the mean and standard deviation of the inputs over the current -mini-batch, from this the name batch normalization. - -## Dropout - -It is a fairly simple algorithm: at every training step, every neuron (including the input neurons but -excluding the output neurons) has a probability $p$ of being temporarily dropped out, meaning it will be -entirely ignored during this training step, but it may be active during the next step. - -The -hyperparameter $p$ is called the dropout rate, and it is typically set to 50%. After training, the neurons are not dropped anymore. - It is viewed as one of the most popular regularization techniques. - -## Gradient Clipping - -A popular technique to lessen the exploding gradients problem is to simply clip the gradients during -backpropagation so that they never exceed some threshold (this is mostly useful for recurrent neural -networks). - -This technique is called Gradient Clipping. - -In general however, Batch -Normalization is preferred. - - -## A top-down perspective on Neural networks - - -The first thing we would like to do is divide the data into two or three -parts. A training set, a validation or dev (development) set, and a -test set. The test set is the data on which we want to make -predictions. The dev set is a subset of the training data we use to -check how well we are doing out-of-sample, after training the model on -the training dataset. We use the validation error as a proxy for the -test error in order to make tweaks to our model. It is crucial that we -do not use any of the test data to train the algorithm. This is a -cardinal sin in ML. Then: - - -* Estimate optimal error rate - -* Minimize underfitting (bias) on training data set. - -* Make sure you are not overfitting. - -If the validation and test sets are drawn from the same distributions, -then a good performance on the validation set should lead to similarly -good performance on the test set. - -However, sometimes -the training data and test data differ in subtle ways because, for -example, they are collected using slightly different methods, or -because it is cheaper to collect data in one way versus another. In -this case, there can be a mismatch between the training and test -data. This can lead to the neural network overfitting these small -differences between the test and training sets, and a poor performance -on the test set despite having a good performance on the validation -set. To rectify this, Andrew Ng suggests making two validation or dev -sets, one constructed from the training data and one constructed from -the test data. The difference between the performance of the algorithm -on these two validation sets quantifies the train-test mismatch. This -can serve as another important diagnostic when using DNNs for -supervised learning. - - -## Limitations of supervised learning with deep networks - -Like all statistical methods, supervised learning using neural -networks has important limitations. This is especially important when -one seeks to apply these methods, especially to physics problems. Like -all tools, DNNs are not a universal solution. Often, the same or -better performance on a task can be achieved by using a few -hand-engineered features (or even a collection of random -features). - -Here we list some of the important limitations of supervised neural network based models. - - - -* **Need labeled data**. All supervised learning methods, DNNs for supervised learning require labeled data. Often, labeled data is harder to acquire than unlabeled data (e.g. one must pay for human experts to label images). - -* **Supervised neural networks are extremely data intensive.** DNNs are data hungry. They perform best when data is plentiful. This is doubly so for supervised methods where the data must also be labeled. The utility of DNNs is extremely limited if data is hard to acquire or the datasets are small (hundreds to a few thousand samples). In this case, the performance of other methods that utilize hand-engineered features can exceed that of DNNs. - -* **Homogeneous data.** Almost all DNNs deal with homogeneous data of one type. It is very hard to design architectures that mix and match data types (i.e. some continuous variables, some discrete variables, some time series). In applications beyond images, video, and language, this is often what is required. In contrast, ensemble models like random forests or gradient-boosted trees have no difficulty handling mixed data types. - -* **Many problems are not about prediction.** In natural science we are often interested in learning something about the underlying distribution that generates the data. In this case, it is often difficult to cast these ideas in a supervised learning setting. While the problems are related, it is possible to make good predictions with a *wrong* model. The model might or might not be useful for understanding the underlying science. - -Some of these remarks are particular to DNNs, others are shared by all supervised learning methods. 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za`*6;xTjE@u}usQxWx{h`e6hew?ZOBH^{YF3bkmVM%YIAb_+m$;3eF6v#g=M1c}rD zNv!8FH)FW3!2>8lsQQ%n&MiyD?@+-1wHp6@WKbrFifLNm+e>eJElfP3P*v8tkahm% G{r>{<=Cs)W diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb deleted file mode 100644 index 3ff3900d1..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter11.ipynb +++ /dev/null @@ -1,3310 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Solving Differential Equations with Deep Learning\n", - "\n", - "The Universal Approximation Theorem states that a neural network can\n", - "approximate any function at a single hidden layer along with one input\n", - "and output layer to any given precision. \n", - "\n", - "\n", - "An ordinary differential equation (ODE) is an equation involving functions having one variable.\n", - "\n", - "In general, an ordinary differential equation looks like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{ode} \\tag{1}\n", - "f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right) = 0\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(x)$ is the function to find, and $g^{(n)}(x)$ is the $n$-th derivative of $g(x)$.\n", - "\n", - "The $f\\left(x, g(x), g'(x), g''(x), \\, \\dots \\, , g^{(n)}(x)\\right)$ is just a way to write that there is an expression involving $x$ and $g(x), \\ g'(x), \\ g''(x), \\, \\dots \\, , \\text{ and } g^{(n)}(x)$ on the left side of the equality sign in ([1](#ode)).\n", - "The highest order of derivative, that is the value of $n$, determines to the order of the equation.\n", - "The equation is referred to as a $n$-th order ODE.\n", - "Along with ([1](#ode)), some additional conditions of the function $g(x)$ are typically given\n", - "for the solution to be unique.\n", - "\n", - "\n", - "\n", - "Let the trial solution $g_t(x)$ be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\tg_t(x) = h_1(x) + h_2(x,N(x,P))\n", - "\\label{_auto1} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $h_1(x)$ is a function that makes $g_t(x)$ satisfy a given set\n", - "of conditions, $N(x,P)$ a neural network with weights and biases\n", - "described by $P$ and $h_2(x, N(x,P))$ some expression involving the\n", - "neural network. The role of the function $h_2(x, N(x,P))$, is to\n", - "ensure that the output from $N(x,P)$ is zero when $g_t(x)$ is\n", - "evaluated at the values of $x$ where the given conditions must be\n", - "satisfied. The function $h_1(x)$ should alone make $g_t(x)$ satisfy\n", - "the conditions.\n", - "\n", - "But what about the network $N(x,P)$?\n", - "\n", - "\n", - "As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.\n", - "\n", - "\n", - "\n", - "For the minimization to be defined, we need to have a cost function at hand to minimize.\n", - "\n", - "It is given that $f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)$ should be equal to zero in ([1](#ode)).\n", - "We can choose to consider the mean squared error as the cost function for an input $x$.\n", - "Since we are looking at one input, the cost function is just $f$ squared.\n", - "The cost function $c\\left(x, P \\right)$ can therefore be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(x, P\\right) = \\big(f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)\\big)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If $N$ inputs are given as a vector $\\boldsymbol{x}$ with elements $x_i$ for $i = 1,\\dots,N$,\n", - "the cost function becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{cost} \\tag{3}\n", - "\tC\\left(\\boldsymbol{x}, P\\right) = \\frac{1}{N} \\sum_{i=1}^N \\big(f\\left(x_i, \\, g(x_i), \\, g'(x_i), \\, g''(x_i), \\, \\dots \\, , \\, g^{(n)}(x_i)\\right)\\big)^2\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The neural net should then find the parameters $P$ that minimizes the cost function in\n", - "([3](#cost)) for a set of $N$ training samples $x_i$.\n", - "\n", - "\n", - "\n", - "To perform the minimization using gradient descent, the gradient of $C\\left(\\boldsymbol{x}, P\\right)$ is needed.\n", - "It might happen so that finding an analytical expression of the gradient of $C(\\boldsymbol{x}, P)$ from ([3](#cost)) gets too messy, depending on which cost function one desires to use.\n", - "\n", - "Luckily, there exists libraries that makes the job for us through automatic differentiation.\n", - "Automatic differentiation is a method of finding the derivatives numerically with very high precision.\n", - "\n", - "\n", - "### Example: Exponential decay\n", - "\n", - "An exponential decay of a quantity $g(x)$ is described by the equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solve_expdec} \\tag{4}\n", - " g'(x) = -\\gamma g(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $g(0) = g_0$ for some chosen initial value $g_0$.\n", - "\n", - "The analytical solution of ([4](#solve_expdec)) is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " g(x) = g_0 \\exp\\left(-\\gamma x\\right)\n", - "\\label{_auto2} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of ([4](#solve_expdec)).\n", - "\n", - "\n", - "\n", - "The program will use a neural network to solve" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solveode} \\tag{6}\n", - "g'(x) = -\\gamma g(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(0) = g_0$ with $\\gamma$ and $g_0$ being some chosen values.\n", - "\n", - "In this example, $\\gamma = 2$ and $g_0 = 10$.\n", - "\n", - "\n", - "To begin with, a trial solution $g_t(t)$ must be chosen. A general trial solution for ordinary differential equations could be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x, P) = h_1(x) + h_2(x, N(x, P))\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and $h_2(x,N(x, P))$ an expression involving $x$ and the output from the neural network $N(x,P)$ with $P $ being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.\n", - "\n", - "\n", - "\n", - "In this network, there are no weights and bias at the input layer, so $P = \\{ P_{\\text{hidden}}, P_{\\text{output}} \\}$.\n", - "If there are $N_{\\text{hidden} }$ neurons in the hidden layer, then $P_{\\text{hidden}}$ is a $N_{\\text{hidden} } \\times (1 + N_{\\text{input}})$ matrix, given that there are $N_{\\text{input}}$ neurons in the input layer.\n", - "\n", - "The first column in $P_{\\text{hidden} }$ represents the bias for each neuron in the hidden layer and the second column represents the weights for each neuron in the hidden layer from the input layer.\n", - "If there are $N_{\\text{output} }$ neurons in the output layer, then $P_{\\text{output}} $ is a $N_{\\text{output} } \\times (1 + N_{\\text{hidden} })$ matrix.\n", - "\n", - "Its first column represents the bias of each neuron and the remaining columns represents the weights to each neuron.\n", - "\n", - "It is given that $g(0) = g_0$. The trial solution must fulfill this condition to be a proper solution of ([6](#solveode)). A possible way to ensure that $g_t(0, P) = g_0$, is to let $F(N(x,P)) = x \\cdot N(x,P)$ and $A(x) = g_0$. This gives the following trial solution:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{trial} \\tag{7}\n", - "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Reformulating the problem\n", - "\n", - "We wish that our neural network manages to minimize a given cost function.\n", - "\n", - "A reformulation of out equation, ([6](#solveode)), must therefore be done,\n", - "such that it describes the problem a neural network can solve for.\n", - "\n", - "The neural network must find the set of weights and biases $P$ such that the trial solution in ([7](#trial)) satisfies ([6](#solveode)).\n", - "\n", - "The trial solution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "has been chosen such that it already solves the condition $g(0) = g_0$. What remains, is to find $P$ such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{nnmin} \\tag{8}\n", - "g_t'(x, P) = - \\gamma g_t(x, P)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "is fulfilled as *best as possible*.\n", - "\n", - "\n", - "The left hand side and right hand side of ([8](#nnmin)) must be computed separately, and then the neural network must choose weights and biases, contained in $P$, such that the sides are equal as best as possible.\n", - "This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.\n", - "In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to $P$ of the neural network.\n", - "\n", - "This gives the following cost function our neural network must solve for:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P}\\Big\\{ \\big(g_t'(x, P) - ( -\\gamma g_t(x, P) \\big)^2 \\Big\\}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "(the notation $\\min_{P}\\{ f(x, P) \\}$ means that we desire to find $P$ that yields the minimum of $f(x, P)$)\n", - "\n", - "or, in terms of weights and biases for the hidden and output layer in our network:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }}\\Big\\{ \\big(g_t'(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) - ( -\\gamma g_t(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) \\big)^2 \\Big\\}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for an input value $x$.\n", - "\n", - "\n", - "\n", - "If the neural network evaluates $g_t(x, P)$ at more values for $x$, say $N$ values $x_i$ for $i = 1, \\dots, N$, then the *total* error to minimize becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{min} \\tag{9}\n", - "\\min_{P}\\Big\\{\\frac{1}{N} \\sum_{i=1}^N \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2 \\Big\\}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Letting $\\boldsymbol{x}$ be a vector with elements $x_i$ and $C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2$ denote the cost function, the minimization problem that our network must solve, becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P} C(\\boldsymbol{x}, P)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In terms of $P_{\\text{hidden} }$ and $P_{\\text{output} }$, this could also be expressed as\n", - "\n", - "$$\n", - "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }} C(\\boldsymbol{x}, \\{P_{\\text{hidden} }, P_{\\text{output} }\\})\n", - "$$\n", - "\n", - "\n", - "For simplicity, it is assumed that the input is an array $\\boldsymbol{x} = (x_1, \\dots, x_N)$ with $N$ elements. It is at these points the neural network should find $P$ such that it fulfills ([9](#min)).\n", - "\n", - "First, the neural network must feed forward the inputs.\n", - "This means that $\\boldsymbol{x}s$ must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further.\n", - "The input layer will consist of $N_{\\text{input} }$ neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be $N_{\\text{hidden} }$.\n", - "\n", - "\n", - "For the $i$-th in the hidden layer with weight $w_i^{\\text{hidden} }$ and bias $b_i^{\\text{hidden} }$, the weighting from the $j$-th neuron at the input layer is:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "z_{i,j}^{\\text{hidden}} &= b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_j \\\\\n", - "&=\n", - "\\begin{pmatrix}\n", - "b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 \\\\\n", - "x_j\n", - "\\end{pmatrix}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The result after weighting the inputs at the $i$-th hidden neuron can be written as a vector:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "\\boldsymbol{z}_{i}^{\\text{hidden}} &= \\Big( b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_1 , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_2, \\ \\dots \\, , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_N\\Big) \\\\\n", - "&=\n", - "\\begin{pmatrix}\n", - " b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 & 1 & \\dots & 1 \\\\\n", - "x_1 & x_2 & \\dots & x_N\n", - "\\end{pmatrix} \\\\\n", - "&= \\boldsymbol{p}_{i, \\text{hidden}}^T X\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The vector $\\boldsymbol{p}_{i, \\text{hidden}}^T$ constitutes each row in $P_{\\text{hidden} }$, which contains the weights for the neural network to minimize according to ([9](#min)).\n", - "\n", - "After having found $\\boldsymbol{z}_{i}^{\\text{hidden}} $ for every $i$-th neuron within the hidden layer, the vector will be sent to an activation function $a_i(\\boldsymbol{z})$.\n", - "\n", - "In this example, the sigmoid function has been chosen to be the activation function for each hidden neuron:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(z) = \\frac{1}{1 + \\exp{(-z)}}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is possible to use other activations functions for the hidden layer also.\n", - "\n", - "The output $\\boldsymbol{x}_i^{\\text{hidden}}$ from each $i$-th hidden neuron is:\n", - "\n", - "$$\n", - "\\boldsymbol{x}_i^{\\text{hidden} } = f\\big( \\boldsymbol{z}_{i}^{\\text{hidden}} \\big)\n", - "$$\n", - "\n", - "The outputs $\\boldsymbol{x}_i^{\\text{hidden} } $ are then sent to the output layer.\n", - "\n", - "The output layer consists of one neuron in this case, and combines the\n", - "output from each of the neurons in the hidden layers. The output layer\n", - "combines the results from the hidden layer using some weights $w_i^{\\text{output}}$\n", - "and biases $b_i^{\\text{output}}$. In this case,\n", - "it is assumes that the number of neurons in the output layer is one.\n", - "\n", - "\n", - "\n", - "The procedure of weighting the output neuron $j$ in the hidden layer to the $i$-th neuron in the output layer is similar as for the hidden layer described previously." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "z_{1,j}^{\\text{output}} & =\n", - "\\begin{pmatrix}\n", - "b_1^{\\text{output}} & \\boldsymbol{w}_1^{\\text{output}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 \\\\\n", - "\\boldsymbol{x}_j^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Expressing $z_{1,j}^{\\text{output}}$ as a vector gives the following way of weighting the inputs from the hidden layer:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{z}_{1}^{\\text{output}} =\n", - "\\begin{pmatrix}\n", - "b_1^{\\text{output}} & \\boldsymbol{w}_1^{\\text{output}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 & 1 & \\dots & 1 \\\\\n", - "\\boldsymbol{x}_1^{\\text{hidden}} & \\boldsymbol{x}_2^{\\text{hidden}} & \\dots & \\boldsymbol{x}_N^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\\boldsymbol{z}_{1}^{\\text{output}}$ the neural network has finished its feed forward step, and $\\boldsymbol{z}_{1}^{\\text{output}}$ is the final output of the network.\n", - "\n", - "\n", - "The next step is to decide how the parameters should be changed such that they minimize the cost function.\n", - "\n", - "The chosen cost function for this problem is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to minimize the cost function, an optimization method must be chosen.\n", - "\n", - "Here, gradient descent with a constant step size has been chosen.\n", - "\n", - "### Gradient descent\n", - "\n", - "The idea of the gradient descent algorithm is to update parameters in\n", - "a direction where the cost function decreases goes to a minimum.\n", - "\n", - "In general, the update of some parameters $\\boldsymbol{\\omega}$ given a cost\n", - "function defined by some weights $\\boldsymbol{\\omega}$, $C(\\boldsymbol{x},\n", - "\\boldsymbol{\\omega})$, goes as follows:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\omega}_{\\text{new} } = \\boldsymbol{\\omega} - \\lambda \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for a number of iterations or until $ \\big|\\big| \\boldsymbol{\\omega}_{\\text{new} } - \\boldsymbol{\\omega} \\big|\\big|$ becomes smaller than some given tolerance.\n", - "\n", - "The value of $\\lambda$ decides how large steps the algorithm must take\n", - "in the direction of $ \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})$.\n", - "The notation $\\nabla_{\\boldsymbol{\\omega}}$ express the gradient with respect\n", - "to the elements in $\\boldsymbol{\\omega}$.\n", - "\n", - "In our case, we have to minimize the cost function $C(\\boldsymbol{x}, P)$ with\n", - "respect to the two sets of weights and biases, that is for the hidden\n", - "layer $P_{\\text{hidden} }$ and for the output layer $P_{\\text{output}\n", - "}$ .\n", - "\n", - "This means that $P_{\\text{hidden} }$ and $P_{\\text{output} }$ is updated by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "P_{\\text{hidden},\\text{new}} &= P_{\\text{hidden}} - \\lambda \\nabla_{P_{\\text{hidden}}} C(\\boldsymbol{x}, P) \\\\\n", - "P_{\\text{output},\\text{new}} &= P_{\\text{output}} - \\lambda \\nabla_{P_{\\text{output}}} C(\\boldsymbol{x}, P)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### The code for solving the ODE" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Initial cost: 367.01\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Final cost: 0.0666807\n", - "Max absolute difference: 0.0437499\n" - ] - }, - { - "data": { - "image/png": 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_47_2.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "%matplotlib inline\n", - "\n", - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# Assuming one input, hidden, and output layer\n", - "def neural_network(params, x):\n", - "\n", - " # Find the weights (including and biases) for the hidden and output layer.\n", - " # Assume that params is a list of parameters for each layer.\n", - " # The biases are the first element for each array in params,\n", - " # and the weights are the remaning elements in each array in params.\n", - "\n", - " w_hidden = params[0]\n", - " w_output = params[1]\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " ## Hidden layer:\n", - "\n", - " # Add a row of ones to include bias\n", - " x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_input)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " ## Output layer:\n", - "\n", - " # Include bias:\n", - " x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_hidden)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial(x,params, g0 = 10):\n", - " return g0 + x*neural_network(params,x)\n", - "\n", - "# The right side of the ODE:\n", - "def g(x, g_trial, gamma = 2):\n", - " return -gamma*g_trial\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the neural network\n", - " d_net_out = elementwise_grad(neural_network,1)(P,x)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = g(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# Solve the exponential decay ODE using neural network with one input, hidden, and output layer\n", - "def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):\n", - " ## Set up initial weights and biases\n", - "\n", - " # For the hidden layer\n", - " p0 = npr.randn(num_neurons_hidden, 2 )\n", - "\n", - " # For the output layer\n", - " p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included\n", - "\n", - " P = [p0, p1]\n", - "\n", - " print('Initial cost: %g'%cost_function(P, x))\n", - "\n", - " ## Start finding the optimal weights using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of two arrays;\n", - " # one for the gradient w.r.t P_hidden and\n", - " # one for the gradient w.r.t P_output\n", - " cost_grad = cost_function_grad(P, x)\n", - "\n", - " P[0] = P[0] - lmb * cost_grad[0]\n", - " P[1] = P[1] - lmb * cost_grad[1]\n", - "\n", - " print('Final cost: %g'%cost_function(P, x))\n", - "\n", - " return P\n", - "\n", - "def g_analytic(x, gamma = 2, g0 = 10):\n", - " return g0*np.exp(-gamma*x)\n", - "\n", - "# Solve the given problem\n", - "if __name__ == '__main__':\n", - " # Set seed such that the weight are initialized\n", - " # with same weights and biases for every run.\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " N = 10\n", - " x = np.linspace(0, 1, N)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = 10\n", - " num_iter = 10000\n", - " lmb = 0.001\n", - "\n", - " # Use the network\n", - " P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " # Print the deviation from the trial solution and true solution\n", - " res = g_trial(x,P)\n", - " res_analytical = g_analytic(x)\n", - "\n", - " print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))\n", - "\n", - " # Plot the results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, res_analytical)\n", - " plt.plot(x, res[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The network with one input layer, specified number of hidden layers, and one output layer\n", - "\n", - "It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.\n", - "\n", - "The number of neurons within each hidden layer are given as a list of integers in the program below." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Initial cost: 324.246\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", - " return array(a, dtype, copy=False, order=order)\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Final cost: 0.119936\n" - ] - }, - { - "data": { - "image/png": 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5Zuh48nzuW2s3A3HAK8aYSGPM+Tjbr7zsBsobY0pnO+9EtvG7gWre7VdBjMWZr7d4t68RxpjWxpiGuVz3pF8PvE50OzMI54MI6dba7F+Rkdt2I7u81qGSOLt9E4BwY8yLQKmCBDfGXGOMqet985OEs6cjixN7fcpv25zfdBXla/1/BG3x8m7MO+G8M03AaepPcGLT/DrOE3sZzgGFi7znFUa+ZJxP/vyMs2uhB847loJ63JtpAc5uh//h7Asv8HRba4/irMhX4oz2fA7caq1dfQIZeuAc8PgN/79x/Q9r7QqcF7cfcN6p7uffu3Y+wpn+P40xyTgHOrbx3rYg0zQYZz/+LpxRkAcLOA05PYUz7D3XOLtCJuEcSIq1di3Oi/0knE+vFeQ7fe4HXvVO04s4y/uUWWvjcD6s8SnOvFyPc5zDMW/hbEgOGGMe9543DWcDeax4zcR5l3jsNNbaScALOO++d+K8uz22Wy+/TFtwytfTxpheuVzlNJwCfhBnV/I0/v/ddnecY1F24Hy1wkveLLk9ThrwK87B3T8UJFsu97EXZ9TpHZxdqY1wnut57XIp8LpegMdeh7O7rCnOcSY7geuBy621s3Jc/YAx5jDOc/0qoKu19rsc1xlj/v09XiPzeOh+OOvfnzjL4FucD0gk4ow+PoYzL54ErvHOo5M1GueTbUtwSu633vNfwfngRpL3/F+PdycFeO73wNlO7MMppYOOc1+rcV5gN3qfF1U4sW38ZJwPb+0yxuQ7b7zb+Mtwnj87cLZNxz6Mldt1T+X14ES3M4NxPoyUs1x8CzTyzp9Rudwu13UI5xCC33E+SLEZZ49MfofyHFMPZ5t6COfYws+ttVNO5PWpANvm/KaryF7rc2OcQxtEApcxZirOgZv93c4igck4uzO2AT2ttSez61K8jDEWZ5fserezSO6MMcVwdlm38L4REB8K2hEvEZHj8e6+KuPdpXrsmK25LscS8YX7gAUqXe7QNwOLSKg6F2dXZSTOd1N19n4iWiRoGWPicd5kdHY3SejSrkYRERERH9GuRhEREREfCYhdjRUqVLC1atVyO4aIiIhIvhYuXLjXWpvrl28HRPGqVasWcXFxbscQERERyZcxJs9fUdCuRhEREREfUfESERER8REVLxEREREfCYhjvEREROTUpaens23bNlJTU92OEhSio6OpVq0aERERBb6NipeIiEiI2LZtGyVLlqRWrVo4v0stJ8taS2JiItu2baN27doFvp12NYqIiISI1NRUypcvr9JVCIwxlC9f/oRHD1W8REREQohKV+E5mXmp4iUiIiLiIypeIiIiEnAGDBhA3759873Ojh07/jndq1cvVq5cecKPNXXqVK655poTvl1uVLxEREQkKOUsXv3796dRo0YuJlLxEhERER/r3LkzLVu2pHHjxnz99dcAlChRgueee46mTZtyzjnnsHv3bgDGjBlDmzZtaN68OR06dPjn/GOSk5OpXbs26enpABw8eJDatWszfPhw4uLi6NmzJ82aNePIkSO0a9fun58g/P3332nRogVNmzalffv2AMyfP59zzz2X5s2bc95557FmzZpCn3Z9nYSIiEgIemXMClbuOFio99moSileurZxvtf77rvvKFeuHEeOHKF169Zcf/31HD58mHPOOYc33niDJ598km+++Ybnn3+e888/n7lz52KMoX///rzzzju8//77/9xXyZIladeuHePGjaNz584MGzaMLl260LVrVz777DPee+89WrVq9a/HT0hI4O6772b69OnUrl2bffv2AXDGGWcwY8YMwsPDmTRpEs8++yy//PJLoc4jFS8RERHxqY8//piRI0cCsHXrVtatW0dkZOQ/x1G1bNmSiRMnAs53j3Xr1o2dO3dy9OjRXL8zq1evXrzzzjt07tyZ77//nm+++ea4jz937lwuvPDCf+6rXLlyACQlJXHbbbexbt06jDH/jKIVJhUvERGREFSQkamiMHXqVCZNmsScOXOIiYmhXbt2pKamEhER8c/XM4SFhZGRkQHAAw88wKOPPkrHjh2ZOnUqL7/88n/us23btsTHxzN16lQyMzNp0qTJSWV74YUXuPjiixk5ciTx8fG0a9fuZCczTzrGS0RERHwmKSmJsmXLEhMTw+rVq5k7d26+169atSoAAwcOzPN6t956Kz169OCOO+7457ySJUuSnJz8n+uec845TJ8+nU2bNgH8s6sx+2MNGDDghKaroFS8RERExGeuuOIKMjIyaNiwIU8//TTnnHPOca//8ssv07VrV1q2bEmFChXyvF7Pnj3Zv38/3bt3/+e822+/nd69e/9zcP0xsbGxfP3113Tp0oWmTZvSrVs3AJ588kmeeeYZmjdv/s+IW2Ez1toiuePC1KpVK3vsUwgiIiJyclatWkXDhg3djlEkRowYwejRoxk8eLBPHze3eWqMWWitbZXb9XWMl4iIiAS0Bx54gAkTJjB+/Hi3o+RLxUtEREQC2ieffOJ2hALTMV4iIiIiPlJkxcsY850xZo8xZnm288oZYyYaY9Z5/y1bVI8vIiIi4m+KcsRrAHBFjvOeBv6y1tYD/vKe9gsZ6UfdjiAiIiJBrsiKl7V2OrAvx9mdgGNfwjEQ6FxUj38i5gx8lo3/a0tWZqbbUURERCSI+foYr0rW2p3e/+8CKuV1RWPMPcaYOGNMXEJCQpGGCi9bg/oZa1k6aWiRPo6IiIj828svv8x7773ndgyfce3geut8gVieXyJmrf3aWtvKWtsqNja2SLM0v/IutpgqlJ7/PjZLo14iIiJSNHxdvHYbYyoDeP/d4+PHz1V4RAS7mj3I6ZnxLJ30g9txREREgtobb7xB/fr1Of/881mzZg0A7dq146mnnuLss8+mfv36zJgxA3B+uqdLly5cccUV1KtXjyeffNLN6KfM19/j9RtwG/C299/RPn78PLW46i62LPmYUvPex3bogfGEuR1JRESk6Ex4Gnb9Xbj3edqZcOXbx73KwoULGTZsGEuWLCEjI4MWLVrQsmVLADIyMpg/fz7jx4/nlVdeYdKkSQAsWbKExYsXExUVRYMGDXjggQeoXr164Wb3kaL8OokfgTlAA2PMNmPMXTiF61JjzDqgg/e0XwiPiGRH0wc4PXMTf/+lUS8REZGiMGPGDK677jpiYmIoVaoUHTt2/OeyLl26ANCyZUvi4+P/Ob99+/aULl2a6OhoGjVqxObNm30du9AU2YiXtbZ7Hhe1L6rHPFUtr+7FlqWfUGLu+9j2GvUSEZEgls/IlBuioqIACAsL+9ePVB87P7fLAo2+uT6biIhIdpzVl9MzN7F8yjC344iIiASdCy+8kFGjRnHkyBGSk5MZM2aM25F8SsUrhxZX380WU5nic97DZmW5HUdERCSotGjRgm7dutG0aVOuvPJKWrdu7XYknzLOtzr4t1atWtm4uDifPd7sXz/jvGXPsvyCL2jSvofPHldERKQorVq1ioYNG7odI6jkNk+NMQutta1yu75GvHLR6hpn1Ctmzrsa9RIREZFCo+KVi8jISLY26cPpGRtZNVXHeomIiEjhUPHKQ6tr72ErlSk2W6NeIiISPALhEKNAcTLzUsUrD1GRUcQ36UPtjI2snvaT23FEREROWXR0NImJiSpfhcBaS2JiItHR0Sd0O19/c31AaX3tPWxd/inRs96FdjeBMW5HEhEROWnVqlVj27ZtJCQkuB0lKERHR1OtWrUTuo2K13FER0WxqXEfLlzxAmum/USDdje5HUlEROSkRUREULt2bbdjhDTtaszH2R3vZQunETnrXdDQrIiIiJwCFa98REdFsanRfdROX8/a6T+7HUdEREQCmIpXAZzd8T62chqRM9/RqJeIiIicNBWvAigWHcWGhvdRK30962dq1EtEREROjopXAZ3dqTdbOY3w6Rr1EhERkZOj4lVAMdHRrD+jN7XS17Nh1gi344iIiEgAUvE6AWd3uo8tVCJs+v806iUiIiInTMXrBBQvFs26Br2pdXQdG2f/4nYcERERCTAqXieoTef72UIlPNPe1qiXiIiInBAVrxNUolg06+rfS62j64jXqJeIiIicABWvk9C6031spRJM07FeIiIiUnAqXiehVPEYVte7l1pH17J5zq9uxxEREZEAoeJ1ks7+Z9RLx3qJiIhIwah4naTSJWJYWfceaqatZeu8kW7HERERkQCg4nUKzul0P9tsRbKmvKVRLxEREcmXitcpKF0yhhXeUa9t80e7HUdERET8nIrXKWrT+X622opkTn5To14iIiJyXCpep6hMyeIsr3M3NdPWsH2BRr1EREQkbypehaBN5z5stRXJ0KiXiIiIHIeKVyEoV6o4f59+NzVT17Aj7je344iIiIifUvEqJG063882G0v6Xxr1EhERkdypeBWS8qVLsPT0u6mZupqdGvUSERGRXKh4FaKzO/XRqJeIiIjkScWrEMWWKcGSWr2okbqa3QvHuB1HRERE/IyKVyE7+zrnWK80jXqJiIhIDipehaximVIsqnUXNY6sYs+isW7HERERET+i4lUEnO/1iiV10hsa9RIREZF/qHgVgUplS7G4xp3UOLKKhMUa9RIRERGHilcRaX1dH7bZChyZpGO9RERExKHiVUQqlytNXI27qJGykr1Lx7sdR0RERPyAilcRat3ZGfVK+fN1jXqJiIiIildRqlq+NPOr30mNlJUkatRLREQk5Kl4FbGzO/dhu0a9REREBBWvIletQhnmVbuD6ikr2bfsd7fjiIiIiItUvHygdee+bLcVOPznaxr1EhERCWEqXj5QPbYMc6veTvXDKzjwt0a9REREQpWKl4+08o56HfpDx3qJiIiEKhUvH6lZsSyzq9xGtcPLObBco14iIiKhSMXLh1p1foAdtjyHfteol4iISChS8fKh2pXKMqvy7VQ7vJykFX+4HUdERER8TMXLx1p07ssOW55kjXqJiIiEHBUvH6tzWjlmnnYr1Q79zcEVf7odR0RERHxIxcsFLTr3Zbstz0GNeomIiIQUFS8X1K1cgRmn3Uq1Q8tIXjnR7TgiIiLiIypeLmnZ6diol77NXkREJFSoeLmkXpUKzKh4C1WTl3FolUa9REREQoGKl4uad+7LDluOpAka9RIREQkFKl4ualA1lqmx3lGv1X+5HUdERESKmIqXy5p3esA76vWqRr1ERESCnIqXyxpWj2VK7C1UPbiUw2s06iUiIhLMVLz8QLOO3mO9xmvUS0REJJipePmBxjUqMrlCT6ocXEqKRr1ERESCloqXn2ja8QF22nIcGK9POIqIiAQrFS8/cWbNSkws35MqB5dwZO1kt+OIiIhIEVDx8iP/P+qlY71ERESCkYqXH2laqxITy/WgcpJGvURERIKRipefaXJNXx3rJSIiEqRUvPxMizqV+bNsDyonLSZt3VS344iIiEghUvHyQ8dGvfaNf0WjXiIiIkFExcsPtaxbmT/K3kTlAxr1EhERCSYqXn6q0dUPsMuWZb8+4SgiIhI0VLz81Nn1qjChTHdOO7CIo+unuR1HRERECoGKlx9r6B312jdOo14iIiLBQMXLj7WpV5lxpbpz2oGFGvUSEREJAipefswYQ8Nr+upYLxERkSCh4uXnzq1fhXGlulFp/0KObtCol4iISCBT8fJzxhgaXP0Au20Z9o97ze04IiIicgpUvAJA2wZV+a3kTVTaH0f6+uluxxEREZGTpOIVAIwxNLiqL7ttGefb7EVERCQgqXgFiAsaVuO3Et2otC+OjA0a9RIREQlEKl4BwhhDvSv7OKNe4151O46IiIicBBWvAHJR4xqMLtGNivsWaNRLREQkAKl4BRBjDHUv9456jdeol4iISKBR8QowF59Zg1HFu1IxcQGZGvUSEREJKCpeAcYYw+lX9GWPRr1EREQCjopXAOpwZk1+jelKbOICMjfOcDuOiIiIFJCKVwByRr36eEe99G32IiIigULFK0B1OLMWvxa7gdi98zTqJSIiEiBUvAKUx2OoeXlfjXqJiIgEEBWvAHZ501qMiL6e2L3zyNo00+04IiIikg8VrwDm8RhqaNRLREQkYKh4Bbgrm9VmePT1VEiYq1EvERERP6fiFeDCPIYal/UhwZbWqJeIiIifU/EKAlc1P53hUcdGvWa5HUdERETyoOIVBMI8hqod7ifBlmb/BI16iYiI+CsVryBxdcs6/BR1PeX3zMHGa9RLRETEH6l4BYnwMA9V29+vY71ERET8mIpXELm2VR1+iuyiUS8RERE/peIVRMLDPFRufz8JtpSO9RIREfFDKl5BpmPrugyL6EK53XOwm2e7HUdERESyUfEKMhFhHiodG/XSsV4iIiJ+RcUrCHVuXc876jVbo14iIiJ+RMUrCEWGe4i9uDcJthQHJrzudhwRERHxUvEKUte1qc+P4ddRdtcs7OY5bscRERERVLyCVlR4GLHtvKNev2vUS0RExB+oeAWxLuc24MfwzpTdOVOjXiIiIn7AleJljHnEGLPCGLPcGPOjMSbajRzBLio8jPIX3cdeW4okjXqJiIi4zufFyxhTFXgQaGWtbQKEATf5OkeouP7cBgwN60yZnTNB32YvIiLiKrd2NYYDxYwx4UAMsMOlHEEvOiKMMhfdxzZbgcOjHoHMDLcjiYiIhCyfFy9r7XbgPWALsBNIstb+mfN6xph7jDFxxpi4hIQEX8cMKt3Oa8AXUb0ofmANGXO/dDuOiIhIyHJjV2NZoBNQG6gCFDfG3Jzzetbar621ray1rWJjY30dM6hER4RxaZc7mZrZlKzJb8LBnW5HEhERCUlu7GrsAGyy1iZYa9OBX4HzXMgRUtqdUYmpdZ6AjDQOjX3G7TgiIiIhyY3itQU4xxgTY4wxQHtglQs5Qs59XS7lOzpRYu1I7MZpbscREREJOW4c4zUPGAEsAv72Zvja1zlCUaVS0ZTo8CRbsmI5NPIRyDjqdiQREZGQ4sqnGq21L1lrz7DWNrHW3mKtTXMjRyjq3rYBA0rfT8nkDaTO+tTtOCIiIiFF31wfYsI8huu63cWkzBaYae9A0na3I4mIiIQMFa8QdGa10iw/61lsZgYHRj3hdhwREZGQoeIVou66th0Dwm6gzKZxZK6d5HYcERGRkKDiFaJKRkdQ89qn2ZRViUOjHoEMHWYnIiJS1FS8QtgVzWoyotJDlE7ZQvKUD9yOIyIiEvRUvEKYMYZuN93B71lnEzW7H+zf7HYkERGRoKbiFeJqlI9h97kvkZEFCSMedTuOiIhIUFPxErpfeh5Do28idvsk0lZOcDuOiIhI0FLxEiLDPZx1wzOsz6rCkdGPQfoRtyOJiIgEJRUvAaBNvcpMrP0EZdK2k/jH/9yOIyIiEpRUvOQfN3btyQTaUjLuU2ziRrfjiIiIBB0VL/lH+RJRpLd/haM2jF0/PQjWuh1JREQkqKh4yb9c07YVI0reQuU9Mzi0dLTbcURERIKKipf8i8djOKf7M6zJqk76uCfhaIrbkURERIKGipf8xxlVy7Og8bOUTd/NjjGvuR1HREQkaKh4Sa66XHcjEzztiP37a47uXut2HBERkaCg4iW5iokMp8S1b3LERrB72AM60F5ERKQQqHhJni5o3pjxsXdRff9c9s7/2e04IiIiAU/FS47rwh5Ps8rWwvPns9i0ZLfjiIiIBDQVLzmuKuVKsq71y5TL3MumX15yO46IiEhAU/GSfF11ZSf+iLyUGmsHkLJtudtxREREApaKl+QrPMxDlevf5pCNZs9POtBeRETkZKl4SYGc2aAuU6vfR63kRWydPsjtOCIiIgFJxUsK7OKbnmAFdSg+9SUyUw64HUdERCTgqHhJgZUuEc3ei96kTNYB1v70nNtxREREAo6Kl5yQC9tdzpQSV1Jv8w8kblzodhwREZGAouIlJ8QYQ52b3uGgLc6B4Q9CVpbbkURERAKGipecsFrVqxNX72HqHFnOmonfuB1HREQkYKh4yUm58MaHWO5pQMU5b5B6MNHtOCIiIgFBxUtOSnRkBOmXv0cpe5DVPzzpdhwREZGAoOIlJ615mwuZWe46ztr5C1tXzHY7joiIiN9T8ZJT0qjH2+wzpUgd9Qg2K9PtOCIiIn5NxUtOSWxsJVaf+ST10lez5LdP3Y4jIiLi11S85JSd1/l+VkQ0pvaSdzmYuNvtOCIiIn5LxUtOmSfMQ1SnDyhhD7Nm6GNuxxEREfFbKl5SKOo2acOC07rRMnEsq+Mmux1HRETEL6l4SaE5s+ebJJoyhE14nIz0dLfjiIiI+B0VLyk0JUqVY3ubF6iXuYF5I953O46IiIjfUfGSQtX08jtYFd2cJqs/ZteOLW7HERER8SsqXlKojMdDmRs+pBipbPjhcbfjiIiI+BUVLyl0les2Y3mNW2h76A/mTx/vdhwRERG/oeIlReLMHq+zx1Sg7JRnSElNdTuOiIiIX1DxkiIRUawkBy96lXo2nlk/vu12HBEREb+g4iVFpu5FPVhT4mzOjf+SdRvWuR1HRETEdSpeUnSM4bSbPibSpLNj+BNkZVm3E4mIiLhKxUuKVOlqDVlf7y4uSp3C1D9+dTuOiIiIq1S8pMg17PoSe8IqUXPeiyQmHXI7joiIiGtUvKTImcjiZFz+NnXYxqyhr7sdR0RExDUqXuITVc7uwvqy59N+93fELfvb7TgiIiKuUPESn6nW/RPCjSX5t6dIy8h0O46IiIjPqXiJz0RXPJ0dZ/bm4oxZTBj9o9txREREfE7FS3yqdsfnSIioQtNlr7N5zz6344iIiPiUipf4VkQ0Yde8T22zk3lDX8VafbeXiIiEDhUv8blyTa9ic8VLuPbAUCbPi3M7joiIiM+oeIkrqnX/CGMMYX88y8HUdLfjiIiI+ISKl7girGwN9rd+mHZ2PmOGD3A7joiIiE+oeIlrKl/+OAlRNWi7/l3+jt/ldhwREZEip+Il7gmPJOa6D6lldrPsp1fI1I9oi4hIkFPxElcVP6M926teyQ0pwxn510y344iIiBQpFS9xXZUb3yfLE07szBfZnXTE7TgiIiJFRsVLXGdKV+XIeU9wkVnEyGH93Y4jIiJSZFS8xC+Uu+RBEmNO55odHzJtxWa344iIiBQJFS/xD2ERlLz+I6qZvcSPfJXUdP2ItoiIBB8VL/EbkXUuJKF2Z25KH8XQcX+5HUdERKTQqXiJX4nt8g5ZYVHUX/Qa63cfdDuOiIhIoVLxEv9SshKZ7Z7lAs8yRv/4pX5EW0REgoqKl/idEm17s79kA7rv/4JR89e6HUdERKTQqHiJ/wkLp/QNH1PF7CPp9zfYf/io24lEREQKhYqX+CVPzXM40OBGemaN5buRE9yOIyIiUihUvMRvlen4FhnhMZy75m0WbEp0O46IiMgpU/ES/1W8AmEdXuS8sJVM/Pkz0jOz3E4kIiJySlS8xK9FtrmLpLJNuCvlWwZN/dvtOCIiIqdExUv8myeM0jd8TKxJImza22zdl+J2IhERkZOm4iX+r2pLjjS5mZvN73wzYoy+20tERAKWipcEhOJXvUpGZCmu2fY+fyzf5XYcERGRk6LiJYEhphzhl7/C2Z41zB/9GYfSMtxOJCIicsJUvCRghLe4lUOxzbk/fSBfTFjodhwREZETpuIlgcPjocR1H1LOHKLSwvdYsSPJ7UQiIiInRMVLAkuVZqQ3v4OeYZPo//MoMrN0oL2IiAQOFS8JOFGXvUh6VFlu3fcxP86LdzuOiIhIgal4SeApVoaoK1+nuWc9a//4koTkNLcTiYiIFIiKlwQk07Q7RyqfzSN2KP1+m+N2HBERkQJR8ZLAZAzFOn9IaZPCmas+Yua6vW4nEhERyZeKlwSuSo3JOvsebgqfwtBffyU1PdPtRCIiIsel4iUBLfySZ0kvFst9hz/ns7/WuB1HRETkuFS8JLBFlyLqqrc4y7OJpBlfMX1tgtuJRERE8qTiJYGvyfVk1m7HsxE/8MmPo9h+4IjbiURERHKl4iWBzxjCunxNeEwZ3sl6n8cGzyQtQ8d7iYiI/1HxkuBQshLhN35PLbObnnve442xK91OJCIi8h8qXhI8ap2Paf8C14bNxS7oz+gl291OJCIi8i8qXhJc2j5MVt1LeSliCEN+Gcna3cluJxIREfmHipcEF48HT5evMSVP48Owj3h88DQOpWW4nUpERARQ8ZJgFFOOsG4DqezZz4NJ7/PU8KVYa91OJSIiouIlQapaKzyXvU6HsEVUXfUN38+KdzuRiIiIipcEsTb3Yht14qmIn/hzwkji4ve5nUhEREKcipcEL2MwHT+BsjX5JPITnh86hb2H0txOJSIiIUzFS4JbdGnCug2mvOcwL6R9wIND48jIzHI7lYiIhCgVLwl+p52J56p3aev5m7O39qffxLVuJxIRkRCl4iWhocWt0LQ7D4aPZNn0UUxaudvtRCIiEoJUvCQ0GANXvw+xDfg06nPe+nkyWxJT3E4lIiIhRsVLQkdkcTw3DqJUeDrv8iF9Bs8jNV0/pi0iIr6j4iWhJbYBno6f0ILVXL23Py+NXuF2IhERCSEqXhJ6zrwBWt1F7/CxJC4axc8LtrqdSEREQoSKl4Smy9/EVm7KR1Ff8dXoySzfnuR2IhERCQEqXhKaIqIxXQdSLNLDJxEf8dCQeSSlpLudSkREgpyKl4SucrXxdP6CRnYDdxz6mseGLyErSz+mLSIiRceV4mWMKWOMGWGMWW2MWWWMOdeNHCI0vAbO7cvNYRMptmYUX07f4HYiEREJYm6NeH0E/G6tPQNoCqxyKYcIdHgZW70N70Z9yy9/TmH2hr1uJxIRkSDl8+JljCkNXAh8C2CtPWqtPeDrHCL/CIvA3PA9UdExfBP9CU/8MJddSalupxIRkSDkxohXbSAB+N4Ys9gY098YUzznlYwx9xhj4owxcQkJCb5PKaGldFXM9d9QO2sLj2d8TZ8fFpGuH9MWEZFC5kbxCgdaAF9Ya5sDh4Gnc17JWvu1tbaVtbZVbGysrzNKKKrbHnPRk1xnpnH6tpG8NX6124lERCTIuFG8tgHbrLXzvKdH4BQxEfdd9BTUvog3Iwcwe/Y0xi3b6XYiEREJIj4vXtbaXcBWY0wD71ntgZW+ziGSK08YXN+f8OLl+C7mY14ZMYf1ew65nUpERIKEW59qfAAYaoxZBjQD3nQph8h/laiIueE7Kmft5vWwr7lvcByH0zLcTiUiIkHAleJlrV3iPX7rLGttZ2vtfjdyiOSpVltM+xe4zM7hvH2/8syvf2OtvlxVREROjb65XiQv5z0E9a/gxcihbF42ncFzN7udSEREApyKl0hePB7o/AWeUpX5NuYzPh47n8VbNDgrIiInT8VL5HhiymG6DqS83cdHUV/Rd0gc+w4fdTuViIgEKBUvkfxUa4m5/A3aZsXR+cgvPDRsMZn6MW0RETkJKl4iBXH2PdCoM4+F/0za+hl89Nc6txOJiEgAKnDxMsaUNcY0NsacboxRYZPQYgx0/ARTthbfFP+cH/6KY8qaPW6nEhGRAHPcAmWMKW2MedYY8zcwF/gK+BnYbIwZboy52BchRfxCdCnMjYMoxWG+KfEFjw1byNZ9KW6nEhGRAJLfyNUIYCtwgbW2gbX2fO/3b1UH3gY6GWPuKvKUIv7itCaYq96jecYy7s4aQZ8fFpGWkel2KhERCRDhx7vQWnvpcS5bCCws9EQi/q7FLbBlDr2X/MDsHXV5dUxp3rjuTLdTiYhIACjQsVo5R7WMMWHGmJeKJpJIALjqPUzFhnwV8yV/zVvMLwu3uZ1IREQCQEEPkm9vjBlvjKlsjGmMc7xXySLMJeLfImPgxkEU82QwsOQXvDRqMat3HXQ7lYiI+LkCFS9rbQ9gIPA3MB542Fr7eFEGE/F7Fephrv2IBukreTriZ+4bsoiDqelupxIRET9W0F2N9YCHgF+AzcAtxpiYogwmEhDOvAFa9+LmrN+ov386Tw5fph/TFhGRPBV0V+MY4AVr7b3ARcA6YEGRpRIJJJe/CZWb8XGxr1ixcin9Z2xyO5GIiPipghavs621fwFYx/vAdUUXSySAhEfBjQOJDA9jaKnP6ff7MuZtTHQ7lYiI+KH8vkD1fABr7X+OGrbWrjXGlDLGNCmqcCIBo2wtTOcvqZG2jrdjhtH3x8XsOZjqdioREfEz+Y14XW+MmW2MedEYc7Ux5mxjzIXGmDuNMYOBsUAxH+QU8X9nXAXnPUCnjAlclDaVvj8uJiMzy+1UIiLiR45bvKy1jwDXADuBrsCrwCNAXeAra+2F1lod6yVyTPuXoMa5vB3Rn8T4v3n3jzVuJxIRET+S7zFe1tp9QClgGTARmAnsBRoYY5oVaTqRQBMWATd8R3hUDD+U/pxB01fy+/JdbqcSERE/UdCD61sCvYHKQBXgXuAK4BtjzJNFlE0kMJWqAtf3p2JqPJ+VHsoTw5ewae9ht1OJiIgfKGjxqga0sNY+bq19DKeIVQQuBG4vomwigavOJZiLnuKStL+43jOF+4Ys5MhR/Zi2iEioK2jxqgikZTudDlSy1h7Jcb6IHHPRk3B6O17wfI9nz3KeG/W3vlxVRCTEFbR4DQXmGWNe8v449izgB2NMcWBlkaUTCWSeMOjSn7CYcgwt9TkTF63jx/lb3U4lIiIuKuhvNb4G3AMc8P71tta+aq09bK3tWXTxRAJciVi44TvKpO3g27IDefm35SzbdsDtVCIi4pKCjnhhrY2z1n7k/YsrylAiQaXmeZj2L3L2kRncV2wS9w1ZxIGUo26nEhERFxS4eInIKTjvQah/JQ9lDaLyoeU88tMSsrJ0vJeISKhR8RLxBY8HrvsCT6nKDCjxOYvXbOSzKevdTiUiIj6m4iXiK8XKQtcBFE9PZGj57/lg0mpmrEtwO5WIiPiQipeIL1Vtibn8TRofnstzpf7koWFL2HHgiNupRETER1S8RHytdS9ofB13Hh3CWRnLuX/oIo5m6Me0RURCgYqXiK8ZAx0/wZQ7nS+Lfcb2rfG8OX6V26lERMQHVLxE3BBVEm4cRHRGMsNjv2XQ7I2MXrLd7VQiIlLEVLxE3FKpMVz9PrWSF/JO+XE8/cvfrNud7HYqEREpQipeIm5qfjM0u5kbDv9Ih4hl9B6ykENpGW6nEhGRIqLiJeK2q96Fio3pF/E5qXu38NQvy/Rj2iIiQUrFS8RtkTFw40AibDq/xH7DH8u2MmB2vNupRESkCKh4ifiDCvWg48ecdnAZn1f6jTfGrWLh5n1upxIRkUKm4iXiL5pcD63v5rKkEdxUcin3D13E3kNpbqcSEZFCpOIl4k8ufwOqtOCVrM8ombKVB39cTKZ+TFtEJGioeIn4k/Ao6DqAsDAPw8t9xcINO+k3cY3bqUREpJCoeIn4m7I14bqvKHtwFQOrjOSzKRv4a9Vut1OJiEghUPES8UcNroS2D3HOvtH0rbCQR35awpbEFLdTiYjIKVLxEvFXl7wANc7l0bQvqM02eg9ZSHJqutupRETkFKh4ifirsAi44Ts8ETH8UOpztuxO4K4BcaQc1Tfbi4gEKhUvEX9Wqgpc35/iBzfwe51fiducyD2DFpKanul2MhEROQkqXiL+rs7FcPGzVNs6hgmNJjFzfQL3D13E0Ywst5OJiMgJUvESCQQXPgGt7qLBhu/5rcl0Jq/ew0PDFpORqfIlIhJIVLxEAoExcNV70Oxmzlr/FcObzGXC8l08NnypvmBVRCSAhLsdQEQKyOOBjh9DRiqtl3/M4MYR3LIEosI9vN3lLDwe43ZCERHJh4qXSCDxhMF1X0JGKhesfp9vGj3D3XEQFR7Gq50aY4zKl4iIP9OuRpFA4/2aCepeSoeNb/Nxo9UMnruZN8evwlrtdhQR8WcqXiKBKDwKug3G1L6Aaze9zv/OWM83MzbxwcS1bicTEZHjUPESCVQRxaD7MEz1Nty45VVeqR/Px5PX89mU9W4nExGRPKh4iQSyyOLQ42fMaWdx6/aXearuNt79Yw3fztzkdjIREcmFipdIoIsuBTf/gqnQgN67XuTB03fx2tiVDJm72e1kIiKSg4qXSDCIKQe3jsKUrcUjCS9wT60Enh+1nBELt7mdTEREslHxEgkWxSvAraMxJSvxzP7nuaXGPp4csZQxS3e4nUxERLxUvESCScnT4NbfMNFlefXg83SpcoCHf1rCHyt2uZ1MRERQ8RIJPmWqw22/YSJieOfIi1xR6SB9f1jElDV73E4mIhLyVLxEglG52nDbb3iM4ZP0l7iwwiF6D17I7PV73U4mIhLSVLxEglWFenDraDyZaXxtX6FV2cPcNTCOuPh9bicTEQlZKl4iwaxSY7hlFGFpyQwMe40mJQ9z+/cLWLr1gNvJRERCkoqXSLCr0gxu/oXwlL38GP0WtWMOc+t381m546DbyUREQo6Kl0goqN4aevxE+MFt/FL8XU6LSOHmb+exbney28lEREKKipdIqKh1Ptw0lMj96/mtTD9KmRR69J/Hpr2H3U4mIhIyVLxEQknd9nDjIKL2rmBChY+JzEyh5zdz2bovxe1kIiIhQcVLJNQ0uBKu/5Ziuxfxx2lfkJ6WQs/+89iVlOp2MhGRoKfiJRKKGneGzl9SYscc/qr2DcmHD9Oj/1wSktPcTiYiEtRUvERCVdNucO1HlNo2jck1BpBw4BA395/HvsNH3U4mIhK0VLxEQlnL2+DKdyi7dSKTT/+BzYnJ3PrdPJKOpLudTEQkKKl4iYS6NvdCh1eI3TyOKXWHs3ZXErd/P59DaRluJxMRCToqXiIC5z8M7Z6hcvxIJjUYw7JtB7hzwAKOHM10O5mISFBR8RIRx0VPQduHqbFxGH82/J0F8YncMziO1HSVLxGRwqLiJSIOY6DDy9CmN3U2DGJ846nMWLeXPkMXcTQjy+10IiJBQcVLRP6fMXDF29DiNhqu/4ZRZ87mr9V7ePinxWRkqnyJiJyqcLcDiIifMQau+RAy0mi27FN+ahJBt79bExW+jPe6NiXMY9xOKCISsFS8ROS/PB7o9BlkpNJm5QcMaPIUty+GqHAPb153Jh6VLxGRk6LiJSK5CwuH6/tDRhrt1v6PLxs/T+8FTvl6uWNjjFH5EhE5UTrGS0TyFhYBXQdAnUu4fMMbfNBwHQPnbObtCaux1rqdTkQk4Kh4icjxRURDt6GYmm3pHP8qb56xia+mb+TDSevcTiYiEnBUvEQkf5Ex0GMYpmpLum95mRfqb+Wjv9bxxdQNbicTEQkoKl4iUjBRJaHncEylRty5/UUer7uT//2+mu9mbnI7mYhIwFDxEpGCK1YGbhmFKV+HPrtfoM/pu3l17Ep+mLfF7WQiIgFBxUtETkxMObh1NKZUVR7f+yJ31krkuVF/88vCbW4nExHxeypeInLiSlSE237DFC/PCwee56ZqB3hixFLGLtvhdjIREb+m4iUiJ6dUFbhtDCayJG8eeoFOVQ7y8LAl/Llil9vJRET8loqXiJy8MjWcka+wCN5PfZEOlQ7R94fFTFub4HYyERG/pOIlIqemfB24dTQem8nnGS9xXvnD3DMojtkb9rqdTETE76h4icipq3iGU77SU/jWvEKLMin0GhjHws373E4mIuJXVLxEpHCcdibc8ithR/YzOOINziiewu3fLWDZtgNuJxMR8RsqXiJSeKq2hJ7DCT+0k59i/keNYke45dv5rNp50O1kIiJ+QcVLRApXzXOh+zAikuIZWfJdYsNTubn/PNbvSXY7mYiI61S8RKTwnX4RdBtCZOIaxpb7gOKk0uObecTvPex2MhERV6l4iUjRqHcpdP2e6D1L+b3iJ4RlHKFn/3ls25/idjIREdeoeIlI0Wl4LXT5mphdC5hY5StSUw/Ts/88diWlup1MRMQVKl4iUrTOvAE6fUaJ7TOYXP17kpIP07P/XPYeSnM7mYiIz6l4iUjRa9YDru5H6a1/8VftIew6cIib+89j/+GjbicTEfEpFS8R8Y3Wd8Hlb1J+8wSm1BlO/N5kbv1uPgdT091OJiLiMypeIuI75/aBS16gYvwo/qo/ktU7D3DH9ws4nJbhdjIREZ9Q8RIR37rwcbjwCapuHM7EhhNYsnU/dw1cwJGjmW4nExEpcipeIuJ7Fz8H5/al1oYh/N74L+ZtSuSewXGkpqt8iUhwU/ESEd8zBi57HVr3ot66bxnbZCYz1u3l5v7z9GlHEQlqKl4i4g5j4Mp3odnNNF73BeNbxvH39iQ6fTpLv+0oIkFLxUtE3OPxQMePockNNFrRj+mt55CZmcH1X8zmzxW73E4nIlLoVLxExF2eMLjuS2h2M5UWf8TUmgNoEhvGvUMW8vnU9Vhr3U4oIlJoXCtexpgwY8xiY8xYtzKIiJ8Ii4BOn8LlbxG9YQLDwl7iljMM7/y+hsd+XqqD7kUkaLg54vUQsMrFxxcRf2IMnHs/9ByB5+A2Xtndl/dbJ/Pr4u10/2Yue5L1+44iEvhcKV7GmGrA1UB/Nx5fRPxY3fZw9xRMTHmuX9GHseeuZfXOZDp/Oovl25PcTicickrcGvH6EHgSyMrrCsaYe4wxccaYuISEBJ8FExE/UL4O9JoEdS6hyeKXmdlkLB6bQdcv5/D78p1upxMROWk+L17GmGuAPdbahce7nrX2a2ttK2ttq9jYWB+lExG/EV0aug+Dtg9RftVgJlf8kFYVs+g9ZBGf/LVOB92LSEByY8SrLdDRGBMPDAMuMcYMcSGHiPg7Txhc+ipc9zWROxcyMOMp7jsjlfcnruWhYUt00L2IBByfFy9r7TPW2mrW2lrATcBka+3Nvs4hIgGkaTe4cwKerHSe3P4An7fcwW9Ld9DtqznsPqiD7kUkcOh7vEQkMFRt6Rx0X/EMrlrxOBNbzmPdnmQ6fjqTZdsOuJ1ORKRAXC1e1tqp1tpr3MwgIgGkVGW4fTycdRP1VnzEnLpDKGGOcuNXcxi7bIfb6URE8qURLxEJLBHRzjfdX/Y6pTeN5/dSb3BRpTT6/rCYDyauJStLB92LiP9S8RKRwGMMnPcA9PiZiINb+DLlMR4/Yx8f/bWOB35czJGjOuheRPyTipeIBK56l0KvvzDFytBnyyMMbLaK8ct30vWr2exMOuJ2OhGR/1DxEpHAFlsfek3C1L6Qi1a/xvQmE9iScJCOn85i8Zb9bqcTEfkXFS8RCXzFykKPn+HcvlRfN5g51T+nQthhun09l9FLtrudTkTkHypeIhIcwsLh8jeg8xcU3zWfsdEvck2lAzw0bAnv/bFGB92LiF9Q8RKR4NKsB9w+nrCMI7yf/AQv19/Cp1PWc9/QhRxOy3A7nYiEOBUvEQk+1Vs7X7Zavg63bXmG4Y3nMHHlLm74cg7bD+igexFxj4qXiASn0lXhzt8xTa6n9YZPmFPvB/bu20+nT2excLMOuhcRd6h4iUjwiigG1/eHDi9Tact4ZlR8l1oR++n+9Vx+XbTN7XQiEoJUvEQkuBkD5z8C3YcRnbSJn80zdKu8g0d/XsrbE1aTqYPuRcSHVLxEJDQ0uAJ6TcITVYJX9z/Ne/WW8+W0Ddw7OI5DOuheRHxExUtEQkfFM+DuyZga53LD1jcZ12A809fs4oYvZrN1X4rb6UQkBKh4iUhoiSkHN/8KbXrTePMQ5tX8kuQDCXT6bBYL4ve5nU5EgpyKl4iEnrBwuPJ/0PETyu6ex5TSr9Ekajc9vpnLz3Fb3U4nIkFMxUtEQleLW+H2sURmHGJgxtPcc9oGnhyxjNfHrtRB9yJSJFS8RCS01TjH+bLVcrV5PPEFvj59Jv1nbqTXwAUkp6a7nU5EgoyKl4hImepw5x+Yxp25bMfnTD79R+au20GXz2ezJVEH3YtI4VHxEhEBiIyBG76HS57n9B1jmV+5H/bgTjp9NpM5GxLdTiciQULFS0TkGGPgwieg21BKJm/g95gXOTc6nlu+nceP87e4nU5EgoCKl4hITg2vgbv+JDwyms/Snufxykt45te/efm3FWRkZrmdTkQCmIqXiEhuKjV2Drqvfja9E//HDzXHMWj2Ru4cGEfSER10LyInR8VLRCQvxcvDLSOhdS/O2z2UmdW/ZNn6LVz3+Sw27T3sdjoRCUAqXiIixxMWAVe/D9d8QJXEucyp8AYlD8XT+bNZzFq/1+10IhJgVLxERAqi1Z1w628Uy0hiZMQLXFVsBbd+N5/Bc+LdTiYiAUTFS0SkoGq1hbun4ClbgzePvMrrlabxwujlvDBqOek66F5ECkDFS0TkRJSt6XzZ6hlX033/l4yu9gM/zV3P7d/PJylFB92LyPGpeImInKioEtB1EFz0NE33jmPOaf3YtGkDnT+fxYaEQ26nExE/puIlInIyPB64+Bm4cRDlD69jaulXqZqyis6fzWL62gS304mIn1LxEhE5FY06wV1/EhkRwSDzMj1j5nP79/P5ftYmrLVupxMRP6PiJSJyqk470znovmoLnk55j08qjuHVMct5dqQOuheRf1PxEhEpDCVi4dbR0PJ2rk76kYmnfcmY+au55dt57D981O10IuInVLxERApLeCRc8yFc9R51k+Ywu8Kb7N2ymk6fzWLd7mS304mIH1DxEhEpTMbA2XfDLSMplbmf32NeonHqYrp8PptJK3e7nU5EXKbiJSJSFE6/CO6eQnjpKnxuX6dPzCR6DVrAE8OX6ke2RUKYipeISFEpVxt6TcTUv5zeR75mXLXBTFy0mss/mM6U1XvcTiciLlDxEhEpSlElodtQaPcMjRMnsqDMc1wdPo87BszncY1+iYQcFS8RkaLm8UC7p+GeqUSUqcILKf9jYuWvmL34by77YBqTV+vYL5FQoeIlIuIrlc+CXpPh0leplzyfGcWfpnvYZO4aMJ/Hfl6q33oUCQEqXiIivhQWDm0fgvtmE1a1GQ8f+YzpFfuxZEkcl32o0S+RYKfiJSLihvJ14LYx0PETqqdtYGKxZ7jHjOaeAXN59OclGv0SCVIqXiIibjEGWtwKfefjqX8Zd6UNYnb511m/dBaXfjBN3/slEoRUvERE3FbyNOg2BG4cTEWTxOjIF3jCM5S+g2bx6E9LOJCinxwSCRYqXiIi/qJRR+gzH9O8J13TfmVumRfYs2wil34wnYka/RIJCipeIiL+pFgZ6PgJ3DaGMsUiGBLxOq+ar3hs0DQeHrZYo18iAU7FS0TEH9W+EO6fA20f4or0v5hT8mmO/j2aDv2m8+eKXW6nE5GTpOIlIuKvIorBpa9i7p5M8fJV+DziAz407/Pc4Ek8PGwx+w9r9Esk0Kh4iYj4uyrN4O4p0OFl2tpFzCj+NMWW/8Cl/abxh0a/RAKKipeISCAIi4DzH8HcN5voak15K/xrvjWv8uaQcTz4o0a/RAKFipeISCA59sWr13zIWWGb+KvYM1RZ8TVX9JvM78s1+iXi71S8REQCjccDre7A9JlPeL0OPB3+A0N4jo+H/sIDPy5mn0a/RPyWipeISKAqVQVuGgpdB1I3+iBjo16g8cp+XNvvT35fvtPtdCKSCxUvEZFAZgw07ozpMw9Ps+70DvuNn+0TfD90KH1/WKTRLxE/o+IlIhIMYspB58/gllFUKRXBT1Gv0XbV61z3/ngm/K3RLxF/oeIlIhJM6lyMuX8OnNuXm8Kn8Kt9hF9//Jo+Pywi8VCa2+lEQp6Kl4hIsIksDpe/gek1iXKxlfkmsh9Xr36am/r9xniNfom4SsVLRCRYVW2JuXcaXPICV4Qv5lf7MJOHfUCfIQvZq9EvEVeoeImIBLOwCLjwcTz3zaJEtSa8F/EVPdY+xB39hjNumUa/RHxNxUtEJBTE1sfcMQGufp9zojYxPOtRFv30Gn2GzNfol4gPqXiJiIQKjwda9yKs73wi613MCxFDuXdtb/r0G8TYZTuw1rqdUCToqXiJiISa0lXx9BgGN3xHo5gDDM16ik0/P8ODg+eSkKzRL5GipOIlIhKKjIEm1xP+YByes7ryQPgoHt5wJ0/2+5Lflmr0S6SoqHiJiISymHJ4unwFN/9KjVJhfG9f5MDwB3lk0AyNfokUARUvERGBuu2J6DuXrDb3cUv4JJ7aeBuv9uvH6CXbNfolUohUvERExBFVAs+VbztfvFo+lk/s25hf7uKJAZPYk5zqdjqRoKDiJSIi/1atFVH3zySr3bNcHR7Hc/G381G/1xi9eJtGv0ROkYqXiIj8V3gknnZPEXbfTKIrn8Eb9lPK/NqdZ78bp9EvkVOg4iUiInmreAbF7plI1pXvcm7kep7fciffv/8UoxZt0eiXyElQ8RIRkePzePC0uYfIB+ZDrbY8xQBqjrqOl/r/wp6DGv0SOREqXiIiUjBlqlP89l/Juu4bzojay/Pb7uHXfn0YHbdRo18iBaTiJSIiBWcMnqY3UuzhRaQ16ERvRtDot2t46+tBGv0SKQAVLxEROXHFy1Oyx/dkdh/OaTGZPL3jIf7qdyvf/xnHkaOZbqcT8VsqXiIictLCGlxGyUfiSD7rDroxkRtnXcVPb/dixIxlpGdmuR1PxO+oeImIyKmJKknp6z/A02ceR2pfxq1ZI7l80mUMe/te/ohbTVaWjv8SOUbFS0RECkdsfSrcPgRz32xSqrfjlvSfOXdMO358tw+zV2x0O52IX1DxEhGRQmUqNaJSr2Fk3juT5Mpt6XlkKI1/bsvwfg/y94atbscTcZWKl4iIFImwymdStfcvHO01lQOxZ9P14ECqDTqHUZ88xsbtu92OJ+IKFS8RESlSkdWaU7PvaFJun8T+cs3onNif0l+3ZPyXT7Nr716344n4lIqXiIj4REyt1pz+0DgO9JzA/tKNuGrXF4R/0oJJ377AgaQDbscT8QkVLxER8aky9c6j7qN/srvrbySWqEuHrR+T/kFTZgx+jZSUQ27HEylSKl4iIuKKSo0vosETk9nScQSJ0TW5YMN7HHrnTOb99D/S0464HU+kSKh4iYiIq2q0uJQznp7Omst/IDGiMm1Wvcm+t5uwdNQHZKWnuR1PpFCpeImIiF9ocO7VnPHMTJa0G0CipwJNl7xMwltNWDPhM2zGUbfjiRQKFS8REfEbxuOhWbvrOOPZOcw+5ysSKU2Dec+y562ziP/ra8jMcDuiyClR8RIREb/jCfNw3hU3UfeZeUxq/gn7MotRa8YT7H77LHbNGABZ+iFuCUwqXiIi4rciI8Lo0OlWajw9nzGN3mf/0XBO++sh9rzdjH1zf1ABk4Cj4iUiIn6veHQE197Yi4pPzGd4nbfYn2Yp9/t9JLzTkkMLf4asLLcjihSIipeIiASMciWi6XrL/ZR4eB6Dq73C/pSjlBhzN3vfa0XqspEqYOL3VLxERCTgVC1bnFt6PYy5bzbfVHyOpEMpRP96O4kfnEP6irFgrdsRRXKl4iUiIgGrXuUy3H3/kyTdOZNPSj/BwaQDRAzvyf4P25K1+ncVMPE7Kl4iIhLwWtSqQN+Hn2NLj2n0i3mY5P178AzrxsFPLsSum6gCJn7D2ABYGVu1amXj4uLcjiEiIgEgK8sybukWVk74ip5pP1HN7OVQbAtKXPEinN4OjHE7ogQ5Y8xCa22rXC9T8RIRkWCUnpnFz/M2snnS19yWOYKqJpGUym2IuewFqH2B2/EkiKl4iYhIyEo5msHAGWtJnN6fXvzKaWY/qdXOI7rD81CrrdvxJAipeImISMjbf/go30xZydF533GPZzQVzQGO1riAyA4vQI02bseTIKLiJSIi4rUz6Qif/bmc6KUD6B02hgomiYzalxDe/jmolutrpcgJUfESERHJYf2eQ3zy+xIqrRnCfRHjKMtBMuteStjFz0LVFm7HkwCm4iUiIpKHpVsP8OH4xZyxZRi9I8ZSmkPY+ldiLn4GKjd1O54EIBUvERGRfMxYl8DH4xfRZs9w7o0YT0kOYxtei2n3DFRq7HY8CSAqXiIiIgWQlWWZsHwXX/yxiPYHfuGeyAkUtynQqDO0exoqNnQ7ogQAFS8REZETkJ6ZxfC4bXw3aRHXHhnFPRF/EG2PYOpfAWf3gtMvAY9+/EVyp+IlIiJyEo4czWTgnHiGTlnMjRljuC1qKqUyD2DL1sa0uhOa3wwx5dyOKX5GxUtEROQUJKWk03/mRn6au4E2qbO4p9hkzsxciQ2LwjS5Hlr3gmot3Y4pfkLFS0REpBCkpmcydtlOBs6O5+iOv7kjajLXeWYSlZUClZs5BazJ9RAZ43ZUcZGKl4iISCGy1rJoywEGzI5nxt8buNbMpHfMFKqmx2OjS2Oa3Qyt7oQKdd2OKi7wq+JljKkODAIqARb42lr70fFuo+IlIiL+avfBVIbO28IPczdzespSehefSrvMOXhsBpx+sTMKVv8KCAt3O6r4iL8Vr8pAZWvtImNMSWAh0NlauzKv26h4iYiIv0vLyGT83zsZMHszO7bGc0vUdG6LnEzp9D1Qqiq0vANa3AolK7kdVYqYXxWv/wQwZjTwqbV2Yl7XUfESEZFAsmTrAQbOjmfCsq1cYBfxYKlpnJm6EOsJxzTs6IyC1TwPjHE7qhQBvy1exphawHSgibX2YI7L7gHuAahRo0bLzZs3+z6giIjIKdiTnMqP87YydN5mYg7Fc1/xaXRmClEZyRDbEFrfBWd1g+hSbkeVQuSXxcsYUwKYBrxhrf31eNfViJeIiASyoxlZ/L5iFwNnx7Ni8y6uj5zH/cWnUPXIGogsAWfd6IyC6aeJgoLfFS9jTAQwFvjDWtsvv+ureImISLBYvj2JAbPj+W3pDs7IXMfjZWfQNm0aYZlpUONcp4A17AjhkW5HlZPkV8XLGGOAgcA+a+3DBbmNipeIiASbxENpDFuwlSFzN3MkKYFeJedwS/gkSh/ZBsVjnQPxW94BZaq7HVVOkL8Vr/OBGcDfQJb37GettePzuo2Kl4iIBKuMzCz+XLmbAbPjWbBpL+0jVvBo2Rk0TJ6NAeerKFrfpd+HDCB+VbxOhoqXiIiEgpU7DjJwdjyjlmynQsZuHq8wh6vSJxKVlghlazsFrFlP/T6kn1PxEhERCSD7Dx/lp7itDJ6zmYQDB+lecim9YyZTOWkJhEc7P0vU+i6oqt+H9EcqXiIiIgEoIzOLSav2MHB2PHM2JnJW+FaeqTibs5MnEZZ+GKo0dw7Gb9xFvw/pR1S8REREAtyaXckMnBPPyEXbCUtP5qGKi7mJPyl5cB1El3F2Qer3If2CipeIiEiQSEpJZ/jCrQyas5kt+w5zRYmNPFp2OvUSp2Cy9PuQ/kDFS0REJMhkZlmmrN7DwDnxzFi3lyphB3m+ygI6pIwn8vBO/T6ki1S8REREgtj6PYcYNCeeEQu3kXb0KHdXWsOdkZOpmDAbPOHOF7Lq9yF9RsVLREQkBBxMTWdE3DYGzYknPjGFliUSea7iHJoljsOTlqTfh/QRFS8REZEQkpVlmbYugYGz45m6JoGSYUd5pvpKOqePJyZxuff3Ibs5JUy/D1noVLxERERC1MaEQwyas5kRC7dxKC2dGyrt5oFS06ix43eMfh+ySKh4iYiIhLhDaRn8umgbA2fHsyHhMHWKp/FitcW0PTCa8KTNzu9DNusJzXpAbAO34wY0FS8REREBwFrLzPV7GTArnslr9hBuLA/X3kZPz5+U3jYVYzOhcjNo2t35hvwSsW5HDjgqXiIiIvIfmxMPM3jOZn6K20pyagbnVsygT+wSWh/8k6iE5WDCoN6lzvFgDa6CiGi3IwcEFS8RERHJU8rRDEYu3s4vC7exaMsBAK45bT93l1pAk31/EHZoJ0SVhsadnJGw6ueAx+NuaD+m4iUiIiIFsm1/CuOW7WTssp38vT0JD1ncdtoWehabzel7J+NJT4EyNeCsm6DpTVC+jtuR/Y6Kl4iIiJyw+L2HGbtsB2OX7WT1rmRiTCp9Kq3muvCZVE6ci7FZUK21syuyyfUQU87tyH5BxUtEREROybrdyYxZtpOxS3ewce9hKnv282DFpVyZOYUyyevAEwH1L3dKWP3LITzK7ciuUfESERGRQmGtZeXOg4xdtpMxS3ewbf8RzgrbSt/yC7gwbSrRaXshugw06eIcD1atdcj9TJGKl4iIiBQ6ay1LtyUxZukOxi3bScLBw1wcsYJ7yiygZcoswjJTodzpzvFgZ90I5Wq7HdknVLxERESkSGVlWeI272fssh2M/3snRw4l0TkyjttKzKNeymIM1vmW/LO6QePOUKys25GLjIqXiIiI+ExGZhbzNu1j7LIdTFi+i2Ipu+gWPYfuUbOolLYZGxaFaXCFMxJWt0PQ/VSRipeIiIi4Ij0zi5nr9zJ26U7+XLGTmkfX0T1qNp3CZ1Mi4wA2pjymyfXOV1NUaREUx4OpeImIiIjrUtMzmb42gTHLdjJ15XZaZS6hR9QsLiaOcHsUW74epqn3eLAyNdyOe9JUvERERMSvHDmayeTVexizdAcL1myig53LTZGzaG5XAmBrnY856yZo1AmiS7mc9sSoeImIiIjfSk5NZ9Kq3YxdupMN61ZyDTO4MXIWNewOssKiMA2vwTTtDqdfDGHhbsfNl4qXiIiIBISklHT+WLGLMUu3c3jTfDqZ6VwXMZdSNpmMYrGEN+3qHA922ll+ezyYipeIiIgEnMRDaUxYvovxSzZTYusUrvPMoEPYYiLI4Gi5BkS26OEcD1aqittR/0XFS0RERALa7oOpjFu2k6lL1lBt5590CZtBK89aLIajNS4gqkUPaHgtRJVwO6qKl4iIiASPbftTGLdsJwsXL6TR3glc55lJTc8e0j3RZDa4huhWPaH2ReAJcyWfipeIiIgEpfi9hxm7dDsbFk2mVdKfXBM2h9ImhZSoipizulKsVU+o1NinmVS8REREJOit3Z3MhMWb2LdkLOcfnkQ7zxIiTCYHSjUgulVPopvfBCUrFXkOFS8REREJGdZaVu48yKS4VWT+PYJL0ibTzLOBLDzs6PQT1ZpfVqSPf7zi5f9fhiEiIiJyAowxNK5SmsYdz8Fe24al25L4fN4colb9yuXVcu1DPqPiJSIiIkHLGEOz6mVoVv1KrL0C4/J3f3lcfXQRERERH3G7dIGKl4iIiIj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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_49_3.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# The neural network with one input layer and one output layer,\n", - "# but with number of hidden layers specified by the user.\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - "\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consists of\n", - " # parameters to all the hidden\n", - " # layers AND the output layer.\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial_deep(x,params, g0 = 10):\n", - " return g0 + x*deep_neural_network(params, x)\n", - "\n", - "# The right side of the ODE:\n", - "def g(x, g_trial, gamma = 2):\n", - " return -gamma*g_trial\n", - "\n", - "# The same cost function as before, but calls deep_neural_network instead.\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the neural network\n", - " d_net_out = elementwise_grad(deep_neural_network,1)(P,x)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial_deep,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = g(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# Solve the exponential decay ODE using neural network with one input and one output layer,\n", - "# but with specified number of hidden layers from the user.\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # The number of elements in the list num_hidden_neurons thus represents\n", - " # the number of hidden layers.\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weights and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weights using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "def g_analytic(x, gamma = 2, g0 = 10):\n", - " return g0*np.exp(-gamma*x)\n", - "\n", - "# Solve the given problem\n", - "if __name__ == '__main__':\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " N = 10\n", - " x = np.linspace(0, 1, N)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = np.array([10,10])\n", - " num_iter = 10000\n", - " lmb = 0.001\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " res = g_trial_deep(x,P)\n", - " res_analytical = g_analytic(x)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, res_analytical)\n", - " plt.plot(x, res[0,:])\n", - " plt.legend(['analytical','dnn'])\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Example: Population growth\n", - "\n", - "A logistic model of population growth assumes that a population converges toward an equilibrium.\n", - "The population growth can be modeled by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{log} \\tag{10}\n", - "\tg'(t) = \\alpha g(t)(A - g(t))\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(t)$ is the population density at time $t$, $\\alpha > 0$ the growth rate and $A > 0$ is the maximum population number in the environment.\n", - "Also, at $t = 0$ the population has the size $g(0) = g_0$, where $g_0$ is some chosen constant.\n", - "\n", - "In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability\n", - "and high execution time (this might be more apparent in the examples solving PDEs),\n", - "using a library like TensorFlow is recommended.\n", - "Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.\n", - "\n", - "\n", - "\n", - "Here, we will model a population $g(t)$ in an environment having carrying capacity $A$.\n", - "The population follows the model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solveode_population} \\tag{11}\n", - "g'(t) = \\alpha g(t)(A - g(t))\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(0) = g_0$.\n", - "\n", - "In this example, we let $\\alpha = 2$, $A = 1$, and $g_0 = 1.2$.\n", - "\n", - "\n", - "We will get a slightly different trial solution, as the boundary conditions are different\n", - "compared to the case for exponential decay.\n", - "\n", - "A possible trial solution satisfying the condition $g(0) = g_0$ could be\n", - "\n", - "$$\n", - "h_1(t) = g_0 + t \\cdot N(t,P)\n", - "$$\n", - "\n", - "with $N(t,P)$ being the output from the neural network with weights and biases for each layer collected in the set $P$.\n", - "\n", - "The analytical solution is\n", - "\n", - "$$\n", - "g(t) = \\frac{Ag_0}{g_0 + (A - g_0)\\exp(-\\alpha A t)}\n", - "$$\n", - "\n", - "\n", - "\n", - "The network will be the similar as for the exponential decay example, but with some small modifications for our problem." - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Initial cost: 0.221805\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", - " return array(a, dtype, copy=False, order=order)\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Final cost: 0.000417932\n", - "The max absolute difference between the solutions is: 0.00424909\n" - ] - }, - { - "data": { - "image/png": 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\n", - "text/plain": [ - "
" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_55_3.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# Function to get the parameters.\n", - "# Done such that one can easily change the paramaters after one's liking.\n", - "def get_parameters():\n", - " alpha = 2\n", - " A = 1\n", - " g0 = 1.2\n", - " return alpha, A, g0\n", - "\n", - "def deep_neural_network(P, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(P) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = P[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = P[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial_deep,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = f(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# The right side of the ODE:\n", - "def f(x, g_trial):\n", - " alpha,A, g0 = get_parameters()\n", - " return alpha*g_trial*(A - g_trial)\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial_deep(x, params):\n", - " alpha,A, g0 = get_parameters()\n", - " return g0 + x*deep_neural_network(params,x)\n", - "\n", - "# The analytical solution:\n", - "def g_analytic(t):\n", - " alpha,A, g0 = get_parameters()\n", - " return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nt = 10\n", - " T = 1\n", - " t = np.linspace(0,T, Nt)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [100, 50, 25]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(t,P)\n", - " g_analytical = g_analytic(t)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%diff_ag)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(t, g_analytical)\n", - " plt.plot(t, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Using forward Euler to solve the ODE\n", - "\n", - "A straightforward way of solving an ODE numerically, is to use Euler's method.\n", - "\n", - "Euler's method uses Taylor series to approximate the value at a function $f$ at a step $\\Delta x$ from $x$:\n", - "\n", - "$$\n", - "f(x + \\Delta x) \\approx f(x) + \\Delta x f'(x)\n", - "$$\n", - "\n", - "In our case, using Euler's method to approximate the value of $g$ at a step $\\Delta t$ from $t$ yields" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - " g(t + \\Delta t) &\\approx g(t) + \\Delta t g'(t) \\\\\n", - " &= g(t) + \\Delta t \\big(\\alpha g(t)(A - g(t))\\big)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "along with the condition that $g(0) = g_0$.\n", - "\n", - "Let $t_i = i \\cdot \\Delta t$ where $\\Delta t = \\frac{T}{N_t-1}$ where $T$ is the final time our solver must solve for and $N_t$ the number of values for $t \\in [0, T]$ for $i = 0, \\dots, N_t-1$.\n", - "\n", - "For $i \\geq 1$, we have that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "t_i &= i\\Delta t \\\\\n", - "&= (i - 1)\\Delta t + \\Delta t \\\\\n", - "&= t_{i-1} + \\Delta t\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now, if $g_i = g(t_i)$ then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\begin{aligned}\n", - " g_i &= g(t_i) \\\\\n", - " &= g(t_{i-1} + \\Delta t) \\\\\n", - " &\\approx g(t_{i-1}) + \\Delta t \\big(\\alpha g(t_{i-1})(A - g(t_{i-1}))\\big) \\\\\n", - " &= g_{i-1} + \\Delta t \\big(\\alpha g_{i-1}(A - g_{i-1})\\big)\n", - " \\end{aligned}\n", - "\\end{equation} \\label{odenum} \\tag{12}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for $i \\geq 1$ and $g_0 = g(t_0) = g(0) = g_0$.\n", - "\n", - "Equation ([12](#odenum)) could be implemented in the following way,\n", - "extending the program that uses the network using Autograd:" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Initial cost: 0.221805\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", - " return array(a, dtype, copy=False, order=order)\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Final cost: 0.000417932\n", - "The max absolute difference between the solutions is: 0.00424909\n", - "Max absolute difference between Euler method and analytical: 0.011225\n", - "Max absolute difference between deep neural network and analytical: 0.00424909\n" - ] - }, - { - "data": { - "image/png": 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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_63_4.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "# Assume that all function definitions from the example program using Autograd\n", - "# are located here.\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nt = 10\n", - " T = 1\n", - " t = np.linspace(0,T, Nt)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [100,50,25]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(t,P)\n", - " g_analytical = g_analytic(t)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%diff_ag)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(t, g_analytical)\n", - " plt.plot(t, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " ## Find an approximation to the funtion using forward Euler\n", - "\n", - " alpha, A, g0 = get_parameters()\n", - " dt = T/(Nt - 1)\n", - "\n", - " # Perform forward Euler to solve the ODE\n", - " g_euler = np.zeros(Nt)\n", - " g_euler[0] = g0\n", - "\n", - " for i in range(1,Nt):\n", - " g_euler[i] = g_euler[i-1] + dt*(alpha*g_euler[i-1]*(A - g_euler[i-1]))\n", - "\n", - " # Print the errors done by each method\n", - " diff1 = np.max(np.abs(g_euler - g_analytical))\n", - " diff2 = np.max(np.abs(g_dnn_ag[0,:] - g_analytical))\n", - "\n", - " print('Max absolute difference between Euler method and analytical: %g'%diff1)\n", - " print('Max absolute difference between deep neural network and analytical: %g'%diff2)\n", - "\n", - " # Plot results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.plot(t,g_euler)\n", - " plt.plot(t,g_analytical)\n", - " plt.plot(t,g_dnn_ag[0,:])\n", - "\n", - " plt.legend(['euler','analytical','dnn'])\n", - " plt.xlabel('Time t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Solving the one dimensional Poisson equation\n", - "\n", - "The Poisson equation for $g(x)$ in one dimension is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{poisson} \\tag{13}\n", - " -g''(x) = f(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f(x)$ is a given function for $x \\in (0,1)$.\n", - "\n", - "The conditions that $g(x)$ is chosen to fulfill, are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " g(0) &= 0 \\\\\n", - " g(1) &= 0\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used.\n", - "The results from the networks can then be compared to the analytical solution.\n", - "In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.\n", - "\n", - "\n", - "Here, the function $g(x)$ to solve for follows the equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "-g''(x) = f(x),\\qquad x \\in (0,1)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f(x)$ is a given function, along with the chosen conditions" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{aligned}\n", - "g(0) = g(1) = 0\n", - "\\end{aligned}\\label{cond} \\tag{14}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this example, we consider the case when $f(x) = (3x + x^2)\\exp(x)$.\n", - "\n", - "For this case, a possible trial solution satisfying the conditions could be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x) = x \\cdot (1-x) \\cdot N(P,x)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The analytical solution for this problem is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g(x) = x(1 - x)\\exp(x)\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Initial cost: 457.256\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", - " return array(a, dtype, copy=False, order=order)\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Final cost: 0.00310113\n", - "The max absolute difference between the solutions is: 0.000464088\n" - ] - }, - { - "data": { - "image/png": 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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_76_3.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "## Set up the cost function specified for this Poisson equation:\n", - "\n", - "# The right side of the ODE\n", - "def f(x):\n", - " return (3*x + x**2)*np.exp(x)\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)\n", - "\n", - " right_side = f(x)\n", - "\n", - " err_sqr = (-d2_g_t - right_side)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum/np.size(err_sqr)\n", - "\n", - "# The trial solution:\n", - "def g_trial_deep(x,P):\n", - " return x*(1-x)*deep_neural_network(P,x)\n", - "\n", - "# The analytic solution;\n", - "def g_analytic(x):\n", - " return x*(1-x)*np.exp(x)\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10\n", - " x = np.linspace(0,1, Nx)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [200,100]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(x,P)\n", - " g_analytical = g_analytic(x)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " max_diff = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%max_diff)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, g_analytical)\n", - " plt.plot(x, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Comparing with a numerical scheme\n", - "\n", - "The Poisson equation is possible to solve using Taylor series to approximate the second derivative.\n", - "\n", - "Using Taylor series, the second derivative can be expressed as\n", - "\n", - "$$\n", - "g''(x) = \\frac{g(x + \\Delta x) - 2g(x) + g(x-\\Delta x)}{\\Delta x^2} + E_{\\Delta x}(x)\n", - "$$\n", - "\n", - "where $\\Delta x$ is a small step size and $E_{\\Delta x}(x)$ being the error term.\n", - "\n", - "Looking away from the error terms gives an approximation to the second derivative:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{approx} \\tag{15}\n", - "g''(x) \\approx \\frac{g(x + \\Delta x) - 2g(x) + g(x-\\Delta x)}{\\Delta x^2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If $x_i = i \\Delta x = x_{i-1} + \\Delta x$ and $g_i = g(x_i)$ for $i = 1,\\dots N_x - 2$ with $N_x$ being the number of values for $x$, ([15](#approx)) becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "g''(x_i) &\\approx \\frac{g(x_i + \\Delta x) - 2g(x_i) + g(x_i -\\Delta x)}{\\Delta x^2} \\\\\n", - "&= \\frac{g_{i+1} - 2g_i + g_{i-1}}{\\Delta x^2}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Since we know from our problem that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "-g''(x) &= f(x) \\\\\n", - "&= (3x + x^2)\\exp(x)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "along with the conditions $g(0) = g(1) = 0$,\n", - "the following scheme can be used to find an approximate solution for $g(x)$ numerically:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\begin{aligned}\n", - " -\\Big( \\frac{g_{i+1} - 2g_i + g_{i-1}}{\\Delta x^2} \\Big) &= f(x_i) \\\\\n", - " -g_{i+1} + 2g_i - g_{i-1} &= \\Delta x^2 f(x_i)\n", - " \\end{aligned}\n", - "\\end{equation} \\label{odesys} \\tag{16}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for $i = 1, \\dots, N_x - 2$ where $g_0 = g_{N_x - 1} = 0$ and $f(x_i) = (3x_i + x_i^2)\\exp(x_i)$, which is given for our specific problem.\n", - "\n", - "The equation can be rewritten into a matrix equation:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "\\begin{pmatrix}\n", - "2 & -1 & 0 & \\dots & 0 \\\\\n", - "-1 & 2 & -1 & \\dots & 0 \\\\\n", - "\\vdots & & \\ddots & & \\vdots \\\\\n", - "0 & \\dots & -1 & 2 & -1 \\\\\n", - "0 & \\dots & 0 & -1 & 2\\\\\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "g_1 \\\\\n", - "g_2 \\\\\n", - "\\vdots \\\\\n", - "g_{N_x - 3} \\\\\n", - "g_{N_x - 2}\n", - "\\end{pmatrix}\n", - "&=\n", - "\\Delta x^2\n", - "\\begin{pmatrix}\n", - "f(x_1) \\\\\n", - "f(x_2) \\\\\n", - "\\vdots \\\\\n", - "f(x_{N_x - 3}) \\\\\n", - "f(x_{N_x - 2})\n", - "\\end{pmatrix} \\\\\n", - "\\boldsymbol{A}\\boldsymbol{g} &= \\boldsymbol{f},\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which makes it possible to solve for the vector $\\boldsymbol{g}$.\n", - "\n", - "\n", - "We can then compare the result from this numerical scheme with the output from our network using Autograd:" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Initial cost: 457.256\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", - " return array(a, dtype, copy=False, order=order)\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Final cost: 0.00310113\n", - "The max absolute difference between the analytical solution and DNN Autograd: 0.000464088\n", - "The max absolute difference between the analytical solution and numerical scheme: 0.00266858\n" - ] - }, - { - "data": { - "image/png": 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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter11_88_4.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "## Set up the cost function specified for this Poisson equation:\n", - "\n", - "# The right side of the ODE\n", - "def f(x):\n", - " return (3*x + x**2)*np.exp(x)\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)\n", - "\n", - " right_side = f(x)\n", - "\n", - " err_sqr = (-d2_g_t - right_side)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum/np.size(err_sqr)\n", - "\n", - "# The trial solution:\n", - "def g_trial_deep(x,P):\n", - " return x*(1-x)*deep_neural_network(P,x)\n", - "\n", - "# The analytic solution;\n", - "def g_analytic(x):\n", - " return x*(1-x)*np.exp(x)\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10\n", - " x = np.linspace(0,1, Nx)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [200,100]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(x,P)\n", - " g_analytical = g_analytic(x)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, g_analytical)\n", - " plt.plot(x, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - "\n", - " ## Perform the computation using the numerical scheme\n", - "\n", - " dx = 1/(Nx - 1)\n", - "\n", - " # Set up the matrix A\n", - " A = np.zeros((Nx-2,Nx-2))\n", - "\n", - " A[0,0] = 2\n", - " A[0,1] = -1\n", - "\n", - " for i in range(1,Nx-3):\n", - " A[i,i-1] = -1\n", - " A[i,i] = 2\n", - " A[i,i+1] = -1\n", - "\n", - " A[Nx - 3, Nx - 4] = -1\n", - " A[Nx - 3, Nx - 3] = 2\n", - "\n", - " # Set up the vector f\n", - " f_vec = dx**2 * f(x[1:-1])\n", - "\n", - " # Solve the equation\n", - " g_res = np.linalg.solve(A,f_vec)\n", - "\n", - " g_vec = np.zeros(Nx)\n", - " g_vec[1:-1] = g_res\n", - "\n", - " # Print the differences between each method\n", - " max_diff1 = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " max_diff2 = np.max(np.abs(g_vec - g_analytical))\n", - " print(\"The max absolute difference between the analytical solution and DNN Autograd: %g\"%max_diff1)\n", - " print(\"The max absolute difference between the analytical solution and numerical scheme: %g\"%max_diff2)\n", - "\n", - " # Plot the results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.plot(x,g_vec)\n", - " plt.plot(x,g_analytical)\n", - " plt.plot(x,g_dnn_ag[0,:])\n", - "\n", - " plt.legend(['numerical scheme','analytical','dnn'])\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Partial Differential Equations\n", - "\n", - "A partial differential equation (PDE) has a solution here the function\n", - "is defined by multiple variables. The equation may involve all kinds\n", - "of combinations of which variables the function is differentiated with\n", - "respect to.\n", - "\n", - "In general, a partial differential equation for a function $g(x_1,\\dots,x_N)$ with $N$ variables may be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{PDE} \\tag{17}\n", - " f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) = 0\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given.\n", - "\n", - "### Type of problem\n", - "\n", - "The problem our network must solve for, is similar to the ODE case.\n", - "We must have a trial solution $g_t$ at hand.\n", - "\n", - "For instance, the trial solution could be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " g_t(x_1,\\dots,x_N) = h_1(x_1,\\dots,x_N) + h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $h_1(x_1,\\dots,x_N)$ is a function that ensures $g_t(x_1,\\dots,x_N)$ satisfies some given conditions.\n", - "The neural network $N(x_1,\\dots,x_N,P)$ has weights and biases described by $P$ and $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$ is an expression using the output from the neural network in some way.\n", - "\n", - "The role of the function $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$, is to ensure that the output of $N(x_1,\\dots,x_N,P)$ is zero when $g_t(x_1,\\dots,x_N)$ is evaluated at the values of $x_1,\\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\\dots,x_N)$ should alone make $g_t(x_1,\\dots,x_N)$ satisfy the conditions.\n", - "\n", - "\n", - "\n", - "### Network requirements\n", - "\n", - "The network tries then the minimize the cost function following the\n", - "same ideas as described for the ODE case, but now with more than one\n", - "variables to consider. The concept still remains the same; find a set\n", - "of parameters $P$ such that the expression $f$ in ([17](#PDE)) is as\n", - "close to zero as possible.\n", - "\n", - "As for the ODE case, the cost function is the mean squared error that\n", - "the network must try to minimize. The cost function for the network to\n", - "minimize is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(x_1, \\dots, x_N, P\\right) = \\left( f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) \\right)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we let $\\boldsymbol{x} = \\big( x_1, \\dots, x_N \\big)$ be an array containing the values for $x_1, \\dots, x_N$ respectively, the cost function can be reformulated into the following:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(\\boldsymbol{x}, P\\right) = f\\left( \\left( \\boldsymbol{x}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}) }{\\partial x_N^n} \\right) \\right)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we also have $M$ different sets of values for $x_1, \\dots, x_N$, that is $\\boldsymbol{x}_i = \\big(x_1^{(i)}, \\dots, x_N^{(i)}\\big)$ for $i = 1,\\dots,M$ being the rows in matrix $X$, the cost function can be generalized into" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(X, P \\right) = \\sum_{i=1}^M f\\left( \\left( \\boldsymbol{x}_i, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}_i) }{\\partial x_N^n} \\right) \\right)^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Example: The diffusion equation\n", - "\n", - "In one spatial dimension, the equation reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where a possible choice of conditions are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "g(0,t) &= 0 ,\\qquad t \\geq 0 \\\\\n", - "g(1,t) &= 0, \\qquad t \\geq 0 \\\\\n", - "g(x,0) &= u(x),\\qquad x\\in [0,1]\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $u(x)$ being some given function.\n", - "\n", - "\n", - "\n", - "For this case, we want to find $g(x,t)$ such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "\\end{equation} \\label{diffonedim} \\tag{18}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "g(0,t) &= 0 ,\\qquad t \\geq 0 \\\\\n", - "g(1,t) &= 0, \\qquad t \\geq 0 \\\\\n", - "g(x,0) &= u(x),\\qquad x\\in [0,1]\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $u(x) = \\sin(\\pi x)$.\n", - "\n", - "First, let us set up the deep neural network.\n", - "The deep neural network will follow the same structure as discussed in the examples solving the ODEs.\n", - "First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions.\n", - "\n", - "\n", - "\n", - "\n", - "The only change to do here, is to extend our network such that\n", - "functions of multiple parameters are correctly handled. In this case\n", - "we have two variables in our function to solve for, that is time $t$\n", - "and position $x$. The variables will be represented by a\n", - "one-dimensional array in the program. The program will evaluate the\n", - "network at each possible pair $(x,t)$, given an array for the desired\n", - "$x$-values and $t$-values to approximate the solution at." - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The cost function must then iterate through the given arrays\n", - "containing values for $x$ and $t$, defines a point $(x,t)$ the deep\n", - "neural network and the trial solution is evaluated at, and then finds\n", - "the Jacobian of the trial solution.\n", - "\n", - "A possible trial solution for this PDE is\n", - "\n", - "$$\n", - "g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P)\n", - "$$\n", - "\n", - "with $A(x,t)$ being a function ensuring that $g_t(x,t)$ satisfies our given conditions, and $N(x,t,P)$ being the output from the deep neural network using weights and biases for each layer from $P$.\n", - "\n", - "To fulfill the conditions, $A(x,t)$ could be:\n", - "\n", - "$$\n", - "h_1(x,t) = (1-t)\\Big(u(x) - \\big((1-x)u(0) + x u(1)\\big)\\Big) = (1-t)u(x) = (1-t)\\sin(\\pi x)\n", - "$$\n", - "since $(0) = u(1) = 0$ and $u(x) = \\sin(\\pi x)$.\n", - "\n", - "\n", - "\n", - "The Jacobian is used because the program must find the derivative of\n", - "the trial solution with respect to $x$ and $t$.\n", - "\n", - "This gives the necessity of computing the Jacobian matrix, as we want\n", - "to evaluate the gradient with respect to $x$ and $t$ (note that the\n", - "Jacobian of a scalar-valued multivariate function is simply its\n", - "gradient).\n", - "\n", - "In Autograd, the differentiation is by default done with respect to\n", - "the first input argument of your Python function. Since the points is\n", - "an array representing $x$ and $t$, the Jacobian is calculated using\n", - "the values of $x$ and $t$.\n", - "\n", - "To find the second derivative with respect to $x$ and $t$, the\n", - "Jacobian can be found for the second time. The result is a Hessian\n", - "matrix, which is the matrix containing all the possible second order\n", - "mixed derivatives of $g(x,t)$." - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Set up the trial function:\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)\n", - "\n", - "# The right side of the ODE:\n", - "def f(point):\n", - " return 0.\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_jacobian_func = jacobian(g_trial)\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t = g_trial(point,P)\n", - " g_t_jacobian = g_t_jacobian_func(point,P)\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_dt = g_t_jacobian[1]\n", - " g_t_d2x = g_t_hessian[0][0]\n", - "\n", - " func = f(point)\n", - "\n", - " err_sqr = ( (g_t_dt - g_t_d2x) - func)**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Setting up the network using Autograd; The full program\n", - "\n", - "Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.\n", - "\n", - "The analytical solution of our problem is\n", - "\n", - "$$\n", - "g(x,t) = \\exp(-\\pi^2 t)\\sin(\\pi x)\n", - "$$\n", - "\n", - "A possible way to implement a neural network solving the PDE, is given below.\n", - "Be aware, though, that it is fairly slow for the parameters used.\n", - "A better result is possible, but requires more iterations, and thus longer time to complete.\n", - "\n", - "\n", - "Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.\n", - "Using TensorFlow results in a much better execution time. Try it!" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/numpy/core/_asarray.py:83: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray\n", - " return array(a, dtype, copy=False, order=order)\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Initial cost: 41.05505310046363\n" - ] - } - ], - "source": [ - "import autograd.numpy as np\n", - "from autograd import jacobian,hessian,grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import cm\n", - "from matplotlib import pyplot as plt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "## Set up the network\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]\n", - "\n", - "## Define the trial solution and cost function\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)\n", - "\n", - "# The right side of the ODE:\n", - "def f(point):\n", - " return 0.\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_jacobian_func = jacobian(g_trial)\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t = g_trial(point,P)\n", - " g_t_jacobian = g_t_jacobian_func(point,P)\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_dt = g_t_jacobian[1]\n", - " g_t_d2x = g_t_hessian[0][0]\n", - "\n", - " func = f(point)\n", - "\n", - " err_sqr = ( (g_t_dt - g_t_d2x) - func)**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum /( np.size(x)*np.size(t) )\n", - "\n", - "## For comparison, define the analytical solution\n", - "def g_analytic(point):\n", - " x,t = point\n", - " return np.exp(-np.pi**2*t)*np.sin(np.pi*x)\n", - "\n", - "## Set up a function for training the network to solve for the equation\n", - "def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):\n", - " ## Set up initial weigths and biases\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: ',cost_function(P, x, t))\n", - "\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " cost_grad = cost_function_grad(P, x , t)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_grad[l]\n", - "\n", - " print('Final cost: ',cost_function(P, x, t))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " ### Use the neural network:\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10; Nt = 10\n", - " x = np.linspace(0, 1, Nx)\n", - " t = np.linspace(0,1,Nt)\n", - "\n", - " ## Set up the parameters for the network\n", - " num_hidden_neurons = [100, 25]\n", - " num_iter = 250\n", - " lmb = 0.01\n", - "\n", - " P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " ## Store the results\n", - " g_dnn_ag = np.zeros((Nx, Nt))\n", - " G_analytical = np.zeros((Nx, Nt))\n", - " for i,x_ in enumerate(x):\n", - " for j, t_ in enumerate(t):\n", - " point = np.array([x_, t_])\n", - " g_dnn_ag[i,j] = g_trial(point,P)\n", - "\n", - " G_analytical[i,j] = g_analytic(point)\n", - "\n", - " # Find the map difference between the analytical and the computed solution\n", - " diff_ag = np.abs(g_dnn_ag - G_analytical)\n", - " print('Max absolute difference between the analytical solution and the network: %g'%np.max(diff_ag))\n", - "\n", - " ## Plot the solutions in two dimensions, that being in position and time\n", - "\n", - " T,X = np.meshgrid(t,x)\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", - " s = ax.plot_surface(T,X,g_dnn_ag,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Analytical solution')\n", - " s = ax.plot_surface(T,X,G_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Difference')\n", - " s = ax.plot_surface(T,X,diff_ag,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " ## Take some slices of the 3D plots just to see the solutions at particular times\n", - " indx1 = 0\n", - " indx2 = int(Nt/2)\n", - " indx3 = Nt-1\n", - "\n", - " t1 = t[indx1]\n", - " t2 = t[indx2]\n", - " t3 = t[indx3]\n", - "\n", - " # Slice the results from the DNN\n", - " res1 = g_dnn_ag[:,indx1]\n", - " res2 = g_dnn_ag[:,indx2]\n", - " res3 = g_dnn_ag[:,indx3]\n", - "\n", - " # Slice the analytical results\n", - " res_analytical1 = G_analytical[:,indx1]\n", - " res_analytical2 = G_analytical[:,indx2]\n", - " res_analytical3 = G_analytical[:,indx3]\n", - "\n", - " # Plot the slices\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t1)\n", - " plt.plot(x, res1)\n", - " plt.plot(x,res_analytical1)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t2)\n", - " plt.plot(x, res2)\n", - " plt.plot(x,res_analytical2)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t3)\n", - " plt.plot(x, res3)\n", - " plt.plot(x,res_analytical3)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Solving the wave equation with Neural Networks\n", - "\n", - "The wave equation is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2\\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $c$ being the specified wave speed.\n", - "\n", - "Here, the chosen conditions are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\tg(0,t) &= 0 \\\\\n", - "\tg(1,t) &= 0 \\\\\n", - "\tg(x,0) &= u(x) \\\\\n", - "\t\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0} &= v(x)\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0}$ means the derivative of $g(x,t)$ with respect to $t$ is evaluated at $t = 0$, and $u(x)$ and $v(x)$ being given functions.\n", - "\n", - "\n", - "The wave equation to solve for, is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{wave} \\tag{19}\n", - "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2 \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $c$ is the given wave speed.\n", - "The chosen conditions for this equation are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{aligned}\n", - "g(0,t) &= 0, &t \\geq 0 \\\\\n", - "g(1,t) &= 0, &t \\geq 0 \\\\\n", - "g(x,0) &= u(x), &x\\in[0,1] \\\\\n", - "\\frac{\\partial g(x,t)}{\\partial t}\\Big |_{t = 0} &= v(x), &x \\in [0,1]\n", - "\\end{aligned} \\label{condwave} \\tag{20}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this example, let $c = 1$ and $u(x) = \\sin(\\pi x)$ and $v(x) = -\\pi\\sin(\\pi x)$.\n", - "\n", - "\n", - "\n", - "Setting up the network is done in similar matter as for the example of solving the diffusion equation.\n", - "The only things we have to change, is the trial solution such that it satisfies the conditions from ([20](#condwave)) and the cost function.\n", - "\n", - "The trial solution becomes slightly different since we have other conditions than in the example of solving the diffusion equation. Here, a possible trial solution $g_t(x,t)$ is\n", - "\n", - "$$\n", - "g_t(x,t) = h_1(x,t) + x(1-x)t^2N(x,t,P)\n", - "$$\n", - "\n", - "where\n", - "\n", - "$$\n", - "h_1(x,t) = (1-t^2)u(x) + tv(x)\n", - "$$\n", - "\n", - "Note that this trial solution satisfies the conditions only if $u(0) = v(0) = u(1) = v(1) = 0$, which is the case in this example.\n", - "\n", - "\n", - "The analytical solution for our specific problem, is\n", - "\n", - "$$\n", - "g(x,t) = \\sin(\\pi x)\\cos(\\pi t) - \\sin(\\pi x)\\sin(\\pi t)\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import hessian,grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import cm\n", - "from matplotlib import pyplot as plt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "## Set up the trial function:\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def v(x):\n", - " return -np.pi*np.sin(np.pi*x)\n", - "\n", - "def h1(point):\n", - " x,t = point\n", - " return (1 - t**2)*u(x) + t*v(x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return h1(point) + x*(1-x)*t**2*deep_neural_network(P,point)\n", - "\n", - "## Define the cost function\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_d2x = g_t_hessian[0][0]\n", - " g_t_d2t = g_t_hessian[1][1]\n", - "\n", - " err_sqr = ( (g_t_d2t - g_t_d2x) )**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum / (np.size(t) * np.size(x))\n", - "\n", - "## The neural network\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]\n", - "\n", - "## The analytical solution\n", - "def g_analytic(point):\n", - " x,t = point\n", - " return np.sin(np.pi*x)*np.cos(np.pi*t) - np.sin(np.pi*x)*np.sin(np.pi*t)\n", - "\n", - "def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):\n", - " ## Set up initial weigths and biases\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: ',cost_function(P, x, t))\n", - "\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " cost_grad = cost_function_grad(P, x , t)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_grad[l]\n", - "\n", - "\n", - " print('Final cost: ',cost_function(P, x, t))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " ### Use the neural network:\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10; Nt = 10\n", - " x = np.linspace(0, 1, Nx)\n", - " t = np.linspace(0,1,Nt)\n", - "\n", - " ## Set up the parameters for the network\n", - " num_hidden_neurons = [50,20]\n", - " num_iter = 1000\n", - " lmb = 0.01\n", - "\n", - " P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " ## Store the results\n", - " res = np.zeros((Nx, Nt))\n", - " res_analytical = np.zeros((Nx, Nt))\n", - " for i,x_ in enumerate(x):\n", - " for j, t_ in enumerate(t):\n", - " point = np.array([x_, t_])\n", - " res[i,j] = g_trial(point,P)\n", - "\n", - " res_analytical[i,j] = g_analytic(point)\n", - "\n", - " diff = np.abs(res - res_analytical)\n", - " print(\"Max difference between analytical and solution from nn: %g\"%np.max(diff))\n", - "\n", - " ## Plot the solutions in two dimensions, that being in position and time\n", - "\n", - " T,X = np.meshgrid(t,x)\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", - " s = ax.plot_surface(T,X,res,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Analytical solution')\n", - " s = ax.plot_surface(T,X,res_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Difference')\n", - " s = ax.plot_surface(T,X,diff,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " ## Take some slices of the 3D plots just to see the solutions at particular times\n", - " indx1 = 0\n", - " indx2 = int(Nt/2)\n", - " indx3 = Nt-1\n", - "\n", - " t1 = t[indx1]\n", - " t2 = t[indx2]\n", - " t3 = t[indx3]\n", - "\n", - " # Slice the results from the DNN\n", - " res1 = res[:,indx1]\n", - " res2 = res[:,indx2]\n", - " res3 = res[:,indx3]\n", - "\n", - " # Slice the analytical results\n", - " res_analytical1 = res_analytical[:,indx1]\n", - " res_analytical2 = res_analytical[:,indx2]\n", - " res_analytical3 = res_analytical[:,indx3]\n", - "\n", - " # Plot the slices\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t1)\n", - " plt.plot(x, res1)\n", - " plt.plot(x,res_analytical1)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t2)\n", - " plt.plot(x, res2)\n", - " plt.plot(x,res_analytical2)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t3)\n", - " plt.plot(x, res3)\n", - " plt.plot(x,res_analytical3)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Resources on differential equations and deep learning\n", - "\n", - "1. [Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al](https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf)\n", - "\n", - "2. [Neural networks for solving differential equations by A. Honchar](https://becominghuman.ai/neural-networks-for-solving-differential-equations-fa230ac5e04c)\n", - "\n", - "3. [Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener](http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf)\n", - "\n", - "4. [Introduction to Partial Differential Equations by A. Tveito, R. Winther](https://www.springer.com/us/book/9783540225515)" - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11.py b/doc/LectureNotes/_build/jupyter_execute/chapter11.py deleted file mode 100644 index 17470c2a0..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter11.py +++ /dev/null @@ -1,2345 +0,0 @@ -# Solving Differential Equations with Deep Learning - -The Universal Approximation Theorem states that a neural network can -approximate any function at a single hidden layer along with one input -and output layer to any given precision. - - -An ordinary differential equation (ODE) is an equation involving functions having one variable. - -In general, an ordinary differential equation looks like - - -
- -$$ -\begin{equation} \label{ode} \tag{1} -f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right) = 0 -\end{equation} -$$ - -where $g(x)$ is the function to find, and $g^{(n)}(x)$ is the $n$-th derivative of $g(x)$. - -The $f\left(x, g(x), g'(x), g''(x), \, \dots \, , g^{(n)}(x)\right)$ is just a way to write that there is an expression involving $x$ and $g(x), \ g'(x), \ g''(x), \, \dots \, , \text{ and } g^{(n)}(x)$ on the left side of the equality sign in ([1](#ode)). -The highest order of derivative, that is the value of $n$, determines to the order of the equation. -The equation is referred to as a $n$-th order ODE. -Along with ([1](#ode)), some additional conditions of the function $g(x)$ are typically given -for the solution to be unique. - - - -Let the trial solution $g_t(x)$ be - - -
- -$$ -\begin{equation} - g_t(x) = h_1(x) + h_2(x,N(x,P)) -\label{_auto1} \tag{2} -\end{equation} -$$ - -where $h_1(x)$ is a function that makes $g_t(x)$ satisfy a given set -of conditions, $N(x,P)$ a neural network with weights and biases -described by $P$ and $h_2(x, N(x,P))$ some expression involving the -neural network. The role of the function $h_2(x, N(x,P))$, is to -ensure that the output from $N(x,P)$ is zero when $g_t(x)$ is -evaluated at the values of $x$ where the given conditions must be -satisfied. The function $h_1(x)$ should alone make $g_t(x)$ satisfy -the conditions. - -But what about the network $N(x,P)$? - - -As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation. - - - -For the minimization to be defined, we need to have a cost function at hand to minimize. - -It is given that $f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right)$ should be equal to zero in ([1](#ode)). -We can choose to consider the mean squared error as the cost function for an input $x$. -Since we are looking at one input, the cost function is just $f$ squared. -The cost function $c\left(x, P \right)$ can therefore be expressed as - -$$ -C\left(x, P\right) = \big(f\left(x, \, g(x), \, g'(x), \, g''(x), \, \dots \, , \, g^{(n)}(x)\right)\big)^2 -$$ - -If $N$ inputs are given as a vector $\boldsymbol{x}$ with elements $x_i$ for $i = 1,\dots,N$, -the cost function becomes - - -
- -$$ -\begin{equation} \label{cost} \tag{3} - C\left(\boldsymbol{x}, P\right) = \frac{1}{N} \sum_{i=1}^N \big(f\left(x_i, \, g(x_i), \, g'(x_i), \, g''(x_i), \, \dots \, , \, g^{(n)}(x_i)\right)\big)^2 -\end{equation} -$$ - -The neural net should then find the parameters $P$ that minimizes the cost function in -([3](#cost)) for a set of $N$ training samples $x_i$. - - - -To perform the minimization using gradient descent, the gradient of $C\left(\boldsymbol{x}, P\right)$ is needed. -It might happen so that finding an analytical expression of the gradient of $C(\boldsymbol{x}, P)$ from ([3](#cost)) gets too messy, depending on which cost function one desires to use. - -Luckily, there exists libraries that makes the job for us through automatic differentiation. -Automatic differentiation is a method of finding the derivatives numerically with very high precision. - - -### Example: Exponential decay - -An exponential decay of a quantity $g(x)$ is described by the equation - - -
- -$$ -\begin{equation} \label{solve_expdec} \tag{4} - g'(x) = -\gamma g(x) -\end{equation} -$$ - -with $g(0) = g_0$ for some chosen initial value $g_0$. - -The analytical solution of ([4](#solve_expdec)) is - - -
- -$$ -\begin{equation} - g(x) = g_0 \exp\left(-\gamma x\right) -\label{_auto2} \tag{5} -\end{equation} -$$ - -Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of ([4](#solve_expdec)). - - - -The program will use a neural network to solve - - -
- -$$ -\begin{equation} \label{solveode} \tag{6} -g'(x) = -\gamma g(x) -\end{equation} -$$ - -where $g(0) = g_0$ with $\gamma$ and $g_0$ being some chosen values. - -In this example, $\gamma = 2$ and $g_0 = 10$. - - -To begin with, a trial solution $g_t(t)$ must be chosen. A general trial solution for ordinary differential equations could be - -$$ -g_t(x, P) = h_1(x) + h_2(x, N(x, P)) -$$ - -with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and $h_2(x,N(x, P))$ an expression involving $x$ and the output from the neural network $N(x,P)$ with $P $ being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer. - - - -In this network, there are no weights and bias at the input layer, so $P = \{ P_{\text{hidden}}, P_{\text{output}} \}$. -If there are $N_{\text{hidden} }$ neurons in the hidden layer, then $P_{\text{hidden}}$ is a $N_{\text{hidden} } \times (1 + N_{\text{input}})$ matrix, given that there are $N_{\text{input}}$ neurons in the input layer. - -The first column in $P_{\text{hidden} }$ represents the bias for each neuron in the hidden layer and the second column represents the weights for each neuron in the hidden layer from the input layer. -If there are $N_{\text{output} }$ neurons in the output layer, then $P_{\text{output}} $ is a $N_{\text{output} } \times (1 + N_{\text{hidden} })$ matrix. - -Its first column represents the bias of each neuron and the remaining columns represents the weights to each neuron. - -It is given that $g(0) = g_0$. The trial solution must fulfill this condition to be a proper solution of ([6](#solveode)). A possible way to ensure that $g_t(0, P) = g_0$, is to let $F(N(x,P)) = x \cdot N(x,P)$ and $A(x) = g_0$. This gives the following trial solution: - - -
- -$$ -\begin{equation} \label{trial} \tag{7} -g_t(x, P) = g_0 + x \cdot N(x, P) -\end{equation} -$$ - -### Reformulating the problem - -We wish that our neural network manages to minimize a given cost function. - -A reformulation of out equation, ([6](#solveode)), must therefore be done, -such that it describes the problem a neural network can solve for. - -The neural network must find the set of weights and biases $P$ such that the trial solution in ([7](#trial)) satisfies ([6](#solveode)). - -The trial solution - -$$ -g_t(x, P) = g_0 + x \cdot N(x, P) -$$ - -has been chosen such that it already solves the condition $g(0) = g_0$. What remains, is to find $P$ such that - - -
- -$$ -\begin{equation} \label{nnmin} \tag{8} -g_t'(x, P) = - \gamma g_t(x, P) -\end{equation} -$$ - -is fulfilled as *best as possible*. - - -The left hand side and right hand side of ([8](#nnmin)) must be computed separately, and then the neural network must choose weights and biases, contained in $P$, such that the sides are equal as best as possible. -This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero. -In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to $P$ of the neural network. - -This gives the following cost function our neural network must solve for: - -$$ -\min_{P}\Big\{ \big(g_t'(x, P) - ( -\gamma g_t(x, P) \big)^2 \Big\} -$$ - -(the notation $\min_{P}\{ f(x, P) \}$ means that we desire to find $P$ that yields the minimum of $f(x, P)$) - -or, in terms of weights and biases for the hidden and output layer in our network: - -$$ -\min_{P_{\text{hidden} }, \ P_{\text{output} }}\Big\{ \big(g_t'(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) - ( -\gamma g_t(x, \{ P_{\text{hidden} }, P_{\text{output} }\}) \big)^2 \Big\} -$$ - -for an input value $x$. - - - -If the neural network evaluates $g_t(x, P)$ at more values for $x$, say $N$ values $x_i$ for $i = 1, \dots, N$, then the *total* error to minimize becomes - - -
- -$$ -\begin{equation} \label{min} \tag{9} -\min_{P}\Big\{\frac{1}{N} \sum_{i=1}^N \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 \Big\} -\end{equation} -$$ - -Letting $\boldsymbol{x}$ be a vector with elements $x_i$ and $C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2$ denote the cost function, the minimization problem that our network must solve, becomes - -$$ -\min_{P} C(\boldsymbol{x}, P) -$$ - -In terms of $P_{\text{hidden} }$ and $P_{\text{output} }$, this could also be expressed as - -$$ -\min_{P_{\text{hidden} }, \ P_{\text{output} }} C(\boldsymbol{x}, \{P_{\text{hidden} }, P_{\text{output} }\}) -$$ - - -For simplicity, it is assumed that the input is an array $\boldsymbol{x} = (x_1, \dots, x_N)$ with $N$ elements. It is at these points the neural network should find $P$ such that it fulfills ([9](#min)). - -First, the neural network must feed forward the inputs. -This means that $\boldsymbol{x}s$ must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further. -The input layer will consist of $N_{\text{input} }$ neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be $N_{\text{hidden} }$. - - -For the $i$-th in the hidden layer with weight $w_i^{\text{hidden} }$ and bias $b_i^{\text{hidden} }$, the weighting from the $j$-th neuron at the input layer is: - -$$ -\begin{aligned} -z_{i,j}^{\text{hidden}} &= b_i^{\text{hidden}} + w_i^{\text{hidden}}x_j \\ -&= -\begin{pmatrix} -b_i^{\text{hidden}} & w_i^{\text{hidden}} -\end{pmatrix} -\begin{pmatrix} -1 \\ -x_j -\end{pmatrix} -\end{aligned} -$$ - -The result after weighting the inputs at the $i$-th hidden neuron can be written as a vector: - -$$ -\begin{aligned} -\boldsymbol{z}_{i}^{\text{hidden}} &= \Big( b_i^{\text{hidden}} + w_i^{\text{hidden}}x_1 , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_2, \ \dots \, , \ b_i^{\text{hidden}} + w_i^{\text{hidden}} x_N\Big) \\ -&= -\begin{pmatrix} - b_i^{\text{hidden}} & w_i^{\text{hidden}} -\end{pmatrix} -\begin{pmatrix} -1 & 1 & \dots & 1 \\ -x_1 & x_2 & \dots & x_N -\end{pmatrix} \\ -&= \boldsymbol{p}_{i, \text{hidden}}^T X -\end{aligned} -$$ - -The vector $\boldsymbol{p}_{i, \text{hidden}}^T$ constitutes each row in $P_{\text{hidden} }$, which contains the weights for the neural network to minimize according to ([9](#min)). - -After having found $\boldsymbol{z}_{i}^{\text{hidden}} $ for every $i$-th neuron within the hidden layer, the vector will be sent to an activation function $a_i(\boldsymbol{z})$. - -In this example, the sigmoid function has been chosen to be the activation function for each hidden neuron: - -$$ -f(z) = \frac{1}{1 + \exp{(-z)}} -$$ - -It is possible to use other activations functions for the hidden layer also. - -The output $\boldsymbol{x}_i^{\text{hidden}}$ from each $i$-th hidden neuron is: - -$$ -\boldsymbol{x}_i^{\text{hidden} } = f\big( \boldsymbol{z}_{i}^{\text{hidden}} \big) -$$ - -The outputs $\boldsymbol{x}_i^{\text{hidden} } $ are then sent to the output layer. - -The output layer consists of one neuron in this case, and combines the -output from each of the neurons in the hidden layers. The output layer -combines the results from the hidden layer using some weights $w_i^{\text{output}}$ -and biases $b_i^{\text{output}}$. In this case, -it is assumes that the number of neurons in the output layer is one. - - - -The procedure of weighting the output neuron $j$ in the hidden layer to the $i$-th neuron in the output layer is similar as for the hidden layer described previously. - -$$ -\begin{aligned} -z_{1,j}^{\text{output}} & = -\begin{pmatrix} -b_1^{\text{output}} & \boldsymbol{w}_1^{\text{output}} -\end{pmatrix} -\begin{pmatrix} -1 \\ -\boldsymbol{x}_j^{\text{hidden}} -\end{pmatrix} -\end{aligned} -$$ - -Expressing $z_{1,j}^{\text{output}}$ as a vector gives the following way of weighting the inputs from the hidden layer: - -$$ -\boldsymbol{z}_{1}^{\text{output}} = -\begin{pmatrix} -b_1^{\text{output}} & \boldsymbol{w}_1^{\text{output}} -\end{pmatrix} -\begin{pmatrix} -1 & 1 & \dots & 1 \\ -\boldsymbol{x}_1^{\text{hidden}} & \boldsymbol{x}_2^{\text{hidden}} & \dots & \boldsymbol{x}_N^{\text{hidden}} -\end{pmatrix} -$$ - -In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\boldsymbol{z}_{1}^{\text{output}}$ the neural network has finished its feed forward step, and $\boldsymbol{z}_{1}^{\text{output}}$ is the final output of the network. - - -The next step is to decide how the parameters should be changed such that they minimize the cost function. - -The chosen cost function for this problem is - -$$ -C(\boldsymbol{x}, P) = \frac{1}{N} \sum_i \big(g_t'(x_i, P) - ( -\gamma g_t(x_i, P) \big)^2 -$$ - -In order to minimize the cost function, an optimization method must be chosen. - -Here, gradient descent with a constant step size has been chosen. - -### Gradient descent - -The idea of the gradient descent algorithm is to update parameters in -a direction where the cost function decreases goes to a minimum. - -In general, the update of some parameters $\boldsymbol{\omega}$ given a cost -function defined by some weights $\boldsymbol{\omega}$, $C(\boldsymbol{x}, -\boldsymbol{\omega})$, goes as follows: - -$$ -\boldsymbol{\omega}_{\text{new} } = \boldsymbol{\omega} - \lambda \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega}) -$$ - -for a number of iterations or until $ \big|\big| \boldsymbol{\omega}_{\text{new} } - \boldsymbol{\omega} \big|\big|$ becomes smaller than some given tolerance. - -The value of $\lambda$ decides how large steps the algorithm must take -in the direction of $ \nabla_{\boldsymbol{\omega}} C(\boldsymbol{x}, \boldsymbol{\omega})$. -The notation $\nabla_{\boldsymbol{\omega}}$ express the gradient with respect -to the elements in $\boldsymbol{\omega}$. - -In our case, we have to minimize the cost function $C(\boldsymbol{x}, P)$ with -respect to the two sets of weights and biases, that is for the hidden -layer $P_{\text{hidden} }$ and for the output layer $P_{\text{output} -}$ . - -This means that $P_{\text{hidden} }$ and $P_{\text{output} }$ is updated by - -$$ -\begin{aligned} -P_{\text{hidden},\text{new}} &= P_{\text{hidden}} - \lambda \nabla_{P_{\text{hidden}}} C(\boldsymbol{x}, P) \\ -P_{\text{output},\text{new}} &= P_{\text{output}} - \lambda \nabla_{P_{\text{output}}} C(\boldsymbol{x}, P) -\end{aligned} -$$ - -### The code for solving the ODE - -%matplotlib inline - -import autograd.numpy as np -from autograd import grad, elementwise_grad -import autograd.numpy.random as npr -from matplotlib import pyplot as plt - -def sigmoid(z): - return 1/(1 + np.exp(-z)) - -# Assuming one input, hidden, and output layer -def neural_network(params, x): - - # Find the weights (including and biases) for the hidden and output layer. - # Assume that params is a list of parameters for each layer. - # The biases are the first element for each array in params, - # and the weights are the remaning elements in each array in params. - - w_hidden = params[0] - w_output = params[1] - - # Assumes input x being an one-dimensional array - num_values = np.size(x) - x = x.reshape(-1, num_values) - - # Assume that the input layer does nothing to the input x - x_input = x - - ## Hidden layer: - - # Add a row of ones to include bias - x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0) - - z_hidden = np.matmul(w_hidden, x_input) - x_hidden = sigmoid(z_hidden) - - ## Output layer: - - # Include bias: - x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0) - - z_output = np.matmul(w_output, x_hidden) - x_output = z_output - - return x_output - -# The trial solution using the deep neural network: -def g_trial(x,params, g0 = 10): - return g0 + x*neural_network(params,x) - -# The right side of the ODE: -def g(x, g_trial, gamma = 2): - return -gamma*g_trial - -# The cost function: -def cost_function(P, x): - - # Evaluate the trial function with the current parameters P - g_t = g_trial(x,P) - - # Find the derivative w.r.t x of the neural network - d_net_out = elementwise_grad(neural_network,1)(P,x) - - # Find the derivative w.r.t x of the trial function - d_g_t = elementwise_grad(g_trial,0)(x,P) - - # The right side of the ODE - func = g(x, g_t) - - err_sqr = (d_g_t - func)**2 - cost_sum = np.sum(err_sqr) - - return cost_sum / np.size(err_sqr) - -# Solve the exponential decay ODE using neural network with one input, hidden, and output layer -def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb): - ## Set up initial weights and biases - - # For the hidden layer - p0 = npr.randn(num_neurons_hidden, 2 ) - - # For the output layer - p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included - - P = [p0, p1] - - print('Initial cost: %g'%cost_function(P, x)) - - ## Start finding the optimal weights using gradient descent - - # Find the Python function that represents the gradient of the cost function - # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer - cost_function_grad = grad(cost_function,0) - - # Let the update be done num_iter times - for i in range(num_iter): - # Evaluate the gradient at the current weights and biases in P. - # The cost_grad consist now of two arrays; - # one for the gradient w.r.t P_hidden and - # one for the gradient w.r.t P_output - cost_grad = cost_function_grad(P, x) - - P[0] = P[0] - lmb * cost_grad[0] - P[1] = P[1] - lmb * cost_grad[1] - - print('Final cost: %g'%cost_function(P, x)) - - return P - -def g_analytic(x, gamma = 2, g0 = 10): - return g0*np.exp(-gamma*x) - -# Solve the given problem -if __name__ == '__main__': - # Set seed such that the weight are initialized - # with same weights and biases for every run. - npr.seed(15) - - ## Decide the vales of arguments to the function to solve - N = 10 - x = np.linspace(0, 1, N) - - ## Set up the initial parameters - num_hidden_neurons = 10 - num_iter = 10000 - lmb = 0.001 - - # Use the network - P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb) - - # Print the deviation from the trial solution and true solution - res = g_trial(x,P) - res_analytical = g_analytic(x) - - print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical))) - - # Plot the results - plt.figure(figsize=(10,10)) - - plt.title('Performance of neural network solving an ODE compared to the analytical solution') - plt.plot(x, res_analytical) - plt.plot(x, res[0,:]) - plt.legend(['analytical','nn']) - plt.xlabel('x') - plt.ylabel('g(x)') - plt.show() - -## The network with one input layer, specified number of hidden layers, and one output layer - -It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers. - -The number of neurons within each hidden layer are given as a list of integers in the program below. - -import autograd.numpy as np -from autograd import grad, elementwise_grad -import autograd.numpy.random as npr -from matplotlib import pyplot as plt - -def sigmoid(z): - return 1/(1 + np.exp(-z)) - -# The neural network with one input layer and one output layer, -# but with number of hidden layers specified by the user. -def deep_neural_network(deep_params, x): - # N_hidden is the number of hidden layers - - N_hidden = np.size(deep_params) - 1 # -1 since params consists of - # parameters to all the hidden - # layers AND the output layer. - - # Assumes input x being an one-dimensional array - num_values = np.size(x) - x = x.reshape(-1, num_values) - - # Assume that the input layer does nothing to the input x - x_input = x - - # Due to multiple hidden layers, define a variable referencing to the - # output of the previous layer: - x_prev = x_input - - ## Hidden layers: - - for l in range(N_hidden): - # From the list of parameters P; find the correct weigths and bias for this layer - w_hidden = deep_params[l] - - # Add a row of ones to include bias - x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0) - - z_hidden = np.matmul(w_hidden, x_prev) - x_hidden = sigmoid(z_hidden) - - # Update x_prev such that next layer can use the output from this layer - x_prev = x_hidden - - ## Output layer: - - # Get the weights and bias for this layer - w_output = deep_params[-1] - - # Include bias: - x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0) - - z_output = np.matmul(w_output, x_prev) - x_output = z_output - - return x_output - -# The trial solution using the deep neural network: -def g_trial_deep(x,params, g0 = 10): - return g0 + x*deep_neural_network(params, x) - -# The right side of the ODE: -def g(x, g_trial, gamma = 2): - return -gamma*g_trial - -# The same cost function as before, but calls deep_neural_network instead. -def cost_function_deep(P, x): - - # Evaluate the trial function with the current parameters P - g_t = g_trial_deep(x,P) - - # Find the derivative w.r.t x of the neural network - d_net_out = elementwise_grad(deep_neural_network,1)(P,x) - - # Find the derivative w.r.t x of the trial function - d_g_t = elementwise_grad(g_trial_deep,0)(x,P) - - # The right side of the ODE - func = g(x, g_t) - - err_sqr = (d_g_t - func)**2 - cost_sum = np.sum(err_sqr) - - return cost_sum / np.size(err_sqr) - -# Solve the exponential decay ODE using neural network with one input and one output layer, -# but with specified number of hidden layers from the user. -def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb): - # num_hidden_neurons is now a list of number of neurons within each hidden layer - - # The number of elements in the list num_hidden_neurons thus represents - # the number of hidden layers. - - # Find the number of hidden layers: - N_hidden = np.size(num_neurons) - - ## Set up initial weights and biases - - # Initialize the list of parameters: - P = [None]*(N_hidden + 1) # + 1 to include the output layer - - P[0] = npr.randn(num_neurons[0], 2 ) - for l in range(1,N_hidden): - P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias - - # For the output layer - P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included - - print('Initial cost: %g'%cost_function_deep(P, x)) - - ## Start finding the optimal weights using gradient descent - - # Find the Python function that represents the gradient of the cost function - # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer - cost_function_deep_grad = grad(cost_function_deep,0) - - # Let the update be done num_iter times - for i in range(num_iter): - # Evaluate the gradient at the current weights and biases in P. - # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases - # in the hidden layers and output layers evaluated at x. - cost_deep_grad = cost_function_deep_grad(P, x) - - for l in range(N_hidden+1): - P[l] = P[l] - lmb * cost_deep_grad[l] - - print('Final cost: %g'%cost_function_deep(P, x)) - - return P - -def g_analytic(x, gamma = 2, g0 = 10): - return g0*np.exp(-gamma*x) - -# Solve the given problem -if __name__ == '__main__': - npr.seed(15) - - ## Decide the vales of arguments to the function to solve - N = 10 - x = np.linspace(0, 1, N) - - ## Set up the initial parameters - num_hidden_neurons = np.array([10,10]) - num_iter = 10000 - lmb = 0.001 - - P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb) - - res = g_trial_deep(x,P) - res_analytical = g_analytic(x) - - plt.figure(figsize=(10,10)) - - plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution') - plt.plot(x, res_analytical) - plt.plot(x, res[0,:]) - plt.legend(['analytical','dnn']) - plt.ylabel('g(x)') - plt.show() - -### Example: Population growth - -A logistic model of population growth assumes that a population converges toward an equilibrium. -The population growth can be modeled by - - -
- -$$ -\begin{equation} \label{log} \tag{10} - g'(t) = \alpha g(t)(A - g(t)) -\end{equation} -$$ - -where $g(t)$ is the population density at time $t$, $\alpha > 0$ the growth rate and $A > 0$ is the maximum population number in the environment. -Also, at $t = 0$ the population has the size $g(0) = g_0$, where $g_0$ is some chosen constant. - -In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability -and high execution time (this might be more apparent in the examples solving PDEs), -using a library like TensorFlow is recommended. -Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method. - - - -Here, we will model a population $g(t)$ in an environment having carrying capacity $A$. -The population follows the model - - -
- -$$ -\begin{equation} \label{solveode_population} \tag{11} -g'(t) = \alpha g(t)(A - g(t)) -\end{equation} -$$ - -where $g(0) = g_0$. - -In this example, we let $\alpha = 2$, $A = 1$, and $g_0 = 1.2$. - - -We will get a slightly different trial solution, as the boundary conditions are different -compared to the case for exponential decay. - -A possible trial solution satisfying the condition $g(0) = g_0$ could be - -$$ -h_1(t) = g_0 + t \cdot N(t,P) -$$ - -with $N(t,P)$ being the output from the neural network with weights and biases for each layer collected in the set $P$. - -The analytical solution is - -$$ -g(t) = \frac{Ag_0}{g_0 + (A - g_0)\exp(-\alpha A t)} -$$ - - - -The network will be the similar as for the exponential decay example, but with some small modifications for our problem. - -import autograd.numpy as np -from autograd import grad, elementwise_grad -import autograd.numpy.random as npr -from matplotlib import pyplot as plt - -def sigmoid(z): - return 1/(1 + np.exp(-z)) - -# Function to get the parameters. -# Done such that one can easily change the paramaters after one's liking. -def get_parameters(): - alpha = 2 - A = 1 - g0 = 1.2 - return alpha, A, g0 - -def deep_neural_network(P, x): - # N_hidden is the number of hidden layers - N_hidden = np.size(P) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer - - # Assumes input x being an one-dimensional array - num_values = np.size(x) - x = x.reshape(-1, num_values) - - # Assume that the input layer does nothing to the input x - x_input = x - - # Due to multiple hidden layers, define a variable referencing to the - # output of the previous layer: - x_prev = x_input - - ## Hidden layers: - - for l in range(N_hidden): - # From the list of parameters P; find the correct weigths and bias for this layer - w_hidden = P[l] - - # Add a row of ones to include bias - x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0) - - z_hidden = np.matmul(w_hidden, x_prev) - x_hidden = sigmoid(z_hidden) - - # Update x_prev such that next layer can use the output from this layer - x_prev = x_hidden - - ## Output layer: - - # Get the weights and bias for this layer - w_output = P[-1] - - # Include bias: - x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0) - - z_output = np.matmul(w_output, x_prev) - x_output = z_output - - return x_output - - -def cost_function_deep(P, x): - - # Evaluate the trial function with the current parameters P - g_t = g_trial_deep(x,P) - - # Find the derivative w.r.t x of the trial function - d_g_t = elementwise_grad(g_trial_deep,0)(x,P) - - # The right side of the ODE - func = f(x, g_t) - - err_sqr = (d_g_t - func)**2 - cost_sum = np.sum(err_sqr) - - return cost_sum / np.size(err_sqr) - -# The right side of the ODE: -def f(x, g_trial): - alpha,A, g0 = get_parameters() - return alpha*g_trial*(A - g_trial) - -# The trial solution using the deep neural network: -def g_trial_deep(x, params): - alpha,A, g0 = get_parameters() - return g0 + x*deep_neural_network(params,x) - -# The analytical solution: -def g_analytic(t): - alpha,A, g0 = get_parameters() - return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t)) - -def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb): - # num_hidden_neurons is now a list of number of neurons within each hidden layer - - # Find the number of hidden layers: - N_hidden = np.size(num_neurons) - - ## Set up initial weigths and biases - - # Initialize the list of parameters: - P = [None]*(N_hidden + 1) # + 1 to include the output layer - - P[0] = npr.randn(num_neurons[0], 2 ) - for l in range(1,N_hidden): - P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias - - # For the output layer - P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included - - print('Initial cost: %g'%cost_function_deep(P, x)) - - ## Start finding the optimal weigths using gradient descent - - # Find the Python function that represents the gradient of the cost function - # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer - cost_function_deep_grad = grad(cost_function_deep,0) - - # Let the update be done num_iter times - for i in range(num_iter): - # Evaluate the gradient at the current weights and biases in P. - # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases - # in the hidden layers and output layers evaluated at x. - cost_deep_grad = cost_function_deep_grad(P, x) - - for l in range(N_hidden+1): - P[l] = P[l] - lmb * cost_deep_grad[l] - - print('Final cost: %g'%cost_function_deep(P, x)) - - return P - -if __name__ == '__main__': - npr.seed(4155) - - ## Decide the vales of arguments to the function to solve - Nt = 10 - T = 1 - t = np.linspace(0,T, Nt) - - ## Set up the initial parameters - num_hidden_neurons = [100, 50, 25] - num_iter = 1000 - lmb = 1e-3 - - P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb) - - g_dnn_ag = g_trial_deep(t,P) - g_analytical = g_analytic(t) - - # Find the maximum absolute difference between the solutons: - diff_ag = np.max(np.abs(g_dnn_ag - g_analytical)) - print("The max absolute difference between the solutions is: %g"%diff_ag) - - plt.figure(figsize=(10,10)) - - plt.title('Performance of neural network solving an ODE compared to the analytical solution') - plt.plot(t, g_analytical) - plt.plot(t, g_dnn_ag[0,:]) - plt.legend(['analytical','nn']) - plt.xlabel('t') - plt.ylabel('g(t)') - - plt.show() - -## Using forward Euler to solve the ODE - -A straightforward way of solving an ODE numerically, is to use Euler's method. - -Euler's method uses Taylor series to approximate the value at a function $f$ at a step $\Delta x$ from $x$: - -$$ -f(x + \Delta x) \approx f(x) + \Delta x f'(x) -$$ - -In our case, using Euler's method to approximate the value of $g$ at a step $\Delta t$ from $t$ yields - -$$ -\begin{aligned} - g(t + \Delta t) &\approx g(t) + \Delta t g'(t) \\ - &= g(t) + \Delta t \big(\alpha g(t)(A - g(t))\big) -\end{aligned} -$$ - -along with the condition that $g(0) = g_0$. - -Let $t_i = i \cdot \Delta t$ where $\Delta t = \frac{T}{N_t-1}$ where $T$ is the final time our solver must solve for and $N_t$ the number of values for $t \in [0, T]$ for $i = 0, \dots, N_t-1$. - -For $i \geq 1$, we have that - -$$ -\begin{aligned} -t_i &= i\Delta t \\ -&= (i - 1)\Delta t + \Delta t \\ -&= t_{i-1} + \Delta t -\end{aligned} -$$ - -Now, if $g_i = g(t_i)$ then - - -
- -$$ -\begin{equation} - \begin{aligned} - g_i &= g(t_i) \\ - &= g(t_{i-1} + \Delta t) \\ - &\approx g(t_{i-1}) + \Delta t \big(\alpha g(t_{i-1})(A - g(t_{i-1}))\big) \\ - &= g_{i-1} + \Delta t \big(\alpha g_{i-1}(A - g_{i-1})\big) - \end{aligned} -\end{equation} \label{odenum} \tag{12} -$$ - -for $i \geq 1$ and $g_0 = g(t_0) = g(0) = g_0$. - -Equation ([12](#odenum)) could be implemented in the following way, -extending the program that uses the network using Autograd: - -# Assume that all function definitions from the example program using Autograd -# are located here. - -if __name__ == '__main__': - npr.seed(4155) - - ## Decide the vales of arguments to the function to solve - Nt = 10 - T = 1 - t = np.linspace(0,T, Nt) - - ## Set up the initial parameters - num_hidden_neurons = [100,50,25] - num_iter = 1000 - lmb = 1e-3 - - P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb) - - g_dnn_ag = g_trial_deep(t,P) - g_analytical = g_analytic(t) - - # Find the maximum absolute difference between the solutons: - diff_ag = np.max(np.abs(g_dnn_ag - g_analytical)) - print("The max absolute difference between the solutions is: %g"%diff_ag) - - plt.figure(figsize=(10,10)) - - plt.title('Performance of neural network solving an ODE compared to the analytical solution') - plt.plot(t, g_analytical) - plt.plot(t, g_dnn_ag[0,:]) - plt.legend(['analytical','nn']) - plt.xlabel('t') - plt.ylabel('g(t)') - - ## Find an approximation to the funtion using forward Euler - - alpha, A, g0 = get_parameters() - dt = T/(Nt - 1) - - # Perform forward Euler to solve the ODE - g_euler = np.zeros(Nt) - g_euler[0] = g0 - - for i in range(1,Nt): - g_euler[i] = g_euler[i-1] + dt*(alpha*g_euler[i-1]*(A - g_euler[i-1])) - - # Print the errors done by each method - diff1 = np.max(np.abs(g_euler - g_analytical)) - diff2 = np.max(np.abs(g_dnn_ag[0,:] - g_analytical)) - - print('Max absolute difference between Euler method and analytical: %g'%diff1) - print('Max absolute difference between deep neural network and analytical: %g'%diff2) - - # Plot results - plt.figure(figsize=(10,10)) - - plt.plot(t,g_euler) - plt.plot(t,g_analytical) - plt.plot(t,g_dnn_ag[0,:]) - - plt.legend(['euler','analytical','dnn']) - plt.xlabel('Time t') - plt.ylabel('g(t)') - - plt.show() - -## Solving the one dimensional Poisson equation - -The Poisson equation for $g(x)$ in one dimension is - - -
- -$$ -\begin{equation} \label{poisson} \tag{13} - -g''(x) = f(x) -\end{equation} -$$ - -where $f(x)$ is a given function for $x \in (0,1)$. - -The conditions that $g(x)$ is chosen to fulfill, are - -$$ -\begin{align*} - g(0) &= 0 \\ - g(1) &= 0 -\end{align*} -$$ - -This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used. -The results from the networks can then be compared to the analytical solution. -In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks. - - -Here, the function $g(x)$ to solve for follows the equation - -$$ --g''(x) = f(x),\qquad x \in (0,1) -$$ - -where $f(x)$ is a given function, along with the chosen conditions - - -
- -$$ -\begin{aligned} -g(0) = g(1) = 0 -\end{aligned}\label{cond} \tag{14} -$$ - -In this example, we consider the case when $f(x) = (3x + x^2)\exp(x)$. - -For this case, a possible trial solution satisfying the conditions could be - -$$ -g_t(x) = x \cdot (1-x) \cdot N(P,x) -$$ - -The analytical solution for this problem is - -$$ -g(x) = x(1 - x)\exp(x) -$$ - -import autograd.numpy as np -from autograd import grad, elementwise_grad -import autograd.numpy.random as npr -from matplotlib import pyplot as plt - -def sigmoid(z): - return 1/(1 + np.exp(-z)) - -def deep_neural_network(deep_params, x): - # N_hidden is the number of hidden layers - N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer - - # Assumes input x being an one-dimensional array - num_values = np.size(x) - x = x.reshape(-1, num_values) - - # Assume that the input layer does nothing to the input x - x_input = x - - # Due to multiple hidden layers, define a variable referencing to the - # output of the previous layer: - x_prev = x_input - - ## Hidden layers: - - for l in range(N_hidden): - # From the list of parameters P; find the correct weigths and bias for this layer - w_hidden = deep_params[l] - - # Add a row of ones to include bias - x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0) - - z_hidden = np.matmul(w_hidden, x_prev) - x_hidden = sigmoid(z_hidden) - - # Update x_prev such that next layer can use the output from this layer - x_prev = x_hidden - - ## Output layer: - - # Get the weights and bias for this layer - w_output = deep_params[-1] - - # Include bias: - x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0) - - z_output = np.matmul(w_output, x_prev) - x_output = z_output - - return x_output - -def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb): - # num_hidden_neurons is now a list of number of neurons within each hidden layer - - # Find the number of hidden layers: - N_hidden = np.size(num_neurons) - - ## Set up initial weigths and biases - - # Initialize the list of parameters: - P = [None]*(N_hidden + 1) # + 1 to include the output layer - - P[0] = npr.randn(num_neurons[0], 2 ) - for l in range(1,N_hidden): - P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias - - # For the output layer - P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included - - print('Initial cost: %g'%cost_function_deep(P, x)) - - ## Start finding the optimal weigths using gradient descent - - # Find the Python function that represents the gradient of the cost function - # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer - cost_function_deep_grad = grad(cost_function_deep,0) - - # Let the update be done num_iter times - for i in range(num_iter): - # Evaluate the gradient at the current weights and biases in P. - # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases - # in the hidden layers and output layers evaluated at x. - cost_deep_grad = cost_function_deep_grad(P, x) - - for l in range(N_hidden+1): - P[l] = P[l] - lmb * cost_deep_grad[l] - - print('Final cost: %g'%cost_function_deep(P, x)) - - return P - -## Set up the cost function specified for this Poisson equation: - -# The right side of the ODE -def f(x): - return (3*x + x**2)*np.exp(x) - -def cost_function_deep(P, x): - - # Evaluate the trial function with the current parameters P - g_t = g_trial_deep(x,P) - - # Find the derivative w.r.t x of the trial function - d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P) - - right_side = f(x) - - err_sqr = (-d2_g_t - right_side)**2 - cost_sum = np.sum(err_sqr) - - return cost_sum/np.size(err_sqr) - -# The trial solution: -def g_trial_deep(x,P): - return x*(1-x)*deep_neural_network(P,x) - -# The analytic solution; -def g_analytic(x): - return x*(1-x)*np.exp(x) - -if __name__ == '__main__': - npr.seed(4155) - - ## Decide the vales of arguments to the function to solve - Nx = 10 - x = np.linspace(0,1, Nx) - - ## Set up the initial parameters - num_hidden_neurons = [200,100] - num_iter = 1000 - lmb = 1e-3 - - P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb) - - g_dnn_ag = g_trial_deep(x,P) - g_analytical = g_analytic(x) - - # Find the maximum absolute difference between the solutons: - max_diff = np.max(np.abs(g_dnn_ag - g_analytical)) - print("The max absolute difference between the solutions is: %g"%max_diff) - - plt.figure(figsize=(10,10)) - - plt.title('Performance of neural network solving an ODE compared to the analytical solution') - plt.plot(x, g_analytical) - plt.plot(x, g_dnn_ag[0,:]) - plt.legend(['analytical','nn']) - plt.xlabel('x') - plt.ylabel('g(x)') - plt.show() - -### Comparing with a numerical scheme - -The Poisson equation is possible to solve using Taylor series to approximate the second derivative. - -Using Taylor series, the second derivative can be expressed as - -$$ -g''(x) = \frac{g(x + \Delta x) - 2g(x) + g(x-\Delta x)}{\Delta x^2} + E_{\Delta x}(x) -$$ - -where $\Delta x$ is a small step size and $E_{\Delta x}(x)$ being the error term. - -Looking away from the error terms gives an approximation to the second derivative: - - -
- -$$ -\begin{equation} \label{approx} \tag{15} -g''(x) \approx \frac{g(x + \Delta x) - 2g(x) + g(x-\Delta x)}{\Delta x^2} -\end{equation} -$$ - -If $x_i = i \Delta x = x_{i-1} + \Delta x$ and $g_i = g(x_i)$ for $i = 1,\dots N_x - 2$ with $N_x$ being the number of values for $x$, ([15](#approx)) becomes - -$$ -\begin{aligned} -g''(x_i) &\approx \frac{g(x_i + \Delta x) - 2g(x_i) + g(x_i -\Delta x)}{\Delta x^2} \\ -&= \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2} -\end{aligned} -$$ - -Since we know from our problem that - -$$ -\begin{aligned} --g''(x) &= f(x) \\ -&= (3x + x^2)\exp(x) -\end{aligned} -$$ - -along with the conditions $g(0) = g(1) = 0$, -the following scheme can be used to find an approximate solution for $g(x)$ numerically: - - -
- -$$ -\begin{equation} - \begin{aligned} - -\Big( \frac{g_{i+1} - 2g_i + g_{i-1}}{\Delta x^2} \Big) &= f(x_i) \\ - -g_{i+1} + 2g_i - g_{i-1} &= \Delta x^2 f(x_i) - \end{aligned} -\end{equation} \label{odesys} \tag{16} -$$ - -for $i = 1, \dots, N_x - 2$ where $g_0 = g_{N_x - 1} = 0$ and $f(x_i) = (3x_i + x_i^2)\exp(x_i)$, which is given for our specific problem. - -The equation can be rewritten into a matrix equation: - -$$ -\begin{aligned} -\begin{pmatrix} -2 & -1 & 0 & \dots & 0 \\ --1 & 2 & -1 & \dots & 0 \\ -\vdots & & \ddots & & \vdots \\ -0 & \dots & -1 & 2 & -1 \\ -0 & \dots & 0 & -1 & 2\\ -\end{pmatrix} -\begin{pmatrix} -g_1 \\ -g_2 \\ -\vdots \\ -g_{N_x - 3} \\ -g_{N_x - 2} -\end{pmatrix} -&= -\Delta x^2 -\begin{pmatrix} -f(x_1) \\ -f(x_2) \\ -\vdots \\ -f(x_{N_x - 3}) \\ -f(x_{N_x - 2}) -\end{pmatrix} \\ -\boldsymbol{A}\boldsymbol{g} &= \boldsymbol{f}, -\end{aligned} -$$ - -which makes it possible to solve for the vector $\boldsymbol{g}$. - - -We can then compare the result from this numerical scheme with the output from our network using Autograd: - -import autograd.numpy as np -from autograd import grad, elementwise_grad -import autograd.numpy.random as npr -from matplotlib import pyplot as plt - -def sigmoid(z): - return 1/(1 + np.exp(-z)) - -def deep_neural_network(deep_params, x): - # N_hidden is the number of hidden layers - N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer - - # Assumes input x being an one-dimensional array - num_values = np.size(x) - x = x.reshape(-1, num_values) - - # Assume that the input layer does nothing to the input x - x_input = x - - # Due to multiple hidden layers, define a variable referencing to the - # output of the previous layer: - x_prev = x_input - - ## Hidden layers: - - for l in range(N_hidden): - # From the list of parameters P; find the correct weigths and bias for this layer - w_hidden = deep_params[l] - - # Add a row of ones to include bias - x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0) - - z_hidden = np.matmul(w_hidden, x_prev) - x_hidden = sigmoid(z_hidden) - - # Update x_prev such that next layer can use the output from this layer - x_prev = x_hidden - - ## Output layer: - - # Get the weights and bias for this layer - w_output = deep_params[-1] - - # Include bias: - x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0) - - z_output = np.matmul(w_output, x_prev) - x_output = z_output - - return x_output - -def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb): - # num_hidden_neurons is now a list of number of neurons within each hidden layer - - # Find the number of hidden layers: - N_hidden = np.size(num_neurons) - - ## Set up initial weigths and biases - - # Initialize the list of parameters: - P = [None]*(N_hidden + 1) # + 1 to include the output layer - - P[0] = npr.randn(num_neurons[0], 2 ) - for l in range(1,N_hidden): - P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias - - # For the output layer - P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included - - print('Initial cost: %g'%cost_function_deep(P, x)) - - ## Start finding the optimal weigths using gradient descent - - # Find the Python function that represents the gradient of the cost function - # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer - cost_function_deep_grad = grad(cost_function_deep,0) - - # Let the update be done num_iter times - for i in range(num_iter): - # Evaluate the gradient at the current weights and biases in P. - # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases - # in the hidden layers and output layers evaluated at x. - cost_deep_grad = cost_function_deep_grad(P, x) - - for l in range(N_hidden+1): - P[l] = P[l] - lmb * cost_deep_grad[l] - - print('Final cost: %g'%cost_function_deep(P, x)) - - return P - -## Set up the cost function specified for this Poisson equation: - -# The right side of the ODE -def f(x): - return (3*x + x**2)*np.exp(x) - -def cost_function_deep(P, x): - - # Evaluate the trial function with the current parameters P - g_t = g_trial_deep(x,P) - - # Find the derivative w.r.t x of the trial function - d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P) - - right_side = f(x) - - err_sqr = (-d2_g_t - right_side)**2 - cost_sum = np.sum(err_sqr) - - return cost_sum/np.size(err_sqr) - -# The trial solution: -def g_trial_deep(x,P): - return x*(1-x)*deep_neural_network(P,x) - -# The analytic solution; -def g_analytic(x): - return x*(1-x)*np.exp(x) - -if __name__ == '__main__': - npr.seed(4155) - - ## Decide the vales of arguments to the function to solve - Nx = 10 - x = np.linspace(0,1, Nx) - - ## Set up the initial parameters - num_hidden_neurons = [200,100] - num_iter = 1000 - lmb = 1e-3 - - P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb) - - g_dnn_ag = g_trial_deep(x,P) - g_analytical = g_analytic(x) - - # Find the maximum absolute difference between the solutons: - - plt.figure(figsize=(10,10)) - - plt.title('Performance of neural network solving an ODE compared to the analytical solution') - plt.plot(x, g_analytical) - plt.plot(x, g_dnn_ag[0,:]) - plt.legend(['analytical','nn']) - plt.xlabel('x') - plt.ylabel('g(x)') - - ## Perform the computation using the numerical scheme - - dx = 1/(Nx - 1) - - # Set up the matrix A - A = np.zeros((Nx-2,Nx-2)) - - A[0,0] = 2 - A[0,1] = -1 - - for i in range(1,Nx-3): - A[i,i-1] = -1 - A[i,i] = 2 - A[i,i+1] = -1 - - A[Nx - 3, Nx - 4] = -1 - A[Nx - 3, Nx - 3] = 2 - - # Set up the vector f - f_vec = dx**2 * f(x[1:-1]) - - # Solve the equation - g_res = np.linalg.solve(A,f_vec) - - g_vec = np.zeros(Nx) - g_vec[1:-1] = g_res - - # Print the differences between each method - max_diff1 = np.max(np.abs(g_dnn_ag - g_analytical)) - max_diff2 = np.max(np.abs(g_vec - g_analytical)) - print("The max absolute difference between the analytical solution and DNN Autograd: %g"%max_diff1) - print("The max absolute difference between the analytical solution and numerical scheme: %g"%max_diff2) - - # Plot the results - plt.figure(figsize=(10,10)) - - plt.plot(x,g_vec) - plt.plot(x,g_analytical) - plt.plot(x,g_dnn_ag[0,:]) - - plt.legend(['numerical scheme','analytical','dnn']) - plt.show() - -## Partial Differential Equations - -A partial differential equation (PDE) has a solution here the function -is defined by multiple variables. The equation may involve all kinds -of combinations of which variables the function is differentiated with -respect to. - -In general, a partial differential equation for a function $g(x_1,\dots,x_N)$ with $N$ variables may be expressed as - - -
- -$$ -\begin{equation} \label{PDE} \tag{17} - f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) = 0 -\end{equation} -$$ - -where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given. - -### Type of problem - -The problem our network must solve for, is similar to the ODE case. -We must have a trial solution $g_t$ at hand. - -For instance, the trial solution could be expressed as - -$$ -\begin{align*} - g_t(x_1,\dots,x_N) = h_1(x_1,\dots,x_N) + h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P)) -\end{align*} -$$ - -where $h_1(x_1,\dots,x_N)$ is a function that ensures $g_t(x_1,\dots,x_N)$ satisfies some given conditions. -The neural network $N(x_1,\dots,x_N,P)$ has weights and biases described by $P$ and $h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))$ is an expression using the output from the neural network in some way. - -The role of the function $h_2(x_1,\dots,x_N,N(x_1,\dots,x_N,P))$, is to ensure that the output of $N(x_1,\dots,x_N,P)$ is zero when $g_t(x_1,\dots,x_N)$ is evaluated at the values of $x_1,\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\dots,x_N)$ should alone make $g_t(x_1,\dots,x_N)$ satisfy the conditions. - - - -### Network requirements - -The network tries then the minimize the cost function following the -same ideas as described for the ODE case, but now with more than one -variables to consider. The concept still remains the same; find a set -of parameters $P$ such that the expression $f$ in ([17](#PDE)) is as -close to zero as possible. - -As for the ODE case, the cost function is the mean squared error that -the network must try to minimize. The cost function for the network to -minimize is - -$$ -C\left(x_1, \dots, x_N, P\right) = \left( f\left(x_1, \, \dots \, , x_N, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1}, \dots , \frac{\partial g(x_1,\dots,x_N) }{\partial x_N}, \frac{\partial g(x_1,\dots,x_N) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(x_1,\dots,x_N) }{\partial x_N^n} \right) \right)^2 -$$ - -If we let $\boldsymbol{x} = \big( x_1, \dots, x_N \big)$ be an array containing the values for $x_1, \dots, x_N$ respectively, the cost function can be reformulated into the following: - -$$ -C\left(\boldsymbol{x}, P\right) = f\left( \left( \boldsymbol{x}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}) }{\partial x_N^n} \right) \right)^2 -$$ - -If we also have $M$ different sets of values for $x_1, \dots, x_N$, that is $\boldsymbol{x}_i = \big(x_1^{(i)}, \dots, x_N^{(i)}\big)$ for $i = 1,\dots,M$ being the rows in matrix $X$, the cost function can be generalized into - -$$ -C\left(X, P \right) = \sum_{i=1}^M f\left( \left( \boldsymbol{x}_i, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1}, \dots , \frac{\partial g(\boldsymbol{x}_i) }{\partial x_N}, \frac{\partial g(\boldsymbol{x}_i) }{\partial x_1\partial x_2}, \, \dots \, , \frac{\partial^n g(\boldsymbol{x}_i) }{\partial x_N^n} \right) \right)^2. -$$ - -## Example: The diffusion equation - -In one spatial dimension, the equation reads - -$$ -\frac{\partial g(x,t)}{\partial t} = \frac{\partial^2 g(x,t)}{\partial x^2} -$$ - -where a possible choice of conditions are - -$$ -\begin{align*} -g(0,t) &= 0 ,\qquad t \geq 0 \\ -g(1,t) &= 0, \qquad t \geq 0 \\ -g(x,0) &= u(x),\qquad x\in [0,1] -\end{align*} -$$ - -with $u(x)$ being some given function. - - - -For this case, we want to find $g(x,t)$ such that - - -
- -$$ -\begin{equation} - \frac{\partial g(x,t)}{\partial t} = \frac{\partial^2 g(x,t)}{\partial x^2} -\end{equation} \label{diffonedim} \tag{18} -$$ - -and - -$$ -\begin{align*} -g(0,t) &= 0 ,\qquad t \geq 0 \\ -g(1,t) &= 0, \qquad t \geq 0 \\ -g(x,0) &= u(x),\qquad x\in [0,1] -\end{align*} -$$ - -with $u(x) = \sin(\pi x)$. - -First, let us set up the deep neural network. -The deep neural network will follow the same structure as discussed in the examples solving the ODEs. -First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions. - - - - -The only change to do here, is to extend our network such that -functions of multiple parameters are correctly handled. In this case -we have two variables in our function to solve for, that is time $t$ -and position $x$. The variables will be represented by a -one-dimensional array in the program. The program will evaluate the -network at each possible pair $(x,t)$, given an array for the desired -$x$-values and $t$-values to approximate the solution at. - -def sigmoid(z): - return 1/(1 + np.exp(-z)) - -def deep_neural_network(deep_params, x): - # x is now a point and a 1D numpy array; make it a column vector - num_coordinates = np.size(x,0) - x = x.reshape(num_coordinates,-1) - - num_points = np.size(x,1) - - # N_hidden is the number of hidden layers - N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer - - # Assume that the input layer does nothing to the input x - x_input = x - x_prev = x_input - - ## Hidden layers: - - for l in range(N_hidden): - # From the list of parameters P; find the correct weigths and bias for this layer - w_hidden = deep_params[l] - - # Add a row of ones to include bias - x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0) - - z_hidden = np.matmul(w_hidden, x_prev) - x_hidden = sigmoid(z_hidden) - - # Update x_prev such that next layer can use the output from this layer - x_prev = x_hidden - - ## Output layer: - - # Get the weights and bias for this layer - w_output = deep_params[-1] - - # Include bias: - x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0) - - z_output = np.matmul(w_output, x_prev) - x_output = z_output - - return x_output[0][0] - -The cost function must then iterate through the given arrays -containing values for $x$ and $t$, defines a point $(x,t)$ the deep -neural network and the trial solution is evaluated at, and then finds -the Jacobian of the trial solution. - -A possible trial solution for this PDE is - -$$ -g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P) -$$ - -with $A(x,t)$ being a function ensuring that $g_t(x,t)$ satisfies our given conditions, and $N(x,t,P)$ being the output from the deep neural network using weights and biases for each layer from $P$. - -To fulfill the conditions, $A(x,t)$ could be: - -$$ -h_1(x,t) = (1-t)\Big(u(x) - \big((1-x)u(0) + x u(1)\big)\Big) = (1-t)u(x) = (1-t)\sin(\pi x) -$$ -since $(0) = u(1) = 0$ and $u(x) = \sin(\pi x)$. - - - -The Jacobian is used because the program must find the derivative of -the trial solution with respect to $x$ and $t$. - -This gives the necessity of computing the Jacobian matrix, as we want -to evaluate the gradient with respect to $x$ and $t$ (note that the -Jacobian of a scalar-valued multivariate function is simply its -gradient). - -In Autograd, the differentiation is by default done with respect to -the first input argument of your Python function. Since the points is -an array representing $x$ and $t$, the Jacobian is calculated using -the values of $x$ and $t$. - -To find the second derivative with respect to $x$ and $t$, the -Jacobian can be found for the second time. The result is a Hessian -matrix, which is the matrix containing all the possible second order -mixed derivatives of $g(x,t)$. - -# Set up the trial function: -def u(x): - return np.sin(np.pi*x) - -def g_trial(point,P): - x,t = point - return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point) - -# The right side of the ODE: -def f(point): - return 0. - -# The cost function: -def cost_function(P, x, t): - cost_sum = 0 - - g_t_jacobian_func = jacobian(g_trial) - g_t_hessian_func = hessian(g_trial) - - for x_ in x: - for t_ in t: - point = np.array([x_,t_]) - - g_t = g_trial(point,P) - g_t_jacobian = g_t_jacobian_func(point,P) - g_t_hessian = g_t_hessian_func(point,P) - - g_t_dt = g_t_jacobian[1] - g_t_d2x = g_t_hessian[0][0] - - func = f(point) - - err_sqr = ( (g_t_dt - g_t_d2x) - func)**2 - cost_sum += err_sqr - - return cost_sum - -### Setting up the network using Autograd; The full program - -Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution. - -The analytical solution of our problem is - -$$ -g(x,t) = \exp(-\pi^2 t)\sin(\pi x) -$$ - -A possible way to implement a neural network solving the PDE, is given below. -Be aware, though, that it is fairly slow for the parameters used. -A better result is possible, but requires more iterations, and thus longer time to complete. - - -Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE. -Using TensorFlow results in a much better execution time. Try it! - -import autograd.numpy as np -from autograd import jacobian,hessian,grad -import autograd.numpy.random as npr -from matplotlib import cm -from matplotlib import pyplot as plt -from mpl_toolkits.mplot3d import axes3d - -## Set up the network - -def sigmoid(z): - return 1/(1 + np.exp(-z)) - -def deep_neural_network(deep_params, x): - # x is now a point and a 1D numpy array; make it a column vector - num_coordinates = np.size(x,0) - x = x.reshape(num_coordinates,-1) - - num_points = np.size(x,1) - - # N_hidden is the number of hidden layers - N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer - - # Assume that the input layer does nothing to the input x - x_input = x - x_prev = x_input - - ## Hidden layers: - - for l in range(N_hidden): - # From the list of parameters P; find the correct weigths and bias for this layer - w_hidden = deep_params[l] - - # Add a row of ones to include bias - x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0) - - z_hidden = np.matmul(w_hidden, x_prev) - x_hidden = sigmoid(z_hidden) - - # Update x_prev such that next layer can use the output from this layer - x_prev = x_hidden - - ## Output layer: - - # Get the weights and bias for this layer - w_output = deep_params[-1] - - # Include bias: - x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0) - - z_output = np.matmul(w_output, x_prev) - x_output = z_output - - return x_output[0][0] - -## Define the trial solution and cost function -def u(x): - return np.sin(np.pi*x) - -def g_trial(point,P): - x,t = point - return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point) - -# The right side of the ODE: -def f(point): - return 0. - -# The cost function: -def cost_function(P, x, t): - cost_sum = 0 - - g_t_jacobian_func = jacobian(g_trial) - g_t_hessian_func = hessian(g_trial) - - for x_ in x: - for t_ in t: - point = np.array([x_,t_]) - - g_t = g_trial(point,P) - g_t_jacobian = g_t_jacobian_func(point,P) - g_t_hessian = g_t_hessian_func(point,P) - - g_t_dt = g_t_jacobian[1] - g_t_d2x = g_t_hessian[0][0] - - func = f(point) - - err_sqr = ( (g_t_dt - g_t_d2x) - func)**2 - cost_sum += err_sqr - - return cost_sum /( np.size(x)*np.size(t) ) - -## For comparison, define the analytical solution -def g_analytic(point): - x,t = point - return np.exp(-np.pi**2*t)*np.sin(np.pi*x) - -## Set up a function for training the network to solve for the equation -def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb): - ## Set up initial weigths and biases - N_hidden = np.size(num_neurons) - - ## Set up initial weigths and biases - - # Initialize the list of parameters: - P = [None]*(N_hidden + 1) # + 1 to include the output layer - - P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias - for l in range(1,N_hidden): - P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias - - # For the output layer - P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included - - print('Initial cost: ',cost_function(P, x, t)) - - cost_function_grad = grad(cost_function,0) - - # Let the update be done num_iter times - for i in range(num_iter): - cost_grad = cost_function_grad(P, x , t) - - for l in range(N_hidden+1): - P[l] = P[l] - lmb * cost_grad[l] - - print('Final cost: ',cost_function(P, x, t)) - - return P - -if __name__ == '__main__': - ### Use the neural network: - npr.seed(15) - - ## Decide the vales of arguments to the function to solve - Nx = 10; Nt = 10 - x = np.linspace(0, 1, Nx) - t = np.linspace(0,1,Nt) - - ## Set up the parameters for the network - num_hidden_neurons = [100, 25] - num_iter = 250 - lmb = 0.01 - - P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb) - - ## Store the results - g_dnn_ag = np.zeros((Nx, Nt)) - G_analytical = np.zeros((Nx, Nt)) - for i,x_ in enumerate(x): - for j, t_ in enumerate(t): - point = np.array([x_, t_]) - g_dnn_ag[i,j] = g_trial(point,P) - - G_analytical[i,j] = g_analytic(point) - - # Find the map difference between the analytical and the computed solution - diff_ag = np.abs(g_dnn_ag - G_analytical) - print('Max absolute difference between the analytical solution and the network: %g'%np.max(diff_ag)) - - ## Plot the solutions in two dimensions, that being in position and time - - T,X = np.meshgrid(t,x) - - fig = plt.figure(figsize=(10,10)) - ax = fig.gca(projection='3d') - ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons)) - s = ax.plot_surface(T,X,g_dnn_ag,linewidth=0,antialiased=False,cmap=cm.viridis) - ax.set_xlabel('Time $t$') - ax.set_ylabel('Position $x$'); - - - fig = plt.figure(figsize=(10,10)) - ax = fig.gca(projection='3d') - ax.set_title('Analytical solution') - s = ax.plot_surface(T,X,G_analytical,linewidth=0,antialiased=False,cmap=cm.viridis) - ax.set_xlabel('Time $t$') - ax.set_ylabel('Position $x$'); - - fig = plt.figure(figsize=(10,10)) - ax = fig.gca(projection='3d') - ax.set_title('Difference') - s = ax.plot_surface(T,X,diff_ag,linewidth=0,antialiased=False,cmap=cm.viridis) - ax.set_xlabel('Time $t$') - ax.set_ylabel('Position $x$'); - - ## Take some slices of the 3D plots just to see the solutions at particular times - indx1 = 0 - indx2 = int(Nt/2) - indx3 = Nt-1 - - t1 = t[indx1] - t2 = t[indx2] - t3 = t[indx3] - - # Slice the results from the DNN - res1 = g_dnn_ag[:,indx1] - res2 = g_dnn_ag[:,indx2] - res3 = g_dnn_ag[:,indx3] - - # Slice the analytical results - res_analytical1 = G_analytical[:,indx1] - res_analytical2 = G_analytical[:,indx2] - res_analytical3 = G_analytical[:,indx3] - - # Plot the slices - plt.figure(figsize=(10,10)) - plt.title("Computed solutions at time = %g"%t1) - plt.plot(x, res1) - plt.plot(x,res_analytical1) - plt.legend(['dnn','analytical']) - - plt.figure(figsize=(10,10)) - plt.title("Computed solutions at time = %g"%t2) - plt.plot(x, res2) - plt.plot(x,res_analytical2) - plt.legend(['dnn','analytical']) - - plt.figure(figsize=(10,10)) - plt.title("Computed solutions at time = %g"%t3) - plt.plot(x, res3) - plt.plot(x,res_analytical3) - plt.legend(['dnn','analytical']) - - plt.show() - -## Solving the wave equation with Neural Networks - -The wave equation is - -$$ -\frac{\partial^2 g(x,t)}{\partial t^2} = c^2\frac{\partial^2 g(x,t)}{\partial x^2} -$$ - -with $c$ being the specified wave speed. - -Here, the chosen conditions are - -$$ -\begin{align*} - g(0,t) &= 0 \\ - g(1,t) &= 0 \\ - g(x,0) &= u(x) \\ - \frac{\partial g(x,t)}{\partial t} \Big |_{t = 0} &= v(x) -\end{align*} -$$ - -where $\frac{\partial g(x,t)}{\partial t} \Big |_{t = 0}$ means the derivative of $g(x,t)$ with respect to $t$ is evaluated at $t = 0$, and $u(x)$ and $v(x)$ being given functions. - - -The wave equation to solve for, is - - -
- -$$ -\begin{equation} \label{wave} \tag{19} -\frac{\partial^2 g(x,t)}{\partial t^2} = c^2 \frac{\partial^2 g(x,t)}{\partial x^2} -\end{equation} -$$ - -where $c$ is the given wave speed. -The chosen conditions for this equation are - - -
- -$$ -\begin{aligned} -g(0,t) &= 0, &t \geq 0 \\ -g(1,t) &= 0, &t \geq 0 \\ -g(x,0) &= u(x), &x\in[0,1] \\ -\frac{\partial g(x,t)}{\partial t}\Big |_{t = 0} &= v(x), &x \in [0,1] -\end{aligned} \label{condwave} \tag{20} -$$ - -In this example, let $c = 1$ and $u(x) = \sin(\pi x)$ and $v(x) = -\pi\sin(\pi x)$. - - - -Setting up the network is done in similar matter as for the example of solving the diffusion equation. -The only things we have to change, is the trial solution such that it satisfies the conditions from ([20](#condwave)) and the cost function. - -The trial solution becomes slightly different since we have other conditions than in the example of solving the diffusion equation. Here, a possible trial solution $g_t(x,t)$ is - -$$ -g_t(x,t) = h_1(x,t) + x(1-x)t^2N(x,t,P) -$$ - -where - -$$ -h_1(x,t) = (1-t^2)u(x) + tv(x) -$$ - -Note that this trial solution satisfies the conditions only if $u(0) = v(0) = u(1) = v(1) = 0$, which is the case in this example. - - -The analytical solution for our specific problem, is - -$$ -g(x,t) = \sin(\pi x)\cos(\pi t) - \sin(\pi x)\sin(\pi t) -$$ - -import autograd.numpy as np -from autograd import hessian,grad -import autograd.numpy.random as npr -from matplotlib import cm -from matplotlib import pyplot as plt -from mpl_toolkits.mplot3d import axes3d - -## Set up the trial function: -def u(x): - return np.sin(np.pi*x) - -def v(x): - return -np.pi*np.sin(np.pi*x) - -def h1(point): - x,t = point - return (1 - t**2)*u(x) + t*v(x) - -def g_trial(point,P): - x,t = point - return h1(point) + x*(1-x)*t**2*deep_neural_network(P,point) - -## Define the cost function -def cost_function(P, x, t): - cost_sum = 0 - - g_t_hessian_func = hessian(g_trial) - - for x_ in x: - for t_ in t: - point = np.array([x_,t_]) - - g_t_hessian = g_t_hessian_func(point,P) - - g_t_d2x = g_t_hessian[0][0] - g_t_d2t = g_t_hessian[1][1] - - err_sqr = ( (g_t_d2t - g_t_d2x) )**2 - cost_sum += err_sqr - - return cost_sum / (np.size(t) * np.size(x)) - -## The neural network -def sigmoid(z): - return 1/(1 + np.exp(-z)) - -def deep_neural_network(deep_params, x): - # x is now a point and a 1D numpy array; make it a column vector - num_coordinates = np.size(x,0) - x = x.reshape(num_coordinates,-1) - - num_points = np.size(x,1) - - # N_hidden is the number of hidden layers - N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer - - # Assume that the input layer does nothing to the input x - x_input = x - x_prev = x_input - - ## Hidden layers: - - for l in range(N_hidden): - # From the list of parameters P; find the correct weigths and bias for this layer - w_hidden = deep_params[l] - - # Add a row of ones to include bias - x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0) - - z_hidden = np.matmul(w_hidden, x_prev) - x_hidden = sigmoid(z_hidden) - - # Update x_prev such that next layer can use the output from this layer - x_prev = x_hidden - - ## Output layer: - - # Get the weights and bias for this layer - w_output = deep_params[-1] - - # Include bias: - x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0) - - z_output = np.matmul(w_output, x_prev) - x_output = z_output - - return x_output[0][0] - -## The analytical solution -def g_analytic(point): - x,t = point - return np.sin(np.pi*x)*np.cos(np.pi*t) - np.sin(np.pi*x)*np.sin(np.pi*t) - -def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb): - ## Set up initial weigths and biases - N_hidden = np.size(num_neurons) - - ## Set up initial weigths and biases - - # Initialize the list of parameters: - P = [None]*(N_hidden + 1) # + 1 to include the output layer - - P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias - for l in range(1,N_hidden): - P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias - - # For the output layer - P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included - - print('Initial cost: ',cost_function(P, x, t)) - - cost_function_grad = grad(cost_function,0) - - # Let the update be done num_iter times - for i in range(num_iter): - cost_grad = cost_function_grad(P, x , t) - - for l in range(N_hidden+1): - P[l] = P[l] - lmb * cost_grad[l] - - - print('Final cost: ',cost_function(P, x, t)) - - return P - -if __name__ == '__main__': - ### Use the neural network: - npr.seed(15) - - ## Decide the vales of arguments to the function to solve - Nx = 10; Nt = 10 - x = np.linspace(0, 1, Nx) - t = np.linspace(0,1,Nt) - - ## Set up the parameters for the network - num_hidden_neurons = [50,20] - num_iter = 1000 - lmb = 0.01 - - P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb) - - ## Store the results - res = np.zeros((Nx, Nt)) - res_analytical = np.zeros((Nx, Nt)) - for i,x_ in enumerate(x): - for j, t_ in enumerate(t): - point = np.array([x_, t_]) - res[i,j] = g_trial(point,P) - - res_analytical[i,j] = g_analytic(point) - - diff = np.abs(res - res_analytical) - print("Max difference between analytical and solution from nn: %g"%np.max(diff)) - - ## Plot the solutions in two dimensions, that being in position and time - - T,X = np.meshgrid(t,x) - - fig = plt.figure(figsize=(10,10)) - ax = fig.gca(projection='3d') - ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons)) - s = ax.plot_surface(T,X,res,linewidth=0,antialiased=False,cmap=cm.viridis) - ax.set_xlabel('Time $t$') - ax.set_ylabel('Position $x$'); - - - fig = plt.figure(figsize=(10,10)) - ax = fig.gca(projection='3d') - ax.set_title('Analytical solution') - s = ax.plot_surface(T,X,res_analytical,linewidth=0,antialiased=False,cmap=cm.viridis) - ax.set_xlabel('Time $t$') - ax.set_ylabel('Position $x$'); - - - fig = plt.figure(figsize=(10,10)) - ax = fig.gca(projection='3d') - ax.set_title('Difference') - s = ax.plot_surface(T,X,diff,linewidth=0,antialiased=False,cmap=cm.viridis) - ax.set_xlabel('Time $t$') - ax.set_ylabel('Position $x$'); - - ## Take some slices of the 3D plots just to see the solutions at particular times - indx1 = 0 - indx2 = int(Nt/2) - indx3 = Nt-1 - - t1 = t[indx1] - t2 = t[indx2] - t3 = t[indx3] - - # Slice the results from the DNN - res1 = res[:,indx1] - res2 = res[:,indx2] - res3 = res[:,indx3] - - # Slice the analytical results - res_analytical1 = res_analytical[:,indx1] - res_analytical2 = res_analytical[:,indx2] - res_analytical3 = res_analytical[:,indx3] - - # Plot the slices - plt.figure(figsize=(10,10)) - plt.title("Computed solutions at time = %g"%t1) - plt.plot(x, res1) - plt.plot(x,res_analytical1) - plt.legend(['dnn','analytical']) - - plt.figure(figsize=(10,10)) - plt.title("Computed solutions at time = %g"%t2) - plt.plot(x, res2) - plt.plot(x,res_analytical2) - plt.legend(['dnn','analytical']) - - plt.figure(figsize=(10,10)) - plt.title("Computed solutions at time = %g"%t3) - plt.plot(x, res3) - plt.plot(x,res_analytical3) - plt.legend(['dnn','analytical']) - - plt.show() - -## Resources on differential equations and deep learning - -1. [Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al](https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf) - -2. [Neural networks for solving differential equations by A. Honchar](https://becominghuman.ai/neural-networks-for-solving-differential-equations-fa230ac5e04c) - -3. [Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener](http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf) - -4. [Introduction to Partial Differential Equations by A. Tveito, R. Winther](https://www.springer.com/us/book/9783540225515) \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter11_47_2.png b/doc/LectureNotes/_build/jupyter_execute/chapter11_47_2.png deleted file mode 100644 index 5667f33510ee727076ad0a2888ec3a6a662e37b1..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 23383 zcmY&=2RPMV{J-%nB%_o)Qe^M!ReV!7t3);-gzRhuCWXs-rU7Krj z`JH>e|NrlKeouMuIp^~}@9}z%^FHSaeW~`0><0Y}JUl!y$n$?S@bCzXz?T8>b?}4} zRTv4rNxD4Mad~ZT;o@%MWR9n5;^OeZ-sOXp={+}dCub{rI}v{HkN4ht7Z(R-DFFf7 z|DC{Z?_??PC#fbK5Fv4RuIr44M@ohL!pBQZr^Un5`wjW`i6%6CYuY{i&hmWwUY~Ph z_VL5d_nuxOy}?~QR{x4OnRrCUOuYIVgcq{;@My+sC(Q8QSB0F!VP6M|4ve-lXIbh` zqy^%$r+R0mS23P@3)>c5?rs*A$n);a51)kYL*P04NpUWC;Co^70y_jA&q^%-ez{o& zg}~X&KeZ~u*|-#Wjo@q$I@7(*7)pgfN`+aVi`!WhTv;~M@E zOKI?^C zT70uNn$Pf6{ey+tm?0q$1>!HR#H_)T`|w^kM5s;OEFz6PBb!%&I3$qrdN3t*Af-hh zW#$dmN?sw%Z%Tz%o(w|rfxFJc^Rbi)A-9us%hFeq&W=Ogt#H=|QU18WDpSWi3VRf| 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Introduction\n", - "\n", - "Resampling methods are an indispensable tool in modern\n", - "statistics. They involve repeatedly drawing samples from a training\n", - "set and refitting a model of interest on each sample in order to\n", - "obtain additional information about the fitted model. For example, in\n", - "order to estimate the variability of a linear regression fit, we can\n", - "repeatedly draw different samples from the training data, fit a linear\n", - "regression to each new sample, and then examine the extent to which\n", - "the resulting fits differ. Such an approach may allow us to obtain\n", - "information that would not be available from fitting the model only\n", - "once using the original training sample.\n", - "\n", - "Two resampling methods are often used in Machine Learning analyses,\n", - "1. The **bootstrap method**\n", - "\n", - "2. and **Cross-Validation**\n", - "\n", - "In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular\n", - "cross-validation and the bootstrap method. \n", - "\n", - "\n", - "Resampling approaches can be computationally expensive, because they\n", - "involve fitting the same statistical method multiple times using\n", - "different subsets of the training data. However, due to recent\n", - "advances in computing power, the computational requirements of\n", - "resampling methods generally are not prohibitive. In this chapter, we\n", - "discuss two of the most commonly used resampling methods,\n", - "cross-validation and the bootstrap. Both methods are important tools\n", - "in the practical application of many statistical learning\n", - "procedures. For example, cross-validation can be used to estimate the\n", - "test error associated with a given statistical learning method in\n", - "order to evaluate its performance, or to select the appropriate level\n", - "of flexibility. The process of evaluating a model’s performance is\n", - "known as model assessment, whereas the process of selecting the proper\n", - "level of flexibility for a model is known as model selection. The\n", - "bootstrap is widely used.\n", - "\n", - "\n", - "* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods\n", - "\n", - "* The results can be analysed with the same statistical tools as we would use analysing experimental data.\n", - "\n", - "* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n", - "\n", - "## Reminder on Statistics\n", - "\n", - "\n", - "* As in other experiments, many numerical experiments have two classes of errors:\n", - "\n", - " * Statistical errors\n", - "\n", - " * Systematical errors\n", - "\n", - "\n", - "* Statistical errors can be estimated using standard tools from statistics\n", - "\n", - "* Systematical errors are method specific and must be treated differently from case to case. \n", - "\n", - "The\n", - "advantage of doing linear regression is that we actually end up with\n", - "analytical expressions for several statistical quantities. \n", - "Standard least squares and Ridge regression allow us to\n", - "derive quantities like the variance and other expectation values in a\n", - "rather straightforward way.\n", - "\n", - "\n", - "It is assumed that $\\varepsilon_i\n", - "\\sim \\mathcal{N}(0, \\sigma^2)$ and the $\\varepsilon_{i}$ are\n", - "independent, i.e.:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*} \n", - "\\mbox{Cov}(\\varepsilon_{i_1},\n", - "\\varepsilon_{i_2}) & = \\left\\{ \\begin{array}{lcc} \\sigma^2 & \\mbox{if}\n", - "& i_1 = i_2, \\\\ 0 & \\mbox{if} & i_1 \\not= i_2. \\end{array} \\right.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The randomness of $\\varepsilon_i$ implies that\n", - "$\\mathbf{y}_i$ is also a random variable. In particular,\n", - "$\\mathbf{y}_i$ is normally distributed, because $\\varepsilon_i \\sim\n", - "\\mathcal{N}(0, \\sigma^2)$ and $\\mathbf{X}_{i,\\ast} \\, \\boldsymbol{\\beta}$ is a\n", - "non-random scalar. To specify the parameters of the distribution of\n", - "$\\mathbf{y}_i$ we need to calculate its first two moments. \n", - "\n", - "Recall that $\\boldsymbol{X}$ is a matrix of dimensionality $n\\times p$. The\n", - "notation above $\\mathbf{X}_{i,\\ast}$ means that we are looking at the\n", - "row number $i$ and perform a sum over all values $p$.\n", - "\n", - "\n", - "The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off)\n", - "that there exists a function $f(\\boldsymbol{x})$ and a normal distributed error $\\boldsymbol{\\varepsilon}\\sim \\mathcal{N}(0, \\sigma^2)$\n", - "which describe our data" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y} = f(\\boldsymbol{x})+\\boldsymbol{\\varepsilon}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We approximate this function with our model from the solution of the linear regression equations, that is our\n", - "function $f$ is approximated by $\\boldsymbol{\\tilde{y}}$ where we want to minimize $(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2$, our MSE, with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can calculate the expectation value of $\\boldsymbol{y}$ for a given element $i$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*} \n", - "\\mathbb{E}(y_i) & =\n", - "\\mathbb{E}(\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}) + \\mathbb{E}(\\varepsilon_i)\n", - "\\, \\, \\, = \\, \\, \\, \\mathbf{X}_{i, \\ast} \\, \\beta, \n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "while\n", - "its variance is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*} \\mbox{Var}(y_i) & = \\mathbb{E} \\{ [y_i\n", - "- \\mathbb{E}(y_i)]^2 \\} \\, \\, \\, = \\, \\, \\, \\mathbb{E} ( y_i^2 ) -\n", - "[\\mathbb{E}(y_i)]^2 \\\\ & = \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\,\n", - "\\beta + \\varepsilon_i )^2] - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \\\\ &\n", - "= \\mathbb{E} [ ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2 \\varepsilon_i\n", - "\\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} + \\varepsilon_i^2 ] - ( \\mathbf{X}_{i,\n", - "\\ast} \\, \\beta)^2 \\\\ & = ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 + 2\n", - "\\mathbb{E}(\\varepsilon_i) \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta} +\n", - "\\mathbb{E}(\\varepsilon_i^2 ) - ( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta})^2 \n", - "\\\\ & = \\mathbb{E}(\\varepsilon_i^2 ) \\, \\, \\, = \\, \\, \\,\n", - "\\mbox{Var}(\\varepsilon_i) \\, \\, \\, = \\, \\, \\, \\sigma^2. \n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Hence, $y_i \\sim \\mathcal{N}( \\mathbf{X}_{i, \\ast} \\, \\boldsymbol{\\beta}, \\sigma^2)$, that is $\\boldsymbol{y}$ follows a normal distribution with \n", - "mean value $\\boldsymbol{X}\\boldsymbol{\\beta}$ and variance $\\sigma^2$ (not be confused with the singular values of the SVD). \n", - "\n", - "\n", - "With the OLS expressions for the parameters $\\boldsymbol{\\beta}$ we can evaluate the expectation value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}(\\boldsymbol{\\beta}) = \\mathbb{E}[ (\\mathbf{X}^{\\top} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1}\\mathbf{X}^{T} \\mathbb{E}[ \\mathbf{Y}]=(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\mathbf{X}^{T}\\mathbf{X}\\boldsymbol{\\beta}=\\boldsymbol{\\beta}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This means that the estimator of the regression parameters is unbiased.\n", - "\n", - "We can also calculate the variance\n", - "\n", - "The variance of $\\boldsymbol{\\beta}$ is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{eqnarray*}\n", - "\\mbox{Var}(\\boldsymbol{\\beta}) & = & \\mathbb{E} \\{ [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})] [\\boldsymbol{\\beta} - \\mathbb{E}(\\boldsymbol{\\beta})]^{T} \\}\n", - "\\\\\n", - "& = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} - \\boldsymbol{\\beta}]^{T} \\}\n", - "\\\\\n", - "% & = & \\mathbb{E} \\{ [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}] \\, [(\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y}]^{T} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "% & = & \\mathbb{E} \\{ (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\mathbf{Y} \\, \\mathbf{Y}^{T} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\mathbb{E} \\{ \\mathbf{Y} \\, \\mathbf{Y}^{T} \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "\\\\\n", - "& = & (\\mathbf{X}^{T} \\mathbf{X})^{-1} \\, \\mathbf{X}^{T} \\, \\{ \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} + \\sigma^2 \\} \\, \\mathbf{X} \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "% \\\\\n", - "% & = & (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^T \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T % \\mathbf{X})^{-1}\n", - "% \\\\\n", - "% & & + \\, \\, \\sigma^2 \\, (\\mathbf{X}^T \\mathbf{X})^{-1} \\, \\mathbf{X}^T \\, \\mathbf{X} \\, (\\mathbf{X}^T \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\boldsymbol{\\beta}^T\n", - "\\\\\n", - "& = & \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} + \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1} - \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T}\n", - "\\, \\, \\, = \\, \\, \\, \\sigma^2 \\, (\\mathbf{X}^{T} \\mathbf{X})^{-1},\n", - "\\end{eqnarray*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have used that $\\mathbb{E} (\\mathbf{Y} \\mathbf{Y}^{T}) =\n", - "\\mathbf{X} \\, \\boldsymbol{\\beta} \\, \\boldsymbol{\\beta}^{T} \\, \\mathbf{X}^{T} +\n", - "\\sigma^2 \\, \\mathbf{I}_{nn}$. From $\\mbox{Var}(\\boldsymbol{\\beta}) = \\sigma^2\n", - "\\, (\\mathbf{X}^{T} \\mathbf{X})^{-1}$, one obtains an estimate of the\n", - "variance of the estimate of the $j$-th regression coefficient:\n", - "$\\boldsymbol{\\sigma}^2 (\\boldsymbol{\\beta}_j ) = \\boldsymbol{\\sigma}^2 \\sqrt{\n", - "[(\\mathbf{X}^{T} \\mathbf{X})^{-1}]_{jj} }$. This may be used to\n", - "construct a confidence interval for the estimates.\n", - "\n", - "\n", - "In a similar way, we can obtain analytical expressions for say the\n", - "expectation values of the parameters $\\boldsymbol{\\beta}$ and their variance\n", - "when we employ Ridge regression, allowing us again to define a confidence interval. \n", - "\n", - "It is rather straightforward to show that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big]=(\\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I}_{pp})^{-1} (\\mathbf{X}^{\\top} \\mathbf{X})\\boldsymbol{\\beta}^{\\mathrm{OLS}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see clearly that \n", - "$\\mathbb{E} \\big[ \\boldsymbol{\\beta}^{\\mathrm{Ridge}} \\big] \\not= \\boldsymbol{\\beta}^{\\mathrm{OLS}}$ for any $\\lambda > 0$. We say then that the ridge estimator is biased.\n", - "\n", - "We can also compute the variance as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{Ridge}}]=\\sigma^2[ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1} \\mathbf{X}^{T} \\mathbf{X} \\{ [ \\mathbf{X}^{\\top} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and it is easy to see that if the parameter $\\lambda$ goes to infinity then the variance of Ridge parameters $\\boldsymbol{\\beta}$ goes to zero. \n", - "\n", - "With this, we can compute the difference" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mbox{Var}[\\boldsymbol{\\beta}^{\\mathrm{OLS}}]-\\mbox{Var}(\\boldsymbol{\\beta}^{\\mathrm{Ridge}})=\\sigma^2 [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}[ 2\\lambda\\mathbf{I} + \\lambda^2 (\\mathbf{X}^{T} \\mathbf{X})^{-1} ] \\{ [ \\mathbf{X}^{T} \\mathbf{X} + \\lambda \\mathbf{I} ]^{-1}\\}^{T}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The difference is non-negative definite since each component of the\n", - "matrix product is non-negative definite. \n", - "This means the variance we obtain with the standard OLS will always for $\\lambda > 0$ be larger than the variance of $\\boldsymbol{\\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. \n", - "\n", - "\n", - "\n", - "## Resampling methods\n", - "\n", - "With all these analytical equations for both the OLS and Ridge\n", - "regression, we will now outline how to assess a given model. This will\n", - "lead us to a discussion of the so-called bias-variance tradeoff (see\n", - "below) and so-called resampling methods.\n", - "\n", - "One of the quantities we have discussed as a way to measure errors is\n", - "the mean-squared error (MSE), mainly used for fitting of continuous\n", - "functions. Another choice is the absolute error.\n", - "\n", - "In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data,\n", - "we discuss the\n", - "1. prediction error or simply the **test error** $\\mathrm{Err_{Test}}$, where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the \n", - "\n", - "2. training error $\\mathrm{Err_{Train}}$, which is the average loss over the training data.\n", - "\n", - "As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error.\n", - "For a certain level of complexity the test error will reach minimum, before starting to increase again. The\n", - "training error reaches a saturation.\n", - "\n", - "\n", - "\n", - "Two famous\n", - "resampling methods are the **independent bootstrap** and **the jackknife**. \n", - "\n", - "The jackknife is a special case of the independent bootstrap. Still, the jackknife was made\n", - "popular prior to the independent bootstrap. And as the popularity of\n", - "the independent bootstrap soared, new variants, such as **the dependent bootstrap**.\n", - "\n", - "The Jackknife and independent bootstrap work for\n", - "independent, identically distributed random variables.\n", - "If these conditions are not\n", - "satisfied, the methods will fail. Yet, it should be said that if the data are\n", - "independent, identically distributed, and we only want to estimate the\n", - "variance of $\\overline{X}$ (which often is the case), then there is no\n", - "need for bootstrapping. \n", - "\n", - "\n", - "The Jackknife works by making many replicas of the estimator $\\widehat{\\theta}$. \n", - "The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\\boldsymbol{x} = (x_1,x_2,\\cdots,X_n)$. \n", - "Let $\\boldsymbol{x}_i$ denote the vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i = (x_1,x_2,\\cdots,x_{i-1},x_{i+1},\\cdots,x_n),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which equals the vector $\\boldsymbol{x}$ with the exception that observation\n", - "number $i$ is left out. Using this notation, define\n", - "$\\widehat{\\theta}_i$ to be the estimator\n", - "$\\widehat{\\theta}$ computed using $\\vec{X}_i$." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Runtime: 0.139172 sec\n", - "Jackknife Statistics :\n", - "original bias std. error\n", - " 100.048 100.038 0.150004\n" - ] - } - ], - "source": [ - "from numpy import *\n", - "from numpy.random import randint, randn\n", - "from time import time\n", - "\n", - "def jackknife(data, stat):\n", - " n = len(data);t = zeros(n); inds = arange(n); t0 = time()\n", - " ## 'jackknifing' by leaving out an observation for each i \n", - " for i in range(n):\n", - " t[i] = stat(delete(data,i) )\n", - "\n", - " # analysis \n", - " print(\"Runtime: %g sec\" % (time()-t0)); print(\"Jackknife Statistics :\")\n", - " print(\"original bias std. error\")\n", - " print(\"%8g %14g %15g\" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))\n", - "\n", - " return t\n", - "\n", - "\n", - "# Returns mean of data samples \n", - "def stat(data):\n", - " return mean(data)\n", - "\n", - "\n", - "mu, sigma = 100, 15\n", - "datapoints = 10000\n", - "x = mu + sigma*random.randn(datapoints)\n", - "# jackknife returns the data sample \n", - "t = jackknife(x, stat)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Bootstrap\n", - "\n", - "Bootstrapping is a nonparametric approach to statistical inference\n", - "that substitutes computation for more traditional distributional\n", - "assumptions and asymptotic results. Bootstrapping offers a number of\n", - "advantages: \n", - "1. The bootstrap is quite general, although there are some cases in which it fails. \n", - "\n", - "2. Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small. \n", - "\n", - "3. It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically. \n", - "\n", - "4. It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).\n", - "\n", - "Since $\\widehat{\\theta} = \\widehat{\\theta}(\\boldsymbol{X})$ is a function of random variables,\n", - "$\\widehat{\\theta}$ itself must be a random variable. Thus it has\n", - "a pdf, call this function $p(\\boldsymbol{t})$. The aim of the bootstrap is to\n", - "estimate $p(\\boldsymbol{t})$ by the relative frequency of\n", - "$\\widehat{\\theta}$. You can think of this as using a histogram\n", - "in the place of $p(\\boldsymbol{t})$. If the relative frequency closely\n", - "resembles $p(\\vec{t})$, then using numerics, it is straight forward to\n", - "estimate all the interesting parameters of $p(\\boldsymbol{t})$ using point\n", - "estimators. \n", - "\n", - "\n", - "\n", - "In the case that $\\widehat{\\theta}$ has\n", - "more than one component, and the components are independent, we use the\n", - "same estimator on each component separately. If the probability\n", - "density function of $X_i$, $p(x)$, had been known, then it would have\n", - "been straight forward to do this by: \n", - "1. Drawing lots of numbers from $p(x)$, suppose we call one such set of numbers $(X_1^*, X_2^*, \\cdots, X_n^*)$. \n", - "\n", - "2. Then using these numbers, we could compute a replica of $\\widehat{\\theta}$ called $\\widehat{\\theta}^*$. \n", - "\n", - "By repeated use of (1) and (2), many\n", - "estimates of $\\widehat{\\theta}$ could have been obtained. The\n", - "idea is to use the relative frequency of $\\widehat{\\theta}^*$\n", - "(think of a histogram) as an estimate of $p(\\boldsymbol{t})$.\n", - "\n", - "\n", - "But\n", - "unless there is enough information available about the process that\n", - "generated $X_1,X_2,\\cdots,X_n$, $p(x)$ is in general\n", - "unknown. Therefore, [Efron in 1979](https://projecteuclid.org/euclid.aos/1176344552) asked the\n", - "question: What if we replace $p(x)$ by the relative frequency\n", - "of the observation $X_i$; if we draw observations in accordance with\n", - "the relative frequency of the observations, will we obtain the same\n", - "result in some asymptotic sense? The answer is yes.\n", - "\n", - "\n", - "Instead of generating the histogram for the relative\n", - "frequency of the observation $X_i$, just draw the values\n", - "$(X_1^*,X_2^*,\\cdots,X_n^*)$ with replacement from the vector\n", - "$\\boldsymbol{X}$. \n", - "\n", - "\n", - "The independent bootstrap works like this: \n", - "\n", - "1. Draw with replacement $n$ numbers for the observed variables $\\boldsymbol{x} = (x_1,x_2,\\cdots,x_n)$. \n", - "\n", - "2. Define a vector $\\boldsymbol{x}^*$ containing the values which were drawn from $\\boldsymbol{x}$. \n", - "\n", - "3. Using the vector $\\boldsymbol{x}^*$ compute $\\widehat{\\theta}^*$ by evaluating $\\widehat \\theta$ under the observations $\\boldsymbol{x}^*$. \n", - "\n", - "4. Repeat this process $k$ times. \n", - "\n", - "When you are done, you can draw a histogram of the relative frequency\n", - "of $\\widehat \\theta^*$. This is your estimate of the probability\n", - "distribution $p(t)$. Using this probability distribution you can\n", - "estimate any statistics thereof. In principle you never draw the\n", - "histogram of the relative frequency of $\\widehat{\\theta}^*$. Instead\n", - "you use the estimators corresponding to the statistic of interest. For\n", - "example, if you are interested in estimating the variance of $\\widehat\n", - "\\theta$, apply the etsimator $\\widehat \\sigma^2$ to the values\n", - "$\\widehat \\theta ^*$.\n", - "\n", - "\n", - "\n", - "The following code starts with a Gaussian distribution with mean value\n", - "$\\mu =100$ and variance $\\sigma=15$. We use this to generate the data\n", - "used in the bootstrap analysis. The bootstrap analysis returns a data\n", - "set after a given number of bootstrap operations (as many as we have\n", - "data points). This data set consists of estimated mean values for each\n", - "bootstrap operation. The histogram generated by the bootstrap method\n", - "shows that the distribution for these mean values is also a Gaussian,\n", - "centered around the mean value $\\mu=100$ but with standard deviation\n", - "$\\sigma/\\sqrt{n}$, where $n$ is the number of bootstrap samples (in\n", - "this case the same as the number of original data points). The value\n", - "of the standard deviation is what we expect from the central limit\n", - "theorem." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Runtime: 1.73359 sec\n", - "Bootstrap Statistics :\n", - "original bias std. error\n", - " 99.9929 15.0315 99.994 0.150978\n" - ] - }, - { - "ename": "AttributeError", - "evalue": "'Rectangle' object has no property 'normed'", - "output_type": "error", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 31\u001b[0m \u001b[0mt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mbootstrap\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mstat\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdatapoints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 32\u001b[0m \u001b[0;31m# the histogram of the bootstrapped data\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 33\u001b[0;31m \u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbinsboot\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpatches\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mhist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m50\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnormed\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfacecolor\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'red'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0malpha\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m0.75\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 34\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 35\u001b[0m \u001b[0;31m# add a 'best fit' line\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;32m~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/pyplot.py\u001b[0m in \u001b[0;36mhist\u001b[0;34m(x, bins, range, density, weights, cumulative, bottom, histtype, align, orientation, rwidth, log, color, label, stacked, data, **kwargs)\u001b[0m\n\u001b[1;32m 2683\u001b[0m \u001b[0morientation\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'vertical'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mrwidth\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlog\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m,\u001b[0m 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6817\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mset_label\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlbl\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;32m~/opt/anaconda3/lib/python3.8/site-packages/matplotlib/artist.py\u001b[0m in \u001b[0;36mupdate\u001b[0;34m(self, props)\u001b[0m\n\u001b[1;32m 994\u001b[0m \u001b[0mfunc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34mf\"set_{k}\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 995\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mcallable\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 996\u001b[0;31m raise AttributeError(f\"{type(self).__name__!r} object \"\n\u001b[0m\u001b[1;32m 997\u001b[0m f\"has no property {k!r}\")\n\u001b[1;32m 998\u001b[0m \u001b[0mret\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mv\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mAttributeError\u001b[0m: 'Rectangle' object has no property 'normed'" - ] - }, - { - "data": { - "image/png": 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter2_25_2.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "%matplotlib inline\n", - "\n", - "from numpy import *\n", - "from numpy.random import randint, randn\n", - "from time import time\n", - "import matplotlib.mlab as mlab\n", - "import matplotlib.pyplot as plt\n", - "\n", - "# Returns mean of bootstrap samples \n", - "def stat(data):\n", - " return mean(data)\n", - "\n", - "# Bootstrap algorithm\n", - "def bootstrap(data, statistic, R):\n", - " t = zeros(R); n = len(data); inds = arange(n); t0 = time()\n", - " # non-parametric bootstrap \n", - " for i in range(R):\n", - " t[i] = statistic(data[randint(0,n,n)])\n", - "\n", - " # analysis \n", - " print(\"Runtime: %g sec\" % (time()-t0)); print(\"Bootstrap Statistics :\")\n", - " print(\"original bias std. error\")\n", - " print(\"%8g %8g %14g %15g\" % (statistic(data), std(data),mean(t),std(t)))\n", - " return t\n", - "\n", - "\n", - "mu, sigma = 100, 15\n", - "datapoints = 10000\n", - "x = mu + sigma*random.randn(datapoints)\n", - "# bootstrap returns the data sample \n", - "t = bootstrap(x, stat, datapoints)\n", - "# the histogram of the bootstrapped data \n", - "n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75)\n", - "\n", - "# add a 'best fit' line \n", - "y = mlab.normpdf( binsboot, mean(t), std(t))\n", - "lt = plt.plot(binsboot, y, 'r--', linewidth=1)\n", - "plt.xlabel('Smarts')\n", - "plt.ylabel('Probability')\n", - "plt.axis([99.5, 100.6, 0, 3.0])\n", - "plt.grid(True)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Various steps in cross-validation\n", - "\n", - "When the repetitive splitting of the data set is done randomly,\n", - "samples may accidently end up in a fast majority of the splits in\n", - "either training or test set. Such samples may have an unbalanced\n", - "influence on either model building or prediction evaluation. To avoid\n", - "this $k$-fold cross-validation structures the data splitting. The\n", - "samples are divided into $k$ more or less equally sized exhaustive and\n", - "mutually exclusive subsets. In turn (at each split) one of these\n", - "subsets plays the role of the test set while the union of the\n", - "remaining subsets constitutes the training set. Such a splitting\n", - "warrants a balanced representation of each sample in both training and\n", - "test set over the splits. Still the division into the $k$ subsets\n", - "involves a degree of randomness. This may be fully excluded when\n", - "choosing $k=n$. This particular case is referred to as leave-one-out\n", - "cross-validation (LOOCV). \n", - "\n", - "\n", - "* Define a range of interest for the penalty parameter.\n", - "\n", - "* Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n", - "\n", - "* Fit the linear regression model by means of ridge estimation for each $\\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\\boldsymbol{\\sigma}_{-i}^2(\\lambda)$, as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\boldsymbol{\\beta}_{-i}(\\lambda) & = ( \\boldsymbol{X}_{-i, \\ast}^{T}\n", - "\\boldsymbol{X}_{-i, \\ast} + \\lambda \\boldsymbol{I}_{pp})^{-1}\n", - "\\boldsymbol{X}_{-i, \\ast}^{T} \\boldsymbol{y}_{-i}\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "* Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n", - "\n", - "* Repeat the first three steps such that each sample plays the role of the test set once.\n", - "\n", - "* Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\frac{1}{n} \\sum_{i = 1}^n \\log\\{L[y_i, \\mathbf{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For the various values of $k$\n", - "\n", - "1. shuffle the dataset randomly.\n", - "\n", - "2. Split the dataset into $k$ groups.\n", - "\n", - "3. For each unique group:\n", - "\n", - "a. Decide which group to use as set for test data\n", - "\n", - "b. Take the remaining groups as a training data set\n", - "\n", - "c. Fit a model on the training set and evaluate it on the test set\n", - "\n", - "d. Retain the evaluation score and discard the model\n", - "\n", - "\n", - "5. Summarize the model using the sample of model evaluation scores\n", - "\n", - "The code here uses Ridge regression with cross-validation (CV) resampling and $k$-fold CV in order to fit a specific polynomial." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.model_selection import KFold\n", - "from sklearn.linear_model import Ridge\n", - "from sklearn.model_selection import cross_val_score\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "\n", - "# A seed just to ensure that the random numbers are the same for every run.\n", - "# Useful for eventual debugging.\n", - "np.random.seed(3155)\n", - "\n", - "# Generate the data.\n", - "nsamples = 100\n", - "x = np.random.randn(nsamples)\n", - "y = 3*x**2 + np.random.randn(nsamples)\n", - "\n", - "## Cross-validation on Ridge regression using KFold only\n", - "\n", - "# Decide degree on polynomial to fit\n", - "poly = PolynomialFeatures(degree = 6)\n", - "\n", - "# Decide which values of lambda to use\n", - "nlambdas = 500\n", - "lambdas = np.logspace(-3, 5, nlambdas)\n", - "\n", - "# Initialize a KFold instance\n", - "k = 5\n", - "kfold = KFold(n_splits = k)\n", - "\n", - "# Perform the cross-validation to estimate MSE\n", - "scores_KFold = np.zeros((nlambdas, k))\n", - "\n", - "i = 0\n", - "for lmb in lambdas:\n", - " ridge = Ridge(alpha = lmb)\n", - " j = 0\n", - " for train_inds, test_inds in kfold.split(x):\n", - " xtrain = x[train_inds]\n", - " ytrain = y[train_inds]\n", - "\n", - " xtest = x[test_inds]\n", - " ytest = y[test_inds]\n", - "\n", - " Xtrain = poly.fit_transform(xtrain[:, np.newaxis])\n", - " ridge.fit(Xtrain, ytrain[:, np.newaxis])\n", - "\n", - " Xtest = poly.fit_transform(xtest[:, np.newaxis])\n", - " ypred = ridge.predict(Xtest)\n", - "\n", - " scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred)\n", - "\n", - " j += 1\n", - " i += 1\n", - "\n", - "\n", - "estimated_mse_KFold = np.mean(scores_KFold, axis = 1)\n", - "\n", - "## Cross-validation using cross_val_score from sklearn along with KFold\n", - "\n", - "# kfold is an instance initialized above as:\n", - "# kfold = KFold(n_splits = k)\n", - "\n", - "estimated_mse_sklearn = np.zeros(nlambdas)\n", - "i = 0\n", - "for lmb in lambdas:\n", - " ridge = Ridge(alpha = lmb)\n", - "\n", - " X = poly.fit_transform(x[:, np.newaxis])\n", - " estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold)\n", - "\n", - " # cross_val_score return an array containing the estimated negative mse for every fold.\n", - " # we have to the the mean of every array in order to get an estimate of the mse of the model\n", - " estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)\n", - "\n", - " i += 1\n", - "\n", - "## Plot and compare the slightly different ways to perform cross-validation\n", - "\n", - "plt.figure()\n", - "\n", - "plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')\n", - "plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold')\n", - "\n", - "plt.xlabel('log10(lambda)')\n", - "plt.ylabel('mse')\n", - "\n", - "plt.legend()\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The bias-variance tradeoff\n", - "\n", - "\n", - "We will discuss the bias-variance tradeoff in the context of\n", - "continuous predictions such as regression. However, many of the\n", - "intuitions and ideas discussed here also carry over to classification\n", - "tasks. Consider a dataset $\\mathcal{L}$ consisting of the data\n", - "$\\mathbf{X}_\\mathcal{L}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$. \n", - "\n", - "Let us assume that the true data is generated from a noisy model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y}=f(\\boldsymbol{x}) + \\boldsymbol{\\epsilon}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\epsilon$ is normally distributed with mean zero and standard deviation $\\sigma^2$.\n", - "\n", - "In our derivation of the ordinary least squares method we defined then\n", - "an approximation to the function $f$ in terms of the parameters\n", - "$\\boldsymbol{\\beta}$ and the design matrix $\\boldsymbol{X}$ which embody our model,\n", - "that is $\\boldsymbol{\\tilde{y}}=\\boldsymbol{X}\\boldsymbol{\\beta}$. \n", - "\n", - "Thereafter we found the parameters $\\boldsymbol{\\beta}$ by optimizing the means squared error via the so-called cost function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{X},\\boldsymbol{\\beta}) =\\frac{1}{n}\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2=\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right].\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can rewrite this as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\frac{1}{n}\\sum_i(f_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\frac{1}{n}\\sum_i(\\tilde{y}_i-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2+\\sigma^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The three terms represent the square of the bias of the learning\n", - "method, which can be thought of as the error caused by the simplifying\n", - "assumptions built into the method. The second term represents the\n", - "variance of the chosen model and finally the last terms is variance of\n", - "the error $\\boldsymbol{\\epsilon}$.\n", - "\n", - "To derive this equation, we need to recall that the variance of $\\boldsymbol{y}$ and $\\boldsymbol{\\epsilon}$ are both equal to $\\sigma^2$. The mean value of $\\boldsymbol{\\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\\boldsymbol{\\tilde{y}}$.\n", - "We use a more compact notation in terms of the expectation value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}})^2\\right],\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and adding and subtracting $\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]$ we get" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{f}+\\boldsymbol{\\epsilon}-\\boldsymbol{\\tilde{y}}+\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right]-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right],\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which, using the abovementioned expectation values can be rewritten as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}\\left[(\\boldsymbol{y}-\\boldsymbol{\\tilde{y}})^2\\right]=\\mathbb{E}\\left[(\\boldsymbol{y}-\\mathbb{E}\\left[\\boldsymbol{\\tilde{y}}\\right])^2\\right]+\\mathrm{Var}\\left[\\boldsymbol{\\tilde{y}}\\right]+\\sigma^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "that is the rewriting in terms of the so-called bias, the variance of the model $\\boldsymbol{\\tilde{y}}$ and the variance of $\\boldsymbol{\\epsilon}$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.utils import resample\n", - "\n", - "np.random.seed(2018)\n", - "\n", - "n = 500\n", - "n_boostraps = 100\n", - "degree = 18 # A quite high value, just to show.\n", - "noise = 0.1\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-1, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape)\n", - "\n", - "# Hold out some test data that is never used in training.\n", - "x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "\n", - "# Combine x transformation and model into one operation.\n", - "# Not neccesary, but convenient.\n", - "model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))\n", - "\n", - "# The following (m x n_bootstraps) matrix holds the column vectors y_pred\n", - "# for each bootstrap iteration.\n", - "y_pred = np.empty((y_test.shape[0], n_boostraps))\n", - "for i in range(n_boostraps):\n", - " x_, y_ = resample(x_train, y_train)\n", - "\n", - " # Evaluate the new model on the same test data each time.\n", - " y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()\n", - "\n", - "# Note: Expectations and variances taken w.r.t. different training\n", - "# data sets, hence the axis=1. Subsequent means are taken across the test data\n", - "# set in order to obtain a total value, but before this we have error/bias/variance\n", - "# calculated per data point in the test set.\n", - "# Note 2: The use of keepdims=True is important in the calculation of bias as this \n", - "# maintains the column vector form. Dropping this yields very unexpected results.\n", - "error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n", - "bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n", - "variance = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n", - "print('Error:', error)\n", - "print('Bias^2:', bias)\n", - "print('Var:', variance)\n", - "print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance))\n", - "\n", - "plt.plot(x[::5, :], y[::5, :], label='f(x)')\n", - "plt.scatter(x_test, y_test, label='Data points')\n", - "plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.utils import resample\n", - "\n", - "np.random.seed(2018)\n", - "\n", - "n = 40\n", - "n_boostraps = 100\n", - "maxdegree = 14\n", - "\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "error = np.zeros(maxdegree)\n", - "bias = np.zeros(maxdegree)\n", - "variance = np.zeros(maxdegree)\n", - "polydegree = np.zeros(maxdegree)\n", - "x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "\n", - "for degree in range(maxdegree):\n", - " model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))\n", - " y_pred = np.empty((y_test.shape[0], n_boostraps))\n", - " for i in range(n_boostraps):\n", - " x_, y_ = resample(x_train, y_train)\n", - " y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel()\n", - "\n", - " polydegree[degree] = degree\n", - " error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n", - " bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n", - " variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n", - " print('Polynomial degree:', degree)\n", - " print('Error:', error[degree])\n", - " print('Bias^2:', bias[degree])\n", - " print('Var:', variance[degree])\n", - " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", - "\n", - "plt.plot(polydegree, error, label='Error')\n", - "plt.plot(polydegree, bias, label='bias')\n", - "plt.plot(polydegree, variance, label='Variance')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The bias-variance tradeoff summarizes the fundamental tension in\n", - "machine learning, particularly supervised learning, between the\n", - "complexity of a model and the amount of training data needed to train\n", - "it. Since data is often limited, in practice it is often useful to\n", - "use a less-complex model with higher bias, that is a model whose asymptotic\n", - "performance is worse than another model because it is easier to\n", - "train and less sensitive to sampling noise arising from having a\n", - "finite-sized training dataset (smaller variance). \n", - "\n", - "\n", - "\n", - "The above equations tell us that in\n", - "order to minimize the expected test error, we need to select a\n", - "statistical learning method that simultaneously achieves low variance\n", - "and low bias. Note that variance is inherently a nonnegative quantity,\n", - "and squared bias is also nonnegative. Hence, we see that the expected\n", - "test MSE can never lie below $Var(\\epsilon)$, the irreducible error.\n", - "\n", - "\n", - "What do we mean by the variance and bias of a statistical learning\n", - "method? The variance refers to the amount by which our model would change if we\n", - "estimated it using a different training data set. Since the training\n", - "data are used to fit the statistical learning method, different\n", - "training data sets will result in a different estimate. But ideally the\n", - "estimate for our model should not vary too much between training\n", - "sets. However, if a method has high variance then small changes in\n", - "the training data can result in large changes in the model. In general, more\n", - "flexible statistical methods have higher variance.\n", - "\n", - "\n", - "You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\"\"\"\n", - "============================\n", - "Underfitting vs. Overfitting\n", - "============================\n", - "\n", - "This example demonstrates the problems of underfitting and overfitting and\n", - "how we can use linear regression with polynomial features to approximate\n", - "nonlinear functions. The plot shows the function that we want to approximate,\n", - "which is a part of the cosine function. In addition, the samples from the\n", - "real function and the approximations of different models are displayed. The\n", - "models have polynomial features of different degrees. We can see that a\n", - "linear function (polynomial with degree 1) is not sufficient to fit the\n", - "training samples. This is called **underfitting**. A polynomial of degree 4\n", - "approximates the true function almost perfectly. However, for higher degrees\n", - "the model will **overfit** the training data, i.e. it learns the noise of the\n", - "training data.\n", - "We evaluate quantitatively **overfitting** / **underfitting** by using\n", - "cross-validation. We calculate the mean squared error (MSE) on the validation\n", - "set, the higher, the less likely the model generalizes correctly from the\n", - "training data.\n", - "\"\"\"\n", - "\n", - "print(__doc__)\n", - "\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.pipeline import Pipeline\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.linear_model import LinearRegression\n", - "from sklearn.model_selection import cross_val_score\n", - "\n", - "\n", - "def true_fun(X):\n", - " return np.cos(1.5 * np.pi * X)\n", - "\n", - "np.random.seed(0)\n", - "\n", - "n_samples = 30\n", - "degrees = [1, 4, 15]\n", - "\n", - "X = np.sort(np.random.rand(n_samples))\n", - "y = true_fun(X) + np.random.randn(n_samples) * 0.1\n", - "\n", - "plt.figure(figsize=(14, 5))\n", - "for i in range(len(degrees)):\n", - " ax = plt.subplot(1, len(degrees), i + 1)\n", - " plt.setp(ax, xticks=(), yticks=())\n", - "\n", - " polynomial_features = PolynomialFeatures(degree=degrees[i],\n", - " include_bias=False)\n", - " linear_regression = LinearRegression()\n", - " pipeline = Pipeline([(\"polynomial_features\", polynomial_features),\n", - " (\"linear_regression\", linear_regression)])\n", - " pipeline.fit(X[:, np.newaxis], y)\n", - "\n", - " # Evaluate the models using crossvalidation\n", - " scores = cross_val_score(pipeline, X[:, np.newaxis], y,\n", - " scoring=\"neg_mean_squared_error\", cv=10)\n", - "\n", - " X_test = np.linspace(0, 1, 100)\n", - " plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label=\"Model\")\n", - " plt.plot(X_test, true_fun(X_test), label=\"True function\")\n", - " plt.scatter(X, y, edgecolor='b', s=20, label=\"Samples\")\n", - " plt.xlabel(\"x\")\n", - " plt.ylabel(\"y\")\n", - " plt.xlim((0, 1))\n", - " plt.ylim((-2, 2))\n", - " plt.legend(loc=\"best\")\n", - " plt.title(\"Degree {}\\nMSE = {:.2e}(+/- {:.2e})\".format(\n", - " degrees[i], -scores.mean(), scores.std()))\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.utils import resample\n", - "from sklearn.metrics import mean_squared_error\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"EoS.csv\"),'r')\n", - "\n", - "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", - "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", - "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", - "EoS = EoS.dropna()\n", - "Energies = EoS['Energy']\n", - "Density = EoS['Density']\n", - "# The design matrix now as function of various polytrops\n", - "\n", - "Maxpolydegree = 30\n", - "X = np.zeros((len(Density),Maxpolydegree))\n", - "X[:,0] = 1.0\n", - "testerror = np.zeros(Maxpolydegree)\n", - "trainingerror = np.zeros(Maxpolydegree)\n", - "polynomial = np.zeros(Maxpolydegree)\n", - "\n", - "trials = 100\n", - "for polydegree in range(1, Maxpolydegree):\n", - " polynomial[polydegree] = polydegree\n", - " for degree in range(polydegree):\n", - " X[:,degree] = Density**(degree/3.0)\n", - "\n", - "# loop over trials in order to estimate the expectation value of the MSE\n", - " testerror[polydegree] = 0.0\n", - " trainingerror[polydegree] = 0.0\n", - " for samples in range(trials):\n", - " x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)\n", - " model = LinearRegression(fit_intercept=True).fit(x_train, y_train)\n", - " ypred = model.predict(x_train)\n", - " ytilde = model.predict(x_test)\n", - " testerror[polydegree] += mean_squared_error(y_test, ytilde)\n", - " trainingerror[polydegree] += mean_squared_error(y_train, ypred) \n", - "\n", - " testerror[polydegree] /= trials\n", - " trainingerror[polydegree] /= trials\n", - " print(\"Degree of polynomial: %3d\"% polynomial[polydegree])\n", - " print(\"Mean squared error on training data: %.8f\" % trainingerror[polydegree])\n", - " print(\"Mean squared error on test data: %.8f\" % testerror[polydegree])\n", - "\n", - "plt.plot(polynomial, np.log10(trainingerror), label='Training Error')\n", - "plt.plot(polynomial, np.log10(testerror), label='Test Error')\n", - "plt.xlabel('Polynomial degree')\n", - "plt.ylabel('log10[MSE]')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.metrics import mean_squared_error\n", - "from sklearn.model_selection import KFold\n", - "from sklearn.model_selection import cross_val_score\n", - "\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"EoS.csv\"),'r')\n", - "\n", - "# Read the EoS data as csv file and organize the data into two arrays with density and energies\n", - "EoS = pd.read_csv(infile, names=('Density', 'Energy'))\n", - "EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')\n", - "EoS = EoS.dropna()\n", - "Energies = EoS['Energy']\n", - "Density = EoS['Density']\n", - "# The design matrix now as function of various polytrops\n", - "\n", - "Maxpolydegree = 30\n", - "X = np.zeros((len(Density),Maxpolydegree))\n", - "X[:,0] = 1.0\n", - "estimated_mse_sklearn = np.zeros(Maxpolydegree)\n", - "polynomial = np.zeros(Maxpolydegree)\n", - "k =5\n", - "kfold = KFold(n_splits = k)\n", - "\n", - "for polydegree in range(1, Maxpolydegree):\n", - " polynomial[polydegree] = polydegree\n", - " for degree in range(polydegree):\n", - " X[:,degree] = Density**(degree/3.0)\n", - " OLS = LinearRegression()\n", - "# loop over trials in order to estimate the expectation value of the MSE\n", - " estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold)\n", - "#[:, np.newaxis]\n", - " estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds)\n", - "\n", - "plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error')\n", - "plt.xlabel('Polynomial degree')\n", - "plt.ylabel('log10[MSE]')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.model_selection import KFold\n", - "from sklearn.linear_model import Ridge\n", - "from sklearn.model_selection import cross_val_score\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "\n", - "# A seed just to ensure that the random numbers are the same for every run.\n", - "np.random.seed(3155)\n", - "# Generate the data.\n", - "n = 100\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "# Decide degree on polynomial to fit\n", - "poly = PolynomialFeatures(degree = 10)\n", - "\n", - "# Decide which values of lambda to use\n", - "nlambdas = 500\n", - "lambdas = np.logspace(-3, 5, nlambdas)\n", - "# Initialize a KFold instance\n", - "k = 5\n", - "kfold = KFold(n_splits = k)\n", - "estimated_mse_sklearn = np.zeros(nlambdas)\n", - "i = 0\n", - "for lmb in lambdas:\n", - " ridge = Ridge(alpha = lmb)\n", - " estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold)\n", - " estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds)\n", - " i += 1\n", - "plt.figure()\n", - "plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score')\n", - "plt.xlabel('log10(lambda)')\n", - "plt.ylabel('MSE')\n", - "plt.legend()\n", - "plt.show()" - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2.py b/doc/LectureNotes/_build/jupyter_execute/chapter2.py deleted file mode 100644 index f0b72ab51..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter2.py +++ /dev/null @@ -1,1042 +0,0 @@ -# Resampling Methods - -[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSept3.mp4?vrtx=view-as-webpage) - - -## Introduction - -Resampling methods are an indispensable tool in modern -statistics. They involve repeatedly drawing samples from a training -set and refitting a model of interest on each sample in order to -obtain additional information about the fitted model. For example, in -order to estimate the variability of a linear regression fit, we can -repeatedly draw different samples from the training data, fit a linear -regression to each new sample, and then examine the extent to which -the resulting fits differ. Such an approach may allow us to obtain -information that would not be available from fitting the model only -once using the original training sample. - -Two resampling methods are often used in Machine Learning analyses, -1. The **bootstrap method** - -2. and **Cross-Validation** - -In addition there are several other methods such as the Jackknife and the Blocking methods. We will discuss in particular -cross-validation and the bootstrap method. - - -Resampling approaches can be computationally expensive, because they -involve fitting the same statistical method multiple times using -different subsets of the training data. However, due to recent -advances in computing power, the computational requirements of -resampling methods generally are not prohibitive. In this chapter, we -discuss two of the most commonly used resampling methods, -cross-validation and the bootstrap. Both methods are important tools -in the practical application of many statistical learning -procedures. For example, cross-validation can be used to estimate the -test error associated with a given statistical learning method in -order to evaluate its performance, or to select the appropriate level -of flexibility. The process of evaluating a model’s performance is -known as model assessment, whereas the process of selecting the proper -level of flexibility for a model is known as model selection. The -bootstrap is widely used. - - -* Our simulations can be treated as *computer experiments*. This is particularly the case for Monte Carlo methods - -* The results can be analysed with the same statistical tools as we would use analysing experimental data. - -* As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors. - -## Reminder on Statistics - - -* As in other experiments, many numerical experiments have two classes of errors: - - * Statistical errors - - * Systematical errors - - -* Statistical errors can be estimated using standard tools from statistics - -* Systematical errors are method specific and must be treated differently from case to case. - -The -advantage of doing linear regression is that we actually end up with -analytical expressions for several statistical quantities. -Standard least squares and Ridge regression allow us to -derive quantities like the variance and other expectation values in a -rather straightforward way. - - -It is assumed that $\varepsilon_i -\sim \mathcal{N}(0, \sigma^2)$ and the $\varepsilon_{i}$ are -independent, i.e.: - -$$ -\begin{align*} -\mbox{Cov}(\varepsilon_{i_1}, -\varepsilon_{i_2}) & = \left\{ \begin{array}{lcc} \sigma^2 & \mbox{if} -& i_1 = i_2, \\ 0 & \mbox{if} & i_1 \not= i_2. \end{array} \right. -\end{align*} -$$ - -The randomness of $\varepsilon_i$ implies that -$\mathbf{y}_i$ is also a random variable. In particular, -$\mathbf{y}_i$ is normally distributed, because $\varepsilon_i \sim -\mathcal{N}(0, \sigma^2)$ and $\mathbf{X}_{i,\ast} \, \boldsymbol{\beta}$ is a -non-random scalar. To specify the parameters of the distribution of -$\mathbf{y}_i$ we need to calculate its first two moments. - -Recall that $\boldsymbol{X}$ is a matrix of dimensionality $n\times p$. The -notation above $\mathbf{X}_{i,\ast}$ means that we are looking at the -row number $i$ and perform a sum over all values $p$. - - -The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off) -that there exists a function $f(\boldsymbol{x})$ and a normal distributed error $\boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2)$ -which describe our data - -$$ -\boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} -$$ - -We approximate this function with our model from the solution of the linear regression equations, that is our -function $f$ is approximated by $\boldsymbol{\tilde{y}}$ where we want to minimize $(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2$, our MSE, with - -$$ -\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. -$$ - -We can calculate the expectation value of $\boldsymbol{y}$ for a given element $i$ - -$$ -\begin{align*} -\mathbb{E}(y_i) & = -\mathbb{E}(\mathbf{X}_{i, \ast} \, \boldsymbol{\beta}) + \mathbb{E}(\varepsilon_i) -\, \, \, = \, \, \, \mathbf{X}_{i, \ast} \, \beta, -\end{align*} -$$ - -while -its variance is - -$$ -\begin{align*} \mbox{Var}(y_i) & = \mathbb{E} \{ [y_i -- \mathbb{E}(y_i)]^2 \} \, \, \, = \, \, \, \mathbb{E} ( y_i^2 ) - -[\mathbb{E}(y_i)]^2 \\ & = \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, -\beta + \varepsilon_i )^2] - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 \\ & -= \mathbb{E} [ ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 \varepsilon_i -\mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + \varepsilon_i^2 ] - ( \mathbf{X}_{i, -\ast} \, \beta)^2 \\ & = ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 + 2 -\mathbb{E}(\varepsilon_i) \mathbf{X}_{i, \ast} \, \boldsymbol{\beta} + -\mathbb{E}(\varepsilon_i^2 ) - ( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta})^2 -\\ & = \mathbb{E}(\varepsilon_i^2 ) \, \, \, = \, \, \, -\mbox{Var}(\varepsilon_i) \, \, \, = \, \, \, \sigma^2. -\end{align*} -$$ - -Hence, $y_i \sim \mathcal{N}( \mathbf{X}_{i, \ast} \, \boldsymbol{\beta}, \sigma^2)$, that is $\boldsymbol{y}$ follows a normal distribution with -mean value $\boldsymbol{X}\boldsymbol{\beta}$ and variance $\sigma^2$ (not be confused with the singular values of the SVD). - - -With the OLS expressions for the parameters $\boldsymbol{\beta}$ we can evaluate the expectation value - -$$ -\mathbb{E}(\boldsymbol{\beta}) = \mathbb{E}[ (\mathbf{X}^{\top} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1}\mathbf{X}^{T} \mathbb{E}[ \mathbf{Y}]=(\mathbf{X}^{T} \mathbf{X})^{-1} \mathbf{X}^{T}\mathbf{X}\boldsymbol{\beta}=\boldsymbol{\beta}. -$$ - -This means that the estimator of the regression parameters is unbiased. - -We can also calculate the variance - -The variance of $\boldsymbol{\beta}$ is - -$$ -\begin{eqnarray*} -\mbox{Var}(\boldsymbol{\beta}) & = & \mathbb{E} \{ [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})] [\boldsymbol{\beta} - \mathbb{E}(\boldsymbol{\beta})]^{T} \} -\\ -& = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} - \boldsymbol{\beta}]^{T} \} -\\ -% & = & \mathbb{E} \{ [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}] \, [(\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y}]^{T} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & \mathbb{E} \{ (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \mathbf{Y} \, \mathbf{Y}^{T} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} \} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \mathbb{E} \{ \mathbf{Y} \, \mathbf{Y}^{T} \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\\ -& = & (\mathbf{X}^{T} \mathbf{X})^{-1} \, \mathbf{X}^{T} \, \{ \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + \sigma^2 \} \, \mathbf{X} \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -% \\ -% & = & (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^T \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T % \mathbf{X})^{-1} -% \\ -% & & + \, \, \sigma^2 \, (\mathbf{X}^T \mathbf{X})^{-1} \, \mathbf{X}^T \, \mathbf{X} \, (\mathbf{X}^T \mathbf{X})^{-1} - \boldsymbol{\beta} \boldsymbol{\beta}^T -\\ -& = & \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} + \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1} - \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} -\, \, \, = \, \, \, \sigma^2 \, (\mathbf{X}^{T} \mathbf{X})^{-1}, -\end{eqnarray*} -$$ - -where we have used that $\mathbb{E} (\mathbf{Y} \mathbf{Y}^{T}) = -\mathbf{X} \, \boldsymbol{\beta} \, \boldsymbol{\beta}^{T} \, \mathbf{X}^{T} + -\sigma^2 \, \mathbf{I}_{nn}$. From $\mbox{Var}(\boldsymbol{\beta}) = \sigma^2 -\, (\mathbf{X}^{T} \mathbf{X})^{-1}$, one obtains an estimate of the -variance of the estimate of the $j$-th regression coefficient: -$\boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 \sqrt{ -[(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} }$. This may be used to -construct a confidence interval for the estimates. - - -In a similar way, we can obtain analytical expressions for say the -expectation values of the parameters $\boldsymbol{\beta}$ and their variance -when we employ Ridge regression, allowing us again to define a confidence interval. - -It is rather straightforward to show that - -$$ -\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\boldsymbol{\beta}^{\mathrm{OLS}}. -$$ - -We see clearly that -$\mathbb{E} \big[ \boldsymbol{\beta}^{\mathrm{Ridge}} \big] \not= \boldsymbol{\beta}^{\mathrm{OLS}}$ for any $\lambda > 0$. We say then that the ridge estimator is biased. - -We can also compute the variance as - -$$ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{Ridge}}]=\sigma^2[ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1} \mathbf{X}^{T} \mathbf{X} \{ [ \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}, -$$ - -and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of Ridge parameters $\boldsymbol{\beta}$ goes to zero. - -With this, we can compute the difference - -$$ -\mbox{Var}[\boldsymbol{\beta}^{\mathrm{OLS}}]-\mbox{Var}(\boldsymbol{\beta}^{\mathrm{Ridge}})=\sigma^2 [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}[ 2\lambda\mathbf{I} + \lambda^2 (\mathbf{X}^{T} \mathbf{X})^{-1} ] \{ [ \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T}. -$$ - -The difference is non-negative definite since each component of the -matrix product is non-negative definite. -This means the variance we obtain with the standard OLS will always for $\lambda > 0$ be larger than the variance of $\boldsymbol{\beta}$ obtained with the Ridge estimator. This has interesting consequences when we discuss the so-called bias-variance trade-off below. - - - -## Resampling methods - -With all these analytical equations for both the OLS and Ridge -regression, we will now outline how to assess a given model. This will -lead us to a discussion of the so-called bias-variance tradeoff (see -below) and so-called resampling methods. - -One of the quantities we have discussed as a way to measure errors is -the mean-squared error (MSE), mainly used for fitting of continuous -functions. Another choice is the absolute error. - -In the discussions below we will focus on the MSE and in particular since we will split the data into test and training data, -we discuss the -1. prediction error or simply the **test error** $\mathrm{Err_{Test}}$, where we have a fixed training set and the test error is the MSE arising from the data reserved for testing. We discuss also the - -2. training error $\mathrm{Err_{Train}}$, which is the average loss over the training data. - -As our model becomes more and more complex, more of the training data tends to used. The training may thence adapt to more complicated structures in the data. This may lead to a decrease in the bias (see below for code example) and a slight increase of the variance for the test error. -For a certain level of complexity the test error will reach minimum, before starting to increase again. The -training error reaches a saturation. - - - -Two famous -resampling methods are the **independent bootstrap** and **the jackknife**. - -The jackknife is a special case of the independent bootstrap. Still, the jackknife was made -popular prior to the independent bootstrap. And as the popularity of -the independent bootstrap soared, new variants, such as **the dependent bootstrap**. - -The Jackknife and independent bootstrap work for -independent, identically distributed random variables. -If these conditions are not -satisfied, the methods will fail. Yet, it should be said that if the data are -independent, identically distributed, and we only want to estimate the -variance of $\overline{X}$ (which often is the case), then there is no -need for bootstrapping. - - -The Jackknife works by making many replicas of the estimator $\widehat{\theta}$. -The jackknife is a resampling method where we systematically leave out one observation from the vector of observed values $\boldsymbol{x} = (x_1,x_2,\cdots,X_n)$. -Let $\boldsymbol{x}_i$ denote the vector - -$$ -\boldsymbol{x}_i = (x_1,x_2,\cdots,x_{i-1},x_{i+1},\cdots,x_n), -$$ - -which equals the vector $\boldsymbol{x}$ with the exception that observation -number $i$ is left out. Using this notation, define -$\widehat{\theta}_i$ to be the estimator -$\widehat{\theta}$ computed using $\vec{X}_i$. - -from numpy import * -from numpy.random import randint, randn -from time import time - -def jackknife(data, stat): - n = len(data);t = zeros(n); inds = arange(n); t0 = time() - ## 'jackknifing' by leaving out an observation for each i - for i in range(n): - t[i] = stat(delete(data,i) ) - - # analysis - print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :") - print("original bias std. error") - print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5)) - - return t - - -# Returns mean of data samples -def stat(data): - return mean(data) - - -mu, sigma = 100, 15 -datapoints = 10000 -x = mu + sigma*random.randn(datapoints) -# jackknife returns the data sample -t = jackknife(x, stat) - -### Bootstrap - -Bootstrapping is a nonparametric approach to statistical inference -that substitutes computation for more traditional distributional -assumptions and asymptotic results. Bootstrapping offers a number of -advantages: -1. The bootstrap is quite general, although there are some cases in which it fails. - -2. Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small. - -3. It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically. - -4. It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples). - -Since $\widehat{\theta} = \widehat{\theta}(\boldsymbol{X})$ is a function of random variables, -$\widehat{\theta}$ itself must be a random variable. Thus it has -a pdf, call this function $p(\boldsymbol{t})$. The aim of the bootstrap is to -estimate $p(\boldsymbol{t})$ by the relative frequency of -$\widehat{\theta}$. You can think of this as using a histogram -in the place of $p(\boldsymbol{t})$. If the relative frequency closely -resembles $p(\vec{t})$, then using numerics, it is straight forward to -estimate all the interesting parameters of $p(\boldsymbol{t})$ using point -estimators. - - - -In the case that $\widehat{\theta}$ has -more than one component, and the components are independent, we use the -same estimator on each component separately. If the probability -density function of $X_i$, $p(x)$, had been known, then it would have -been straight forward to do this by: -1. Drawing lots of numbers from $p(x)$, suppose we call one such set of numbers $(X_1^*, X_2^*, \cdots, X_n^*)$. - -2. Then using these numbers, we could compute a replica of $\widehat{\theta}$ called $\widehat{\theta}^*$. - -By repeated use of (1) and (2), many -estimates of $\widehat{\theta}$ could have been obtained. The -idea is to use the relative frequency of $\widehat{\theta}^*$ -(think of a histogram) as an estimate of $p(\boldsymbol{t})$. - - -But -unless there is enough information available about the process that -generated $X_1,X_2,\cdots,X_n$, $p(x)$ is in general -unknown. Therefore, [Efron in 1979](https://projecteuclid.org/euclid.aos/1176344552) asked the -question: What if we replace $p(x)$ by the relative frequency -of the observation $X_i$; if we draw observations in accordance with -the relative frequency of the observations, will we obtain the same -result in some asymptotic sense? The answer is yes. - - -Instead of generating the histogram for the relative -frequency of the observation $X_i$, just draw the values -$(X_1^*,X_2^*,\cdots,X_n^*)$ with replacement from the vector -$\boldsymbol{X}$. - - -The independent bootstrap works like this: - -1. Draw with replacement $n$ numbers for the observed variables $\boldsymbol{x} = (x_1,x_2,\cdots,x_n)$. - -2. Define a vector $\boldsymbol{x}^*$ containing the values which were drawn from $\boldsymbol{x}$. - -3. Using the vector $\boldsymbol{x}^*$ compute $\widehat{\theta}^*$ by evaluating $\widehat \theta$ under the observations $\boldsymbol{x}^*$. - -4. Repeat this process $k$ times. - -When you are done, you can draw a histogram of the relative frequency -of $\widehat \theta^*$. This is your estimate of the probability -distribution $p(t)$. Using this probability distribution you can -estimate any statistics thereof. In principle you never draw the -histogram of the relative frequency of $\widehat{\theta}^*$. Instead -you use the estimators corresponding to the statistic of interest. For -example, if you are interested in estimating the variance of $\widehat -\theta$, apply the etsimator $\widehat \sigma^2$ to the values -$\widehat \theta ^*$. - - - -The following code starts with a Gaussian distribution with mean value -$\mu =100$ and variance $\sigma=15$. We use this to generate the data -used in the bootstrap analysis. The bootstrap analysis returns a data -set after a given number of bootstrap operations (as many as we have -data points). This data set consists of estimated mean values for each -bootstrap operation. The histogram generated by the bootstrap method -shows that the distribution for these mean values is also a Gaussian, -centered around the mean value $\mu=100$ but with standard deviation -$\sigma/\sqrt{n}$, where $n$ is the number of bootstrap samples (in -this case the same as the number of original data points). The value -of the standard deviation is what we expect from the central limit -theorem. - -%matplotlib inline - -from numpy import * -from numpy.random import randint, randn -from time import time -import matplotlib.mlab as mlab -import matplotlib.pyplot as plt - -# Returns mean of bootstrap samples -def stat(data): - return mean(data) - -# Bootstrap algorithm -def bootstrap(data, statistic, R): - t = zeros(R); n = len(data); inds = arange(n); t0 = time() - # non-parametric bootstrap - for i in range(R): - t[i] = statistic(data[randint(0,n,n)]) - - # analysis - print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :") - print("original bias std. error") - print("%8g %8g %14g %15g" % (statistic(data), std(data),mean(t),std(t))) - return t - - -mu, sigma = 100, 15 -datapoints = 10000 -x = mu + sigma*random.randn(datapoints) -# bootstrap returns the data sample -t = bootstrap(x, stat, datapoints) -# the histogram of the bootstrapped data -n, binsboot, patches = plt.hist(t, 50, normed=1, facecolor='red', alpha=0.75) - -# add a 'best fit' line -y = mlab.normpdf( binsboot, mean(t), std(t)) -lt = plt.plot(binsboot, y, 'r--', linewidth=1) -plt.xlabel('Smarts') -plt.ylabel('Probability') -plt.axis([99.5, 100.6, 0, 3.0]) -plt.grid(True) - -plt.show() - -## Various steps in cross-validation - -When the repetitive splitting of the data set is done randomly, -samples may accidently end up in a fast majority of the splits in -either training or test set. Such samples may have an unbalanced -influence on either model building or prediction evaluation. To avoid -this $k$-fold cross-validation structures the data splitting. The -samples are divided into $k$ more or less equally sized exhaustive and -mutually exclusive subsets. In turn (at each split) one of these -subsets plays the role of the test set while the union of the -remaining subsets constitutes the training set. Such a splitting -warrants a balanced representation of each sample in both training and -test set over the splits. Still the division into the $k$ subsets -involves a degree of randomness. This may be fully excluded when -choosing $k=n$. This particular case is referred to as leave-one-out -cross-validation (LOOCV). - - -* Define a range of interest for the penalty parameter. - -* Divide the data set into training and test set comprising samples $\{1, \ldots, n\} \setminus i$ and $\{ i \}$, respectively. - -* Fit the linear regression model by means of ridge estimation for each $\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\boldsymbol{\sigma}_{-i}^2(\lambda)$, as - -$$ -\begin{align*} -\boldsymbol{\beta}_{-i}(\lambda) & = ( \boldsymbol{X}_{-i, \ast}^{T} -\boldsymbol{X}_{-i, \ast} + \lambda \boldsymbol{I}_{pp})^{-1} -\boldsymbol{X}_{-i, \ast}^{T} \boldsymbol{y}_{-i} -\end{align*} -$$ - -* Evaluate the prediction performance of these models on the test set by $\log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}$. Or, by the prediction error $|y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)|$, the relative error, the error squared or the R2 score function. - -* Repeat the first three steps such that each sample plays the role of the test set once. - -* Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as - -$$ -\begin{align*} -\frac{1}{n} \sum_{i = 1}^n \log\{L[y_i, \mathbf{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\}. -\end{align*} -$$ - -For the various values of $k$ - -1. shuffle the dataset randomly. - -2. Split the dataset into $k$ groups. - -3. For each unique group: - -a. Decide which group to use as set for test data - -b. Take the remaining groups as a training data set - -c. Fit a model on the training set and evaluate it on the test set - -d. Retain the evaluation score and discard the model - - -5. Summarize the model using the sample of model evaluation scores - -The code here uses Ridge regression with cross-validation (CV) resampling and $k$-fold CV in order to fit a specific polynomial. - -import numpy as np -import matplotlib.pyplot as plt -from sklearn.model_selection import KFold -from sklearn.linear_model import Ridge -from sklearn.model_selection import cross_val_score -from sklearn.preprocessing import PolynomialFeatures - -# A seed just to ensure that the random numbers are the same for every run. -# Useful for eventual debugging. -np.random.seed(3155) - -# Generate the data. -nsamples = 100 -x = np.random.randn(nsamples) -y = 3*x**2 + np.random.randn(nsamples) - -## Cross-validation on Ridge regression using KFold only - -# Decide degree on polynomial to fit -poly = PolynomialFeatures(degree = 6) - -# Decide which values of lambda to use -nlambdas = 500 -lambdas = np.logspace(-3, 5, nlambdas) - -# Initialize a KFold instance -k = 5 -kfold = KFold(n_splits = k) - -# Perform the cross-validation to estimate MSE -scores_KFold = np.zeros((nlambdas, k)) - -i = 0 -for lmb in lambdas: - ridge = Ridge(alpha = lmb) - j = 0 - for train_inds, test_inds in kfold.split(x): - xtrain = x[train_inds] - ytrain = y[train_inds] - - xtest = x[test_inds] - ytest = y[test_inds] - - Xtrain = poly.fit_transform(xtrain[:, np.newaxis]) - ridge.fit(Xtrain, ytrain[:, np.newaxis]) - - Xtest = poly.fit_transform(xtest[:, np.newaxis]) - ypred = ridge.predict(Xtest) - - scores_KFold[i,j] = np.sum((ypred - ytest[:, np.newaxis])**2)/np.size(ypred) - - j += 1 - i += 1 - - -estimated_mse_KFold = np.mean(scores_KFold, axis = 1) - -## Cross-validation using cross_val_score from sklearn along with KFold - -# kfold is an instance initialized above as: -# kfold = KFold(n_splits = k) - -estimated_mse_sklearn = np.zeros(nlambdas) -i = 0 -for lmb in lambdas: - ridge = Ridge(alpha = lmb) - - X = poly.fit_transform(x[:, np.newaxis]) - estimated_mse_folds = cross_val_score(ridge, X, y[:, np.newaxis], scoring='neg_mean_squared_error', cv=kfold) - - # cross_val_score return an array containing the estimated negative mse for every fold. - # we have to the the mean of every array in order to get an estimate of the mse of the model - estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds) - - i += 1 - -## Plot and compare the slightly different ways to perform cross-validation - -plt.figure() - -plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score') -plt.plot(np.log10(lambdas), estimated_mse_KFold, 'r--', label = 'KFold') - -plt.xlabel('log10(lambda)') -plt.ylabel('mse') - -plt.legend() - -plt.show() - -## The bias-variance tradeoff - - -We will discuss the bias-variance tradeoff in the context of -continuous predictions such as regression. However, many of the -intuitions and ideas discussed here also carry over to classification -tasks. Consider a dataset $\mathcal{L}$ consisting of the data -$\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}$. - -Let us assume that the true data is generated from a noisy model - -$$ -\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon} -$$ - -where $\epsilon$ is normally distributed with mean zero and standard deviation $\sigma^2$. - -In our derivation of the ordinary least squares method we defined then -an approximation to the function $f$ in terms of the parameters -$\boldsymbol{\beta}$ and the design matrix $\boldsymbol{X}$ which embody our model, -that is $\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta}$. - -Thereafter we found the parameters $\boldsymbol{\beta}$ by optimizing the means squared error via the so-called cost function - -$$ -C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]. -$$ - -We can rewrite this as - -$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2. -$$ - -The three terms represent the square of the bias of the learning -method, which can be thought of as the error caused by the simplifying -assumptions built into the method. The second term represents the -variance of the chosen model and finally the last terms is variance of -the error $\boldsymbol{\epsilon}$. - -To derive this equation, we need to recall that the variance of $\boldsymbol{y}$ and $\boldsymbol{\epsilon}$ are both equal to $\sigma^2$. The mean value of $\boldsymbol{\epsilon}$ is by definition equal to zero. Furthermore, the function $f$ is not a stochastics variable, idem for $\boldsymbol{\tilde{y}}$. -We use a more compact notation in terms of the expectation value - -$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}})^2\right], -$$ - -and adding and subtracting $\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]$ we get - -$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{f}+\boldsymbol{\epsilon}-\boldsymbol{\tilde{y}}+\mathbb{E}\left[\boldsymbol{\tilde{y}}\right]-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right], -$$ - -which, using the abovementioned expectation values can be rewritten as - -$$ -\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\mathbb{E}\left[(\boldsymbol{y}-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2\right]+\mathrm{Var}\left[\boldsymbol{\tilde{y}}\right]+\sigma^2, -$$ - -that is the rewriting in terms of the so-called bias, the variance of the model $\boldsymbol{\tilde{y}}$ and the variance of $\boldsymbol{\epsilon}$. - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.linear_model import LinearRegression, Ridge, Lasso -from sklearn.preprocessing import PolynomialFeatures -from sklearn.model_selection import train_test_split -from sklearn.pipeline import make_pipeline -from sklearn.utils import resample - -np.random.seed(2018) - -n = 500 -n_boostraps = 100 -degree = 18 # A quite high value, just to show. -noise = 0.1 - -# Make data set. -x = np.linspace(-1, 3, n).reshape(-1, 1) -y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2) + np.random.normal(0, 0.1, x.shape) - -# Hold out some test data that is never used in training. -x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2) - -# Combine x transformation and model into one operation. -# Not neccesary, but convenient. -model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False)) - -# The following (m x n_bootstraps) matrix holds the column vectors y_pred -# for each bootstrap iteration. -y_pred = np.empty((y_test.shape[0], n_boostraps)) -for i in range(n_boostraps): - x_, y_ = resample(x_train, y_train) - - # Evaluate the new model on the same test data each time. - y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel() - -# Note: Expectations and variances taken w.r.t. different training -# data sets, hence the axis=1. Subsequent means are taken across the test data -# set in order to obtain a total value, but before this we have error/bias/variance -# calculated per data point in the test set. -# Note 2: The use of keepdims=True is important in the calculation of bias as this -# maintains the column vector form. Dropping this yields very unexpected results. -error = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) ) -bias = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 ) -variance = np.mean( np.var(y_pred, axis=1, keepdims=True) ) -print('Error:', error) -print('Bias^2:', bias) -print('Var:', variance) -print('{} >= {} + {} = {}'.format(error, bias, variance, bias+variance)) - -plt.plot(x[::5, :], y[::5, :], label='f(x)') -plt.scatter(x_test, y_test, label='Data points') -plt.scatter(x_test, np.mean(y_pred, axis=1), label='Pred') -plt.legend() -plt.show() - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.linear_model import LinearRegression, Ridge, Lasso -from sklearn.preprocessing import PolynomialFeatures -from sklearn.model_selection import train_test_split -from sklearn.pipeline import make_pipeline -from sklearn.utils import resample - -np.random.seed(2018) - -n = 40 -n_boostraps = 100 -maxdegree = 14 - - -# Make data set. -x = np.linspace(-3, 3, n).reshape(-1, 1) -y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape) -error = np.zeros(maxdegree) -bias = np.zeros(maxdegree) -variance = np.zeros(maxdegree) -polydegree = np.zeros(maxdegree) -x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2) - -for degree in range(maxdegree): - model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False)) - y_pred = np.empty((y_test.shape[0], n_boostraps)) - for i in range(n_boostraps): - x_, y_ = resample(x_train, y_train) - y_pred[:, i] = model.fit(x_, y_).predict(x_test).ravel() - - polydegree[degree] = degree - error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) ) - bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 ) - variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) ) - print('Polynomial degree:', degree) - print('Error:', error[degree]) - print('Bias^2:', bias[degree]) - print('Var:', variance[degree]) - print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree])) - -plt.plot(polydegree, error, label='Error') -plt.plot(polydegree, bias, label='bias') -plt.plot(polydegree, variance, label='Variance') -plt.legend() -plt.show() - -The bias-variance tradeoff summarizes the fundamental tension in -machine learning, particularly supervised learning, between the -complexity of a model and the amount of training data needed to train -it. Since data is often limited, in practice it is often useful to -use a less-complex model with higher bias, that is a model whose asymptotic -performance is worse than another model because it is easier to -train and less sensitive to sampling noise arising from having a -finite-sized training dataset (smaller variance). - - - -The above equations tell us that in -order to minimize the expected test error, we need to select a -statistical learning method that simultaneously achieves low variance -and low bias. Note that variance is inherently a nonnegative quantity, -and squared bias is also nonnegative. Hence, we see that the expected -test MSE can never lie below $Var(\epsilon)$, the irreducible error. - - -What do we mean by the variance and bias of a statistical learning -method? The variance refers to the amount by which our model would change if we -estimated it using a different training data set. Since the training -data are used to fit the statistical learning method, different -training data sets will result in a different estimate. But ideally the -estimate for our model should not vary too much between training -sets. However, if a method has high variance then small changes in -the training data can result in large changes in the model. In general, more -flexible statistical methods have higher variance. - - -You may also find this recent [article](https://www.pnas.org/content/116/32/15849) of interest. - -""" -============================ -Underfitting vs. Overfitting -============================ - -This example demonstrates the problems of underfitting and overfitting and -how we can use linear regression with polynomial features to approximate -nonlinear functions. The plot shows the function that we want to approximate, -which is a part of the cosine function. In addition, the samples from the -real function and the approximations of different models are displayed. The -models have polynomial features of different degrees. We can see that a -linear function (polynomial with degree 1) is not sufficient to fit the -training samples. This is called **underfitting**. A polynomial of degree 4 -approximates the true function almost perfectly. However, for higher degrees -the model will **overfit** the training data, i.e. it learns the noise of the -training data. -We evaluate quantitatively **overfitting** / **underfitting** by using -cross-validation. We calculate the mean squared error (MSE) on the validation -set, the higher, the less likely the model generalizes correctly from the -training data. -""" - -print(__doc__) - -import numpy as np -import matplotlib.pyplot as plt -from sklearn.pipeline import Pipeline -from sklearn.preprocessing import PolynomialFeatures -from sklearn.linear_model import LinearRegression -from sklearn.model_selection import cross_val_score - - -def true_fun(X): - return np.cos(1.5 * np.pi * X) - -np.random.seed(0) - -n_samples = 30 -degrees = [1, 4, 15] - -X = np.sort(np.random.rand(n_samples)) -y = true_fun(X) + np.random.randn(n_samples) * 0.1 - -plt.figure(figsize=(14, 5)) -for i in range(len(degrees)): - ax = plt.subplot(1, len(degrees), i + 1) - plt.setp(ax, xticks=(), yticks=()) - - polynomial_features = PolynomialFeatures(degree=degrees[i], - include_bias=False) - linear_regression = LinearRegression() - pipeline = Pipeline([("polynomial_features", polynomial_features), - ("linear_regression", linear_regression)]) - pipeline.fit(X[:, np.newaxis], y) - - # Evaluate the models using crossvalidation - scores = cross_val_score(pipeline, X[:, np.newaxis], y, - scoring="neg_mean_squared_error", cv=10) - - X_test = np.linspace(0, 1, 100) - plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model") - plt.plot(X_test, true_fun(X_test), label="True function") - plt.scatter(X, y, edgecolor='b', s=20, label="Samples") - plt.xlabel("x") - plt.ylabel("y") - plt.xlim((0, 1)) - plt.ylim((-2, 2)) - plt.legend(loc="best") - plt.title("Degree {}\nMSE = {:.2e}(+/- {:.2e})".format( - degrees[i], -scores.mean(), scores.std())) -plt.show() - -# Common imports -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression, Ridge, Lasso -from sklearn.model_selection import train_test_split -from sklearn.utils import resample -from sklearn.metrics import mean_squared_error -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("EoS.csv"),'r') - -# Read the EoS data as csv file and organize the data into two arrays with density and energies -EoS = pd.read_csv(infile, names=('Density', 'Energy')) -EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') -EoS = EoS.dropna() -Energies = EoS['Energy'] -Density = EoS['Density'] -# The design matrix now as function of various polytrops - -Maxpolydegree = 30 -X = np.zeros((len(Density),Maxpolydegree)) -X[:,0] = 1.0 -testerror = np.zeros(Maxpolydegree) -trainingerror = np.zeros(Maxpolydegree) -polynomial = np.zeros(Maxpolydegree) - -trials = 100 -for polydegree in range(1, Maxpolydegree): - polynomial[polydegree] = polydegree - for degree in range(polydegree): - X[:,degree] = Density**(degree/3.0) - -# loop over trials in order to estimate the expectation value of the MSE - testerror[polydegree] = 0.0 - trainingerror[polydegree] = 0.0 - for samples in range(trials): - x_train, x_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2) - model = LinearRegression(fit_intercept=True).fit(x_train, y_train) - ypred = model.predict(x_train) - ytilde = model.predict(x_test) - testerror[polydegree] += mean_squared_error(y_test, ytilde) - trainingerror[polydegree] += mean_squared_error(y_train, ypred) - - testerror[polydegree] /= trials - trainingerror[polydegree] /= trials - print("Degree of polynomial: %3d"% polynomial[polydegree]) - print("Mean squared error on training data: %.8f" % trainingerror[polydegree]) - print("Mean squared error on test data: %.8f" % testerror[polydegree]) - -plt.plot(polynomial, np.log10(trainingerror), label='Training Error') -plt.plot(polynomial, np.log10(testerror), label='Test Error') -plt.xlabel('Polynomial degree') -plt.ylabel('log10[MSE]') -plt.legend() -plt.show() - -# Common imports -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression, Ridge, Lasso -from sklearn.metrics import mean_squared_error -from sklearn.model_selection import KFold -from sklearn.model_selection import cross_val_score - - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("EoS.csv"),'r') - -# Read the EoS data as csv file and organize the data into two arrays with density and energies -EoS = pd.read_csv(infile, names=('Density', 'Energy')) -EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce') -EoS = EoS.dropna() -Energies = EoS['Energy'] -Density = EoS['Density'] -# The design matrix now as function of various polytrops - -Maxpolydegree = 30 -X = np.zeros((len(Density),Maxpolydegree)) -X[:,0] = 1.0 -estimated_mse_sklearn = np.zeros(Maxpolydegree) -polynomial = np.zeros(Maxpolydegree) -k =5 -kfold = KFold(n_splits = k) - -for polydegree in range(1, Maxpolydegree): - polynomial[polydegree] = polydegree - for degree in range(polydegree): - X[:,degree] = Density**(degree/3.0) - OLS = LinearRegression() -# loop over trials in order to estimate the expectation value of the MSE - estimated_mse_folds = cross_val_score(OLS, X, Energies, scoring='neg_mean_squared_error', cv=kfold) -#[:, np.newaxis] - estimated_mse_sklearn[polydegree] = np.mean(-estimated_mse_folds) - -plt.plot(polynomial, np.log10(estimated_mse_sklearn), label='Test Error') -plt.xlabel('Polynomial degree') -plt.ylabel('log10[MSE]') -plt.legend() -plt.show() - -import numpy as np -import matplotlib.pyplot as plt -from sklearn.model_selection import KFold -from sklearn.linear_model import Ridge -from sklearn.model_selection import cross_val_score -from sklearn.preprocessing import PolynomialFeatures - -# A seed just to ensure that the random numbers are the same for every run. -np.random.seed(3155) -# Generate the data. -n = 100 -x = np.linspace(-3, 3, n).reshape(-1, 1) -y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape) -# Decide degree on polynomial to fit -poly = PolynomialFeatures(degree = 10) - -# Decide which values of lambda to use -nlambdas = 500 -lambdas = np.logspace(-3, 5, nlambdas) -# Initialize a KFold instance -k = 5 -kfold = KFold(n_splits = k) -estimated_mse_sklearn = np.zeros(nlambdas) -i = 0 -for lmb in lambdas: - ridge = Ridge(alpha = lmb) - estimated_mse_folds = cross_val_score(ridge, x, y, scoring='neg_mean_squared_error', cv=kfold) - estimated_mse_sklearn[i] = np.mean(-estimated_mse_folds) - i += 1 -plt.figure() -plt.plot(np.log10(lambdas), estimated_mse_sklearn, label = 'cross_val_score') -plt.xlabel('log10(lambda)') -plt.ylabel('MSE') -plt.legend() -plt.show() \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2_25_2.png b/doc/LectureNotes/_build/jupyter_execute/chapter2_25_2.png deleted file mode 100644 index dfbfcf4b0e1f66cb6e4be62963eb33399771a88d..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 4789 zcmai&2~-p3;>8mppg^FmxFBd5q{^m5b_FeNV+yzp$iB-aB(kp|sa2m=MJE-MwLDOC zLSzqN5eBVdL`Vw;WDkO%>_Q=dK=OjGwfg@3zjtzy$vHFMO!D3N-Fv^ee9ZPRUSXpG z27|$aM-H68V6air@51GB(ofan_YO-pX5j}^ z4GSQJ5cH)l-7Ws%;p8xL1B2jC2k4VR{S2O`l&4EivVwfXF${xIT)p(hVp7xBV=$|} z1`q5%6`elTosi$7> zys`Px{mzRNL6RBma-7CsPX$5HLl+P$Q>QIzr_;>p!zTQ$a~B3p9_#?XgqXV6sC{th ztWp#>e4|ev+-%sT^}zaK6%g@mbY>ibUQl&AT@^XzG!b&7`Ch1?J4$j8YSb+yS^pq< z=vs90N09%^c-641RS((F5C7S3hX0pMeslIlClFE#eaL&##7G$q?LOOHY3|2d0nBo&c zp-^TtZvcy{bAAxDqjym}G;@?s+Mrt(e-c`IOecnqn7cqFkdXy}Py$y-@RZK_W#TL z>0bIvOarSe%%Z3ImnWZRo7hxn$66A)?rz`pyOE84!;#N!kS|BZ?8r4ock3 z11tFjG!lk;igy;CrX|T#Q4doY$S?=MKEyr5WXe%0k($C;*S6`)y|;ESUmi{NrQ{;| z50G|N!dL7PL|<mddh&k6W2n6@3dj9?hGjK528`nHmoz8-hF1sf_jpMvpf+*a?(a`j|)cI>n@ge^A_7Z(QKic&tfcff*{+u(Eg|%#Xp@J)L`?;|fOL9D4 z1H$&4zwrFRy+09`y*6V=pZUs5BRgfcW7D7J{%QArnDbBtG3}~*ZQgOy|AlqK#W+}1 zYpw-IOdhBKCIV(uwJLNt9{>%TP+LZ#XQyS)>_qD1dI0oW?>i`#P#cipwWt=Fq}7gh zQLL^-+~oGG7$uDfaCWSDL3t#cyWPpSD~M0(7%VT#lF`S71k4Nv-cI`573(t#kJ zC)hl7$SKS&)EvxkHF-B9&tJ(d0R~Bq8(N>-!p5;I=6Q*hBrYGD++PZ zMw93JEy*wr9E4vsa=A%yQE={79dqY_sDwZuFh(8$0sC{(^$2tyqE4d4WDbf``c~Y?Y`5gM+FN$80z_62RCX~T8PN0o zi6{=iT?fp5toyO95sGkzMYJR!Mq~0I|5>Y4QWLtGym(dg$g=sR@K|zhflPtR&tgxx zkqN(6$>zyqqxTuPH)ArQ@YAc2KcaJ>Z8X3}mj!GINcd8ffz&GdU&5P_E|uiwpr#t50GLT z`xTr!v{5Rs(&RKW6Y60?wt-V{@-%6_dhwfBc(NJ!oX8B*RXF%aIz0jHbuw&de^)s(BR44$~gW;%^NUHo@v+P&g$QnMy| z$9uE*oHCXF2K$oFX&enT=M-gEPSd+?4rE8hxiReE5xSaPCXsb z$dY<**q`w0kF=mhJRacr_SE7 zd45t|ORxTpv`9a{QR|6FPS=RXHVSUHPE2 zBkA&MaaIcH%Pg}7{GsmJ)+8gni)F9M)!l(@yTTHU$koy+aAazwhr!O9WcT-UuVchT zXGZPxtsd?P1GyFo_Raye)KZm#vzEDqoyh^@skq>25*!~Rn{;mLq$<(qd6^Eoe5P#l zwDHRN;t#(YD!*Rl?YMtDK1|p93{RlX^6s|1kdHh;`*><;J`a&IZEWJmHi!zmsCq^uM6N_))0 zF{r!znMw)rHXZ-svs2wX#mm)@-Xp$$sb%cK){Z$0dxPV|sU~~D6(A8FAc7PtPL$rc z1xK*v_5kZaMC^x}{8(L(1ypEp&V%YlraB%(iYpQ}_^C;XHb?1lF43#~kqnLB`>a7* zZ8PIjQJ{xB|#P!HPTM?=KU_3QRgQx7i9R4 z*9wB1Bp6Mx<|A45Ie(D@)g6=h#+f7@UpDSOAl!^*puYKu?1ETbl<4%$8G@B|{7B1- zAwsk#8z0?oe$BgWLt?zU9II@~nBTx$2tqs81)7{H$48U>MZsu6FzcKyxmRkJ6oRET zQ*1%y${6@v>9fo(O2;&9&a>MT>^Qi7l1q2FG4NE$?QNutU>DmGuk`rL(AV{e)cunS z{XdAzuAF-ox^kz=LaejbyaybX_uG5f{8%+8LP2PNN@Ue>nFwNMd$+Bj^OL{0y_?$5 z$u*6PW4I%YrB?~pNk@xig)8LfIWvair+X9^9Qf}q&yd3z>clvGOi$W?B1kde*xUPC zUx#Wx(oJWOipZaa)pCS-(f2T1i#Lt(9{9(H`cH+**%&xa&;dLN+6{l^MLk zw7L_`-cRINW3@+%tFDa;Xs==n#%}-e?HDSza%kbw7 z(&lzdYI#Jyql|CZ!Da1ASSy6?O%4j2lFrxV7p<@6_q0$bXG$!!ts|;#kGC7rg^TlU z{G^UvNd_v&FLnR8#47De_65LxX$buL`O4o^aIuy3KWFng_TQ\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\boldsymbol{\\beta} = (\\boldsymbol{X}^{T} \\boldsymbol{X})^{-1} \\boldsymbol{X}^{T} \\boldsymbol{y},\n", - "\\label{_auto1} \\tag{1}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "has linearly dependent column vectors, we will not be able to compute the inverse\n", - "of $\\boldsymbol{X}^T\\boldsymbol{X}$ and we cannot find the parameters (estimators) $\\beta_i$. \n", - "The estimators are only well-defined if $(\\boldsymbol{X}^{T}\\boldsymbol{X})^{-1}$ exits. \n", - "This is more likely to happen when the matrix $\\boldsymbol{X}$ is high-dimensional. In this case it is likely to encounter a situation where \n", - "the regression parameters $\\beta_i$ cannot be estimated.\n", - "\n", - "A cheap *ad hoc* approach is simply to add a small diagonal component to the matrix to invert, that is we change" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^{T} \\boldsymbol{X} \\rightarrow \\boldsymbol{X}^{T} \\boldsymbol{X}+\\lambda \\boldsymbol{I},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\boldsymbol{I}$ is the identity matrix. When we discuss **Ridge** regression this is actually what we end up evaluating. The parameter $\\lambda$ is called a hyperparameter. More about this later. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "From standard linear algebra we know that a square matrix $\\boldsymbol{X}$ can be diagonalized if and only it is \n", - "a so-called [normal matrix](https://en.wikipedia.org/wiki/Normal_matrix), that is if $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times n}$\n", - "we have $\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{X}^T\\boldsymbol{X}$ or if $\\boldsymbol{X}\\in {\\mathbb{C}}^{n\\times n}$ we have $\\boldsymbol{X}\\boldsymbol{X}^{\\dagger}=\\boldsymbol{X}^{\\dagger}\\boldsymbol{X}$.\n", - "The matrix has then a set of eigenpairs" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\lambda_1,\\boldsymbol{u}_1),\\dots, (\\lambda_n,\\boldsymbol{u}_n),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the eigenvalues are given by the diagonal matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\Sigma}=\\mathrm{Diag}(\\lambda_1, \\dots,\\lambda_n).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The matrix $\\boldsymbol{X}$ can be written in terms of an orthogonal/unitary transformation $\\boldsymbol{U}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{U}\\boldsymbol{U}^T=\\boldsymbol{I}$ or $\\boldsymbol{U}\\boldsymbol{U}^{\\dagger}=\\boldsymbol{I}$.\n", - "\n", - "Not all square matrices are diagonalizable. A matrix like the one discussed above" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\begin{bmatrix} \n", - "1& -1 \\\\\n", - "1& -1\\\\\n", - "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "is not diagonalizable, it is a so-called [defective matrix](https://en.wikipedia.org/wiki/Defective_matrix). It is easy to see that the condition\n", - "$\\boldsymbol{X}\\boldsymbol{X}^T=\\boldsymbol{X}^T\\boldsymbol{X}$ is not fulfilled. \n", - "\n", - "\n", - "\n", - "## The SVD, a Fantastic Algorithm\n", - "\n", - "\n", - "However, and this is the strength of the SVD algorithm, any general\n", - "matrix $\\boldsymbol{X}$ can be decomposed in terms of a diagonal matrix and\n", - "two orthogonal/unitary matrices. The [Singular Value Decompostion\n", - "(SVD) theorem](https://en.wikipedia.org/wiki/Singular_value_decomposition)\n", - "states that a general $m\\times n$ matrix $\\boldsymbol{X}$ can be written in\n", - "terms of a diagonal matrix $\\boldsymbol{\\Sigma}$ of dimensionality $m\\times n$\n", - "and two orthognal matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$, where the first has\n", - "dimensionality $m \\times m$ and the last dimensionality $n\\times n$.\n", - "We have then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As an example, the above defective matrix can be decomposed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& 1 \\\\ 1& -1\\\\ \\end{bmatrix} \\begin{bmatrix} 2& 0 \\\\ 0& 0\\\\ \\end{bmatrix} \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& -1 \\\\ 1& 1\\\\ \\end{bmatrix}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with eigenvalues $\\sigma_1=2$ and $\\sigma_2=0$. \n", - "The SVD exits always! \n", - "\n", - "The SVD\n", - "decomposition (singular values) gives eigenvalues \n", - "$\\sigma_i\\geq\\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=p$, the\n", - "eigenvalues (singular values) are zero.\n", - "\n", - "In the general case, where our design matrix $\\boldsymbol{X}$ has dimension\n", - "$n\\times p$, the matrix is thus decomposed into an $n\\times n$\n", - "orthogonal matrix $\\boldsymbol{U}$, a $p\\times p$ orthogonal matrix $\\boldsymbol{V}$\n", - "and a diagonal matrix $\\boldsymbol{\\Sigma}$ with $r=\\mathrm{min}(n,p)$\n", - "singular values $\\sigma_i\\geq 0$ on the main diagonal and zeros filling\n", - "the rest of the matrix. There are at most $p$ singular values\n", - "assuming that $n > p$. In our regression examples for the nuclear\n", - "masses and the equation of state this is indeed the case, while for\n", - "the Ising model we have $p > n$. These are often cases that lead to\n", - "near singular or singular matrices.\n", - "\n", - "The columns of $\\boldsymbol{U}$ are called the left singular vectors while the columns of $\\boldsymbol{V}$ are the right singular vectors.\n", - "\n", - "## Economy-size SVD\n", - "\n", - "If we assume that $n > p$, then our matrix $\\boldsymbol{U}$ has dimension $n\n", - "\\times n$. The last $n-p$ columns of $\\boldsymbol{U}$ become however\n", - "irrelevant in our calculations since they are multiplied with the\n", - "zeros in $\\boldsymbol{\\Sigma}$.\n", - "\n", - "The economy-size decomposition removes extra rows or columns of zeros\n", - "from the diagonal matrix of singular values, $\\boldsymbol{\\Sigma}$, along with the columns\n", - "in either $\\boldsymbol{U}$ or $\\boldsymbol{V}$ that multiply those zeros in the expression. \n", - "Removing these zeros and columns can improve execution time\n", - "and reduce storage requirements without compromising the accuracy of\n", - "the decomposition.\n", - "\n", - "If $n > p$, we keep only the first $p$ columns of $\\boldsymbol{U}$ and $\\boldsymbol{\\Sigma}$ has dimension $p\\times p$. \n", - "If $p > n$, then only the first $n$ columns of $\\boldsymbol{V}$ are computed and $\\boldsymbol{\\Sigma}$ has dimension $n\\times n$.\n", - "The $n=p$ case is obvious, we retain the full SVD. \n", - "In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[ 1. -1. 2.]\n", - " [ 1. 0. 1.]\n", - " [ 1. 2. -1.]\n", - " [ 1. 1. 0.]]\n", - "[[ 4. 2. 2.]\n", - " [ 2. 6. -4.]\n", - " [ 2. -4. 6.]]\n", - "[[-1.18404906e-16 8.16496581e-01 -5.77350269e-01]\n", - " [-7.07106781e-01 4.08248290e-01 5.77350269e-01]\n", - " [ 7.07106781e-01 4.08248290e-01 5.77350269e-01]]\n", - "[1.00000000e+01 6.00000000e+00 9.10898112e-32]\n", - "[[ 3.33066907e-17 -7.07106781e-01 7.07106781e-01]\n", - " [ 8.16496581e-01 4.08248290e-01 4.08248290e-01]\n", - " [ 5.77350269e-01 -5.77350269e-01 -5.77350269e-01]]\n", - "[[-3.65939208e+30 3.65939208e+30 3.65939208e+30]\n", - " [ 3.65939208e+30 -3.65939208e+30 -3.65939208e+30]\n", - " [ 3.65939208e+30 -3.65939208e+30 -3.65939208e+30]]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "# SVD inversion\n", - "def SVDinv(A):\n", - " ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).\n", - " SVD is numerically more stable than the inversion algorithms provided by\n", - " numpy and scipy.linalg at the cost of being slower.\n", - " '''\n", - " U, s, VT = np.linalg.svd(A)\n", - "# print('test U')\n", - "# print( (np.transpose(U) @ U - U @np.transpose(U)))\n", - "# print('test VT')\n", - "# print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))\n", - " print(U)\n", - " print(s)\n", - " print(VT)\n", - "\n", - " D = np.zeros((len(U),len(VT)))\n", - " for i in range(0,len(VT)):\n", - " D[i,i]=s[i]\n", - " UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)\n", - " return np.matmul(V,np.matmul(invD,UT))\n", - "\n", - "\n", - "X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])\n", - "print(X)\n", - "A = np.transpose(X) @ X\n", - "print(A)\n", - "# Brute force inversion of super-collinear matrix\n", - "#B = np.linalg.inv(A)\n", - "#print(B)\n", - "C = SVDinv(A)\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The matrix $\\boldsymbol{X}$ has columns that are linearly dependent. The first\n", - "column is the row-wise sum of the other two columns. The rank of a\n", - "matrix (the column rank) is the dimension of space spanned by the\n", - "column vectors. The rank of the matrix is the number of linearly\n", - "independent columns, in this case just $2$. We see this from the\n", - "singular values when running the above code. Running the standard\n", - "inversion algorithm for matrix inversion with $\\boldsymbol{X}^T\\boldsymbol{X}$ results\n", - "in the program terminating due to a singular matrix.\n", - "\n", - "\n", - "\n", - "\n", - "There are several interesting mathematical properties which will be\n", - "relevant when we are going to discuss the differences between say\n", - "ordinary least squares (OLS) and **Ridge** regression.\n", - "\n", - "We have from OLS that the parameters of the linear approximation are given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\tilde{y}} = \\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The matrix to invert can be rewritten in terms of our SVD decomposition as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{X} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Using the orthogonality properties of $\\boldsymbol{U}$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{X} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{\\Sigma}\\boldsymbol{V}^T = \\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{D}$ being a diagonal matrix with values along the diagonal given by the singular values squared. \n", - "\n", - "This means that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{X}^T\\boldsymbol{X})\\boldsymbol{V} = \\boldsymbol{V}\\boldsymbol{D},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "that is the eigenvectors of $(\\boldsymbol{X}^T\\boldsymbol{X})$ are given by the columns of the right singular matrix of $\\boldsymbol{X}$ and the eigenvalues are the squared singular values. It is easy to show (show this) that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U} = \\boldsymbol{U}\\boldsymbol{D},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "that is, the eigenvectors of $(\\boldsymbol{X}\\boldsymbol{X})^T$ are the columns of the left singular matrix and the eigenvalues are the same. \n", - "\n", - "Going back to our OLS equation we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We will come back to this expression when we discuss Ridge regression. \n", - "\n", - "\n", - "$$ \\tilde{y}^{OLS}=\\boldsymbol{X}\\hat{\\beta}^{OLS}=\\sum_{j=1}^p \\boldsymbol{u}_j\\boldsymbol{u}_j^T\\boldsymbol{y}$$ and for Ridge we have \n", - "\n", - "$$ \\tilde{y}^{Ridge}=\\boldsymbol{X}\\hat{\\beta}^{Ridge}=\\sum_{j=1}^p \\boldsymbol{u}_j\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{u}_j^T\\boldsymbol{y}$$ . \n", - "\n", - "It is indeed the economy-sized SVD, note the summation runs up tp $$p$$ only and not $$n$$. \n", - "\n", - "Here we have that $$\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T$$, with $$\\Sigma$$ being an $$ n\\times p$$ matrix and $$\\boldsymbol{V}$$ being a $$ p\\times p$$ matrix. We also have assumed here that $$ n > p$$. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Ridge and LASSO Regression\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember11.mp4?vrtx=view-as-webpage)\n", - "\n", - "Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is \n", - "our optimization problem is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in {\\mathbb{R}}^{p}}}\\frac{1}{n}\\left\\{\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)^T\\left(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\right)\\right\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or we can state it as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", - "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\sum_{i=0}^{n-1}\\left(y_i-\\tilde{y}_i\\right)^2=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have used the definition of a norm-2 vector, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\vert\\vert \\boldsymbol{x}\\vert\\vert_2 = \\sqrt{\\sum_i x_i^2}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "By minimizing the above equation with respect to the parameters\n", - "$\\boldsymbol{\\beta}$ we could then obtain an analytical expression for the\n", - "parameters $\\boldsymbol{\\beta}$. We can add a regularization parameter $\\lambda$ by\n", - "defining a new cost function to be optimized, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", - "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to the Ridge regression minimization problem where we\n", - "require that $\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_2^2\\le t$, where $t$ is\n", - "a finite number larger than zero. By defining" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we have a new optimization equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\displaystyle \\min_{\\boldsymbol{\\beta}\\in\n", - "{\\mathbb{R}}^{p}}}\\frac{1}{n}\\vert\\vert \\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta}\\vert\\vert_2^2+\\lambda\\vert\\vert \\boldsymbol{\\beta}\\vert\\vert_1\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. \n", - "\n", - "Here we have defined the norm-1 as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\vert\\vert \\boldsymbol{x}\\vert\\vert_1 = \\sum_i \\vert x_i\\vert.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Using the matrix-vector expression for Ridge regression," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{X},\\boldsymbol{\\beta})=\\frac{1}{n}\\left\\{(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})^T(\\boldsymbol{y}-\\boldsymbol{X}\\boldsymbol{\\beta})\\right\\}+\\lambda\\boldsymbol{\\beta}^T\\boldsymbol{\\beta},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "by taking the derivatives with respect to $\\boldsymbol{\\beta}$ we obtain then\n", - "a slightly modified matrix inversion problem which for finite values\n", - "of $\\lambda$ does not suffer from singularity problems. We obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{X}^T\\boldsymbol{X}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{I}$ being a $p\\times p$ identity matrix with the constraint that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sum_{i=0}^{p-1} \\beta_i^2 \\leq t,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $t$ a finite positive number. \n", - "\n", - "We see that Ridge regression is nothing but the standard\n", - "OLS with a modified diagonal term added to $\\boldsymbol{X}^T\\boldsymbol{X}$. The\n", - "consequences, in particular for our discussion of the bias-variance tradeoff \n", - "are rather interesting.\n", - "\n", - "Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{X}\\boldsymbol{X}^T)\\boldsymbol{U} = \\boldsymbol{U}\\boldsymbol{D}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\\boldsymbol{U}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}\\boldsymbol{\\beta} = \\boldsymbol{X}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\boldsymbol{U}\\boldsymbol{U}^T\\boldsymbol{y}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For Ridge regression this becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\boldsymbol{U\\Sigma V^T}\\left(\\boldsymbol{V}\\boldsymbol{D}\\boldsymbol{V}^T+\\lambda\\boldsymbol{I} \\right)^{-1}(\\boldsymbol{U\\Sigma V^T})^T\\boldsymbol{y}=\\sum_{j=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}\\boldsymbol{y},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the vectors $\\boldsymbol{u}_j$ being the columns of $\\boldsymbol{U}$. \n", - "\n", - "\n", - "Since $\\lambda \\geq 0$, it means that compared to OLS, we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda} \\leq 1.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Ridge regression finds the coordinates of $\\boldsymbol{y}$ with respect to the\n", - "orthonormal basis $\\boldsymbol{U}$, it then shrinks the coordinates by\n", - "$\\frac{\\sigma_j^2}{\\sigma_j^2+\\lambda}$. Recall that the SVD has\n", - "eigenvalues ordered in a descending way, that is $\\sigma_i \\geq\n", - "\\sigma_{i+1}$.\n", - "\n", - "For small eigenvalues $\\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom.\n", - "Actually, calculating the variance of $\\boldsymbol{X}\\boldsymbol{v}_j$ shows that this quantity is equal to $\\sigma_j^2/n$.\n", - "With a parameter $\\lambda$ we can thus shrink the role of specific parameters. \n", - "\n", - "\n", - "\n", - "For the sake of simplicity, let us assume that the design matrix is orthonormal, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}^T\\boldsymbol{X}=(\\boldsymbol{X}^T\\boldsymbol{X})^{-1} =\\boldsymbol{I}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case the standard OLS results in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{OLS}} = \\boldsymbol{X}^T\\boldsymbol{y}=\\sum_{i=0}^{p-1}\\boldsymbol{u}_j\\boldsymbol{u}_j^T\\boldsymbol{y},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{Ridge}} = \\left(\\boldsymbol{I}+\\lambda\\boldsymbol{I}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}=\\left(1+\\lambda\\right)^{-1}\\boldsymbol{\\beta}^{\\mathrm{OLS}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\\lambda$, and\n", - "the Ridge estimator converges to zero when the hyperparameter goes to\n", - "infinity.\n", - "\n", - "We will come back to more interpreations after we have gone through some of the statistical analysis part. \n", - "\n", - "For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended.\n", - "Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended.\n", - "\n", - "\n", - "\n", - "## A better understanding of regularization\n", - "\n", - "The parameter $\\lambda$ that we have introduced in the Ridge (and\n", - "Lasso as well) regression is often called a regularization parameter\n", - "or shrinkage parameter. It is common to call it a hyperparameter. What does it mean mathemtically?\n", - "\n", - "Here we will first look at how to analyze the difference between the\n", - "standard OLS equations and the Ridge expressions in terms of a linear\n", - "algebra analysis using the SVD algorithm. Thereafter, we will link\n", - "(see the material on the bias-variance tradeoff below) these\n", - "observation to the statisical analysis of the results. In particular\n", - "we consider how the variance of the parameters $\\boldsymbol{\\beta}$ is\n", - "affected by changing the parameter $\\lambda$.\n", - "\n", - "\n", - "We have our design matrix\n", - " $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$. With the SVD we decompose it as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X} = \\boldsymbol{U\\Sigma V^T},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{U}\\in {\\mathbb{R}}^{n\\times n}$, $\\boldsymbol{\\Sigma}\\in {\\mathbb{R}}^{n\\times p}$\n", - "and $\\boldsymbol{V}\\in {\\mathbb{R}}^{p\\times p}$.\n", - "\n", - "The matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\\boldsymbol{U}^T\\boldsymbol{U}=\\boldsymbol{U}\\boldsymbol{U}^T=\\boldsymbol{I}$ and $\\boldsymbol{V}^T\\boldsymbol{V}=\\boldsymbol{V}\\boldsymbol{V}^T=\\boldsymbol{I}$.\n", - "\n", - "\n", - "\n", - "## Introducing the Covariance and Correlation functions\n", - "\n", - "Before we discuss the link between for example Ridge regression and the singular value decomposition, we need to remind ourselves about\n", - "the definition of the covariance and the correlation function. These are quantities \n", - "\n", - "Suppose we have defined two vectors\n", - "$\\hat{x}$ and $\\hat{y}$ with $n$ elements each. The covariance matrix $\\boldsymbol{C}$ is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{y}] \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where for example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With this definition and recalling that the variance is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{var}[\\boldsymbol{x}]=\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we can rewrite the covariance matrix as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] & \\mathrm{var}[\\boldsymbol{y}] \\\\\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The covariance takes values between zero and infinity and may thus\n", - "lead to problems with loss of numerical precision for particularly\n", - "large values. It is common to scale the covariance matrix by\n", - "introducing instead the correlation matrix defined via the so-called\n", - "correlation function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]=\\frac{\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}]}{\\sqrt{\\mathrm{var}[\\boldsymbol{x}] \\mathrm{var}[\\boldsymbol{y}]}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The correlation function is then given by values $\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]\n", - "\\in [-1,1]$. This avoids eventual problems with too large values. We\n", - "can then define the correlation matrix for the two vectors $\\boldsymbol{x}$\n", - "and $\\boldsymbol{y}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{K}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} 1 & \\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{corr}[\\boldsymbol{y},\\boldsymbol{x}] & 1 \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the above example this is the function we constructed using **pandas**.\n", - "\n", - "\n", - "\n", - "In our derivation of the various regression algorithms like **Ordinary Least Squares** or **Ridge regression**\n", - "we defined the design/feature matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n", - "x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n", - "x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n", - "\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n", - "x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n", - "x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$ refering to the column numbers and the\n", - "entries $n$ being the row elements.\n", - "We can rewrite the design/feature matrix in terms of its column vectors as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix} \\boldsymbol{x}_0 & \\boldsymbol{x}_1 & \\boldsymbol{x}_2 & \\dots & \\dots & \\boldsymbol{x}_{p-1}\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with a given vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i^T = \\begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \\dots & \\dots x_{n-1,i}\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With these definitions, we can now rewrite our $2\\times 2$\n", - "correaltion/covariance matrix in terms of a moe general design/feature\n", - "matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$. This leads to a $p\\times p$\n", - "covariance matrix for the vectors $\\boldsymbol{x}_i$ with $i=0,1,\\dots,p-1$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\begin{bmatrix}\n", - "\\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & \\mathrm{var}[\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & \\mathrm{var}[\\boldsymbol{x}_{p-1}]\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the correlation matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{K}[\\boldsymbol{x}] = \\begin{bmatrix}\n", - "1 & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & 1 & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & 1 & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & 1\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The Numpy function **np.cov** calculates the covariance elements using\n", - "the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have\n", - "the exact mean values. The following simple function uses the\n", - "**np.vstack** function which takes each vector of dimension $1\\times n$\n", - "and produces a $2\\times n$ matrix $\\boldsymbol{W}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{W} = \\begin{bmatrix} x_0 & y_0 \\\\\n", - " x_1 & y_1 \\\\\n", - " x_2 & y_2\\\\\n", - " \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2}\\\\\n", - " x_{n-1} & y_{n-1} & \n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in turn is converted into into the $2\\times 2$ covariance matrix\n", - "$\\boldsymbol{C}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", - "the mean value of each set of samples $\\boldsymbol{x}$ etc using the Numpy\n", - "function **np.mean(x)**. We can also extract the eigenvalues of the\n", - "covariance matrix through the **np.linalg.eig()** function." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0.062127739929490035\n", - "4.226441441217558\n", - "[[0.95977893 2.78652875]\n", - " [2.78652875 9.09409124]]\n" - ] - } - ], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "W = np.vstack((x, y))\n", - "C = np.cov(W)\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The previous example can be converted into the correlation matrix by\n", - "simply scaling the matrix elements with the variances. We should also\n", - "subtract the mean values for each column. This leads to the following\n", - "code which sets up the correlations matrix for the previous example in\n", - "a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\\times 2$ correlation matrix (since we have only two vectors)." - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0.09053391104887817\n", - "1.9755272664385481\n", - "[[1. 0.64723729]\n", - " [0.64723729 1. ]]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "n = 100\n", - "# define two vectors \n", - "x = np.random.random(size=n)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "#scaling the x and y vectors \n", - "x = x - np.mean(x)\n", - "y = y - np.mean(y)\n", - "variance_x = np.sum(x@x)/n\n", - "variance_y = np.sum(y@y)/n\n", - "print(variance_x)\n", - "print(variance_y)\n", - "cov_xy = np.sum(x@y)/n\n", - "cov_xx = np.sum(x@x)/n\n", - "cov_yy = np.sum(y@y)/n\n", - "C = np.zeros((2,2))\n", - "C[0,0]= cov_xx/variance_x\n", - "C[1,1]= cov_yy/variance_y\n", - "C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)\n", - "C[1,0]= C[0,1]\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see that the matrix elements along the diagonal are one as they\n", - "should be and that the matrix is symmetric. Furthermore, diagonalizing\n", - "this matrix we easily see that it is a positive definite matrix.\n", - "\n", - "The above procedure with **numpy** can be made more compact if we use **pandas**.\n", - "\n", - "\n", - "We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[ 0.52374318 0.7528421 ]\n", - " [-1.07892554 -3.5697027 ]\n", - " [ 0.65536057 2.84854 ]\n", - " [-0.9936011 -1.75368597]\n", - " [-0.22233456 -1.63866932]\n", - " [ 1.10799046 4.51410028]\n", - " [ 1.30401938 5.04686521]\n", - " [ 0.70055962 1.28566384]\n", - " [-1.69423925 -6.23684061]\n", - " [-0.30257276 -1.24911284]]\n", - " 0 1\n", - "0 0.523743 0.752842\n", - "1 -1.078926 -3.569703\n", - "2 0.655361 2.848540\n", - "3 -0.993601 -1.753686\n", - "4 -0.222335 -1.638669\n", - "5 1.107990 4.514100\n", - "6 1.304019 5.046865\n", - "7 0.700560 1.285664\n", - "8 -1.694239 -6.236841\n", - "9 -0.302573 -1.249113\n", - " 0 1\n", - "0 1.000000 0.967871\n", - "1 0.967871 1.000000\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "import pandas as pd\n", - "n = 10\n", - "x = np.random.normal(size=n)\n", - "x = x - np.mean(x)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "y = y - np.mean(y)\n", - "X = (np.vstack((x, y))).T\n", - "print(X)\n", - "Xpd = pd.DataFrame(X)\n", - "print(Xpd)\n", - "correlation_matrix = Xpd.corr()\n", - "print(correlation_matrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We expand this model to the Franke function discussed above." - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - " 0 1 2 3 4 5 6 7 \\\n", - "0 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 \n", - "1 0.0 0.076075 0.081429 0.075275 0.076780 0.077999 0.067453 0.067971 \n", - "2 0.0 0.081429 0.088214 0.081300 0.083371 0.085063 0.072811 0.073594 \n", - "3 0.0 0.075275 0.081300 0.080335 0.082127 0.083567 0.075400 0.075990 \n", - "4 0.0 0.076780 0.083371 0.082127 0.084184 0.085857 0.076996 0.077729 \n", - "5 0.0 0.077999 0.085063 0.083567 0.085857 0.087738 0.078264 0.079128 \n", - "6 0.0 0.067453 0.072811 0.075400 0.076996 0.078264 0.072961 0.073444 \n", - "7 0.0 0.067971 0.073594 0.075990 0.077729 0.079128 0.073444 0.074016 \n", - "8 0.0 0.068431 0.074291 0.076495 0.078367 0.079889 0.073843 0.074498 \n", - "9 0.0 0.068860 0.074936 0.076947 0.078943 0.080582 0.074186 0.074922 \n", - "10 0.0 0.059693 0.064192 0.068842 0.070144 0.071159 0.068084 0.068427 \n", - "11 0.0 0.059875 0.064519 0.069009 0.070400 0.071499 0.068172 0.068575 \n", - "12 0.0 0.060056 0.064837 0.069164 0.070641 0.071822 0.068246 0.068709 \n", - "13 0.0 0.060243 0.065156 0.069319 0.070878 0.072139 0.068315 0.068837 \n", - "14 0.0 0.060442 0.065483 0.069478 0.071119 0.072459 0.068387 0.068966 \n", - "\n", - " 8 9 10 11 12 13 14 \n", - "0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 \n", - "1 0.068431 0.068860 0.059693 0.059875 0.060056 0.060243 0.060442 \n", - "2 0.074291 0.074936 0.064192 0.064519 0.064837 0.065156 0.065483 \n", - "3 0.076495 0.076947 0.068842 0.069009 0.069164 0.069319 0.069478 \n", - "4 0.078367 0.078943 0.070144 0.070400 0.070641 0.070878 0.071119 \n", - "5 0.079889 0.080582 0.071159 0.071499 0.071822 0.072139 0.072459 \n", - "6 0.073843 0.074186 0.068084 0.068172 0.068246 0.068315 0.068387 \n", - "7 0.074498 0.074922 0.068427 0.068575 0.068709 0.068837 0.068966 \n", - "8 0.075062 0.075564 0.068693 0.068901 0.069093 0.069278 0.069465 \n", - "9 0.075564 0.076143 0.068909 0.069174 0.069423 0.069665 0.069908 \n", - "10 0.068693 0.068909 0.064578 0.064582 0.064574 0.064559 0.064545 \n", - "11 0.068901 0.069174 0.064582 0.064632 0.064668 0.064698 0.064728 \n", - "12 0.069093 0.069423 0.064574 0.064668 0.064748 0.064822 0.064896 \n", - "13 0.069278 0.069665 0.064559 0.064698 0.064822 0.064940 0.065058 \n", - "14 0.069465 0.069908 0.064545 0.064728 0.064896 0.065058 0.065220 \n" - ] - } - ], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "\n", - "\n", - "def FrankeFunction(x,y):\n", - "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", - "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", - "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", - "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", - "\treturn term1 + term2 + term3 + term4\n", - "\n", - "\n", - "def create_X(x, y, n ):\n", - "\tif len(x.shape) > 1:\n", - "\t\tx = np.ravel(x)\n", - "\t\ty = np.ravel(y)\n", - "\n", - "\tN = len(x)\n", - "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", - "\tX = np.ones((N,l))\n", - "\n", - "\tfor i in range(1,n+1):\n", - "\t\tq = int((i)*(i+1)/2)\n", - "\t\tfor k in range(i+1):\n", - "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", - "\n", - "\treturn X\n", - "\n", - "\n", - "# Making meshgrid of datapoints and compute Franke's function\n", - "n = 4\n", - "N = 100\n", - "x = np.sort(np.random.uniform(0, 1, N))\n", - "y = np.sort(np.random.uniform(0, 1, N))\n", - "z = FrankeFunction(x, y)\n", - "X = create_X(x, y, n=n) \n", - "\n", - "Xpd = pd.DataFrame(X)\n", - "# subtract the mean values and set up the covariance matrix\n", - "Xpd = Xpd - Xpd.mean()\n", - "covariance_matrix = Xpd.cov()\n", - "print(covariance_matrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note here that the covariance is zero for the first rows and\n", - "columns since all matrix elements in the design matrix were set to one\n", - "(we are fitting the function in terms of a polynomial of degree $n$).\n", - "\n", - "This means that the variance for these elements will be zero and will\n", - "cause problems when we set up the correlation matrix. We can simply\n", - "drop these elements and construct a correlation\n", - "matrix without these elements. \n", - "\n", - "\n", - "\n", - "\n", - "We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}= \\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}].\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To see this let us simply look at a design matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{2\\times 2}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{00} & x_{01}\\\\\n", - "x_{10} & x_{11}\\\\\n", - "\\end{bmatrix}=\\begin{bmatrix}\n", - "\\boldsymbol{x}_{0} & \\boldsymbol{x}_{1}\\\\\n", - "\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we then compute the expectation value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\\\\n", - "x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is just" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]=\\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we wrote $$\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]$$ to indicate that this the covariance of the vectors $\\boldsymbol{x}$ of the design/feature matrix $\\boldsymbol{X}$.\n", - "\n", - "It is easy to generalize this to a matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$.\n", - "\n", - "\n", - "## Linking with SVD" - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter3.py b/doc/LectureNotes/_build/jupyter_execute/chapter3.py deleted file mode 100644 index 9469762fc..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter3.py +++ /dev/null @@ -1,790 +0,0 @@ -# Ridge and Lasso Regression - -[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember10.mp4?vrtx=view-as-webpage) - - -## The singular value decomposition - -The examples we have looked at so far are cases where we normally can -invert the matrix $\boldsymbol{X}^T\boldsymbol{X}$. Using a polynomial expansion as we -did both for the masses and the fitting of the equation of state, -leads to row vectors of the design matrix which are essentially -orthogonal due to the polynomial character of our model. Obtaining the inverse of the design matrix is then often done via a so-called LU, QR or Cholesky decomposition. - - - -This may -however not the be case in general and a standard matrix inversion -algorithm based on say LU, QR or Cholesky decomposition may lead to singularities. We will see examples of this below. - -There is however a way to partially circumvent this problem and also gain some insights about the ordinary least squares approach, and later shrinkage methods like Ridge and Lasso regressions. - -This is given by the **Singular Value Decomposition** algorithm, perhaps -the most powerful linear algebra algorithm. Let us look at a -different example where we may have problems with the standard matrix -inversion algorithm. Thereafter we dive into the math of the SVD. - - - -One of the typical problems we encounter with linear regression, in particular -when the matrix $\boldsymbol{X}$ (our so-called design matrix) is high-dimensional, -are problems with near singular or singular matrices. The column vectors of $\boldsymbol{X}$ -may be linearly dependent, normally referred to as super-collinearity. -This means that the matrix may be rank deficient and it is basically impossible to -to model the data using linear regression. As an example, consider the matrix - -$$ -\begin{align*} -\mathbf{X} & = \left[ -\begin{array}{rrr} -1 & -1 & 2 -\\ -1 & 0 & 1 -\\ -1 & 2 & -1 -\\ -1 & 1 & 0 -\end{array} \right] -\end{align*} -$$ - -The columns of $\boldsymbol{X}$ are linearly dependent. We see this easily since the -the first column is the row-wise sum of the other two columns. The rank (more correct, -the column rank) of a matrix is the dimension of the space spanned by the -column vectors. Hence, the rank of $\mathbf{X}$ is equal to the number -of linearly independent columns. In this particular case the matrix has rank 2. - -Super-collinearity of an $(n \times p)$-dimensional design matrix $\mathbf{X}$ implies -that the inverse of the matrix $\boldsymbol{X}^T\boldsymbol{X}$ (the matrix we need to invert to solve the linear regression equations) is non-invertible. If we have a square matrix that does not have an inverse, we say this matrix singular. The example here demonstrates this - -$$ -\begin{align*} -\boldsymbol{X} & = \left[ -\begin{array}{rr} -1 & -1 -\\ -1 & -1 -\end{array} \right]. -\end{align*} -$$ - -We see easily that $\mbox{det}(\boldsymbol{X}) = x_{11} x_{22} - x_{12} x_{21} = 1 \times (-1) - 1 \times (-1) = 0$. Hence, $\mathbf{X}$ is singular and its inverse is undefined. -This is equivalent to saying that the matrix $\boldsymbol{X}$ has at least an eigenvalue which is zero. - - -If our design matrix $\boldsymbol{X}$ which enters the linear regression problem - - -
- -$$ -\begin{equation} -\boldsymbol{\beta} = (\boldsymbol{X}^{T} \boldsymbol{X})^{-1} \boldsymbol{X}^{T} \boldsymbol{y}, -\label{_auto1} \tag{1} -\end{equation} -$$ - -has linearly dependent column vectors, we will not be able to compute the inverse -of $\boldsymbol{X}^T\boldsymbol{X}$ and we cannot find the parameters (estimators) $\beta_i$. -The estimators are only well-defined if $(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}$ exits. -This is more likely to happen when the matrix $\boldsymbol{X}$ is high-dimensional. In this case it is likely to encounter a situation where -the regression parameters $\beta_i$ cannot be estimated. - -A cheap *ad hoc* approach is simply to add a small diagonal component to the matrix to invert, that is we change - -$$ -\boldsymbol{X}^{T} \boldsymbol{X} \rightarrow \boldsymbol{X}^{T} \boldsymbol{X}+\lambda \boldsymbol{I}, -$$ - -where $\boldsymbol{I}$ is the identity matrix. When we discuss **Ridge** regression this is actually what we end up evaluating. The parameter $\lambda$ is called a hyperparameter. More about this later. - - - - - -From standard linear algebra we know that a square matrix $\boldsymbol{X}$ can be diagonalized if and only it is -a so-called [normal matrix](https://en.wikipedia.org/wiki/Normal_matrix), that is if $\boldsymbol{X}\in {\mathbb{R}}^{n\times n}$ -we have $\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X}$ or if $\boldsymbol{X}\in {\mathbb{C}}^{n\times n}$ we have $\boldsymbol{X}\boldsymbol{X}^{\dagger}=\boldsymbol{X}^{\dagger}\boldsymbol{X}$. -The matrix has then a set of eigenpairs - -$$ -(\lambda_1,\boldsymbol{u}_1),\dots, (\lambda_n,\boldsymbol{u}_n), -$$ - -and the eigenvalues are given by the diagonal matrix - -$$ -\boldsymbol{\Sigma}=\mathrm{Diag}(\lambda_1, \dots,\lambda_n). -$$ - -The matrix $\boldsymbol{X}$ can be written in terms of an orthogonal/unitary transformation $\boldsymbol{U}$ - -$$ -\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, -$$ - -with $\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I}$ or $\boldsymbol{U}\boldsymbol{U}^{\dagger}=\boldsymbol{I}$. - -Not all square matrices are diagonalizable. A matrix like the one discussed above - -$$ -\boldsymbol{X} = \begin{bmatrix} -1& -1 \\ -1& -1\\ -\end{bmatrix} -$$ - -is not diagonalizable, it is a so-called [defective matrix](https://en.wikipedia.org/wiki/Defective_matrix). It is easy to see that the condition -$\boldsymbol{X}\boldsymbol{X}^T=\boldsymbol{X}^T\boldsymbol{X}$ is not fulfilled. - - - -## The SVD, a Fantastic Algorithm - - -However, and this is the strength of the SVD algorithm, any general -matrix $\boldsymbol{X}$ can be decomposed in terms of a diagonal matrix and -two orthogonal/unitary matrices. The [Singular Value Decompostion -(SVD) theorem](https://en.wikipedia.org/wiki/Singular_value_decomposition) -states that a general $m\times n$ matrix $\boldsymbol{X}$ can be written in -terms of a diagonal matrix $\boldsymbol{\Sigma}$ of dimensionality $m\times n$ -and two orthognal matrices $\boldsymbol{U}$ and $\boldsymbol{V}$, where the first has -dimensionality $m \times m$ and the last dimensionality $n\times n$. -We have then - -$$ -\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T -$$ - -As an example, the above defective matrix can be decomposed as - -$$ -\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T, -$$ - -with eigenvalues $\sigma_1=2$ and $\sigma_2=0$. -The SVD exits always! - -The SVD -decomposition (singular values) gives eigenvalues -$\sigma_i\geq\sigma_{i+1}$ for all $i$ and for dimensions larger than $i=p$, the -eigenvalues (singular values) are zero. - -In the general case, where our design matrix $\boldsymbol{X}$ has dimension -$n\times p$, the matrix is thus decomposed into an $n\times n$ -orthogonal matrix $\boldsymbol{U}$, a $p\times p$ orthogonal matrix $\boldsymbol{V}$ -and a diagonal matrix $\boldsymbol{\Sigma}$ with $r=\mathrm{min}(n,p)$ -singular values $\sigma_i\geq 0$ on the main diagonal and zeros filling -the rest of the matrix. There are at most $p$ singular values -assuming that $n > p$. In our regression examples for the nuclear -masses and the equation of state this is indeed the case, while for -the Ising model we have $p > n$. These are often cases that lead to -near singular or singular matrices. - -The columns of $\boldsymbol{U}$ are called the left singular vectors while the columns of $\boldsymbol{V}$ are the right singular vectors. - -## Economy-size SVD - -If we assume that $n > p$, then our matrix $\boldsymbol{U}$ has dimension $n -\times n$. The last $n-p$ columns of $\boldsymbol{U}$ become however -irrelevant in our calculations since they are multiplied with the -zeros in $\boldsymbol{\Sigma}$. - -The economy-size decomposition removes extra rows or columns of zeros -from the diagonal matrix of singular values, $\boldsymbol{\Sigma}$, along with the columns -in either $\boldsymbol{U}$ or $\boldsymbol{V}$ that multiply those zeros in the expression. -Removing these zeros and columns can improve execution time -and reduce storage requirements without compromising the accuracy of -the decomposition. - -If $n > p$, we keep only the first $p$ columns of $\boldsymbol{U}$ and $\boldsymbol{\Sigma}$ has dimension $p\times p$. -If $p > n$, then only the first $n$ columns of $\boldsymbol{V}$ are computed and $\boldsymbol{\Sigma}$ has dimension $n\times n$. -The $n=p$ case is obvious, we retain the full SVD. -In general the economy-size SVD leads to less FLOPS and still conserving the desired accuracy. - -import numpy as np -# SVD inversion -def SVDinv(A): - ''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD). - SVD is numerically more stable than the inversion algorithms provided by - numpy and scipy.linalg at the cost of being slower. - ''' - U, s, VT = np.linalg.svd(A) -# print('test U') -# print( (np.transpose(U) @ U - U @np.transpose(U))) -# print('test VT') -# print( (np.transpose(VT) @ VT - VT @np.transpose(VT))) - print(U) - print(s) - print(VT) - - D = np.zeros((len(U),len(VT))) - for i in range(0,len(VT)): - D[i,i]=s[i] - UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D) - return np.matmul(V,np.matmul(invD,UT)) - - -X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ]) -print(X) -A = np.transpose(X) @ X -print(A) -# Brute force inversion of super-collinear matrix -#B = np.linalg.inv(A) -#print(B) -C = SVDinv(A) -print(C) - -The matrix $\boldsymbol{X}$ has columns that are linearly dependent. The first -column is the row-wise sum of the other two columns. The rank of a -matrix (the column rank) is the dimension of space spanned by the -column vectors. The rank of the matrix is the number of linearly -independent columns, in this case just $2$. We see this from the -singular values when running the above code. Running the standard -inversion algorithm for matrix inversion with $\boldsymbol{X}^T\boldsymbol{X}$ results -in the program terminating due to a singular matrix. - - - - -There are several interesting mathematical properties which will be -relevant when we are going to discuss the differences between say -ordinary least squares (OLS) and **Ridge** regression. - -We have from OLS that the parameters of the linear approximation are given by - -$$ -\boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. -$$ - -The matrix to invert can be rewritten in terms of our SVD decomposition as - -$$ -\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T. -$$ - -Using the orthogonality properties of $\boldsymbol{U}$ we have - -$$ -\boldsymbol{X}^T\boldsymbol{X} = \boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T = \boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T, -$$ - -with $\boldsymbol{D}$ being a diagonal matrix with values along the diagonal given by the singular values squared. - -This means that - -$$ -(\boldsymbol{X}^T\boldsymbol{X})\boldsymbol{V} = \boldsymbol{V}\boldsymbol{D}, -$$ - -that is the eigenvectors of $(\boldsymbol{X}^T\boldsymbol{X})$ are given by the columns of the right singular matrix of $\boldsymbol{X}$ and the eigenvalues are the squared singular values. It is easy to show (show this) that - -$$ -(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}, -$$ - -that is, the eigenvectors of $(\boldsymbol{X}\boldsymbol{X})^T$ are the columns of the left singular matrix and the eigenvalues are the same. - -Going back to our OLS equation we have - -$$ -\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}. -$$ - -We will come back to this expression when we discuss Ridge regression. - - -$$ \tilde{y}^{OLS}=\boldsymbol{X}\hat{\beta}^{OLS}=\sum_{j=1}^p \boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}$$ and for Ridge we have  - -$$ \tilde{y}^{Ridge}=\boldsymbol{X}\hat{\beta}^{Ridge}=\sum_{j=1}^p \boldsymbol{u}_j\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{u}_j^T\boldsymbol{y}$$ .  - -It is indeed the economy-sized SVD, note the summation runs up tp $$p$$ only and not $$n$$.  - -Here we have that $$\boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T$$, with $$\Sigma$$ being an $$ n\times p$$ matrix and $$\boldsymbol{V}$$ being a $$ p\times p$$ matrix. We also have assumed here that $$ n > p$$.  - - - - - - - - -## Ridge and LASSO Regression - -[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember11.mp4?vrtx=view-as-webpage) - -Let us remind ourselves about the expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is -our optimization problem is - -$$ -{\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. -$$ - -or we can state it as - -$$ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2, -$$ - -where we have used the definition of a norm-2 vector, that is - -$$ -\vert\vert \boldsymbol{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. -$$ - -By minimizing the above equation with respect to the parameters -$\boldsymbol{\beta}$ we could then obtain an analytical expression for the -parameters $\boldsymbol{\beta}$. We can add a regularization parameter $\lambda$ by -defining a new cost function to be optimized, that is - -$$ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_2^2 -$$ - -which leads to the Ridge regression minimization problem where we -require that $\vert\vert \boldsymbol{\beta}\vert\vert_2^2\le t$, where $t$ is -a finite number larger than zero. By defining - -$$ -C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1, -$$ - -we have a new optimization equation - -$$ -{\displaystyle \min_{\boldsymbol{\beta}\in -{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\vert\vert_2^2+\lambda\vert\vert \boldsymbol{\beta}\vert\vert_1 -$$ - -which leads to Lasso regression. Lasso stands for least absolute shrinkage and selection operator. - -Here we have defined the norm-1 as - -$$ -\vert\vert \boldsymbol{x}\vert\vert_1 = \sum_i \vert x_i\vert. -$$ - -Using the matrix-vector expression for Ridge regression, - -$$ -C(\boldsymbol{X},\boldsymbol{\beta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})^T(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta})\right\}+\lambda\boldsymbol{\beta}^T\boldsymbol{\beta}, -$$ - -by taking the derivatives with respect to $\boldsymbol{\beta}$ we obtain then -a slightly modified matrix inversion problem which for finite values -of $\lambda$ does not suffer from singularity problems. We obtain - -$$ -\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{X}^T\boldsymbol{X}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}, -$$ - -with $\boldsymbol{I}$ being a $p\times p$ identity matrix with the constraint that - -$$ -\sum_{i=0}^{p-1} \beta_i^2 \leq t, -$$ - -with $t$ a finite positive number. - -We see that Ridge regression is nothing but the standard -OLS with a modified diagonal term added to $\boldsymbol{X}^T\boldsymbol{X}$. The -consequences, in particular for our discussion of the bias-variance tradeoff -are rather interesting. - -Furthermore, if we use the result above in terms of the SVD decomposition (our analysis was done for the OLS method), we had - -$$ -(\boldsymbol{X}\boldsymbol{X}^T)\boldsymbol{U} = \boldsymbol{U}\boldsymbol{D}. -$$ - -We can analyse the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\boldsymbol{U}$ as - -$$ -\boldsymbol{X}\boldsymbol{\beta} = \boldsymbol{X}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y} -$$ - -For Ridge regression this becomes - -$$ -\boldsymbol{X}\boldsymbol{\beta}^{\mathrm{Ridge}} = \boldsymbol{U\Sigma V^T}\left(\boldsymbol{V}\boldsymbol{D}\boldsymbol{V}^T+\lambda\boldsymbol{I} \right)^{-1}(\boldsymbol{U\Sigma V^T})^T\boldsymbol{y}=\sum_{j=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\boldsymbol{y}, -$$ - -with the vectors $\boldsymbol{u}_j$ being the columns of $\boldsymbol{U}$. - - -Since $\lambda \geq 0$, it means that compared to OLS, we have - -$$ -\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1. -$$ - -Ridge regression finds the coordinates of $\boldsymbol{y}$ with respect to the -orthonormal basis $\boldsymbol{U}$, it then shrinks the coordinates by -$\frac{\sigma_j^2}{\sigma_j^2+\lambda}$. Recall that the SVD has -eigenvalues ordered in a descending way, that is $\sigma_i \geq -\sigma_{i+1}$. - -For small eigenvalues $\sigma_i$ it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. -Actually, calculating the variance of $\boldsymbol{X}\boldsymbol{v}_j$ shows that this quantity is equal to $\sigma_j^2/n$. -With a parameter $\lambda$ we can thus shrink the role of specific parameters. - - - -For the sake of simplicity, let us assume that the design matrix is orthonormal, that is - -$$ -\boldsymbol{X}^T\boldsymbol{X}=(\boldsymbol{X}^T\boldsymbol{X})^{-1} =\boldsymbol{I}. -$$ - -In this case the standard OLS results in - -$$ -\boldsymbol{\beta}^{\mathrm{OLS}} = \boldsymbol{X}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_j\boldsymbol{u}_j^T\boldsymbol{y}, -$$ - -and - -$$ -\boldsymbol{\beta}^{\mathrm{Ridge}} = \left(\boldsymbol{I}+\lambda\boldsymbol{I}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}=\left(1+\lambda\right)^{-1}\boldsymbol{\beta}^{\mathrm{OLS}}, -$$ - -that is the Ridge estimator scales the OLS estimator by the inverse of a factor $1+\lambda$, and -the Ridge estimator converges to zero when the hyperparameter goes to -infinity. - -We will come back to more interpreations after we have gone through some of the statistical analysis part. - -For more discussions of Ridge and Lasso regression, [Wessel van Wieringen's](https://arxiv.org/abs/1509.09169) article is highly recommended. -Similarly, [Mehta et al's article](https://arxiv.org/abs/1803.08823) is also recommended. - - - -## A better understanding of regularization - -The parameter $\lambda$ that we have introduced in the Ridge (and -Lasso as well) regression is often called a regularization parameter -or shrinkage parameter. It is common to call it a hyperparameter. What does it mean mathemtically? - -Here we will first look at how to analyze the difference between the -standard OLS equations and the Ridge expressions in terms of a linear -algebra analysis using the SVD algorithm. Thereafter, we will link -(see the material on the bias-variance tradeoff below) these -observation to the statisical analysis of the results. In particular -we consider how the variance of the parameters $\boldsymbol{\beta}$ is -affected by changing the parameter $\lambda$. - - -We have our design matrix - $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$. With the SVD we decompose it as - -$$ -\boldsymbol{X} = \boldsymbol{U\Sigma V^T}, -$$ - -with $\boldsymbol{U}\in {\mathbb{R}}^{n\times n}$, $\boldsymbol{\Sigma}\in {\mathbb{R}}^{n\times p}$ -and $\boldsymbol{V}\in {\mathbb{R}}^{p\times p}$. - -The matrices $\boldsymbol{U}$ and $\boldsymbol{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I}$ and $\boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I}$. - - - -## Introducing the Covariance and Correlation functions - -Before we discuss the link between for example Ridge regression and the singular value decomposition, we need to remind ourselves about -the definition of the covariance and the correlation function. These are quantities - -Suppose we have defined two vectors -$\hat{x}$ and $\hat{y}$ with $n$ elements each. The covariance matrix $\boldsymbol{C}$ is defined as - -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ - \end{bmatrix}, -$$ - -where for example - -$$ -\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -$$ - -With this definition and recalling that the variance is defined as - -$$ -\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, -$$ - -we can rewrite the covariance matrix as - -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ - \end{bmatrix}. -$$ - -The covariance takes values between zero and infinity and may thus -lead to problems with loss of numerical precision for particularly -large values. It is common to scale the covariance matrix by -introducing instead the correlation matrix defined via the so-called -correlation function - -$$ -\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. -$$ - -The correlation function is then given by values $\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] -\in [-1,1]$. This avoids eventual problems with too large values. We -can then define the correlation matrix for the two vectors $\boldsymbol{x}$ -and $\boldsymbol{y}$ as - -$$ -\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\ - \end{bmatrix}, -$$ - -In the above example this is the function we constructed using **pandas**. - - - -In our derivation of the various regression algorithms like **Ordinary Least Squares** or **Ridge regression** -we defined the design/feature matrix $\boldsymbol{X}$ as - -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -$$ - -with $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$, with the predictors/features $p$ refering to the column numbers and the -entries $n$ being the row elements. -We can rewrite the design/feature matrix in terms of its column vectors as - -$$ -\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, -$$ - -with a given vector - -$$ -\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. -$$ - -With these definitions, we can now rewrite our $2\times 2$ -correaltion/covariance matrix in terms of a moe general design/feature -matrix $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$. This leads to a $p\times p$ -covariance matrix for the vectors $\boldsymbol{x}_i$ with $i=0,1,\dots,p-1$ - -$$ -\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} -\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ -\end{bmatrix}, -$$ - -and the correlation matrix - -$$ -\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} -1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ -\end{bmatrix}, -$$ - -The Numpy function **np.cov** calculates the covariance elements using -the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have -the exact mean values. The following simple function uses the -**np.vstack** function which takes each vector of dimension $1\times n$ -and produces a $2\times n$ matrix $\boldsymbol{W}$ - -$$ -\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ - x_1 & y_1 \\ - x_2 & y_2\\ - \dots & \dots \\ - x_{n-2} & y_{n-2}\\ - x_{n-1} & y_{n-1} & - \end{bmatrix}, -$$ - -which in turn is converted into into the $2\times 2$ covariance matrix -$\boldsymbol{C}$ via the Numpy function **np.cov()**. We note that we can also calculate -the mean value of each set of samples $\boldsymbol{x}$ etc using the Numpy -function **np.mean(x)**. We can also extract the eigenvalues of the -covariance matrix through the **np.linalg.eig()** function. - -# Importing various packages -import numpy as np -n = 100 -x = np.random.normal(size=n) -print(np.mean(x)) -y = 4+3*x+np.random.normal(size=n) -print(np.mean(y)) -W = np.vstack((x, y)) -C = np.cov(W) -print(C) - -The previous example can be converted into the correlation matrix by -simply scaling the matrix elements with the variances. We should also -subtract the mean values for each column. This leads to the following -code which sets up the correlations matrix for the previous example in -a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\times 2$ correlation matrix (since we have only two vectors). - -import numpy as np -n = 100 -# define two vectors -x = np.random.random(size=n) -y = 4+3*x+np.random.normal(size=n) -#scaling the x and y vectors -x = x - np.mean(x) -y = y - np.mean(y) -variance_x = np.sum(x@x)/n -variance_y = np.sum(y@y)/n -print(variance_x) -print(variance_y) -cov_xy = np.sum(x@y)/n -cov_xx = np.sum(x@x)/n -cov_yy = np.sum(y@y)/n -C = np.zeros((2,2)) -C[0,0]= cov_xx/variance_x -C[1,1]= cov_yy/variance_y -C[0,1]= cov_xy/np.sqrt(variance_y*variance_x) -C[1,0]= C[0,1] -print(C) - -We see that the matrix elements along the diagonal are one as they -should be and that the matrix is symmetric. Furthermore, diagonalizing -this matrix we easily see that it is a positive definite matrix. - -The above procedure with **numpy** can be made more compact if we use **pandas**. - - -We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code - -import numpy as np -import pandas as pd -n = 10 -x = np.random.normal(size=n) -x = x - np.mean(x) -y = 4+3*x+np.random.normal(size=n) -y = y - np.mean(y) -X = (np.vstack((x, y))).T -print(X) -Xpd = pd.DataFrame(X) -print(Xpd) -correlation_matrix = Xpd.corr() -print(correlation_matrix) - -We expand this model to the Franke function discussed above. - -# Common imports -import numpy as np -import pandas as pd - - -def FrankeFunction(x,y): - term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2)) - term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1)) - term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2)) - term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2) - return term1 + term2 + term3 + term4 - - -def create_X(x, y, n ): - if len(x.shape) > 1: - x = np.ravel(x) - y = np.ravel(y) - - N = len(x) - l = int((n+1)*(n+2)/2) # Number of elements in beta - X = np.ones((N,l)) - - for i in range(1,n+1): - q = int((i)*(i+1)/2) - for k in range(i+1): - X[:,q+k] = (x**(i-k))*(y**k) - - return X - - -# Making meshgrid of datapoints and compute Franke's function -n = 4 -N = 100 -x = np.sort(np.random.uniform(0, 1, N)) -y = np.sort(np.random.uniform(0, 1, N)) -z = FrankeFunction(x, y) -X = create_X(x, y, n=n) - -Xpd = pd.DataFrame(X) -# subtract the mean values and set up the covariance matrix -Xpd = Xpd - Xpd.mean() -covariance_matrix = Xpd.cov() -print(covariance_matrix) - -We note here that the covariance is zero for the first rows and -columns since all matrix elements in the design matrix were set to one -(we are fitting the function in terms of a polynomial of degree $n$). - -This means that the variance for these elements will be zero and will -cause problems when we set up the correlation matrix. We can simply -drop these elements and construct a correlation -matrix without these elements. - - - - -We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\boldsymbol{X}$ as - -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}]. -$$ - -To see this let us simply look at a design matrix $\boldsymbol{X}\in {\mathbb{R}}^{2\times 2}$ - -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{00} & x_{01}\\ -x_{10} & x_{11}\\ -\end{bmatrix}=\begin{bmatrix} -\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\ -\end{bmatrix}. -$$ - -If we then compute the expectation value - -$$ -\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix} -x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ -x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ -\end{bmatrix}, -$$ - -which is just - -$$ -\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\ - \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\ - \end{bmatrix}, -$$ - -where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors $\boldsymbol{x}$ of the design/feature matrix $\boldsymbol{X}$. - -It is easy to generalize this to a matrix $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$. - - -## Linking with SVD \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter4.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter4.ipynb deleted file mode 100644 index 83057b090..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter4.ipynb +++ /dev/null @@ -1,2900 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Logistic Regression\n", - "\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureSeptember18.mp4?vrtx=view-as-webpage)\n", - "\n", - "\n", - "## Logistic Regression\n", - "\n", - "In linear regression our main interest was centered on learning the\n", - "coefficients of a functional fit (say a polynomial) in order to be\n", - "able to predict the response of a continuous variable on some unseen\n", - "data. The fit to the continuous variable $y_i$ is based on some\n", - "independent variables $\\hat{x}_i$. Linear regression resulted in\n", - "analytical expressions for standard ordinary Least Squares or Ridge\n", - "regression (in terms of matrices to invert) for several quantities,\n", - "ranging from the variance and thereby the confidence intervals of the\n", - "parameters $\\hat{\\beta}$ to the mean squared error. If we can invert\n", - "the product of the design matrices, linear regression gives then a\n", - "simple recipe for fitting our data.\n", - "\n", - "\n", - "Classification problems, however, are concerned with outcomes taking\n", - "the form of discrete variables (i.e. categories). We may for example,\n", - "on the basis of DNA sequencing for a number of patients, like to find\n", - "out which mutations are important for a certain disease; or based on\n", - "scans of various patients' brains, figure out if there is a tumor or\n", - "not; or given a specific physical system, we'd like to identify its\n", - "state, say whether it is an ordered or disordered system (typical\n", - "situation in solid state physics); or classify the status of a\n", - "patient, whether she/he has a stroke or not and many other similar\n", - "situations.\n", - "\n", - "The most common situation we encounter when we apply logistic\n", - "regression is that of two possible outcomes, normally denoted as a\n", - "binary outcome, true or false, positive or negative, success or\n", - "failure etc.\n", - "\n", - "\n", - "Logistic regression will also serve as our stepping stone towards\n", - "neural network algorithms and supervised deep learning. For logistic\n", - "learning, the minimization of the cost function leads to a non-linear\n", - "equation in the parameters $\\hat{\\beta}$. The optimization of the\n", - "problem calls therefore for minimization algorithms. This forms the\n", - "bottle neck of all machine learning algorithms, namely how to find\n", - "reliable minima of a multi-variable function. This leads us to the\n", - "family of gradient descent methods. The latter are the working horses\n", - "of basically all modern machine learning algorithms.\n", - "\n", - "We note also that many of the topics discussed here on logistic \n", - "regression are also commonly used in modern supervised Deep Learning\n", - "models, as we will see later.\n", - "\n", - "\n", - "\n", - "## Basics\n", - "\n", - "We consider the case where the dependent variables, also called the\n", - "responses or the outcomes, $y_i$ are discrete and only take values\n", - "from $k=0,\\dots,K-1$ (i.e. $K$ classes).\n", - "\n", - "The goal is to predict the\n", - "output classes from the design matrix $\\hat{X}\\in\\mathbb{R}^{n\\times p}$\n", - "made of $n$ samples, each of which carries $p$ features or predictors. The\n", - "primary goal is to identify the classes to which new unseen samples\n", - "belong.\n", - "\n", - "Let us specialize to the case of two classes only, with outputs\n", - "$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a\n", - "credit card user that could default or not on her/his credit card\n", - "debt. That is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\ 1 & \\mathrm{yes} \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Before moving to the logistic model, let us try to use our linear\n", - "regression model to classify these two outcomes. We could for example\n", - "fit a linear model to the default case if $y_i > 0.5$ and the no\n", - "default case $y_i \\leq 0.5$.\n", - "\n", - "We would then have our \n", - "weighted linear combination, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\hat{y} = \\hat{X}^T\\hat{\\beta} + \\hat{\\epsilon},\n", - "\\label{_auto1} \\tag{1}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\hat{y}$ is a vector representing the possible outcomes, $\\hat{X}$ is our\n", - "$n\\times p$ design matrix and $\\hat{\\beta}$ represents our estimators/predictors.\n", - "\n", - "\n", - "The main problem with our function is that it takes values on the\n", - "entire real axis. In the case of logistic regression, however, the\n", - "labels $y_i$ are discrete variables. A typical example is the credit\n", - "card data discussed below here, where we can set the state of\n", - "defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons\n", - "in the data set (see the full example below).\n", - "\n", - "One simple way to get a discrete output is to have sign\n", - "functions that map the output of a linear regressor to values $\\{0,1\\}$,\n", - "$f(s_i)=sign(s_i)=1$ if $s_i\\ge 0$ and 0 if otherwise. \n", - "We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron\" model in the machine learning\n", - "literature. This model is extremely simple. However, in many cases it is more\n", - "favorable to use a ``soft\" classifier that outputs\n", - "the probability of a given category. This leads us to the logistic function.\n", - "\n", - "\n", - "The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "ename": "FileNotFoundError", - "evalue": "[Errno 2] No such file or directory: 'DataFiles/chddata.csv'", - "output_type": "error", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mFileNotFoundError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 38\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msavefig\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mimage_path\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfig_id\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m\".png\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mformat\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'png'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 39\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 40\u001b[0;31m \u001b[0minfile\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mopen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdata_path\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"chddata.csv\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m'r'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 41\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 42\u001b[0m \u001b[0;31m# Read the chd data as csv file and organize the data into arrays with age group, age, and chd\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mFileNotFoundError\u001b[0m: [Errno 2] No such file or directory: 'DataFiles/chddata.csv'" - ] - } - ], - "source": [ - "%matplotlib inline\n", - "\n", - "# Common imports\n", - "import os\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import LinearRegression, Ridge, Lasso\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.utils import resample\n", - "from sklearn.metrics import mean_squared_error\n", - "from IPython.display import display\n", - "from pylab import plt, mpl\n", - "plt.style.use('seaborn')\n", - "mpl.rcParams['font.family'] = 'serif'\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"chddata.csv\"),'r')\n", - "\n", - "# Read the chd data as csv file and organize the data into arrays with age group, age, and chd\n", - "chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD'))\n", - "chd.columns = ['ID', 'Age', 'Agegroup', 'CHD']\n", - "output = chd['CHD']\n", - "age = chd['Age']\n", - "agegroup = chd['Agegroup']\n", - "numberID = chd['ID'] \n", - "display(chd)\n", - "\n", - "plt.scatter(age, output, marker='o')\n", - "plt.axis([18,70.0,-0.1, 1.2])\n", - "plt.xlabel(r'Age')\n", - "plt.ylabel(r'CHD')\n", - "plt.title(r'Age distribution and Coronary heart disease')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "What we could attempt however is to plot the mean value for each group." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800])\n", - "group = np.array([1, 2, 3, 4, 5, 6, 7, 8])\n", - "plt.plot(group, agegroupmean, \"r-\")\n", - "plt.axis([0,9,0, 1.0])\n", - "plt.xlabel(r'Age group')\n", - "plt.ylabel(r'CHD mean values')\n", - "plt.title(r'Mean values for each age group')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We are now trying to find a function $f(y\\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$.\n", - "In standard linear regression with a linear dependence on $x$, we would write this in terms of our model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(y_i\\vert x_i)=\\beta_0+\\beta_1 x_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This expression implies however that $f(y_i\\vert x_i)$ could take any\n", - "value from minus infinity to plus infinity. If we however let\n", - "$f(y\\vert y)$ be represented by the mean value, the above example\n", - "shows us that we can constrain the function to take values between\n", - "zero and one, that is we have $0 \\le f(y_i\\vert x_i) \\le 1$. Looking\n", - "at our last curve we see also that it has an S-shaped form. This leads\n", - "us to a very popular model for the function $f$, namely the so-called\n", - "Sigmoid function or logistic model. We will consider this function as\n", - "representing the probability for finding a value of $y_i$ with a given\n", - "$x_i$.\n", - "\n", - "\n", - "## The logistic function\n", - "\n", - "Another widely studied model, is the so-called \n", - "perceptron model, which is an example of a \"hard classification\" model. We\n", - "will encounter this model when we discuss neural networks as\n", - "well. Each datapoint is deterministically assigned to a category (i.e\n", - "$y_i=0$ or $y_i=1$). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a \"soft\"\n", - "classifier that outputs the probability of a given category rather\n", - "than a single value. For example, given $x_i$, the classifier\n", - "outputs the probability of being in a category $k$. Logistic regression\n", - "is the most common example of a so-called soft classifier. In logistic\n", - "regression, the probability that a data point $x_i$\n", - "belongs to a category $y_i=\\{0,1\\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that $1-p(t)= p(-t)$.\n", - "\n", - "## Examples of likelihood functions used in logistic regression and nueral networks\n", - "\n", - "\n", - "The following code plots the logistic function, the step function and other functions we will encounter from here and on." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\"\"\"The sigmoid function (or the logistic curve) is a\n", - "function that takes any real number, z, and outputs a number (0,1).\n", - "It is useful in neural networks for assigning weights on a relative scale.\n", - "The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n", - "\n", - "import numpy\n", - "import matplotlib.pyplot as plt\n", - "import math as mt\n", - "\n", - "z = numpy.arange(-5, 5, .1)\n", - "sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n", - "sigma = sigma_fn(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, sigma)\n", - "ax.set_ylim([-0.1, 1.1])\n", - "ax.set_xlim([-5,5])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('sigmoid function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"Step Function\"\"\"\n", - "z = numpy.arange(-5, 5, .02)\n", - "step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n", - "step = step_fn(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, step)\n", - "ax.set_ylim([-0.5, 1.5])\n", - "ax.set_xlim([-5,5])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('step function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"tanh Function\"\"\"\n", - "z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n", - "t = numpy.tanh(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, t)\n", - "ax.set_ylim([-1.0, 1.0])\n", - "ax.set_xlim([-2*mt.pi,2*mt.pi])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('tanh function')\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\\beta$ in our fitting of the Sigmoid function, that is we define probabilities" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "p(y_i=1|x_i,\\hat{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n", - "p(y_i=0|x_i,\\hat{\\beta}) &= 1 - p(y_i=1|x_i,\\hat{\\beta}),\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\hat{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", - "\n", - "Note that we used" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(y_i=0\\vert x_i, \\hat{\\beta}) = 1-p(y_i=1\\vert x_i, \\hat{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to define the total likelihood for all possible outcomes from a \n", - "dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n", - "$y_i\\in\\{0,1\\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. \n", - "We aim thus at maximizing \n", - "the probability of seeing the observed data. We can then approximate the \n", - "likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "P(\\mathcal{D}|\\hat{\\beta})& = \\prod_{i=1}^n \\left[p(y_i=1|x_i,\\hat{\\beta})\\right]^{y_i}\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]^{1-y_i}\\nonumber \\\\\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "from which we obtain the log-likelihood and our **cost/loss** function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\hat{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Reordering the logarithms, we can rewrite the **cost/loss** function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\\beta$.\n", - "Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathcal{C}(\\hat{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n", - "in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n", - "\n", - "\n", - "The cross entropy is a convex function of the weights $\\hat{\\beta}$ and,\n", - "therefore, any local minimizer is a global minimizer. \n", - "\n", - "\n", - "Minimizing this\n", - "cost function with respect to the two parameters $\\beta_0$ and $\\beta_1$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_1} = -\\sum_{i=1}^n \\left(y_ix_i -x_i\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us now define a vector $\\hat{y}$ with $n$ elements $y_i$, an\n", - "$n\\times p$ matrix $\\hat{X}$ which contains the $x_i$ values and a\n", - "vector $\\hat{p}$ of fitted probabilities $p(y_i\\vert x_i,\\hat{\\beta})$. We can rewrite in a more compact form the first\n", - "derivative of cost function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}} = -\\hat{X}^T\\left(\\hat{y}-\\hat{p}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we in addition define a diagonal matrix $\\hat{W}$ with elements \n", - "$p(y_i\\vert x_i,\\hat{\\beta})(1-p(y_i\\vert x_i,\\hat{\\beta})$, we can obtain a compact expression of the second derivative as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T} = \\hat{X}^T\\hat{W}\\hat{X}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\log{ \\frac{p(\\hat{\\beta}\\hat{x})}{1-p(\\hat{\\beta}\\hat{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we defined $\\hat{x}=[1,x_1,x_2,\\dots,x_p]$ and $\\hat{\\beta}=[\\beta_0, \\beta_1, \\dots, \\beta_p]$ leading to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(\\hat{\\beta}\\hat{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Till now we have mainly focused on two classes, the so-called binary\n", - "system. Suppose we wish to extend to $K$ classes. Let us for the sake\n", - "of simplicity assume we have only two predictors. We have then following model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\log{\\frac{p(C=1\\vert x)}{p(K\\vert x)}} = \\beta_{10}+\\beta_{11}x_1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and so on till the class $C=K-1$ class" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the model is specified in term of $K-1$ so-called log-odds or\n", - "**logit** transformations.\n", - "\n", - "\n", - "\n", - "In our discussion of neural networks we will encounter the above again\n", - "in terms of a slightly modified function, the so-called **Softmax** function.\n", - "\n", - "The softmax function is used in various multiclass classification\n", - "methods, such as multinomial logistic regression (also known as\n", - "softmax regression), multiclass linear discriminant analysis, naive\n", - "Bayes classifiers, and artificial neural networks. Specifically, in\n", - "multinomial logistic regression and linear discriminant analysis, the\n", - "input to the function is the result of $K$ distinct linear functions,\n", - "and the predicted probability for the $k$-th class given a sample\n", - "vector $\\hat{x}$ and a weighting vector $\\hat{\\beta}$ is (with two\n", - "predictors):" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is easy to extend to more predictors. The final class is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and they sum to one. Our earlier discussions were all specialized to\n", - "the case with two classes only. It is easy to see from the above that\n", - "what we derived earlier is compatible with these equations.\n", - "\n", - "To find the optimal parameters we would typically use a gradient\n", - "descent method. Newton's method and gradient descent methods are\n", - "discussed in the material on [optimization\n", - "methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html).\n", - "\n", - "## Wisconsin Cancer Data\n", - "\n", - "We show here how we can use a simple regression case on the breast\n", - "cancer data using Logistic regression as our algorithm for\n", - "classification." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.linear_model import LogisticRegression\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "# Logistic Regression\n", - "logreg = LogisticRegression(solver='lbfgs')\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Logistic Regression\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In addition to the above scores, we could also study the covariance (and the correlation matrix).\n", - "We use **Pandas** to compute the correlation matrix." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.linear_model import LogisticRegression\n", - "cancer = load_breast_cancer()\n", - "import pandas as pd\n", - "# Making a data frame\n", - "cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)\n", - "\n", - "fig, axes = plt.subplots(15,2,figsize=(10,20))\n", - "malignant = cancer.data[cancer.target == 0]\n", - "benign = cancer.data[cancer.target == 1]\n", - "ax = axes.ravel()\n", - "\n", - "for i in range(30):\n", - " _, bins = np.histogram(cancer.data[:,i], bins =50)\n", - " ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5)\n", - " ax[i].hist(benign[:,i], bins = bins, alpha = 0.5)\n", - " ax[i].set_title(cancer.feature_names[i])\n", - " ax[i].set_yticks(())\n", - "ax[0].set_xlabel(\"Feature magnitude\")\n", - "ax[0].set_ylabel(\"Frequency\")\n", - "ax[0].legend([\"Malignant\", \"Benign\"], loc =\"best\")\n", - "fig.tight_layout()\n", - "plt.show()\n", - "\n", - "import seaborn as sns\n", - "correlation_matrix = cancerpd.corr().round(1)\n", - "# use the heatmap function from seaborn to plot the correlation matrix\n", - "# annot = True to print the values inside the square\n", - "plt.figure(figsize=(15,8))\n", - "sns.heatmap(data=correlation_matrix, annot=True)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the above example we note two things. In the first plot we display\n", - "the overlap of benign and malignant tumors as functions of the various\n", - "features in the Wisconsing breast cancer data set. We see that for\n", - "some of the features we can distinguish clearly the benign and\n", - "malignant cases while for other features we cannot. This can point to\n", - "us which features may be of greater interest when we wish to classify\n", - "a benign or not benign tumour.\n", - "\n", - "In the second figure we have computed the so-called correlation\n", - "matrix, which in our case with thirty features becomes a $30\\times 30$\n", - "matrix.\n", - "\n", - "We constructed this matrix using **pandas** via the statements" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and then" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "correlation_matrix = cancerpd.corr().round(1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Diagonalizing this matrix we can in turn say something about which\n", - "features are of relevance and which are not. This leads us to\n", - "the classical Principal Component Analysis (PCA) theorem with\n", - "applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html))." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.linear_model import LogisticRegression\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "# Logistic Regression\n", - "logreg = LogisticRegression(solver='lbfgs')\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Logistic Regression\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "\n", - "\n", - "from sklearn.preprocessing import LabelEncoder\n", - "from sklearn.model_selection import cross_validate\n", - "#Cross validation\n", - "accuracy = cross_validate(logreg,X_test_scaled,y_test,cv=10)['test_score']\n", - "print(accuracy)\n", - "print(\"Test set accuracy with Logistic Regression and scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "\n", - "\n", - "import scikitplot as skplt\n", - "y_pred = logreg.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "plt.show()\n", - "y_probas = logreg.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Optimization, the central part of any Machine Learning algortithm\n", - "\n", - "Almost every problem in machine learning and data science starts with\n", - "a dataset $X$, a model $g(\\beta)$, which is a function of the\n", - "parameters $\\beta$ and a cost function $C(X, g(\\beta))$ that allows\n", - "us to judge how well the model $g(\\beta)$ explains the observations\n", - "$X$. The model is fit by finding the values of $\\beta$ that minimize\n", - "the cost function. Ideally we would be able to solve for $\\beta$\n", - "analytically, however this is not possible in general and we must use\n", - "some approximative/numerical method to compute the minimum.\n", - "\n", - "\n", - "\n", - "## Revisiting our Logistic Regression case\n", - "\n", - "In our discussion on Logistic Regression we studied the \n", - "case of\n", - "two classes, with $y_i$ either\n", - "$0$ or $1$. Furthermore we assumed also that we have only two\n", - "parameters $\\beta$ in our fitting, that is we\n", - "defined probabilities" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "p(y_i=1|x_i,\\boldsymbol{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n", - "p(y_i=0|x_i,\\boldsymbol{\\beta}) &= 1 - p(y_i=1|x_i,\\boldsymbol{\\beta}),\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\boldsymbol{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", - "\n", - "\n", - "## The equations to solve\n", - "\n", - "Our compact equations used a definition of a vector $\\boldsymbol{y}$ with $n$\n", - "elements $y_i$, an $n\\times p$ matrix $\\boldsymbol{X}$ which contains the\n", - "$x_i$ values and a vector $\\boldsymbol{p}$ of fitted probabilities\n", - "$p(y_i\\vert x_i,\\boldsymbol{\\beta})$. We rewrote in a more compact form\n", - "the first derivative of the cost function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}} = -\\boldsymbol{X}^T\\left(\\boldsymbol{y}-\\boldsymbol{p}\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we in addition define a diagonal matrix $\\boldsymbol{W}$ with elements \n", - "$p(y_i\\vert x_i,\\boldsymbol{\\beta})(1-p(y_i\\vert x_i,\\boldsymbol{\\beta})$, we can obtain a compact expression of the second derivative as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T} = \\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This defines what is called the Hessian matrix.\n", - "\n", - "\n", - "## Solving using Newton-Raphson's method\n", - "\n", - "If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. \n", - "\n", - "Our iterative scheme is then given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{new}} = \\boldsymbol{\\beta}^{\\mathrm{old}}-\\left(\\frac{\\partial^2 \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}\\partial \\boldsymbol{\\beta}^T}\\right)^{-1}_{\\boldsymbol{\\beta}^{\\mathrm{old}}}\\times \\left(\\frac{\\partial \\mathcal{C}(\\boldsymbol{\\beta})}{\\partial \\boldsymbol{\\beta}}\\right)_{\\boldsymbol{\\beta}^{\\mathrm{old}}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or in matrix form as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}^{\\mathrm{new}} = \\boldsymbol{\\beta}^{\\mathrm{old}}-\\left(\\boldsymbol{X}^T\\boldsymbol{W}\\boldsymbol{X} \\right)^{-1}\\times \\left(-\\boldsymbol{X}^T(\\boldsymbol{y}-\\boldsymbol{p}) \\right)_{\\boldsymbol{\\beta}^{\\mathrm{old}}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The right-hand side is computed with the old values of $\\beta$. \n", - "\n", - "If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. \n", - "\n", - "\n", - "\n", - "## Brief reminder on Newton-Raphson's method\n", - "\n", - "Let us quickly remind ourselves how we derive the above method.\n", - "\n", - "Perhaps the most celebrated of all one-dimensional root-finding\n", - "routines is Newton's method, also called the Newton-Raphson\n", - "method. This method requires the evaluation of both the\n", - "function $f$ and its derivative $f'$ at arbitrary points. \n", - "If you can only calculate the derivative\n", - "numerically and/or your function is not of the smooth type, we\n", - "normally discourage the use of this method.\n", - "\n", - "\n", - "## The equations\n", - "\n", - "The Newton-Raphson formula consists geometrically of extending the\n", - "tangent line at a current point until it crosses zero, then setting\n", - "the next guess to the abscissa of that zero-crossing. The mathematics\n", - "behind this method is rather simple. Employing a Taylor expansion for\n", - "$x$ sufficiently close to the solution $s$, we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "f(s)=0=f(x)+(s-x)f'(x)+\\frac{(s-x)^2}{2}f''(x) +\\dots.\n", - " \\label{eq:taylornr} \\tag{2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For small enough values of the function and for well-behaved\n", - "functions, the terms beyond linear are unimportant, hence we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(x)+(s-x)f'(x)\\approx 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "yielding" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "s\\approx x-\\frac{f(x)}{f'(x)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Having in mind an iterative procedure, it is natural to start iterating with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Simple geometric interpretation\n", - "\n", - "The above is Newton-Raphson's method. It has a simple geometric\n", - "interpretation, namely $x_{n+1}$ is the point where the tangent from\n", - "$(x_n,f(x_n))$ crosses the $x$-axis. Close to the solution,\n", - "Newton-Raphson converges fast to the desired result. However, if we\n", - "are far from a root, where the higher-order terms in the series are\n", - "important, the Newton-Raphson formula can give grossly inaccurate\n", - "results. For instance, the initial guess for the root might be so far\n", - "from the true root as to let the search interval include a local\n", - "maximum or minimum of the function. If an iteration places a trial\n", - "guess near such a local extremum, so that the first derivative nearly\n", - "vanishes, then Newton-Raphson may fail totally\n", - "\n", - "\n", - "\n", - "## Extending to more than one variable\n", - "\n", - "Newton's method can be generalized to systems of several non-linear equations\n", - "and variables. Consider the case with two equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{array}{cc} f_1(x_1,x_2) &=0\\\\\n", - " f_2(x_1,x_2) &=0,\\end{array}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which we Taylor expand to obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1\n", - " \\partial f_1/\\partial x_1+h_2\n", - " \\partial f_1/\\partial x_2+\\dots\\\\\n", - " 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1\n", - " \\partial f_2/\\partial x_1+h_2\n", - " \\partial f_2/\\partial x_2+\\dots\n", - " \\end{array}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Defining the Jacobian matrix ${\\bf \\boldsymbol{J}}$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\bf \\boldsymbol{J}}=\\left( \\begin{array}{cc}\n", - " \\partial f_1/\\partial x_1 & \\partial f_1/\\partial x_2 \\\\\n", - " \\partial f_2/\\partial x_1 &\\partial f_2/\\partial x_2\n", - " \\end{array} \\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we can rephrase Newton's method as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\left(\\begin{array}{c} x_1^{n+1} \\\\ x_2^{n+1} \\end{array} \\right)=\n", - "\\left(\\begin{array}{c} x_1^{n} \\\\ x_2^{n} \\end{array} \\right)+\n", - "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\left(\\begin{array}{c} h_1^{n} \\\\ h_2^{n} \\end{array} \\right)=\n", - " -{\\bf \\boldsymbol{J}}^{-1}\n", - " \\left(\\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\\\ f_2(x_1^{n},x_2^{n}) \\end{array} \\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We need thus to compute the inverse of the Jacobian matrix and it\n", - "is to understand that difficulties may\n", - "arise in case ${\\bf \\boldsymbol{J}}$ is nearly singular.\n", - "\n", - "It is rather straightforward to extend the above scheme to systems of\n", - "more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. \n", - "\n", - "\n", - "\n", - "\n", - "## Steepest descent\n", - "\n", - "The basic idea of gradient descent is\n", - "that a function $F(\\mathbf{x})$, \n", - "$\\mathbf{x} \\equiv (x_1,\\cdots,x_n)$, decreases fastest if one goes from $\\bf {x}$ in the\n", - "direction of the negative gradient $-\\nabla F(\\mathbf{x})$.\n", - "\n", - "It can be shown that if" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\gamma_k > 0$.\n", - "\n", - "For $\\gamma_k$ small enough, then $F(\\mathbf{x}_{k+1}) \\leq\n", - "F(\\mathbf{x}_k)$. This means that for a sufficiently small $\\gamma_k$\n", - "we are always moving towards smaller function values, i.e a minimum.\n", - "\n", - "\n", - "## More on Steepest descent\n", - "\n", - "The previous observation is the basis of the method of steepest\n", - "descent, which is also referred to as just gradient descent (GD). One\n", - "starts with an initial guess $\\mathbf{x}_0$ for a minimum of $F$ and\n", - "computes new approximations according to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{x}_{k+1} = \\mathbf{x}_k - \\gamma_k \\nabla F(\\mathbf{x}_k), \\ \\ k \\geq 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The parameter $\\gamma_k$ is often referred to as the step length or\n", - "the learning rate within the context of Machine Learning.\n", - "\n", - "\n", - "## The ideal\n", - "\n", - "Ideally the sequence $\\{\\mathbf{x}_k \\}_{k=0}$ converges to a global\n", - "minimum of the function $F$. In general we do not know if we are in a\n", - "global or local minimum. In the special case when $F$ is a convex\n", - "function, all local minima are also global minima, so in this case\n", - "gradient descent can converge to the global solution. The advantage of\n", - "this scheme is that it is conceptually simple and straightforward to\n", - "implement. However the method in this form has some severe\n", - "limitations:\n", - "\n", - "In machine learing we are often faced with non-convex high dimensional\n", - "cost functions with many local minima. Since GD is deterministic we\n", - "will get stuck in a local minimum, if the method converges, unless we\n", - "have a very good intial guess. This also implies that the scheme is\n", - "sensitive to the chosen initial condition.\n", - "\n", - "Note that the gradient is a function of $\\mathbf{x} =\n", - "(x_1,\\cdots,x_n)$ which makes it expensive to compute numerically.\n", - "\n", - "\n", - "\n", - "## The sensitiveness of the gradient descent\n", - "\n", - "The gradient descent method \n", - "is sensitive to the choice of learning rate $\\gamma_k$. This is due\n", - "to the fact that we are only guaranteed that $F(\\mathbf{x}_{k+1}) \\leq\n", - "F(\\mathbf{x}_k)$ for sufficiently small $\\gamma_k$. The problem is to\n", - "determine an optimal learning rate. If the learning rate is chosen too\n", - "small the method will take a long time to converge and if it is too\n", - "large we can experience erratic behavior.\n", - "\n", - "Many of these shortcomings can be alleviated by introducing\n", - "randomness. One such method is that of Stochastic Gradient Descent\n", - "(SGD), see below.\n", - "\n", - "\n", - "\n", - "## Convex functions\n", - "\n", - "Ideally we want our cost/loss function to be convex(concave).\n", - "\n", - "First we give the definition of a convex set: A set $C$ in\n", - "$\\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and\n", - "all $t \\in (0,1)$ , the point $(1 − t)x + ty$ also belongs to\n", - "C. Geometrically this means that every point on the line segment\n", - "connecting $x$ and $y$ is in $C$ as discussed below.\n", - "\n", - "The convex subsets of $\\mathbb{R}$ are the intervals of\n", - "$\\mathbb{R}$. Examples of convex sets of $\\mathbb{R}^2$ are the\n", - "regular polygons (triangles, rectangles, pentagons, etc...).\n", - "\n", - "\n", - "## Convex function\n", - "\n", - "**Convex function**: Let $X \\subset \\mathbb{R}^n$ be a convex set. Assume that the function $f: X \\rightarrow \\mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \\leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \\in X$ and for all $t \\in [0,1]$. If $\\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \\neq x_2$ and $t\\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below.\n", - "\n", - "\n", - "## Conditions on convex functions\n", - "\n", - "In the following we state first and second-order conditions which\n", - "ensures convexity of a function $f$. We write $D_f$ to denote the\n", - "domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more\n", - "details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/, 2004).\n", - "\n", - "**First order condition.**\n", - "\n", - "Suppose $f$ is differentiable (i.e $\\nabla f(x)$ is well defined for\n", - "all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$\n", - "is a convex set and $$f(y) \\geq f(x) + \\nabla f(x)^T (y-x) $$ holds\n", - "for all $x,y \\in D_f$. This condition means that for a convex function\n", - "the first order Taylor expansion (right hand side above) at any point\n", - "a global under estimator of the function. To convince yourself you can\n", - "make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and\n", - "note that it is always below the graph.\n", - "\n", - "\n", - "\n", - "**Second order condition.**\n", - "\n", - "Assume that $f$ is twice\n", - "differentiable, i.e the Hessian matrix exists at each point in\n", - "$D_f$. Then $f$ is convex if and only if $D_f$ is a convex set and its\n", - "Hessian is positive semi-definite for all $x\\in D_f$. For a\n", - "single-variable function this reduces to $f''(x) \\geq 0$. Geometrically this means that $f$ has nonnegative curvature\n", - "everywhere.\n", - "\n", - "\n", - "\n", - "This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition.\n", - "\n", - "\n", - "## More on convex functions\n", - "\n", - "The next result is of great importance to us and the reason why we are\n", - "going on about convex functions. In machine learning we frequently\n", - "have to minimize a loss/cost function in order to find the best\n", - "parameters for the model we are considering. \n", - "\n", - "Ideally we want the\n", - "global minimum (for high-dimensional models it is hard to know\n", - "if we have local or global minimum). However, if the cost/loss function\n", - "is convex the following result provides invaluable information:\n", - "\n", - "**Any minimum is global for convex functions.**\n", - "\n", - "Consider the problem of finding $x \\in \\mathbb{R}^n$ such that $f(x)$\n", - "is minimal, where $f$ is convex and differentiable. Then, any point\n", - "$x^*$ that satisfies $\\nabla f(x^*) = 0$ is a global minimum.\n", - "\n", - "\n", - "\n", - "This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum.\n", - "\n", - "\n", - "## Some simple problems\n", - "\n", - "1. Show that $f(x)=x^2$ is convex for $x \\in \\mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \\in D_f$ and any $\\lambda \\in [0,1]$ $\\lambda f(x)+(1-\\lambda)f(y)-f(\\lambda x + (1-\\lambda) y ) \\geq 0$.\n", - "\n", - "2. Using the second order condition show that the following functions are convex on the specified domain.\n", - "\n", - " * $f(x) = e^x$ is convex for $x \\in \\mathbb{R}$.\n", - "\n", - " * $g(x) = -\\ln(x)$ is convex for $x \\in (0,\\infty)$.\n", - "\n", - "\n", - "3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \\in \\mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex.\n", - "\n", - "4. A norm is any function that satisfy the following properties\n", - "\n", - " * $f(\\alpha x) = |\\alpha| f(x)$ for all $\\alpha \\in \\mathbb{R}$.\n", - "\n", - " * $f(x+y) \\leq f(x) + f(y)$\n", - "\n", - " * $f(x) \\leq 0$ for all $x \\in \\mathbb{R}^n$ with equality if and only if $x = 0$\n", - "\n", - "\n", - "Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this).\n", - "\n", - "\n", - "\n", - "## Friday September 25\n", - "\n", - "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember25.mp4?vrtx=view-as-webpage) and [link to handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember25.pdf).\n", - "\n", - "\n", - "\n", - "## Standard steepest descent\n", - "\n", - "\n", - "Before we proceed, we would like to discuss the approach called the\n", - "**standard Steepest descent** (different from the above steepest descent discussion), which again leads to us having to be able\n", - "to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG).\n", - "\n", - "[The success of the CG method](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf)\n", - "for finding solutions of non-linear problems is based on the theory\n", - "of conjugate gradients for linear systems of equations. It belongs to\n", - "the class of iterative methods for solving problems from linear\n", - "algebra of the type" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{x} = \\boldsymbol{b}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the iterative process we end up with a problem like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}= \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\boldsymbol{r}$ is the so-called residual or error in the iterative process.\n", - "\n", - "When we have found the exact solution, $\\boldsymbol{r}=0$.\n", - "\n", - "\n", - "## Gradient method\n", - "\n", - "The residual is zero when we reach the minimum of the quadratic equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T\\boldsymbol{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the constraint that the matrix $\\boldsymbol{A}$ is positive definite and\n", - "symmetric. This defines also the Hessian and we want it to be positive definite. \n", - "\n", - "\n", - "\n", - "## Steepest descent method\n", - "\n", - "We denote the initial guess for $\\boldsymbol{x}$ as $\\boldsymbol{x}_0$. \n", - "We can assume without loss of generality that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_0=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or consider the system" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{z} = \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "instead.\n", - "\n", - "\n", - "\n", - "## Steepest descent method\n", - "One can show that the solution $\\boldsymbol{x}$ is also the unique minimizer of the quadratic form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(\\boldsymbol{x}) = \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T \\boldsymbol{x} , \\quad \\boldsymbol{x}\\in\\mathbf{R}^n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This suggests taking the first basis vector $\\boldsymbol{r}_1$ (see below for definition) \n", - "to be the gradient of $f$ at $\\boldsymbol{x}=\\boldsymbol{x}_0$, \n", - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{x}_0-\\boldsymbol{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and \n", - "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", - "\n", - "\n", - "\n", - "\n", - "## Final expressions\n", - "We can compute the residual iteratively as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}_{k+1}=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_{k+1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{b}-\\boldsymbol{A}(\\boldsymbol{x}_k+\\alpha_k\\boldsymbol{r}_k),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k)-\\alpha_k\\boldsymbol{A}\\boldsymbol{r}_k,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\alpha_k = \\frac{\\boldsymbol{r}_k^T\\boldsymbol{r}_k}{\\boldsymbol{r}_k^T\\boldsymbol{A}\\boldsymbol{r}_k}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "leading to the iterative scheme" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_{k+1}=\\boldsymbol{x}_k-\\alpha_k\\boldsymbol{r}_{k},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Steepest descent example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "import numpy.linalg as la\n", - "\n", - "import scipy.optimize as sopt\n", - "\n", - "import matplotlib.pyplot as pt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "def f(x):\n", - " return 0.5*x[0]**2 + 2.5*x[1]**2\n", - "\n", - "def df(x):\n", - " return np.array([x[0], 5*x[1]])\n", - "\n", - "fig = pt.figure()\n", - "ax = fig.gca(projection=\"3d\")\n", - "\n", - "xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j]\n", - "fmesh = f(np.array([xmesh, ymesh]))\n", - "ax.plot_surface(xmesh, ymesh, fmesh)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "And then as countor plot" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pt.axis(\"equal\")\n", - "pt.contour(xmesh, ymesh, fmesh)\n", - "guesses = [np.array([2, 2./5])]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Find guesses" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "x = guesses[-1]\n", - "s = -df(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Run it!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def f1d(alpha):\n", - " return f(x + alpha*s)\n", - "\n", - "alpha_opt = sopt.golden(f1d)\n", - "next_guess = x + alpha_opt * s\n", - "guesses.append(next_guess)\n", - "print(next_guess)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "What happened?" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pt.axis(\"equal\")\n", - "pt.contour(xmesh, ymesh, fmesh, 50)\n", - "it_array = np.array(guesses)\n", - "pt.plot(it_array.T[0], it_array.T[1], \"x-\")" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "In the CG method we define so-called conjugate directions and two vectors \n", - "$\\boldsymbol{s}$ and $\\boldsymbol{t}$\n", - "are said to be\n", - "conjugate if" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{s}^T\\boldsymbol{A}\\boldsymbol{t}= 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The philosophy of the CG method is to perform searches in various conjugate directions\n", - "of our vectors $\\boldsymbol{x}_i$ obeying the above criterion, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i^T\\boldsymbol{A}\\boldsymbol{x}_j= 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Two vectors are conjugate if they are orthogonal with respect to \n", - "this inner product. Being conjugate is a symmetric relation: if $\\boldsymbol{s}$ is conjugate to $\\boldsymbol{t}$, then $\\boldsymbol{t}$ is conjugate to $\\boldsymbol{s}$.\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "An example is given by the eigenvectors of the matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{v}_i^T\\boldsymbol{A}\\boldsymbol{v}_j= \\lambda\\boldsymbol{v}_i^T\\boldsymbol{v}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is zero unless $i=j$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "Assume now that we have a symmetric positive-definite matrix $\\boldsymbol{A}$ of size\n", - "$n\\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_{i+1}=\\boldsymbol{x}_{i}+\\alpha_i\\boldsymbol{p}_{i}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We assume that $\\boldsymbol{p}_{i}$ is a sequence of $n$ mutually conjugate directions. \n", - "Then the $\\boldsymbol{p}_{i}$ form a basis of $R^n$ and we can expand the solution \n", - "$ \\boldsymbol{A}\\boldsymbol{x} = \\boldsymbol{b}$ in this basis, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x} = \\sum^{n}_{i=1} \\alpha_i \\boldsymbol{p}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "The coefficients are given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}\\mathbf{x} = \\sum^{n}_{i=1} \\alpha_i \\mathbf{A} \\mathbf{p}_i = \\mathbf{b}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Multiplying with $\\boldsymbol{p}_k^T$ from the left gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{x} = \\sum^{n}_{i=1} \\alpha_i\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{p}_i= \\boldsymbol{p}_k^T \\boldsymbol{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we can define the coefficients $\\alpha_k$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\alpha_k = \\frac{\\boldsymbol{p}_k^T \\boldsymbol{b}}{\\boldsymbol{p}_k^T \\boldsymbol{A} \\boldsymbol{p}_k}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method and iterations\n", - "\n", - "If we choose the conjugate vectors $\\boldsymbol{p}_k$ carefully, \n", - "then we may not need all of them to obtain a good approximation to the solution \n", - "$\\boldsymbol{x}$. \n", - "We want to regard the conjugate gradient method as an iterative method. \n", - "This will us to solve systems where $n$ is so large that the direct \n", - "method would take too much time.\n", - "\n", - "We denote the initial guess for $\\boldsymbol{x}$ as $\\boldsymbol{x}_0$. \n", - "We can assume without loss of generality that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_0=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or consider the system" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{z} = \\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "instead.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "One can show that the solution $\\boldsymbol{x}$ is also the unique minimizer of the quadratic form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(\\boldsymbol{x}) = \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T \\boldsymbol{x} , \\quad \\boldsymbol{x}\\in\\mathbf{R}^n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This suggests taking the first basis vector $\\boldsymbol{p}_1$ \n", - "to be the gradient of $f$ at $\\boldsymbol{x}=\\boldsymbol{x}_0$, \n", - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{A}\\boldsymbol{x}_0-\\boldsymbol{b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and \n", - "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", - "The other vectors in the basis will be conjugate to the gradient, \n", - "hence the name conjugate gradient method.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Conjugate gradient method\n", - "Let $\\boldsymbol{r}_k$ be the residual at the $k$-th step:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}_k=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that $\\boldsymbol{r}_k$ is the negative gradient of $f$ at \n", - "$\\boldsymbol{x}=\\boldsymbol{x}_k$, \n", - "so the gradient descent method would be to move in the direction $\\boldsymbol{r}_k$. \n", - "Here, we insist that the directions $\\boldsymbol{p}_k$ are conjugate to each other, \n", - "so we take the direction closest to the gradient $\\boldsymbol{r}_k$ \n", - "under the conjugacy constraint. \n", - "This gives the following expression" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{p}_{k+1}=\\boldsymbol{r}_k-\\frac{\\boldsymbol{p}_k^T \\boldsymbol{A}\\boldsymbol{r}_k}{\\boldsymbol{p}_k^T\\boldsymbol{A}\\boldsymbol{p}_k} \\boldsymbol{p}_k.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Conjugate gradient method\n", - "We can also compute the residual iteratively as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}_{k+1}=\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_{k+1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which equals" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{b}-\\boldsymbol{A}(\\boldsymbol{x}_k+\\alpha_k\\boldsymbol{p}_k),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\boldsymbol{b}-\\boldsymbol{A}\\boldsymbol{x}_k)-\\alpha_k\\boldsymbol{A}\\boldsymbol{p}_k,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{r}_{k+1}=\\boldsymbol{r}_k-\\boldsymbol{A}\\boldsymbol{p}_{k},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Revisiting our first homework\n", - "\n", - "We will use linear regression as a case study for the gradient descent\n", - "methods. Linear regression is a great test case for the gradient\n", - "descent methods discussed in the lectures since it has several\n", - "desirable properties such as:\n", - "\n", - "1. An analytical solution (recall homework set 1).\n", - "\n", - "2. The gradient can be computed analytically.\n", - "\n", - "3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates\n", - "\n", - "We revisit an example similar to what we had in the first homework set. We had a function of the type" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "x = 2*np.random.rand(m,1)\n", - "y = 4+3*x+np.random.randn(m,1)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $x_i \\in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\\cal {N}(0,1)$. \n", - "The linear regression model is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "h_\\beta(x) = \\boldsymbol{y} = \\beta_0 + \\beta_1 x,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{y}_i = \\beta_0 + \\beta_1 x_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gradient descent example\n", - "\n", - "Let $\\mathbf{y} = (y_1,\\cdots,y_n)^T$, $\\mathbf{\\boldsymbol{y}} = (\\boldsymbol{y}_1,\\cdots,\\boldsymbol{y}_n)^T$ and $\\beta = (\\beta_0, \\beta_1)^T$\n", - "\n", - "It is convenient to write $\\mathbf{\\boldsymbol{y}} = X\\beta$ where $X \\in \\mathbb{R}^{100 \\times 2} $ is the design matrix given by (we keep the intercept here)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "X \\equiv \\begin{bmatrix}\n", - "1 & x_1 \\\\\n", - "\\vdots & \\vdots \\\\\n", - "1 & x_{100} & \\\\\n", - "\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The cost/loss/risk function is given by (" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\beta) = \\frac{1}{n}||X\\beta-\\mathbf{y}||_{2}^{2} = \\frac{1}{n}\\sum_{i=1}^{100}\\left[ (\\beta_0 + \\beta_1 x_i)^2 - 2 y_i (\\beta_0 + \\beta_1 x_i) + y_i^2\\right]\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we want to find $\\beta$ such that $C(\\beta)$ is minimized.\n", - "\n", - "\n", - "## The derivative of the cost/loss function\n", - "\n", - "Computing $\\partial C(\\beta) / \\partial \\beta_0$ and $\\partial C(\\beta) / \\partial \\beta_1$ we can show that the gradient can be written as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\nabla_{\\beta} C(\\beta) = \\frac{2}{n}\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n", - "\\sum_{i=1}^{100}\\left( x_i (\\beta_0+\\beta_1x_i)-y_ix_i\\right) \\\\\n", - "\\end{bmatrix} = \\frac{2}{n}X^T(X\\beta - \\mathbf{y}),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $X$ is the design matrix defined above.\n", - "\n", - "\n", - "## The Hessian matrix\n", - "The Hessian matrix of $C(\\beta)$ is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{H} \\equiv \\begin{bmatrix}\n", - "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0^2} & \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} \\\\\n", - "\\frac{\\partial^2 C(\\beta)}{\\partial \\beta_0 \\partial \\beta_1} & \\frac{\\partial^2 C(\\beta)}{\\partial \\beta_1^2} & \\\\\n", - "\\end{bmatrix} = \\frac{2}{n}X^T X.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This result implies that $C(\\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Simple program\n", - "\n", - "We can now write a program that minimizes $C(\\beta)$ using the gradient descent method with a constant learning rate $\\gamma$ according to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_{k+1} = \\beta_k - \\gamma \\nabla_\\beta C(\\beta_k), \\ k=0,1,\\cdots\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can use the expression we computed for the gradient and let use a\n", - "$\\beta_0$ be chosen randomly and let $\\gamma = 0.001$. Stop iterating\n", - "when $||\\nabla_\\beta C(\\beta_k) || \\leq \\epsilon = 10^{-8}$. **Note that the code below does not include the latter stop criterion**.\n", - "\n", - "And finally we can compare our solution for $\\beta$ with the analytic result given by \n", - "$\\beta= (X^TX)^{-1} X^T \\mathbf{y}$.\n", - "\n", - "\n", - "## Gradient Descent Example\n", - "\n", - "Here our simple example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "# Importing various packages\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from mpl_toolkits.mplot3d import Axes3D\n", - "from matplotlib import cm\n", - "from matplotlib.ticker import LinearLocator, FormatStrFormatter\n", - "import sys\n", - "\n", - "# the number of datapoints\n", - "n = 100\n", - "x = 2*np.random.rand(n,1)\n", - "y = 4+3*x+np.random.randn(n,1)\n", - "\n", - "X = np.c_[np.ones((n,1)), x]\n", - "# Hessian matrix\n", - "H = (2.0/n)* X.T @ X\n", - "# Get the eigenvalues\n", - "EigValues, EigVectors = np.linalg.eig(H)\n", - "print(EigValues)\n", - "\n", - "beta_linreg = np.linalg.inv(X.T @ X) @ X.T @ y\n", - "print(beta_linreg)\n", - "beta = np.random.randn(2,1)\n", - "\n", - "eta = 1.0/np.max(EigValues)\n", - "Niterations = 1000\n", - "\n", - "for iter in range(Niterations):\n", - " gradient = (2.0/n)*X.T @ (X @ beta-y)\n", - " beta -= eta*gradient\n", - "\n", - "print(beta)\n", - "xnew = np.array([[0],[2]])\n", - "xbnew = np.c_[np.ones((2,1)), xnew]\n", - "ypredict = xbnew.dot(beta)\n", - "ypredict2 = xbnew.dot(beta_linreg)\n", - "plt.plot(xnew, ypredict, \"r-\")\n", - "plt.plot(xnew, ypredict2, \"b-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0,2.0,0, 15.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Gradient descent example')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## And a corresponding example using **scikit-learn**" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import SGDRegressor\n", - "\n", - "n = 100\n", - "x = 2*np.random.rand(n,1)\n", - "y = 4+3*x+np.random.randn(n,1)\n", - "\n", - "X = np.c_[np.ones((n,1)), x]\n", - "beta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n", - "print(beta_linreg)\n", - "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n", - "sgdreg.fit(x,y.ravel())\n", - "print(sgdreg.intercept_, sgdreg.coef_)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gradient descent and Ridge\n", - "\n", - "We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\\beta$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C_{\\text{ridge}}(\\beta) = \\frac{1}{n}||X\\beta -\\mathbf{y}||^2 + \\lambda ||\\beta||^2, \\ \\lambda \\geq 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to minimize $C_{\\text{ridge}}(\\beta)$ using GD we only have adjust the gradient as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\nabla_\\beta C_{\\text{ridge}}(\\beta) = \\frac{2}{n}\\begin{bmatrix} \\sum_{i=1}^{100} \\left(\\beta_0+\\beta_1x_i-y_i\\right) \\\\\n", - "\\sum_{i=1}^{100}\\left( x_i (\\beta_0+\\beta_1x_i)-y_ix_i\\right) \\\\\n", - "\\end{bmatrix} + 2\\lambda\\begin{bmatrix} \\beta_0 \\\\ \\beta_1\\end{bmatrix} = 2 (X^T(X\\beta - \\mathbf{y})+\\lambda \\beta).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can easily extend our program to minimize $C_{\\text{ridge}}(\\beta)$ using gradient descent and compare with the analytical solution given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_{\\text{ridge}} = \\left(X^T X + \\lambda I_{2 \\times 2} \\right)^{-1} X^T \\mathbf{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Program example for gradient descent with Ridge Regression" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from mpl_toolkits.mplot3d import Axes3D\n", - "from matplotlib import cm\n", - "from matplotlib.ticker import LinearLocator, FormatStrFormatter\n", - "import sys\n", - "\n", - "# the number of datapoints\n", - "n = 100\n", - "x = 2*np.random.rand(n,1)\n", - "y = 4+3*x+np.random.randn(n,1)\n", - "\n", - "X = np.c_[np.ones((n,1)), x]\n", - "XT_X = X.T @ X\n", - "\n", - "#Ridge parameter lambda\n", - "lmbda = 0.001\n", - "Id = lmbda* np.eye(XT_X.shape[0])\n", - "\n", - "beta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y\n", - "print(beta_linreg)\n", - "# Start plain gradient descent\n", - "beta = np.random.randn(2,1)\n", - "\n", - "eta = 0.1\n", - "Niterations = 100\n", - "\n", - "for iter in range(Niterations):\n", - " gradients = 2.0/n*X.T @ (X @ (beta)-y)+2*lmbda*beta\n", - " beta -= eta*gradients\n", - "\n", - "print(beta)\n", - "ypredict = X @ beta\n", - "ypredict2 = X @ beta_linreg\n", - "plt.plot(x, ypredict, \"r-\")\n", - "plt.plot(x, ypredict2, \"b-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0,2.0,0, 15.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Gradient descent example for Ridge')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Using gradient descent methods, limitations\n", - "\n", - "* **Gradient descent (GD) finds local minima of our function**. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance.\n", - "\n", - "* **GD is sensitive to initial conditions**. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD.\n", - "\n", - "* **Gradients are computationally expensive to calculate for large datasets**. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \\propto \\sum_{i=1}^n (y_i - \\mathbf{w}^T\\cdot\\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called \"mini batches\". This has the added benefit of introducing stochasticity into our algorithm.\n", - "\n", - "* **GD is very sensitive to choices of learning rates**. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape.\n", - "\n", - "* **GD treats all directions in parameter space uniformly.** Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. \n", - "\n", - "* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points.\n", - "\n", - "## Stochastic Gradient Descent\n", - "\n", - "Stochastic gradient descent (SGD) and variants thereof address some of\n", - "the shortcomings of the Gradient descent method discussed above.\n", - "\n", - "The underlying idea of SGD comes from the observation that the cost\n", - "function, which we want to minimize, can almost always be written as a\n", - "sum over $n$ data points $\\{\\mathbf{x}_i\\}_{i=1}^n$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\mathbf{\\beta}) = \\sum_{i=1}^n c_i(\\mathbf{x}_i,\n", - "\\mathbf{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Computation of gradients\n", - "\n", - "This in turn means that the gradient can be\n", - "computed as a sum over $i$-gradients" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\nabla_\\beta C(\\mathbf{\\beta}) = \\sum_i^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", - "\\mathbf{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Stochasticity/randomness is introduced by only taking the\n", - "gradient on a subset of the data called minibatches. If there are $n$\n", - "data points and the size of each minibatch is $M$, there will be $n/M$\n", - "minibatches. We denote these minibatches by $B_k$ where\n", - "$k=1,\\cdots,n/M$.\n", - "\n", - "\n", - "## SGD example\n", - "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n", - "and we choose to have $M=5$ minibathces,\n", - "then each minibatch contains two data points. In particular we have\n", - "$B_1 = (\\mathbf{x}_1,\\mathbf{x}_2), \\cdots, B_5 =\n", - "(\\mathbf{x}_9,\\mathbf{x}_{10})$. Note that if you choose $M=1$ you\n", - "have only a single batch with all data points and on the other extreme,\n", - "you may choose $M=n$ resulting in a minibatch for each datapoint, i.e\n", - "$B_k = \\mathbf{x}_k$.\n", - "\n", - "The idea is now to approximate the gradient by replacing the sum over\n", - "all data points with a sum over the data points in one the minibatches\n", - "picked at random in each gradient descent step" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\nabla_{\\beta}\n", - "C(\\mathbf{\\beta}) = \\sum_{i=1}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", - "\\mathbf{\\beta}) \\rightarrow \\sum_{i \\in B_k}^n \\nabla_\\beta\n", - "c_i(\\mathbf{x}_i, \\mathbf{\\beta}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The gradient step\n", - "\n", - "Thus a gradient descent step now looks like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_{j+1} = \\beta_j - \\gamma_j \\sum_{i \\in B_k}^n \\nabla_\\beta c_i(\\mathbf{x}_i,\n", - "\\mathbf{\\beta})\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $k$ is picked at random with equal\n", - "probability from $[1,n/M]$. An iteration over the number of\n", - "minibathces (n/M) is commonly referred to as an epoch. Thus it is\n", - "typical to choose a number of epochs and for each epoch iterate over\n", - "the number of minibatches, as exemplified in the code below.\n", - "\n", - "\n", - "## Simple example code" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np \n", - "\n", - "n = 100 #100 datapoints \n", - "M = 5 #size of each minibatch\n", - "m = int(n/M) #number of minibatches\n", - "n_epochs = 10 #number of epochs\n", - "\n", - "j = 0\n", - "for epoch in range(1,n_epochs+1):\n", - " for i in range(m):\n", - " k = np.random.randint(m) #Pick the k-th minibatch at random\n", - " #Compute the gradient using the data in minibatch Bk\n", - " #Compute new suggestion for \n", - " j += 1" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Taking the gradient only on a subset of the data has two important\n", - "benefits. First, it introduces randomness which decreases the chance\n", - "that our opmization scheme gets stuck in a local minima. Second, if\n", - "the size of the minibatches are small relative to the number of\n", - "datapoints ($M < n$), the computation of the gradient is much\n", - "cheaper since we sum over the datapoints in the $k-th$ minibatch and not\n", - "all $n$ datapoints.\n", - "\n", - "\n", - "## When do we stop?\n", - "\n", - "A natural question is when do we stop the search for a new minimum?\n", - "One possibility is to compute the full gradient after a given number\n", - "of epochs and check if the norm of the gradient is smaller than some\n", - "threshold and stop if true. However, the condition that the gradient\n", - "is zero is valid also for local minima, so this would only tell us\n", - "that we are close to a local/global minimum. However, we could also\n", - "evaluate the cost function at this point, store the result and\n", - "continue the search. If the test kicks in at a later stage we can\n", - "compare the values of the cost function and keep the $\\beta$ that\n", - "gave the lowest value.\n", - "\n", - "\n", - "## Slightly different approach\n", - "\n", - "Another approach is to let the step length $\\gamma_j$ depend on the\n", - "number of epochs in such a way that it becomes very small after a\n", - "reasonable time such that we do not move at all.\n", - "\n", - "As an example, let $e = 0,1,2,3,\\cdots$ denote the current epoch and let $t_0, t_1 > 0$ be two fixed numbers. Furthermore, let $t = e \\cdot m + i$ where $m$ is the number of minibatches and $i=0,\\cdots,m-1$. Then the function $$\\gamma_j(t; t_0, t_1) = \\frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length $\\gamma_j (0; t_0, t_1) = t_0/t_1$ which decays in *time* $t$.\n", - "\n", - "In this way we can fix the number of epochs, compute $\\beta$ and\n", - "evaluate the cost function at the end. Repeating the computation will\n", - "give a different result since the scheme is random by design. Then we\n", - "pick the final $\\beta$ that gives the lowest value of the cost\n", - "function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np \n", - "\n", - "def step_length(t,t0,t1):\n", - " return t0/(t+t1)\n", - "\n", - "n = 100 #100 datapoints \n", - "M = 5 #size of each minibatch\n", - "m = int(n/M) #number of minibatches\n", - "n_epochs = 500 #number of epochs\n", - "t0 = 1.0\n", - "t1 = 10\n", - "\n", - "gamma_j = t0/t1\n", - "j = 0\n", - "for epoch in range(1,n_epochs+1):\n", - " for i in range(m):\n", - " k = np.random.randint(m) #Pick the k-th minibatch at random\n", - " #Compute the gradient using the data in minibatch Bk\n", - " #Compute new suggestion for beta\n", - " t = epoch*m+i\n", - " gamma_j = step_length(t,t0,t1)\n", - " j += 1\n", - "\n", - "print(\"gamma_j after %d epochs: %g\" % (n_epochs,gamma_j))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Program for stochastic gradient" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "from math import exp, sqrt\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.linear_model import SGDRegressor\n", - "\n", - "m = 100\n", - "x = 2*np.random.rand(m,1)\n", - "y = 4+3*x+np.random.randn(m,1)\n", - "\n", - "X = np.c_[np.ones((m,1)), x]\n", - "theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y)\n", - "print(\"Own inversion\")\n", - "print(theta_linreg)\n", - "sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1)\n", - "sgdreg.fit(x,y.ravel())\n", - "print(\"sgdreg from scikit\")\n", - "print(sgdreg.intercept_, sgdreg.coef_)\n", - "\n", - "\n", - "theta = np.random.randn(2,1)\n", - "eta = 0.1\n", - "Niterations = 1000\n", - "\n", - "\n", - "for iter in range(Niterations):\n", - " gradients = 2.0/m*X.T @ ((X @ theta)-y)\n", - " theta -= eta*gradients\n", - "print(\"theta from own gd\")\n", - "print(theta)\n", - "\n", - "xnew = np.array([[0],[2]])\n", - "Xnew = np.c_[np.ones((2,1)), xnew]\n", - "ypredict = Xnew.dot(theta)\n", - "ypredict2 = Xnew.dot(theta_linreg)\n", - "\n", - "\n", - "n_epochs = 50\n", - "t0, t1 = 5, 50\n", - "def learning_schedule(t):\n", - " return t0/(t+t1)\n", - "\n", - "theta = np.random.randn(2,1)\n", - "\n", - "for epoch in range(n_epochs):\n", - " for i in range(m):\n", - " random_index = np.random.randint(m)\n", - " xi = X[random_index:random_index+1]\n", - " yi = y[random_index:random_index+1]\n", - " gradients = 2 * xi.T @ ((xi @ theta)-yi)\n", - " eta = learning_schedule(epoch*m+i)\n", - " theta = theta - eta*gradients\n", - "print(\"theta from own sdg\")\n", - "print(theta)\n", - "\n", - "plt.plot(xnew, ypredict, \"r-\")\n", - "plt.plot(xnew, ypredict2, \"b-\")\n", - "plt.plot(x, y ,'ro')\n", - "plt.axis([0,2.0,0, 15.0])\n", - "plt.xlabel(r'$x$')\n", - "plt.ylabel(r'$y$')\n", - "plt.title(r'Random numbers ')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**Challenge**: try to write a similar code for a Logistic Regression case." - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter4.py b/doc/LectureNotes/_build/jupyter_execute/chapter4.py deleted file mode 100644 index a01b04c67..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter4.py +++ /dev/null @@ -1,1752 +0,0 @@ -# Logistic Regression - - -[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/LectureSeptember18.mp4?vrtx=view-as-webpage) - - -## Logistic Regression - -In linear regression our main interest was centered on learning the -coefficients of a functional fit (say a polynomial) in order to be -able to predict the response of a continuous variable on some unseen -data. The fit to the continuous variable $y_i$ is based on some -independent variables $\hat{x}_i$. Linear regression resulted in -analytical expressions for standard ordinary Least Squares or Ridge -regression (in terms of matrices to invert) for several quantities, -ranging from the variance and thereby the confidence intervals of the -parameters $\hat{\beta}$ to the mean squared error. If we can invert -the product of the design matrices, linear regression gives then a -simple recipe for fitting our data. - - -Classification problems, however, are concerned with outcomes taking -the form of discrete variables (i.e. categories). We may for example, -on the basis of DNA sequencing for a number of patients, like to find -out which mutations are important for a certain disease; or based on -scans of various patients' brains, figure out if there is a tumor or -not; or given a specific physical system, we'd like to identify its -state, say whether it is an ordered or disordered system (typical -situation in solid state physics); or classify the status of a -patient, whether she/he has a stroke or not and many other similar -situations. - -The most common situation we encounter when we apply logistic -regression is that of two possible outcomes, normally denoted as a -binary outcome, true or false, positive or negative, success or -failure etc. - - -Logistic regression will also serve as our stepping stone towards -neural network algorithms and supervised deep learning. For logistic -learning, the minimization of the cost function leads to a non-linear -equation in the parameters $\hat{\beta}$. The optimization of the -problem calls therefore for minimization algorithms. This forms the -bottle neck of all machine learning algorithms, namely how to find -reliable minima of a multi-variable function. This leads us to the -family of gradient descent methods. The latter are the working horses -of basically all modern machine learning algorithms. - -We note also that many of the topics discussed here on logistic -regression are also commonly used in modern supervised Deep Learning -models, as we will see later. - - - -## Basics - -We consider the case where the dependent variables, also called the -responses or the outcomes, $y_i$ are discrete and only take values -from $k=0,\dots,K-1$ (i.e. $K$ classes). - -The goal is to predict the -output classes from the design matrix $\hat{X}\in\mathbb{R}^{n\times p}$ -made of $n$ samples, each of which carries $p$ features or predictors. The -primary goal is to identify the classes to which new unseen samples -belong. - -Let us specialize to the case of two classes only, with outputs -$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a -credit card user that could default or not on her/his credit card -debt. That is - -$$ -y_i = \begin{bmatrix} 0 & \mathrm{no}\\ 1 & \mathrm{yes} \end{bmatrix}. -$$ - -Before moving to the logistic model, let us try to use our linear -regression model to classify these two outcomes. We could for example -fit a linear model to the default case if $y_i > 0.5$ and the no -default case $y_i \leq 0.5$. - -We would then have our -weighted linear combination, namely - - -
- -$$ -\begin{equation} -\hat{y} = \hat{X}^T\hat{\beta} + \hat{\epsilon}, -\label{_auto1} \tag{1} -\end{equation} -$$ - -where $\hat{y}$ is a vector representing the possible outcomes, $\hat{X}$ is our -$n\times p$ design matrix and $\hat{\beta}$ represents our estimators/predictors. - - -The main problem with our function is that it takes values on the -entire real axis. In the case of logistic regression, however, the -labels $y_i$ are discrete variables. A typical example is the credit -card data discussed below here, where we can set the state of -defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons -in the data set (see the full example below). - -One simple way to get a discrete output is to have sign -functions that map the output of a linear regressor to values $\{0,1\}$, -$f(s_i)=sign(s_i)=1$ if $s_i\ge 0$ and 0 if otherwise. -We will encounter this model in our first demonstration of neural networks. Historically it is called the ``perceptron" model in the machine learning -literature. This model is extremely simple. However, in many cases it is more -favorable to use a ``soft" classifier that outputs -the probability of a given category. This leads us to the logistic function. - - -The following example on data for coronary heart disease (CHD) as function of age may serve as an illustration. In the code here we read and plot whether a person has had CHD (output = 1) or not (output = 0). This ouput is plotted the person's against age. Clearly, the figure shows that attempting to make a standard linear regression fit may not be very meaningful. - -%matplotlib inline - -# Common imports -import os -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.linear_model import LinearRegression, Ridge, Lasso -from sklearn.model_selection import train_test_split -from sklearn.utils import resample -from sklearn.metrics import mean_squared_error -from IPython.display import display -from pylab import plt, mpl -plt.style.use('seaborn') -mpl.rcParams['font.family'] = 'serif' - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("chddata.csv"),'r') - -# Read the chd data as csv file and organize the data into arrays with age group, age, and chd -chd = pd.read_csv(infile, names=('ID', 'Age', 'Agegroup', 'CHD')) -chd.columns = ['ID', 'Age', 'Agegroup', 'CHD'] -output = chd['CHD'] -age = chd['Age'] -agegroup = chd['Agegroup'] -numberID = chd['ID'] -display(chd) - -plt.scatter(age, output, marker='o') -plt.axis([18,70.0,-0.1, 1.2]) -plt.xlabel(r'Age') -plt.ylabel(r'CHD') -plt.title(r'Age distribution and Coronary heart disease') -plt.show() - -What we could attempt however is to plot the mean value for each group. - -agegroupmean = np.array([0.1, 0.133, 0.250, 0.333, 0.462, 0.625, 0.765, 0.800]) -group = np.array([1, 2, 3, 4, 5, 6, 7, 8]) -plt.plot(group, agegroupmean, "r-") -plt.axis([0,9,0, 1.0]) -plt.xlabel(r'Age group') -plt.ylabel(r'CHD mean values') -plt.title(r'Mean values for each age group') -plt.show() - -We are now trying to find a function $f(y\vert x)$, that is a function which gives us an expected value for the output $y$ with a given input $x$. -In standard linear regression with a linear dependence on $x$, we would write this in terms of our model - -$$ -f(y_i\vert x_i)=\beta_0+\beta_1 x_i. -$$ - -This expression implies however that $f(y_i\vert x_i)$ could take any -value from minus infinity to plus infinity. If we however let -$f(y\vert y)$ be represented by the mean value, the above example -shows us that we can constrain the function to take values between -zero and one, that is we have $0 \le f(y_i\vert x_i) \le 1$. Looking -at our last curve we see also that it has an S-shaped form. This leads -us to a very popular model for the function $f$, namely the so-called -Sigmoid function or logistic model. We will consider this function as -representing the probability for finding a value of $y_i$ with a given -$x_i$. - - -## The logistic function - -Another widely studied model, is the so-called -perceptron model, which is an example of a "hard classification" model. We -will encounter this model when we discuss neural networks as -well. Each datapoint is deterministically assigned to a category (i.e -$y_i=0$ or $y_i=1$). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a "soft" -classifier that outputs the probability of a given category rather -than a single value. For example, given $x_i$, the classifier -outputs the probability of being in a category $k$. Logistic regression -is the most common example of a so-called soft classifier. In logistic -regression, the probability that a data point $x_i$ -belongs to a category $y_i=\{0,1\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event, - -$$ -p(t) = \frac{1}{1+\mathrm \exp{-t}}=\frac{\exp{t}}{1+\mathrm \exp{t}}. -$$ - -Note that $1-p(t)= p(-t)$. - -## Examples of likelihood functions used in logistic regression and nueral networks - - -The following code plots the logistic function, the step function and other functions we will encounter from here and on. - -"""The sigmoid function (or the logistic curve) is a -function that takes any real number, z, and outputs a number (0,1). -It is useful in neural networks for assigning weights on a relative scale. -The value z is the weighted sum of parameters involved in the learning algorithm.""" - -import numpy -import matplotlib.pyplot as plt -import math as mt - -z = numpy.arange(-5, 5, .1) -sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z))) -sigma = sigma_fn(z) - -fig = plt.figure() -ax = fig.add_subplot(111) -ax.plot(z, sigma) -ax.set_ylim([-0.1, 1.1]) -ax.set_xlim([-5,5]) -ax.grid(True) -ax.set_xlabel('z') -ax.set_title('sigmoid function') - -plt.show() - -"""Step Function""" -z = numpy.arange(-5, 5, .02) -step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0) -step = step_fn(z) - -fig = plt.figure() -ax = fig.add_subplot(111) -ax.plot(z, step) -ax.set_ylim([-0.5, 1.5]) -ax.set_xlim([-5,5]) -ax.grid(True) -ax.set_xlabel('z') -ax.set_title('step function') - -plt.show() - -"""tanh Function""" -z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1) -t = numpy.tanh(z) - -fig = plt.figure() -ax = fig.add_subplot(111) -ax.plot(z, t) -ax.set_ylim([-1.0, 1.0]) -ax.set_xlim([-2*mt.pi,2*mt.pi]) -ax.grid(True) -ax.set_xlabel('z') -ax.set_title('tanh function') - -plt.show() - -We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\beta$ in our fitting of the Sigmoid function, that is we define probabilities - -$$ -\begin{align*} -p(y_i=1|x_i,\hat{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ -p(y_i=0|x_i,\hat{\beta}) &= 1 - p(y_i=1|x_i,\hat{\beta}), -\end{align*} -$$ - -where $\hat{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$. - -Note that we used - -$$ -p(y_i=0\vert x_i, \hat{\beta}) = 1-p(y_i=1\vert x_i, \hat{\beta}). -$$ - -In order to define the total likelihood for all possible outcomes from a -dataset $\mathcal{D}=\{(y_i,x_i)\}$, with the binary labels -$y_i\in\{0,1\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. -We aim thus at maximizing -the probability of seeing the observed data. We can then approximate the -likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is - -$$ -\begin{align*} -P(\mathcal{D}|\hat{\beta})& = \prod_{i=1}^n \left[p(y_i=1|x_i,\hat{\beta})\right]^{y_i}\left[1-p(y_i=1|x_i,\hat{\beta}))\right]^{1-y_i}\nonumber \\ -\end{align*} -$$ - -from which we obtain the log-likelihood and our **cost/loss** function - -$$ -\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left( y_i\log{p(y_i=1|x_i,\hat{\beta})} + (1-y_i)\log\left[1-p(y_i=1|x_i,\hat{\beta}))\right]\right). -$$ - -Reordering the logarithms, we can rewrite the **cost/loss** function as - -$$ -\mathcal{C}(\hat{\beta}) = \sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). -$$ - -The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\beta$. -Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that - -$$ -\mathcal{C}(\hat{\beta})=-\sum_{i=1}^n \left(y_i(\beta_0+\beta_1x_i) -\log{(1+\exp{(\beta_0+\beta_1x_i)})}\right). -$$ - -This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, -in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression. - - -The cross entropy is a convex function of the weights $\hat{\beta}$ and, -therefore, any local minimizer is a global minimizer. - - -Minimizing this -cost function with respect to the two parameters $\beta_0$ and $\beta_1$ we obtain - -$$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right), -$$ - -and - -$$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \beta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}}\right). -$$ - -Let us now define a vector $\hat{y}$ with $n$ elements $y_i$, an -$n\times p$ matrix $\hat{X}$ which contains the $x_i$ values and a -vector $\hat{p}$ of fitted probabilities $p(y_i\vert x_i,\hat{\beta})$. We can rewrite in a more compact form the first -derivative of cost function as - -$$ -\frac{\partial \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}} = -\hat{X}^T\left(\hat{y}-\hat{p}\right). -$$ - -If we in addition define a diagonal matrix $\hat{W}$ with elements -$p(y_i\vert x_i,\hat{\beta})(1-p(y_i\vert x_i,\hat{\beta})$, we can obtain a compact expression of the second derivative as - -$$ -\frac{\partial^2 \mathcal{C}(\hat{\beta})}{\partial \hat{\beta}\partial \hat{\beta}^T} = \hat{X}^T\hat{W}\hat{X}. -$$ - -Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors - -$$ -\log{ \frac{p(\hat{\beta}\hat{x})}{1-p(\hat{\beta}\hat{x})}} = \beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p. -$$ - -Here we defined $\hat{x}=[1,x_1,x_2,\dots,x_p]$ and $\hat{\beta}=[\beta_0, \beta_1, \dots, \beta_p]$ leading to - -$$ -p(\hat{\beta}\hat{x})=\frac{ \exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}{1+\exp{(\beta_0+\beta_1x_1+\beta_2x_2+\dots+\beta_px_p)}}. -$$ - -Till now we have mainly focused on two classes, the so-called binary -system. Suppose we wish to extend to $K$ classes. Let us for the sake -of simplicity assume we have only two predictors. We have then following model - -$$ -\log{\frac{p(C=1\vert x)}{p(K\vert x)}} = \beta_{10}+\beta_{11}x_1, -$$ - -and - -$$ -\log{\frac{p(C=2\vert x)}{p(K\vert x)}} = \beta_{20}+\beta_{21}x_1, -$$ - -and so on till the class $C=K-1$ class - -$$ -\log{\frac{p(C=K-1\vert x)}{p(K\vert x)}} = \beta_{(K-1)0}+\beta_{(K-1)1}x_1, -$$ - -and the model is specified in term of $K-1$ so-called log-odds or -**logit** transformations. - - - -In our discussion of neural networks we will encounter the above again -in terms of a slightly modified function, the so-called **Softmax** function. - -The softmax function is used in various multiclass classification -methods, such as multinomial logistic regression (also known as -softmax regression), multiclass linear discriminant analysis, naive -Bayes classifiers, and artificial neural networks. Specifically, in -multinomial logistic regression and linear discriminant analysis, the -input to the function is the result of $K$ distinct linear functions, -and the predicted probability for the $k$-th class given a sample -vector $\hat{x}$ and a weighting vector $\hat{\beta}$ is (with two -predictors): - -$$ -p(C=k\vert \mathbf {x} )=\frac{\exp{(\beta_{k0}+\beta_{k1}x_1)}}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}. -$$ - -It is easy to extend to more predictors. The final class is - -$$ -p(C=K\vert \mathbf {x} )=\frac{1}{1+\sum_{l=1}^{K-1}\exp{(\beta_{l0}+\beta_{l1}x_1)}}, -$$ - -and they sum to one. Our earlier discussions were all specialized to -the case with two classes only. It is easy to see from the above that -what we derived earlier is compatible with these equations. - -To find the optimal parameters we would typically use a gradient -descent method. Newton's method and gradient descent methods are -discussed in the material on [optimization -methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html). - -## Wisconsin Cancer Data - -We show here how we can use a simple regression case on the breast -cancer data using Logistic regression as our algorithm for -classification. - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.datasets import load_breast_cancer -from sklearn.linear_model import LogisticRegression - -# Load the data -cancer = load_breast_cancer() - -X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) -print(X_train.shape) -print(X_test.shape) -# Logistic Regression -logreg = LogisticRegression(solver='lbfgs') -logreg.fit(X_train, y_train) -print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test))) -#now scale the data -from sklearn.preprocessing import StandardScaler -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) -# Logistic Regression -logreg.fit(X_train_scaled, y_train) -print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) - -In addition to the above scores, we could also study the covariance (and the correlation matrix). -We use **Pandas** to compute the correlation matrix. - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.datasets import load_breast_cancer -from sklearn.linear_model import LogisticRegression -cancer = load_breast_cancer() -import pandas as pd -# Making a data frame -cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names) - -fig, axes = plt.subplots(15,2,figsize=(10,20)) -malignant = cancer.data[cancer.target == 0] -benign = cancer.data[cancer.target == 1] -ax = axes.ravel() - -for i in range(30): - _, bins = np.histogram(cancer.data[:,i], bins =50) - ax[i].hist(malignant[:,i], bins = bins, alpha = 0.5) - ax[i].hist(benign[:,i], bins = bins, alpha = 0.5) - ax[i].set_title(cancer.feature_names[i]) - ax[i].set_yticks(()) -ax[0].set_xlabel("Feature magnitude") -ax[0].set_ylabel("Frequency") -ax[0].legend(["Malignant", "Benign"], loc ="best") -fig.tight_layout() -plt.show() - -import seaborn as sns -correlation_matrix = cancerpd.corr().round(1) -# use the heatmap function from seaborn to plot the correlation matrix -# annot = True to print the values inside the square -plt.figure(figsize=(15,8)) -sns.heatmap(data=correlation_matrix, annot=True) -plt.show() - -In the above example we note two things. In the first plot we display -the overlap of benign and malignant tumors as functions of the various -features in the Wisconsing breast cancer data set. We see that for -some of the features we can distinguish clearly the benign and -malignant cases while for other features we cannot. This can point to -us which features may be of greater interest when we wish to classify -a benign or not benign tumour. - -In the second figure we have computed the so-called correlation -matrix, which in our case with thirty features becomes a $30\times 30$ -matrix. - -We constructed this matrix using **pandas** via the statements - -cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names) - -and then - -correlation_matrix = cancerpd.corr().round(1) - -Diagonalizing this matrix we can in turn say something about which -features are of relevance and which are not. This leads us to -the classical Principal Component Analysis (PCA) theorem with -applications. This will be discussed later this semester ([week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week43-bs.html)). - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.datasets import load_breast_cancer -from sklearn.linear_model import LogisticRegression - -# Load the data -cancer = load_breast_cancer() - -X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) -print(X_train.shape) -print(X_test.shape) -# Logistic Regression -logreg = LogisticRegression(solver='lbfgs') -logreg.fit(X_train, y_train) -print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test))) -#now scale the data -from sklearn.preprocessing import StandardScaler -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) -# Logistic Regression -logreg.fit(X_train_scaled, y_train) -print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) - - -from sklearn.preprocessing import LabelEncoder -from sklearn.model_selection import cross_validate -#Cross validation -accuracy = cross_validate(logreg,X_test_scaled,y_test,cv=10)['test_score'] -print(accuracy) -print("Test set accuracy with Logistic Regression and scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) - - -import scikitplot as skplt -y_pred = logreg.predict(X_test_scaled) -skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) -plt.show() -y_probas = logreg.predict_proba(X_test_scaled) -skplt.metrics.plot_roc(y_test, y_probas) -plt.show() -skplt.metrics.plot_cumulative_gain(y_test, y_probas) -plt.show() - -## Optimization, the central part of any Machine Learning algortithm - -Almost every problem in machine learning and data science starts with -a dataset $X$, a model $g(\beta)$, which is a function of the -parameters $\beta$ and a cost function $C(X, g(\beta))$ that allows -us to judge how well the model $g(\beta)$ explains the observations -$X$. The model is fit by finding the values of $\beta$ that minimize -the cost function. Ideally we would be able to solve for $\beta$ -analytically, however this is not possible in general and we must use -some approximative/numerical method to compute the minimum. - - - -## Revisiting our Logistic Regression case - -In our discussion on Logistic Regression we studied the -case of -two classes, with $y_i$ either -$0$ or $1$. Furthermore we assumed also that we have only two -parameters $\beta$ in our fitting, that is we -defined probabilities - -$$ -\begin{align*} -p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ -p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}), -\end{align*} -$$ - -where $\boldsymbol{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$. - - -## The equations to solve - -Our compact equations used a definition of a vector $\boldsymbol{y}$ with $n$ -elements $y_i$, an $n\times p$ matrix $\boldsymbol{X}$ which contains the -$x_i$ values and a vector $\boldsymbol{p}$ of fitted probabilities -$p(y_i\vert x_i,\boldsymbol{\beta})$. We rewrote in a more compact form -the first derivative of the cost function as - -$$ -\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = -\boldsymbol{X}^T\left(\boldsymbol{y}-\boldsymbol{p}\right). -$$ - -If we in addition define a diagonal matrix $\boldsymbol{W}$ with elements -$p(y_i\vert x_i,\boldsymbol{\beta})(1-p(y_i\vert x_i,\boldsymbol{\beta})$, we can obtain a compact expression of the second derivative as - -$$ -\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T} = \boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X}. -$$ - -This defines what is called the Hessian matrix. - - -## Solving using Newton-Raphson's method - -If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. - -Our iterative scheme is then given by - -$$ -\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\frac{\partial^2 \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}\partial \boldsymbol{\beta}^T}\right)^{-1}_{\boldsymbol{\beta}^{\mathrm{old}}}\times \left(\frac{\partial \mathcal{C}(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}}\right)_{\boldsymbol{\beta}^{\mathrm{old}}}, -$$ - -or in matrix form as - -$$ -\boldsymbol{\beta}^{\mathrm{new}} = \boldsymbol{\beta}^{\mathrm{old}}-\left(\boldsymbol{X}^T\boldsymbol{W}\boldsymbol{X} \right)^{-1}\times \left(-\boldsymbol{X}^T(\boldsymbol{y}-\boldsymbol{p}) \right)_{\boldsymbol{\beta}^{\mathrm{old}}}. -$$ - -The right-hand side is computed with the old values of $\beta$. - -If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. - - - -## Brief reminder on Newton-Raphson's method - -Let us quickly remind ourselves how we derive the above method. - -Perhaps the most celebrated of all one-dimensional root-finding -routines is Newton's method, also called the Newton-Raphson -method. This method requires the evaluation of both the -function $f$ and its derivative $f'$ at arbitrary points. -If you can only calculate the derivative -numerically and/or your function is not of the smooth type, we -normally discourage the use of this method. - - -## The equations - -The Newton-Raphson formula consists geometrically of extending the -tangent line at a current point until it crosses zero, then setting -the next guess to the abscissa of that zero-crossing. The mathematics -behind this method is rather simple. Employing a Taylor expansion for -$x$ sufficiently close to the solution $s$, we have - - -
- -$$ -f(s)=0=f(x)+(s-x)f'(x)+\frac{(s-x)^2}{2}f''(x) +\dots. - \label{eq:taylornr} \tag{2} -$$ - -For small enough values of the function and for well-behaved -functions, the terms beyond linear are unimportant, hence we obtain - -$$ -f(x)+(s-x)f'(x)\approx 0, -$$ - -yielding - -$$ -s\approx x-\frac{f(x)}{f'(x)}. -$$ - -Having in mind an iterative procedure, it is natural to start iterating with - -$$ -x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. -$$ - -## Simple geometric interpretation - -The above is Newton-Raphson's method. It has a simple geometric -interpretation, namely $x_{n+1}$ is the point where the tangent from -$(x_n,f(x_n))$ crosses the $x$-axis. Close to the solution, -Newton-Raphson converges fast to the desired result. However, if we -are far from a root, where the higher-order terms in the series are -important, the Newton-Raphson formula can give grossly inaccurate -results. For instance, the initial guess for the root might be so far -from the true root as to let the search interval include a local -maximum or minimum of the function. If an iteration places a trial -guess near such a local extremum, so that the first derivative nearly -vanishes, then Newton-Raphson may fail totally - - - -## Extending to more than one variable - -Newton's method can be generalized to systems of several non-linear equations -and variables. Consider the case with two equations - -$$ -\begin{array}{cc} f_1(x_1,x_2) &=0\\ - f_2(x_1,x_2) &=0,\end{array} -$$ - -which we Taylor expand to obtain - -$$ -\begin{array}{cc} 0=f_1(x_1+h_1,x_2+h_2)=&f_1(x_1,x_2)+h_1 - \partial f_1/\partial x_1+h_2 - \partial f_1/\partial x_2+\dots\\ - 0=f_2(x_1+h_1,x_2+h_2)=&f_2(x_1,x_2)+h_1 - \partial f_2/\partial x_1+h_2 - \partial f_2/\partial x_2+\dots - \end{array}. -$$ - -Defining the Jacobian matrix ${\bf \boldsymbol{J}}$ we have - -$$ -{\bf \boldsymbol{J}}=\left( \begin{array}{cc} - \partial f_1/\partial x_1 & \partial f_1/\partial x_2 \\ - \partial f_2/\partial x_1 &\partial f_2/\partial x_2 - \end{array} \right), -$$ - -we can rephrase Newton's method as - -$$ -\left(\begin{array}{c} x_1^{n+1} \\ x_2^{n+1} \end{array} \right)= -\left(\begin{array}{c} x_1^{n} \\ x_2^{n} \end{array} \right)+ -\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right), -$$ - -where we have defined - -$$ -\left(\begin{array}{c} h_1^{n} \\ h_2^{n} \end{array} \right)= - -{\bf \boldsymbol{J}}^{-1} - \left(\begin{array}{c} f_1(x_1^{n},x_2^{n}) \\ f_2(x_1^{n},x_2^{n}) \end{array} \right). -$$ - -We need thus to compute the inverse of the Jacobian matrix and it -is to understand that difficulties may -arise in case ${\bf \boldsymbol{J}}$ is nearly singular. - -It is rather straightforward to extend the above scheme to systems of -more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. - - - - -## Steepest descent - -The basic idea of gradient descent is -that a function $F(\mathbf{x})$, -$\mathbf{x} \equiv (x_1,\cdots,x_n)$, decreases fastest if one goes from $\bf {x}$ in the -direction of the negative gradient $-\nabla F(\mathbf{x})$. - -It can be shown that if - -$$ -\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), -$$ - -with $\gamma_k > 0$. - -For $\gamma_k$ small enough, then $F(\mathbf{x}_{k+1}) \leq -F(\mathbf{x}_k)$. This means that for a sufficiently small $\gamma_k$ -we are always moving towards smaller function values, i.e a minimum. - - -## More on Steepest descent - -The previous observation is the basis of the method of steepest -descent, which is also referred to as just gradient descent (GD). One -starts with an initial guess $\mathbf{x}_0$ for a minimum of $F$ and -computes new approximations according to - -$$ -\mathbf{x}_{k+1} = \mathbf{x}_k - \gamma_k \nabla F(\mathbf{x}_k), \ \ k \geq 0. -$$ - -The parameter $\gamma_k$ is often referred to as the step length or -the learning rate within the context of Machine Learning. - - -## The ideal - -Ideally the sequence $\{\mathbf{x}_k \}_{k=0}$ converges to a global -minimum of the function $F$. In general we do not know if we are in a -global or local minimum. In the special case when $F$ is a convex -function, all local minima are also global minima, so in this case -gradient descent can converge to the global solution. The advantage of -this scheme is that it is conceptually simple and straightforward to -implement. However the method in this form has some severe -limitations: - -In machine learing we are often faced with non-convex high dimensional -cost functions with many local minima. Since GD is deterministic we -will get stuck in a local minimum, if the method converges, unless we -have a very good intial guess. This also implies that the scheme is -sensitive to the chosen initial condition. - -Note that the gradient is a function of $\mathbf{x} = -(x_1,\cdots,x_n)$ which makes it expensive to compute numerically. - - - -## The sensitiveness of the gradient descent - -The gradient descent method -is sensitive to the choice of learning rate $\gamma_k$. This is due -to the fact that we are only guaranteed that $F(\mathbf{x}_{k+1}) \leq -F(\mathbf{x}_k)$ for sufficiently small $\gamma_k$. The problem is to -determine an optimal learning rate. If the learning rate is chosen too -small the method will take a long time to converge and if it is too -large we can experience erratic behavior. - -Many of these shortcomings can be alleviated by introducing -randomness. One such method is that of Stochastic Gradient Descent -(SGD), see below. - - - -## Convex functions - -Ideally we want our cost/loss function to be convex(concave). - -First we give the definition of a convex set: A set $C$ in -$\mathbb{R}^n$ is said to be convex if, for all $x$ and $y$ in $C$ and -all $t \in (0,1)$ , the point $(1 − t)x + ty$ also belongs to -C. Geometrically this means that every point on the line segment -connecting $x$ and $y$ is in $C$ as discussed below. - -The convex subsets of $\mathbb{R}$ are the intervals of -$\mathbb{R}$. Examples of convex sets of $\mathbb{R}^2$ are the -regular polygons (triangles, rectangles, pentagons, etc...). - - -## Convex function - -**Convex function**: Let $X \subset \mathbb{R}^n$ be a convex set. Assume that the function $f: X \rightarrow \mathbb{R}$ is continuous, then $f$ is said to be convex if $$f(tx_1 + (1-t)x_2) \leq tf(x_1) + (1-t)f(x_2) $$ for all $x_1, x_2 \in X$ and for all $t \in [0,1]$. If $\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \neq x_2$ and $t\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below. - - -## Conditions on convex functions - -In the following we state first and second-order conditions which -ensures convexity of a function $f$. We write $D_f$ to denote the -domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more -details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/, 2004). - -**First order condition.** - -Suppose $f$ is differentiable (i.e $\nabla f(x)$ is well defined for -all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$ -is a convex set and $$f(y) \geq f(x) + \nabla f(x)^T (y-x) $$ holds -for all $x,y \in D_f$. This condition means that for a convex function -the first order Taylor expansion (right hand side above) at any point -a global under estimator of the function. To convince yourself you can -make a drawing of $f(x) = x^2+1$ and draw the tangent line to $f(x)$ and -note that it is always below the graph. - - - -**Second order condition.** - -Assume that $f$ is twice -differentiable, i.e the Hessian matrix exists at each point in -$D_f$. Then $f$ is convex if and only if $D_f$ is a convex set and its -Hessian is positive semi-definite for all $x\in D_f$. For a -single-variable function this reduces to $f''(x) \geq 0$. Geometrically this means that $f$ has nonnegative curvature -everywhere. - - - -This condition is particularly useful since it gives us an procedure for determining if the function under consideration is convex, apart from using the definition. - - -## More on convex functions - -The next result is of great importance to us and the reason why we are -going on about convex functions. In machine learning we frequently -have to minimize a loss/cost function in order to find the best -parameters for the model we are considering. - -Ideally we want the -global minimum (for high-dimensional models it is hard to know -if we have local or global minimum). However, if the cost/loss function -is convex the following result provides invaluable information: - -**Any minimum is global for convex functions.** - -Consider the problem of finding $x \in \mathbb{R}^n$ such that $f(x)$ -is minimal, where $f$ is convex and differentiable. Then, any point -$x^*$ that satisfies $\nabla f(x^*) = 0$ is a global minimum. - - - -This result means that if we know that the cost/loss function is convex and we are able to find a minimum, we are guaranteed that it is a global minimum. - - -## Some simple problems - -1. Show that $f(x)=x^2$ is convex for $x \in \mathbb{R}$ using the definition of convexity. Hint: If you re-write the definition, $f$ is convex if the following holds for all $x,y \in D_f$ and any $\lambda \in [0,1]$ $\lambda f(x)+(1-\lambda)f(y)-f(\lambda x + (1-\lambda) y ) \geq 0$. - -2. Using the second order condition show that the following functions are convex on the specified domain. - - * $f(x) = e^x$ is convex for $x \in \mathbb{R}$. - - * $g(x) = -\ln(x)$ is convex for $x \in (0,\infty)$. - - -3. Let $f(x) = x^2$ and $g(x) = e^x$. Show that $f(g(x))$ and $g(f(x))$ is convex for $x \in \mathbb{R}$. Also show that if $f(x)$ is any convex function than $h(x) = e^{f(x)}$ is convex. - -4. A norm is any function that satisfy the following properties - - * $f(\alpha x) = |\alpha| f(x)$ for all $\alpha \in \mathbb{R}$. - - * $f(x+y) \leq f(x) + f(y)$ - - * $f(x) \leq 0$ for all $x \in \mathbb{R}^n$ with equality if and only if $x = 0$ - - -Using the definition of convexity, try to show that a function satisfying the properties above is convex (the third condition is not needed to show this). - - - -## Friday September 25 - -[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember25.mp4?vrtx=view-as-webpage) and [link to handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember25.pdf). - - - -## Standard steepest descent - - -Before we proceed, we would like to discuss the approach called the -**standard Steepest descent** (different from the above steepest descent discussion), which again leads to us having to be able -to compute a matrix. It belongs to the class of Conjugate Gradient methods (CG). - -[The success of the CG method](https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf) -for finding solutions of non-linear problems is based on the theory -of conjugate gradients for linear systems of equations. It belongs to -the class of iterative methods for solving problems from linear -algebra of the type - -$$ -\boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}. -$$ - -In the iterative process we end up with a problem like - -$$ -\boldsymbol{r}= \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}, -$$ - -where $\boldsymbol{r}$ is the so-called residual or error in the iterative process. - -When we have found the exact solution, $\boldsymbol{r}=0$. - - -## Gradient method - -The residual is zero when we reach the minimum of the quadratic equation - -$$ -P(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T\boldsymbol{b}, -$$ - -with the constraint that the matrix $\boldsymbol{A}$ is positive definite and -symmetric. This defines also the Hessian and we want it to be positive definite. - - - -## Steepest descent method - -We denote the initial guess for $\boldsymbol{x}$ as $\boldsymbol{x}_0$. -We can assume without loss of generality that - -$$ -\boldsymbol{x}_0=0, -$$ - -or consider the system - -$$ -\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0, -$$ - -instead. - - - -## Steepest descent method -One can show that the solution $\boldsymbol{x}$ is also the unique minimizer of the quadratic form - -$$ -f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n. -$$ - -This suggests taking the first basis vector $\boldsymbol{r}_1$ (see below for definition) -to be the gradient of $f$ at $\boldsymbol{x}=\boldsymbol{x}_0$, -which equals - -$$ -\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b}, -$$ - -and -$\boldsymbol{x}_0=0$ it is equal $-\boldsymbol{b}$. - - - - -## Final expressions -We can compute the residual iteratively as - -$$ -\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1}, -$$ - -which equals - -$$ -\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{r}_k), -$$ - -or - -$$ -(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{r}_k, -$$ - -which gives - -$$ -\alpha_k = \frac{\boldsymbol{r}_k^T\boldsymbol{r}_k}{\boldsymbol{r}_k^T\boldsymbol{A}\boldsymbol{r}_k} -$$ - -leading to the iterative scheme - -$$ -\boldsymbol{x}_{k+1}=\boldsymbol{x}_k-\alpha_k\boldsymbol{r}_{k}, -$$ - -## Steepest descent example - -import numpy as np -import numpy.linalg as la - -import scipy.optimize as sopt - -import matplotlib.pyplot as pt -from mpl_toolkits.mplot3d import axes3d - -def f(x): - return 0.5*x[0]**2 + 2.5*x[1]**2 - -def df(x): - return np.array([x[0], 5*x[1]]) - -fig = pt.figure() -ax = fig.gca(projection="3d") - -xmesh, ymesh = np.mgrid[-2:2:50j,-2:2:50j] -fmesh = f(np.array([xmesh, ymesh])) -ax.plot_surface(xmesh, ymesh, fmesh) - -And then as countor plot - -pt.axis("equal") -pt.contour(xmesh, ymesh, fmesh) -guesses = [np.array([2, 2./5])] - -Find guesses - -x = guesses[-1] -s = -df(x) - -Run it! - -def f1d(alpha): - return f(x + alpha*s) - -alpha_opt = sopt.golden(f1d) -next_guess = x + alpha_opt * s -guesses.append(next_guess) -print(next_guess) - -What happened? - -pt.axis("equal") -pt.contour(xmesh, ymesh, fmesh, 50) -it_array = np.array(guesses) -pt.plot(it_array.T[0], it_array.T[1], "x-") - -## Conjugate gradient method -In the CG method we define so-called conjugate directions and two vectors -$\boldsymbol{s}$ and $\boldsymbol{t}$ -are said to be -conjugate if - -$$ -\boldsymbol{s}^T\boldsymbol{A}\boldsymbol{t}= 0. -$$ - -The philosophy of the CG method is to perform searches in various conjugate directions -of our vectors $\boldsymbol{x}_i$ obeying the above criterion, namely - -$$ -\boldsymbol{x}_i^T\boldsymbol{A}\boldsymbol{x}_j= 0. -$$ - -Two vectors are conjugate if they are orthogonal with respect to -this inner product. Being conjugate is a symmetric relation: if $\boldsymbol{s}$ is conjugate to $\boldsymbol{t}$, then $\boldsymbol{t}$ is conjugate to $\boldsymbol{s}$. - - - - -## Conjugate gradient method -An example is given by the eigenvectors of the matrix - -$$ -\boldsymbol{v}_i^T\boldsymbol{A}\boldsymbol{v}_j= \lambda\boldsymbol{v}_i^T\boldsymbol{v}_j, -$$ - -which is zero unless $i=j$. - - - - - -## Conjugate gradient method -Assume now that we have a symmetric positive-definite matrix $\boldsymbol{A}$ of size -$n\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector - -$$ -\boldsymbol{x}_{i+1}=\boldsymbol{x}_{i}+\alpha_i\boldsymbol{p}_{i}. -$$ - -We assume that $\boldsymbol{p}_{i}$ is a sequence of $n$ mutually conjugate directions. -Then the $\boldsymbol{p}_{i}$ form a basis of $R^n$ and we can expand the solution -$ \boldsymbol{A}\boldsymbol{x} = \boldsymbol{b}$ in this basis, namely - -$$ -\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i \boldsymbol{p}_i. -$$ - -## Conjugate gradient method -The coefficients are given by - -$$ -\mathbf{A}\mathbf{x} = \sum^{n}_{i=1} \alpha_i \mathbf{A} \mathbf{p}_i = \mathbf{b}. -$$ - -Multiplying with $\boldsymbol{p}_k^T$ from the left gives - -$$ -\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{x} = \sum^{n}_{i=1} \alpha_i\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{p}_i= \boldsymbol{p}_k^T \boldsymbol{b}, -$$ - -and we can define the coefficients $\alpha_k$ as - -$$ -\alpha_k = \frac{\boldsymbol{p}_k^T \boldsymbol{b}}{\boldsymbol{p}_k^T \boldsymbol{A} \boldsymbol{p}_k} -$$ - -## Conjugate gradient method and iterations - -If we choose the conjugate vectors $\boldsymbol{p}_k$ carefully, -then we may not need all of them to obtain a good approximation to the solution -$\boldsymbol{x}$. -We want to regard the conjugate gradient method as an iterative method. -This will us to solve systems where $n$ is so large that the direct -method would take too much time. - -We denote the initial guess for $\boldsymbol{x}$ as $\boldsymbol{x}_0$. -We can assume without loss of generality that - -$$ -\boldsymbol{x}_0=0, -$$ - -or consider the system - -$$ -\boldsymbol{A}\boldsymbol{z} = \boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_0, -$$ - -instead. - - - - - -## Conjugate gradient method -One can show that the solution $\boldsymbol{x}$ is also the unique minimizer of the quadratic form - -$$ -f(\boldsymbol{x}) = \frac{1}{2}\boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x} - \boldsymbol{x}^T \boldsymbol{x} , \quad \boldsymbol{x}\in\mathbf{R}^n. -$$ - -This suggests taking the first basis vector $\boldsymbol{p}_1$ -to be the gradient of $f$ at $\boldsymbol{x}=\boldsymbol{x}_0$, -which equals - -$$ -\boldsymbol{A}\boldsymbol{x}_0-\boldsymbol{b}, -$$ - -and -$\boldsymbol{x}_0=0$ it is equal $-\boldsymbol{b}$. -The other vectors in the basis will be conjugate to the gradient, -hence the name conjugate gradient method. - - - - - -## Conjugate gradient method -Let $\boldsymbol{r}_k$ be the residual at the $k$-th step: - -$$ -\boldsymbol{r}_k=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k. -$$ - -Note that $\boldsymbol{r}_k$ is the negative gradient of $f$ at -$\boldsymbol{x}=\boldsymbol{x}_k$, -so the gradient descent method would be to move in the direction $\boldsymbol{r}_k$. -Here, we insist that the directions $\boldsymbol{p}_k$ are conjugate to each other, -so we take the direction closest to the gradient $\boldsymbol{r}_k$ -under the conjugacy constraint. -This gives the following expression - -$$ -\boldsymbol{p}_{k+1}=\boldsymbol{r}_k-\frac{\boldsymbol{p}_k^T \boldsymbol{A}\boldsymbol{r}_k}{\boldsymbol{p}_k^T\boldsymbol{A}\boldsymbol{p}_k} \boldsymbol{p}_k. -$$ - -## Conjugate gradient method -We can also compute the residual iteratively as - -$$ -\boldsymbol{r}_{k+1}=\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_{k+1}, -$$ - -which equals - -$$ -\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_k+\alpha_k\boldsymbol{p}_k), -$$ - -or - -$$ -(\boldsymbol{b}-\boldsymbol{A}\boldsymbol{x}_k)-\alpha_k\boldsymbol{A}\boldsymbol{p}_k, -$$ - -which gives - -$$ -\boldsymbol{r}_{k+1}=\boldsymbol{r}_k-\boldsymbol{A}\boldsymbol{p}_{k}, -$$ - -## Revisiting our first homework - -We will use linear regression as a case study for the gradient descent -methods. Linear regression is a great test case for the gradient -descent methods discussed in the lectures since it has several -desirable properties such as: - -1. An analytical solution (recall homework set 1). - -2. The gradient can be computed analytically. - -3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates - -We revisit an example similar to what we had in the first homework set. We had a function of the type - -x = 2*np.random.rand(m,1) -y = 4+3*x+np.random.randn(m,1) - -with $x_i \in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\cal {N}(0,1)$. -The linear regression model is given by - -$$ -h_\beta(x) = \boldsymbol{y} = \beta_0 + \beta_1 x, -$$ - -such that - -$$ -\boldsymbol{y}_i = \beta_0 + \beta_1 x_i. -$$ - -## Gradient descent example - -Let $\mathbf{y} = (y_1,\cdots,y_n)^T$, $\mathbf{\boldsymbol{y}} = (\boldsymbol{y}_1,\cdots,\boldsymbol{y}_n)^T$ and $\beta = (\beta_0, \beta_1)^T$ - -It is convenient to write $\mathbf{\boldsymbol{y}} = X\beta$ where $X \in \mathbb{R}^{100 \times 2} $ is the design matrix given by (we keep the intercept here) - -$$ -X \equiv \begin{bmatrix} -1 & x_1 \\ -\vdots & \vdots \\ -1 & x_{100} & \\ -\end{bmatrix}. -$$ - -The cost/loss/risk function is given by ( - -$$ -C(\beta) = \frac{1}{n}||X\beta-\mathbf{y}||_{2}^{2} = \frac{1}{n}\sum_{i=1}^{100}\left[ (\beta_0 + \beta_1 x_i)^2 - 2 y_i (\beta_0 + \beta_1 x_i) + y_i^2\right] -$$ - -and we want to find $\beta$ such that $C(\beta)$ is minimized. - - -## The derivative of the cost/loss function - -Computing $\partial C(\beta) / \partial \beta_0$ and $\partial C(\beta) / \partial \beta_1$ we can show that the gradient can be written as - -$$ -\nabla_{\beta} C(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} = \frac{2}{n}X^T(X\beta - \mathbf{y}), -$$ - -where $X$ is the design matrix defined above. - - -## The Hessian matrix -The Hessian matrix of $C(\beta)$ is given by - -$$ -\boldsymbol{H} \equiv \begin{bmatrix} -\frac{\partial^2 C(\beta)}{\partial \beta_0^2} & \frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} \\ -\frac{\partial^2 C(\beta)}{\partial \beta_0 \partial \beta_1} & \frac{\partial^2 C(\beta)}{\partial \beta_1^2} & \\ -\end{bmatrix} = \frac{2}{n}X^T X. -$$ - -This result implies that $C(\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite. - - - - - -## Simple program - -We can now write a program that minimizes $C(\beta)$ using the gradient descent method with a constant learning rate $\gamma$ according to - -$$ -\beta_{k+1} = \beta_k - \gamma \nabla_\beta C(\beta_k), \ k=0,1,\cdots -$$ - -We can use the expression we computed for the gradient and let use a -$\beta_0$ be chosen randomly and let $\gamma = 0.001$. Stop iterating -when $||\nabla_\beta C(\beta_k) || \leq \epsilon = 10^{-8}$. **Note that the code below does not include the latter stop criterion**. - -And finally we can compare our solution for $\beta$ with the analytic result given by -$\beta= (X^TX)^{-1} X^T \mathbf{y}$. - - -## Gradient Descent Example - -Here our simple example - - -# Importing various packages -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt -from mpl_toolkits.mplot3d import Axes3D -from matplotlib import cm -from matplotlib.ticker import LinearLocator, FormatStrFormatter -import sys - -# the number of datapoints -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -# Hessian matrix -H = (2.0/n)* X.T @ X -# Get the eigenvalues -EigValues, EigVectors = np.linalg.eig(H) -print(EigValues) - -beta_linreg = np.linalg.inv(X.T @ X) @ X.T @ y -print(beta_linreg) -beta = np.random.randn(2,1) - -eta = 1.0/np.max(EigValues) -Niterations = 1000 - -for iter in range(Niterations): - gradient = (2.0/n)*X.T @ (X @ beta-y) - beta -= eta*gradient - -print(beta) -xnew = np.array([[0],[2]]) -xbnew = np.c_[np.ones((2,1)), xnew] -ypredict = xbnew.dot(beta) -ypredict2 = xbnew.dot(beta_linreg) -plt.plot(xnew, ypredict, "r-") -plt.plot(xnew, ypredict2, "b-") -plt.plot(x, y ,'ro') -plt.axis([0,2.0,0, 15.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Gradient descent example') -plt.show() - -## And a corresponding example using **scikit-learn** - -# Importing various packages -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import SGDRegressor - -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -beta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y) -print(beta_linreg) -sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) -sgdreg.fit(x,y.ravel()) -print(sgdreg.intercept_, sgdreg.coef_) - -## Gradient descent and Ridge - -We have also discussed Ridge regression where the loss function contains a regularized term given by the $L_2$ norm of $\beta$, - -$$ -C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta||^2, \ \lambda \geq 0. -$$ - -In order to minimize $C_{\text{ridge}}(\beta)$ using GD we only have adjust the gradient as follows - -$$ -\nabla_\beta C_{\text{ridge}}(\beta) = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\ -\sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\ -\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta). -$$ - -We can easily extend our program to minimize $C_{\text{ridge}}(\beta)$ using gradient descent and compare with the analytical solution given by - -$$ -\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}. -$$ - -## Program example for gradient descent with Ridge Regression - -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt -from mpl_toolkits.mplot3d import Axes3D -from matplotlib import cm -from matplotlib.ticker import LinearLocator, FormatStrFormatter -import sys - -# the number of datapoints -n = 100 -x = 2*np.random.rand(n,1) -y = 4+3*x+np.random.randn(n,1) - -X = np.c_[np.ones((n,1)), x] -XT_X = X.T @ X - -#Ridge parameter lambda -lmbda = 0.001 -Id = lmbda* np.eye(XT_X.shape[0]) - -beta_linreg = np.linalg.inv(XT_X+Id) @ X.T @ y -print(beta_linreg) -# Start plain gradient descent -beta = np.random.randn(2,1) - -eta = 0.1 -Niterations = 100 - -for iter in range(Niterations): - gradients = 2.0/n*X.T @ (X @ (beta)-y)+2*lmbda*beta - beta -= eta*gradients - -print(beta) -ypredict = X @ beta -ypredict2 = X @ beta_linreg -plt.plot(x, ypredict, "r-") -plt.plot(x, ypredict2, "b-") -plt.plot(x, y ,'ro') -plt.axis([0,2.0,0, 15.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Gradient descent example for Ridge') -plt.show() - -## Using gradient descent methods, limitations - -* **Gradient descent (GD) finds local minima of our function**. Since the GD algorithm is deterministic, if it converges, it will converge to a local minimum of our cost/loss/risk function. Because in ML we are often dealing with extremely rugged landscapes with many local minima, this can lead to poor performance. - -* **GD is sensitive to initial conditions**. One consequence of the local nature of GD is that initial conditions matter. Depending on where one starts, one will end up at a different local minima. Therefore, it is very important to think about how one initializes the training process. This is true for GD as well as more complicated variants of GD. - -* **Gradients are computationally expensive to calculate for large datasets**. In many cases in statistics and ML, the cost/loss/risk function is a sum of terms, with one term for each data point. For example, in linear regression, $E \propto \sum_{i=1}^n (y_i - \mathbf{w}^T\cdot\mathbf{x}_i)^2$; for logistic regression, the square error is replaced by the cross entropy. To calculate the gradient we have to sum over *all* $n$ data points. Doing this at every GD step becomes extremely computationally expensive. An ingenious solution to this, is to calculate the gradients using small subsets of the data called "mini batches". This has the added benefit of introducing stochasticity into our algorithm. - -* **GD is very sensitive to choices of learning rates**. GD is extremely sensitive to the choice of learning rates. If the learning rate is very small, the training process take an extremely long time. For larger learning rates, GD can diverge and give poor results. Furthermore, depending on what the local landscape looks like, we have to modify the learning rates to ensure convergence. Ideally, we would *adaptively* choose the learning rates to match the landscape. - -* **GD treats all directions in parameter space uniformly.** Another major drawback of GD is that unlike Newton's method, the learning rate for GD is the same in all directions in parameter space. For this reason, the maximum learning rate is set by the behavior of the steepest direction and this can significantly slow down training. Ideally, we would like to take large steps in flat directions and small steps in steep directions. Since we are exploring rugged landscapes where curvatures change, this requires us to keep track of not only the gradient but second derivatives. The ideal scenario would be to calculate the Hessian but this proves to be too computationally expensive. - -* GD can take exponential time to escape saddle points, even with random initialization. As we mentioned, GD is extremely sensitive to initial condition since it determines the particular local minimum GD would eventually reach. However, even with a good initialization scheme, through the introduction of randomness, GD can still take exponential time to escape saddle points. - -## Stochastic Gradient Descent - -Stochastic gradient descent (SGD) and variants thereof address some of -the shortcomings of the Gradient descent method discussed above. - -The underlying idea of SGD comes from the observation that the cost -function, which we want to minimize, can almost always be written as a -sum over $n$ data points $\{\mathbf{x}_i\}_{i=1}^n$, - -$$ -C(\mathbf{\beta}) = \sum_{i=1}^n c_i(\mathbf{x}_i, -\mathbf{\beta}). -$$ - -## Computation of gradients - -This in turn means that the gradient can be -computed as a sum over $i$-gradients - -$$ -\nabla_\beta C(\mathbf{\beta}) = \sum_i^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}). -$$ - -Stochasticity/randomness is introduced by only taking the -gradient on a subset of the data called minibatches. If there are $n$ -data points and the size of each minibatch is $M$, there will be $n/M$ -minibatches. We denote these minibatches by $B_k$ where -$k=1,\cdots,n/M$. - - -## SGD example -As an example, suppose we have $10$ data points $(\mathbf{x}_1,\cdots, \mathbf{x}_{10})$ -and we choose to have $M=5$ minibathces, -then each minibatch contains two data points. In particular we have -$B_1 = (\mathbf{x}_1,\mathbf{x}_2), \cdots, B_5 = -(\mathbf{x}_9,\mathbf{x}_{10})$. Note that if you choose $M=1$ you -have only a single batch with all data points and on the other extreme, -you may choose $M=n$ resulting in a minibatch for each datapoint, i.e -$B_k = \mathbf{x}_k$. - -The idea is now to approximate the gradient by replacing the sum over -all data points with a sum over the data points in one the minibatches -picked at random in each gradient descent step - -$$ -\nabla_{\beta} -C(\mathbf{\beta}) = \sum_{i=1}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) \rightarrow \sum_{i \in B_k}^n \nabla_\beta -c_i(\mathbf{x}_i, \mathbf{\beta}). -$$ - -## The gradient step - -Thus a gradient descent step now looks like - -$$ -\beta_{j+1} = \beta_j - \gamma_j \sum_{i \in B_k}^n \nabla_\beta c_i(\mathbf{x}_i, -\mathbf{\beta}) -$$ - -where $k$ is picked at random with equal -probability from $[1,n/M]$. An iteration over the number of -minibathces (n/M) is commonly referred to as an epoch. Thus it is -typical to choose a number of epochs and for each epoch iterate over -the number of minibatches, as exemplified in the code below. - - -## Simple example code - -import numpy as np - -n = 100 #100 datapoints -M = 5 #size of each minibatch -m = int(n/M) #number of minibatches -n_epochs = 10 #number of epochs - -j = 0 -for epoch in range(1,n_epochs+1): - for i in range(m): - k = np.random.randint(m) #Pick the k-th minibatch at random - #Compute the gradient using the data in minibatch Bk - #Compute new suggestion for - j += 1 - -Taking the gradient only on a subset of the data has two important -benefits. First, it introduces randomness which decreases the chance -that our opmization scheme gets stuck in a local minima. Second, if -the size of the minibatches are small relative to the number of -datapoints ($M < n$), the computation of the gradient is much -cheaper since we sum over the datapoints in the $k-th$ minibatch and not -all $n$ datapoints. - - -## When do we stop? - -A natural question is when do we stop the search for a new minimum? -One possibility is to compute the full gradient after a given number -of epochs and check if the norm of the gradient is smaller than some -threshold and stop if true. However, the condition that the gradient -is zero is valid also for local minima, so this would only tell us -that we are close to a local/global minimum. However, we could also -evaluate the cost function at this point, store the result and -continue the search. If the test kicks in at a later stage we can -compare the values of the cost function and keep the $\beta$ that -gave the lowest value. - - -## Slightly different approach - -Another approach is to let the step length $\gamma_j$ depend on the -number of epochs in such a way that it becomes very small after a -reasonable time such that we do not move at all. - -As an example, let $e = 0,1,2,3,\cdots$ denote the current epoch and let $t_0, t_1 > 0$ be two fixed numbers. Furthermore, let $t = e \cdot m + i$ where $m$ is the number of minibatches and $i=0,\cdots,m-1$. Then the function $$\gamma_j(t; t_0, t_1) = \frac{t_0}{t+t_1} $$ goes to zero as the number of epochs gets large. I.e. we start with a step length $\gamma_j (0; t_0, t_1) = t_0/t_1$ which decays in *time* $t$. - -In this way we can fix the number of epochs, compute $\beta$ and -evaluate the cost function at the end. Repeating the computation will -give a different result since the scheme is random by design. Then we -pick the final $\beta$ that gives the lowest value of the cost -function. - -import numpy as np - -def step_length(t,t0,t1): - return t0/(t+t1) - -n = 100 #100 datapoints -M = 5 #size of each minibatch -m = int(n/M) #number of minibatches -n_epochs = 500 #number of epochs -t0 = 1.0 -t1 = 10 - -gamma_j = t0/t1 -j = 0 -for epoch in range(1,n_epochs+1): - for i in range(m): - k = np.random.randint(m) #Pick the k-th minibatch at random - #Compute the gradient using the data in minibatch Bk - #Compute new suggestion for beta - t = epoch*m+i - gamma_j = step_length(t,t0,t1) - j += 1 - -print("gamma_j after %d epochs: %g" % (n_epochs,gamma_j)) - -## Program for stochastic gradient - -# Importing various packages -from math import exp, sqrt -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt -from sklearn.linear_model import SGDRegressor - -m = 100 -x = 2*np.random.rand(m,1) -y = 4+3*x+np.random.randn(m,1) - -X = np.c_[np.ones((m,1)), x] -theta_linreg = np.linalg.inv(X.T @ X) @ (X.T @ y) -print("Own inversion") -print(theta_linreg) -sgdreg = SGDRegressor(max_iter = 50, penalty=None, eta0=0.1) -sgdreg.fit(x,y.ravel()) -print("sgdreg from scikit") -print(sgdreg.intercept_, sgdreg.coef_) - - -theta = np.random.randn(2,1) -eta = 0.1 -Niterations = 1000 - - -for iter in range(Niterations): - gradients = 2.0/m*X.T @ ((X @ theta)-y) - theta -= eta*gradients -print("theta from own gd") -print(theta) - -xnew = np.array([[0],[2]]) -Xnew = np.c_[np.ones((2,1)), xnew] -ypredict = Xnew.dot(theta) -ypredict2 = Xnew.dot(theta_linreg) - - -n_epochs = 50 -t0, t1 = 5, 50 -def learning_schedule(t): - return t0/(t+t1) - -theta = np.random.randn(2,1) - -for epoch in range(n_epochs): - for i in range(m): - random_index = np.random.randint(m) - xi = X[random_index:random_index+1] - yi = y[random_index:random_index+1] - gradients = 2 * xi.T @ ((xi @ theta)-yi) - eta = learning_schedule(epoch*m+i) - theta = theta - eta*gradients -print("theta from own sdg") -print(theta) - -plt.plot(xnew, ypredict, "r-") -plt.plot(xnew, ypredict2, "b-") -plt.plot(x, y ,'ro') -plt.axis([0,2.0,0, 15.0]) -plt.xlabel(r'$x$') -plt.ylabel(r'$y$') -plt.title(r'Random numbers ') -plt.show() - -**Challenge**: try to write a similar code for a Logistic Regression case. \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter5.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter5.ipynb deleted file mode 100644 index e0728e970..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter5.ipynb +++ /dev/null @@ -1,2022 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Support Vector Machines, overarching aims\n", - "\n", - "A Support Vector Machine (SVM) is a very powerful and versatile\n", - "Machine Learning method, capable of performing linear or nonlinear\n", - "classification, regression, and even outlier detection. It is one of\n", - "the most popular models in Machine Learning, and anyone interested in\n", - "Machine Learning should have it in their toolbox. SVMs are\n", - "particularly well suited for classification of complex but small-sized or\n", - "medium-sized datasets. \n", - "\n", - "The case with two well-separated classes only can be understood in an\n", - "intuitive way in terms of lines in a two-dimensional space separating\n", - "the two classes (see figure below).\n", - "\n", - "The basic mathematics behind the SVM is however less familiar to most of us. \n", - "It relies on the definition of hyperplanes and the\n", - "definition of a **margin** which separates classes (in case of\n", - "classification problems) of variables. It is also used for regression\n", - "problems.\n", - "\n", - "With SVMs we distinguish between hard margin and soft margins. The\n", - "latter introduces a so-called softening parameter to be discussed\n", - "below. We distinguish also between linear and non-linear\n", - "approaches. The latter are the most frequent ones since it is rather\n", - "unlikely that we can separate classes easily by say straight lines.\n", - "\n", - "\n", - "## Hyperplanes and all that\n", - "\n", - "The theory behind support vector machines (SVM hereafter) is based on\n", - "the mathematical description of so-called hyperplanes. Let us start\n", - "with a two-dimensional case. This will also allow us to introduce our\n", - "first SVM examples. These will be tailored to the case of two specific\n", - "classes, as displayed in the figure here based on the usage of the petal data.\n", - "\n", - "We assume here that our data set can be well separated into two\n", - "domains, where a straight line does the job in the separating the two\n", - "classes. Here the two classes are represented by either squares or\n", - "circles." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "LinearSVC: [0.28475098] [[1.05364854 1.09903804]]\n", - "SVC: [0.31896852] [[1.1203284 1.02625193]]\n", - "SGDClassifier(alpha=0.00200): [0.117] [[0.77714169 0.72981762]]\n" - ] - }, - { - "data": { - "image/png": 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\n", - "text/plain": [ - "
" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter5_1_1.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "%matplotlib inline\n", - "\n", - "from sklearn import datasets\n", - "from sklearn.svm import SVC, LinearSVC\n", - "from sklearn.linear_model import SGDClassifier\n", - "from sklearn.preprocessing import StandardScaler\n", - "import matplotlib\n", - "import matplotlib.pyplot as plt\n", - "plt.rcParams['axes.labelsize'] = 14\n", - "plt.rcParams['xtick.labelsize'] = 12\n", - "plt.rcParams['ytick.labelsize'] = 12\n", - "\n", - "\n", - "iris = datasets.load_iris()\n", - "X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n", - "y = iris[\"target\"]\n", - "\n", - "setosa_or_versicolor = (y == 0) | (y == 1)\n", - "X = X[setosa_or_versicolor]\n", - "y = y[setosa_or_versicolor]\n", - "\n", - "\n", - "\n", - "C = 5\n", - "alpha = 1 / (C * len(X))\n", - "\n", - "lin_clf = LinearSVC(loss=\"hinge\", C=C, random_state=42)\n", - "svm_clf = SVC(kernel=\"linear\", C=C)\n", - "sgd_clf = SGDClassifier(loss=\"hinge\", learning_rate=\"constant\", eta0=0.001, alpha=alpha,\n", - " max_iter=100000, random_state=42)\n", - "\n", - "scaler = StandardScaler()\n", - "X_scaled = scaler.fit_transform(X)\n", - "\n", - "lin_clf.fit(X_scaled, y)\n", - "svm_clf.fit(X_scaled, y)\n", - "sgd_clf.fit(X_scaled, y)\n", - "\n", - "print(\"LinearSVC: \", lin_clf.intercept_, lin_clf.coef_)\n", - "print(\"SVC: \", svm_clf.intercept_, svm_clf.coef_)\n", - "print(\"SGDClassifier(alpha={:.5f}):\".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)\n", - "\n", - "# Compute the slope and bias of each decision boundary\n", - "w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]\n", - "b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]\n", - "w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]\n", - "b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]\n", - "w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]\n", - "b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]\n", - "\n", - "# Transform the decision boundary lines back to the original scale\n", - "line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])\n", - "line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])\n", - "line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])\n", - "\n", - "# Plot all three decision boundaries\n", - "plt.figure(figsize=(11, 4))\n", - "plt.plot(line1[:, 0], line1[:, 1], \"k:\", label=\"LinearSVC\")\n", - "plt.plot(line2[:, 0], line2[:, 1], \"b--\", linewidth=2, label=\"SVC\")\n", - "plt.plot(line3[:, 0], line3[:, 1], \"r-\", label=\"SGDClassifier\")\n", - "plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\") # label=\"Iris-Versicolor\"\n", - "plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\") # label=\"Iris-Setosa\"\n", - "plt.xlabel(\"Petal length\", fontsize=14)\n", - "plt.ylabel(\"Petal width\", fontsize=14)\n", - "plt.legend(loc=\"upper center\", fontsize=14)\n", - "plt.axis([0, 5.5, 0, 2])\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The aim of the SVM algorithm is to find a hyperplane in a\n", - "$p$-dimensional space, where $p$ is the number of features that\n", - "distinctly classifies the data points.\n", - "\n", - "In a $p$-dimensional space, a hyperplane is what we call an affine subspace of dimension of $p-1$.\n", - "As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is \n", - "a two-dimensional subspace, or stated simply, a plane. \n", - "\n", - "In two dimensions, with the variables $x_1$ and $x_2$, the hyperplane is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b+w_1x_1+w_2x_2=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $b$ is the intercept and $w_1$ and $w_2$ define the elements of a vector orthogonal to the line \n", - "$b+w_1x_1+w_2x_2=0$. \n", - "In two dimensions we define the vectors $\\boldsymbol{x} =[x1,x2]$ and $\\boldsymbol{w}=[w1,w2]$. \n", - "We can then rewrite the above equation as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}^T\\boldsymbol{w}+b=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We limit ourselves to two classes of outputs $y_i$ and assign these classes the values $y_i = \\pm 1$. \n", - "In a $p$-dimensional space of say $p$ features we have a hyperplane defines as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b+wx_1+w_2x_2+\\dots +w_px_p=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we define a \n", - "matrix $\\boldsymbol{X}=\\left[\\boldsymbol{x}_1,\\boldsymbol{x}_2,\\dots, \\boldsymbol{x}_p\\right]$\n", - "of dimension $n\\times p$, where $n$ represents the observations for each feature and each vector $x_i$ is a column vector of the matrix $\\boldsymbol{X}$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i = \\begin{bmatrix} x_{i1} \\\\ x_{i2} \\\\ \\dots \\\\ \\dots \\\\ x_{ip} \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If the above condition is not met for a given vector $\\boldsymbol{x}_i$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip} >0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "if our output $y_i=1$.\n", - "In this case we say that $\\boldsymbol{x}_i$ lies on one of the sides of the hyperplane and if" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip} < 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for the class of observations $y_i=-1$, \n", - "then $\\boldsymbol{x}_i$ lies on the other side. \n", - "\n", - "Equivalently, for the two classes of observations we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i\\left(b+w_1x_{i1}+w_2x_{i2}+\\dots +w_px_{ip}\\right) > 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.\n", - "\n", - "\n", - "### The two-dimensional case\n", - "\n", - "Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional\n", - "plane. To separate the two classes of data points, there are many\n", - "possible lines (hyperplanes if you prefer a more strict naming) \n", - "that could be chosen. Our objective is to find a\n", - "plane that has the maximum margin, i.e the maximum distance between\n", - "data points of both classes. Maximizing the margin distance provides\n", - "some reinforcement so that future data points can be classified with\n", - "more confidence.\n", - "\n", - "What a linear classifier attempts to accomplish is to split the\n", - "feature space into two half spaces by placing a hyperplane between the\n", - "data points. This hyperplane will be our decision boundary. All\n", - "points on one side of the plane will belong to class one and all points\n", - "on the other side of the plane will belong to the second class two.\n", - "\n", - "Unfortunately there are many ways in which we can place a hyperplane\n", - "to divide the data. Below is an example of two candidate hyperplanes\n", - "for our data sample.\n", - "\n", - "\n", - "Let us define the function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(x) = \\boldsymbol{w}^T\\boldsymbol{x}+b = 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "as the function that determines the line $L$ that separates two classes (our two features), see the figure here. \n", - "\n", - "\n", - "Any point defined by $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_2$ on the line $L$ will satisfy $\\boldsymbol{w}^T(\\boldsymbol{x}_1-\\boldsymbol{x}_2)=0$. \n", - "\n", - "The signed distance $\\delta$ from any point defined by a vector $\\boldsymbol{x}$ and a point $\\boldsymbol{x}_0$ on the line $L$ is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta = \\frac{1}{\\vert\\vert \\boldsymbol{w}\\vert\\vert}(\\boldsymbol{w}^T\\boldsymbol{x}+b).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "How do we find the parameter $b$ and the vector $\\boldsymbol{w}$? What we could\n", - "do is to define a cost function which now contains the set of all\n", - "misclassified points $M$ and attempt to minimize this function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{w},b) = -\\sum_{i\\in M} y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We could now for example define all values $y_i =1$ as misclassified in case we have $\\boldsymbol{w}^T\\boldsymbol{x}_i+b < 0$ and the opposite if we have $y_i=-1$. Taking the derivatives gives us" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C}{\\partial b} = -\\sum_{i\\in M} y_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial C}{\\partial \\boldsymbol{w}} = -\\sum_{i\\in M} y_ix_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b \\leftarrow b +\\eta \\frac{\\partial C}{\\partial b},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{w} \\leftarrow \\boldsymbol{w} +\\eta \\frac{\\partial C}{\\partial \\boldsymbol{w}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\eta$ is our by now well-known learning rate. \n", - "\n", - "\n", - "\n", - "The equations we discussed above can be coded rather easily (the\n", - "framework is similar to what we developed for logistic\n", - "regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There are however problems with this approach, although it looks\n", - "pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes.\n", - "\n", - "\n", - "For small\n", - "gaps between the entries, we may also end up needing many iterations\n", - "before the solutions converge and if the data cannot be separated\n", - "properly into two distinct classes, we may not experience a converge\n", - "at all.\n", - "\n", - "\n", - "### A better approach\n", - "\n", - "A better approach is rather to try to define a large margin between\n", - "the two classes (if they are well separated from the beginning).\n", - "\n", - "Thus, we wish to find a margin $M$ with $\\boldsymbol{w}$ normalized to\n", - "$\\vert\\vert \\boldsymbol{w}\\vert\\vert =1$ subject to the condition" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M \\hspace{0.1cm}\\forall i=1,2,\\dots, p.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "All points are thus at a signed distance from the decision boundary defined by the line $L$. The parameters $b$ and $w_1$ and $w_2$ define this line. \n", - "\n", - "We seek thus the largest value $M$ defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{\\vert \\vert \\boldsymbol{w}\\vert\\vert}y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M \\hspace{0.1cm}\\forall i=1,2,\\dots, n,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or just" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq M\\vert \\vert \\boldsymbol{w}\\vert\\vert \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we scale the equation so that $\\vert \\vert \\boldsymbol{w}\\vert\\vert = 1/M$, we have to find the minimum of \n", - "$\\boldsymbol{w}^T\\boldsymbol{w}=\\vert \\vert \\boldsymbol{w}\\vert\\vert$ (the norm) subject to the condition" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) \\geq 1 \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We have thus defined our margin as the invers of the norm of\n", - "$\\boldsymbol{w}$. We want to minimize the norm in order to have a as large as\n", - "possible margin $M$. Before we proceed, we need to remind ourselves\n", - "about Lagrangian multipliers.\n", - "\n", - "\n", - "## A quick Reminder on Lagrangian Multipliers\n", - "\n", - "Consider a function of three independent variables $f(x,y,z)$ . For the function $f$ to be an\n", - "extreme we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "df=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "A necessary and sufficient condition is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial x} =\\frac{\\partial f}{\\partial y}=\\frac{\\partial f}{\\partial z}=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "due to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "df = \\frac{\\partial f}{\\partial x}dx+\\frac{\\partial f}{\\partial y}dy+\\frac{\\partial f}{\\partial z}dz.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In many problems the variables $x,y,z$ are often subject to constraints (such as those above for the margin)\n", - "so that they are no longer all independent. It is possible at least in principle to use each \n", - "constraint to eliminate one variable\n", - "and to proceed with a new and smaller set of independent varables.\n", - "\n", - "The use of so-called Lagrangian multipliers is an alternative technique when the elimination\n", - "of variables is incovenient or undesirable. Assume that we have an equation of constraint on \n", - "the variables $x,y,z$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\phi(x,y,z) = 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "d\\phi = \\frac{\\partial \\phi}{\\partial x}dx+\\frac{\\partial \\phi}{\\partial y}dy+\\frac{\\partial \\phi}{\\partial z}dz =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now we cannot set anymore" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial x} =\\frac{\\partial f}{\\partial y}=\\frac{\\partial f}{\\partial z}=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "if $df=0$ is wanted\n", - "because there are now only two independent variables! Assume $x$ and $y$ are the independent \n", - "variables.\n", - "Then $dz$ is no longer arbitrary.\n", - "\n", - "\n", - "However, we can add to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "df = \\frac{\\partial f}{\\partial x}dx+\\frac{\\partial f}{\\partial y}dy+\\frac{\\partial f}{\\partial z}dz,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "a multiplum of $d\\phi$, viz. $\\lambda d\\phi$, resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "df+\\lambda d\\phi = (\\frac{\\partial f}{\\partial z}+\\lambda\n", - "\\frac{\\partial \\phi}{\\partial x})dx+(\\frac{\\partial f}{\\partial y}+\\lambda\\frac{\\partial \\phi}{\\partial y})dy+\n", - "(\\frac{\\partial f}{\\partial z}+\\lambda\\frac{\\partial \\phi}{\\partial z})dz =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Our multiplier is chosen so that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial z}+\\lambda\\frac{\\partial \\phi}{\\partial z} =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We need to remember that we took $dx$ and $dy$ to be arbitrary and thus we must have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial x}+\\lambda\\frac{\\partial \\phi}{\\partial x} =0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial y}+\\lambda\\frac{\\partial \\phi}{\\partial y} =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "When all these equations are satisfied, $df=0$. We have four unknowns, $x,y,z$ and\n", - "$\\lambda$. Actually we want only $x,y,z$, $\\lambda$ needs not to be determined, \n", - "it is therefore often called\n", - "Lagrange's undetermined multiplier.\n", - "If we have a set of constraints $\\phi_k$ we have the equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial f}{\\partial x_i}+\\sum_k\\lambda_k\\frac{\\partial \\phi_k}{\\partial x_i} =0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to solve the above problem, we define the following Lagrangian function to be minimized" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}(\\lambda,b,\\boldsymbol{w})=\\frac{1}{2}\\boldsymbol{w}^T\\boldsymbol{w}-\\sum_{i=1}^n\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)-1\\right],\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\lambda_i$ is a so-called Lagrange multiplier subject to the condition $\\lambda_i \\geq 0$.\n", - "\n", - "Taking the derivatives with respect to $b$ and $\\boldsymbol{w}$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{L}}{\\partial b} = -\\sum_{i} \\lambda_iy_i=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{L}}{\\partial \\boldsymbol{w}} = 0 = \\boldsymbol{w}-\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Inserting these constraints into the equation for $\\cal{L}$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to the constraints $\\lambda_i\\geq 0$ and $\\sum_i\\lambda_iy_i=0$. \n", - "We must in addition satisfy the [Karush-Kuhn-Tucker](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions) (KKT) condition" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) -1\\right] \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "1. If $\\lambda_i > 0$, then $y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1$ and we say that $x_i$ is on the boundary.\n", - "\n", - "2. If $y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)> 1$, we say $x_i$ is not on the boundary and we set $\\lambda_i=0$. \n", - "\n", - "When $\\lambda_i > 0$, the vectors $\\boldsymbol{x}_i$ are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin $M$. \n", - "\n", - "\n", - "We can rewrite" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and its constraints in terms of a matrix-vector problem where we minimize w.r.t. $\\lambda$ the following problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{2} \\boldsymbol{\\lambda}^T\\begin{bmatrix} y_1y_1\\boldsymbol{x}_1^T\\boldsymbol{x}_1 & y_1y_2\\boldsymbol{x}_1^T\\boldsymbol{x}_2 & \\dots & \\dots & y_1y_n\\boldsymbol{x}_1^T\\boldsymbol{x}_n \\\\\n", - "y_2y_1\\boldsymbol{x}_2^T\\boldsymbol{x}_1 & y_2y_2\\boldsymbol{x}_2^T\\boldsymbol{x}_2 & \\dots & \\dots & y_1y_n\\boldsymbol{x}_2^T\\boldsymbol{x}_n \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "y_ny_1\\boldsymbol{x}_n^T\\boldsymbol{x}_1 & y_ny_2\\boldsymbol{x}_n^T\\boldsymbol{x}_2 & \\dots & \\dots & y_ny_n\\boldsymbol{x}_n^T\\boldsymbol{x}_n \\\\\n", - "\\end{bmatrix}\\boldsymbol{\\lambda}-\\mathbb{1}\\boldsymbol{\\lambda},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$. Here we defined the vectors $\\boldsymbol{\\lambda} =[\\lambda_1,\\lambda_2,\\dots,\\lambda_n]$ and \n", - "$\\boldsymbol{y}=[y_1,y_2,\\dots,y_n]$. \n", - "\n", - "\n", - "\n", - "Solving the above problem, yields the values of $\\lambda_i$.\n", - "To find the coefficients of your hyperplane we need simply to compute" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{w}=\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With our vector $\\boldsymbol{w}$ we can in turn find the value of the intercept $b$ (here in two dimensions) via" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b = \\frac{1}{y_i}-\\boldsymbol{w}^T\\boldsymbol{x}_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or if we write it out in terms of the support vectors only, with $N_s$ being their number, we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b = \\frac{1}{N_s}\\sum_{j\\in N_s}\\left(y_j-\\sum_{i=1}^n\\lambda_iy_i\\boldsymbol{x}_i^T\\boldsymbol{x}_j\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With our hyperplane coefficients we can use our classifier to assign any observation by simply using" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i = \\mathrm{sign}(\\boldsymbol{w}^T\\boldsymbol{x}_i+b).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Below we discuss how to find the optimal values of $\\lambda_i$. Before we proceed however, we discuss now the so-called soft classifier. \n", - "\n", - "\n", - "## A soft classifier\n", - "\n", - "Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined.\n", - "\n", - "Suppose now that classes overlap in feature space, as shown in the\n", - "figure here. One way to deal with this problem before we define the\n", - "so-called **kernel approach**, is to allow a kind of slack in the sense\n", - "that we allow some points to be on the wrong side of the margin.\n", - "\n", - "We introduce thus the so-called **slack** variables $\\boldsymbol{\\xi} =[\\xi_1,x_2,\\dots,x_n]$ and \n", - "modify our previous equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1-\\xi_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the requirement $\\xi_i\\geq 0$. The total violation is now $\\sum_i\\xi$. \n", - "The value $\\xi_i$ in the constraint the last constraint corresponds to the amount by which the prediction\n", - "$y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1$ is on the wrong side of its margin. Hence by bounding the sum $\\sum_i \\xi_i$,\n", - "we bound the total amount by which predictions fall on the wrong side of their margins.\n", - "\n", - "Misclassifications occur when $\\xi_i > 1$. Thus bounding the total sum by some value $C$ bounds in turn the total number of\n", - "misclassifications.\n", - "\n", - "\n", - "This has in turn the consequences that we change our optmization problem to finding the minimum of" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\frac{1}{2}\\boldsymbol{w}^T\\boldsymbol{w}-\\sum_{i=1}^n\\lambda_i\\left[y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)-(1-\\xi_)\\right]+C\\sum_{i=1}^n\\xi_i-\\sum_{i=1}^n\\gamma_i\\xi_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b)=1-\\xi_i \\hspace{0.1cm}\\forall i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with the requirement $\\xi_i\\geq 0$.\n", - "\n", - "Taking the derivatives with respect to $b$ and $\\boldsymbol{w}$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{L}}{\\partial b} = -\\sum_{i} \\lambda_iy_i=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{L}}{\\partial \\boldsymbol{w}} = 0 = \\boldsymbol{w}-\\sum_{i} \\lambda_iy_i\\boldsymbol{x}_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\lambda_i = C-\\gamma_i \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Inserting these constraints into the equation for $\\cal{L}$ we obtain the same equation as before" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{x}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "but now subject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$ and $0\\leq\\lambda_i \\leq C$. \n", - "We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "5\n", - "0\n", - " \n", - "<\n", - "<\n", - "<\n", - "!\n", - "!\n", - "M\n", - "A\n", - "T\n", - "H\n", - "_\n", - "B\n", - "L\n", - "O\n", - "C\n", - "K" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma_i\\xi_i = 0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{x}_i+b) -(1-\\xi_) \\geq 0 \\hspace{0.1cm}\\forall i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Kernels and non-linearity\n", - "\n", - "The cases we have studied till now, were all characterized by two classes\n", - "with a close to linear separability. The classifiers we have described\n", - "so far find linear boundaries in our input feature space. It is\n", - "possible to make our procedure more flexible by exploring the feature\n", - "space using other basis expansions such as higher-order polynomials,\n", - "wavelets, splines etc.\n", - "\n", - "If our feature space is not easy to separate, as shown in the figure\n", - "here, we can achieve a better separation by introducing more complex\n", - "basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to \n", - "obtain a separation between the classes which is almost linear. \n", - "\n", - "The change of basis, from $x\\rightarrow z=\\phi(x)$ leads to the same type of equations to be solved, except that\n", - "we need to introduce for example a polynomial transformation to a two-dimensional training set." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "data": { - "image/png": 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter5_109_0.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "import numpy as np\n", - "import os\n", - "\n", - "np.random.seed(42)\n", - "\n", - "# To plot pretty figures\n", - "import matplotlib\n", - "import matplotlib.pyplot as plt\n", - "plt.rcParams['axes.labelsize'] = 14\n", - "plt.rcParams['xtick.labelsize'] = 12\n", - "plt.rcParams['ytick.labelsize'] = 12\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "from sklearn import datasets\n", - "\n", - "\n", - "\n", - "X1D = np.linspace(-4, 4, 9).reshape(-1, 1)\n", - "X2D = np.c_[X1D, X1D**2]\n", - "y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "\n", - "plt.subplot(121)\n", - "plt.grid(True, which='both')\n", - "plt.axhline(y=0, color='k')\n", - "plt.plot(X1D[:, 0][y==0], np.zeros(4), \"bs\")\n", - "plt.plot(X1D[:, 0][y==1], np.zeros(5), \"g^\")\n", - "plt.gca().get_yaxis().set_ticks([])\n", - "plt.xlabel(r\"$x_1$\", fontsize=20)\n", - "plt.axis([-4.5, 4.5, -0.2, 0.2])\n", - "\n", - "plt.subplot(122)\n", - "plt.grid(True, which='both')\n", - "plt.axhline(y=0, color='k')\n", - "plt.axvline(x=0, color='k')\n", - "plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], \"bs\")\n", - "plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], \"g^\")\n", - "plt.xlabel(r\"$x_1$\", fontsize=20)\n", - "plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n", - "plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])\n", - "plt.plot([-4.5, 4.5], [6.5, 6.5], \"r--\", linewidth=3)\n", - "plt.axis([-4.5, 4.5, -1, 17])\n", - "plt.subplots_adjust(right=1)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with $x_i$ and $y_i$ as variables)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "z = \\phi(x_i) =\\left(x_i^2, y_i^2, \\sqrt{2}x_iy_i\\right).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{z}_i^T\\boldsymbol{z}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$, and for the support vectors" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_i(\\boldsymbol{w}^T\\boldsymbol{z}_i+b)= 1 \\hspace{0.1cm}\\forall i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "from which we also find $b$.\n", - "To compute $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we define the kernel $K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=\\boldsymbol{z}_i^T\\boldsymbol{z}_j= \\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For the above example, the kernel reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=[x_i^2, y_i^2, \\sqrt{2}x_iy_i]^T\\begin{bmatrix} x_j^2 \\\\ y_j^2 \\\\ \\sqrt{2}x_jy_j \\end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note that this is nothing but the dot product of the two original\n", - "vectors $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$. Instead of thus computing the\n", - "product in the Lagrangian of $\\boldsymbol{z}_i^T\\boldsymbol{z}_j$ we simply compute\n", - "the dot product $(\\boldsymbol{x}_i^T\\boldsymbol{x}_j)^2$.\n", - "\n", - "\n", - "This leads to the so-called\n", - "kernel trick and the result leads to the same as if we went through\n", - "the trouble of performing the transformation\n", - "$\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j)$ during the SVM calculations.\n", - "\n", - "\n", - "\n", - "Using our definition of the kernel We can rewrite again the Lagrangian" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\cal{L}=\\sum_i\\lambda_i-\\frac{1}{2}\\sum_{ij}^n\\lambda_i\\lambda_jy_iy_j\\boldsymbol{x}_i^T\\boldsymbol{z}_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to the constraints $\\lambda_i\\geq 0$, $\\sum_i\\lambda_iy_i=0$ in terms of a convex optimization problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{2} \\boldsymbol{\\lambda}^T\\begin{bmatrix} y_1y_1K(\\boldsymbol{x}_1,\\boldsymbol{x}_1) & y_1y_2K(\\boldsymbol{x}_1,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_1,\\boldsymbol{x}_n) \\\\\n", - "y_2y_1K(\\boldsymbol{x}_2,\\boldsymbol{x}_1) & y_2y_2(\\boldsymbol{x}_2,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_2,\\boldsymbol{x}_n) \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "y_ny_1K(\\boldsymbol{x}_n,\\boldsymbol{x}_1) & y_ny_2K(\\boldsymbol{x}_n\\boldsymbol{x}_2) & \\dots & \\dots & y_ny_nK(\\boldsymbol{x}_n,\\boldsymbol{x}_n) \\\\\n", - "\\end{bmatrix}\\boldsymbol{\\lambda}-\\mathbb{1}\\boldsymbol{\\lambda},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$. Here we defined the vectors $\\boldsymbol{\\lambda} =[\\lambda_1,\\lambda_2,\\dots,\\lambda_n]$ and \n", - "$\\boldsymbol{y}=[y_1,y_2,\\dots,y_n]$. \n", - "If we add the slack constants this leads to the additional constraint $0\\leq \\lambda_i \\leq C$.\n", - "\n", - "We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " &\\mathrm{min}_{\\lambda}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{\\lambda}^T\\boldsymbol{P}\\boldsymbol{\\lambda}+\\boldsymbol{q}^T\\boldsymbol{\\lambda},\\\\ \\nonumber\n", - " &\\mathrm{subject\\hspace{0.1cm}to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{\\lambda} \\preceq \\boldsymbol{h} \\hspace{0.2cm} \\wedge \\boldsymbol{A}\\boldsymbol{\\lambda}=f.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Below we discuss how to solve these equations. Here we note that the matrix $\\boldsymbol{P}$ has matrix elements $p_{ij}=y_iy_jK(\\boldsymbol{x}_i,\\boldsymbol{x}_j)$.\n", - "Given a kernel $K$ and the targets $y_i$ this matrix is easy to set up. The constraint $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$ leads to $f=0$ and $\\boldsymbol{A}=\\boldsymbol{y}$. How to set up the matrix $\\boldsymbol{G}$ is discussed later. Here note that the inequalities $0\\leq \\lambda_i \\leq C$ can be split up into\n", - "$0\\leq \\lambda_i$ and $\\lambda_i \\leq C$. These two inequalities define then the matrix $\\boldsymbol{G}$ and the vector $\\boldsymbol{h}$.\n", - "\n", - "\n", - "\n", - "## Different kernels and Mercer's theorem\n", - "\n", - "There are several popular kernels being used. These are\n", - "1. Linear: $K(\\boldsymbol{x},\\boldsymbol{y})=\\boldsymbol{x}^T\\boldsymbol{y}$,\n", - "\n", - "2. Polynomial: $K(\\boldsymbol{x},\\boldsymbol{y})=(\\boldsymbol{x}^T\\boldsymbol{y}+\\gamma)^d$,\n", - "\n", - "3. Gaussian Radial Basis Function: $K(\\boldsymbol{x},\\boldsymbol{y})=\\exp{\\left(-\\gamma\\vert\\vert\\boldsymbol{x}-\\boldsymbol{y}\\vert\\vert^2\\right)}$,\n", - "\n", - "4. Tanh: $K(\\boldsymbol{x},\\boldsymbol{y})=\\tanh{(\\boldsymbol{x}^T\\boldsymbol{y}+\\gamma)}$,\n", - "\n", - "and many other ones.\n", - "\n", - "An important theorem for us is [Mercer's\n", - "theorem](https://en.wikipedia.org/wiki/Mercer%27s_theorem). The\n", - "theorem states that if a kernel function $K$ is symmetric, continuous\n", - "and leads to a positive semi-definite matrix $\\boldsymbol{P}$ then there\n", - "exists a function $\\phi$ that maps $\\boldsymbol{x}_i$ and $\\boldsymbol{x}_j$ into\n", - "another space (possibly with much higher dimensions) such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "K(\\boldsymbol{x}_i,\\boldsymbol{x}_j)=\\phi(\\boldsymbol{x}_i)^T\\phi(\\boldsymbol{x}_j).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "So you can use $K$ as a kernel since you know $\\phi$ exists, even if\n", - "you don’t know what $\\phi$ is. \n", - "\n", - "Note that some frequently used kernels (such as the Sigmoid kernel)\n", - "don’t respect all of Mercer’s conditions, yet they generally work well\n", - "in practice.\n", - "\n", - "\n", - "## The moons example" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - 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X1UkCxoRZbELGuCO9fqVz6W5mz5lJ05WraO14h99lOa69pQMyg7IDji/tJDnrRdcnCRTDzocxQWUhY8qWDpieZTtpXfI25r/rVr9L8lR7Swd8sGPSJAG/WjV2ZIAJqsCcJyMis0XkCRE5KyIHRGRdnuvuFZFhEenP+LjU63pNyoK3wYyQBExioJfbf/I5Tgz0Tvp3pRZ1rKFh7S2MvaeG/fOepm7T/SQ2bnOgYmPCLzAhA3wLSAJzgT8B7heRZXmufURVGzM+3vKsSpMyOjKx3iU19hJ8mTvlZv7bCe0tHVzywTtouHkFiRVdDB96kMENT0yaaWfCR052M9rU4ncZoRaI7jIRmQl8GFiuqv3AiyLyFPAx4C99La5MUe4j73t9N2NzTyG1z7Ll6ioa2m6Y/od8lr1Trqq6smtuepJActaLdPf9G60vddkMNBNrgQgZ4ApgRFX3ZHxtK7A6z/W3ikgvcBS4T1Xvz3WRiNwJ3AnQ1tZGV3K3gyUXluvc9tTXqwvWkdQhupK7WXf7avpOXjDl+7NazrHhu79wrM5SjZw6Q1X7WYZnNdB703+k9oJmzh2DrmNJ32oqJDmkdO1Kct9b32d0LLXHVHL0/Pb+o2Nj/O3z/8zdl37SwXtdBFcsQs8NcHzpAFVDJxg49u+MNTRRUzf1jUf6d17xfebh1PPemTrd52idF4wwshx2VF/JyMAYXbuce56nn5tRF5SQaYSJ2aFpp4CmHNc+CqwHuoF3Aj8UkZOq+nD2haq6fvxarrj8CnV7645iFaojvcVIroAB6Dt5getbkOTS09nHzBd/xLmW7fStrqeq7g+4dNkcz+vIJ9/JhF27ktQt7Oe53/ycEU3tMaUZuz+M6AjPndjI5278qAu75tYix+DQ/hdYvruNPaevp23NVVOucmJbmULbxjj1fPFi+xsnOFmnnOzm+PajXNG0ky3vHGKBg2u9unYlWbA0zzkUERKUkOkntQA8UzNwJvtCVX0j49OXReQfgNuAKSETB150y6U3sTy2ogtZuYzFHWsC9w6s0MmEuXbKzeTmrrk6rwX2gw5MeSo7KuxdsCa6gjLwvweoEZHMFXvXADuK+FmF+K60LDR1dfkti1h+yyJWr7uk7NtPbNzGvLMv0Pf+kzSsvSWQq/azx1uyZ4zl2ik3k9u75o60NrClYScNe39ss85M7ASiJaOqZ0XkceDLIvIJ4FpgLbAy+1oRWQv8EjgJ3AB8CviCZ8WGUKVrJZouHGH0yuCu2p/uZEK/d8pd1LGGA0ACf9fSGOOHoLRkAP4MqAeOk+r6uktVd4jIe0WkP+O6/wy8Saor7XvA11T1u55XO418W6iHaWv1ns4+mg6+SoJuv0vJK9/JhE6sf3FS5lqaZP1LNGx+3qY3m1gouiUjIv8GfAC4TVV/mPF1AR4Ebif1gl/WlGNV7QU+lOPrL5CaGJD+/L+Uc/teq7SP3O/zP9LjMPsu30v91RfTkOesF7+F6WTC9pYOjr8PWs7touFYNQf9LsgYD5TSXfY54DXgKyLypKqmX+2+Tipg1pcbMGYqvwZy05tdDlW9ytkVPTSsDPbuyWE9mVAHT/ldgilC776TNB57k+6mo4B1b5aj6JBR1a0i8n1SgfIx4CER+QLwGVLTiu9yp0RTSL4WTyXmzK2mYV4zJ669MvCbXfo93lKWltQOCVX9x4nLC1cYFyenW/OdKw7SvLgtsK35oCt1TOYeYAj4KxG5G/gfwM+Aj6kWmCNqXPOLDYfY/swBtj9zwPFxoNG2XMuUjCOaGqe/JkLCtIFnT2cfgxueYCjxCIkVXVy4cjmLV62reOKLk/vlOcWLmkqaXaaqXSLyv0lt9fKPwMvAH6rqpEUTIvJ54A+BJcA54NfA51V1uxNFR0G+d3azWubxwsPHyrpNp94RWleOu/bVd3O6YTuXbd5J7+GV8B8u8rukgs4/VyfvKhDkVkil5sytpqFjPicuc641X2gtl1+8qKmc2WWJjH//qaoO5LjmRuD/kJqC/H5gBHhORJxeUh1a+d7B5Vvp77mWcGx6GTbtLR0sXrWOC1cu5+hN3Qz1PMVY70nXZ5qtXnfJxLqpzI9i1lCFqRUSVNOt5cp1vdstjFJrKldJITO+/f7XgfRb7T/PdZ2q3qyqD6rqdlXdRmoMpw14TyXFGhMVizrW0Lx6DWPvqaFKhhg7WFmLYLoQsaDwV661XNNd7+Qu4U7UVK6iQ0ZEfg94CNgOvB3YDXxCRIrZJKhp/L5sYUDAVfUfj914gV/aWzoYvXIh1TWVL1ezEHHBGWe2Asq3lqs3mfvl0IsWhpfry4p6dovIKuAx4BBws6omgC+SGtP5WhE38Q/AFuBX5ZVpvKKHjzJ0tot99cFdgBlF6bN5oiqsi5OdmPySby3XhkOPTHu9Wy2MQuvLnDbtwL+IXAs8TWpX5A+o6lEAVX1MRF4B1orIe8cXTeb6+b8DVgGrMtbWmIBJr4+prn+Jt1bX07D8hsBuIxM51VVAtP80ojpBoBj51nLtPLNryrX5WhhOnnlUqCY31pcVDBkRuRz4KalNKG9W1b1Zl3weeBb4W+BdOX7+70ltA3OTnV45Wb71LbNaznleSzpgaua/Qu+KdhaH4CjlKBlhmIbEfno6+wK5n5nfu0+EXXot11devo9Hd/+Ejyz5Pe5ZeXfOncy92sHCy/VlBUNGVd8E5hX4/nPk2QF5fAv+PyYVMFMjO+byvbNLHbbk/Zkdc+ZWMzDnQmYsXer5fTst82yZjB2JAqm+bSGnTyl7lv+ayzYdIXHwVldO0awkKNLP1bCcJxNE2eMsd127jlzPzbDuYFGIK7swi8i3SM0o+xDQJyLpoOofP17ZGNdkzsz5+Ow7/S6noPaWDs41JpmxcjkJdtC6YxOJjZQcNNOFSJy7q8rh9FqxXOMsuZ6bodzBYhpubfX/Z+P/3Zj19S8B97p0n8ZMecf4wWv/iAXM9busaS3qWMPhngEWDEBn/qNv8rIQcYFDa8XyjbOE5blZKVdCRlVje4hYtjDu2RRm2e8YNxx6hGve/imfqyrOSGsDYLszRU2h2WVheW5WIkjnyUSSrV/wTq53jM8mNgZqrygTP6XMLouiQJyMaYwTwnS2TD7S1ED1kXCumYlKqz21O7Zz8o2z5JpdFkXWkjGRkesd44iGe2ZOmESh1d7T2Uf9ll9xuHaXLUh2iLVkTGTkesfYtSvJgqW1PlRTGTnZjbZEf1A4KNJrxWqTXRxbvhNZuYzFAT6sL0wsZIwJkM1N25jbd4aeF2H2H1jIeGXs4CEaG49y4KpDNLzvFtvtwkHWXeaysO7ZZLyX3pm55/d7qR5+msENT7h+BIA5r/mieqR9jgWMw6wl47IwDXga/7W3dMCqDg7X/Jj211+hYTP0cGMgt5uJGh1wZtdlM5mFTMRFZcZP3MxYupTarl20jFRz0O9iihSFPc5Sa5WMkyxkIi4KM36mk7lXmZM71QaBE9ubePVGI8xvWqoHe1KnXhnH2ZiMCT0vThH0hUPbmsThjUa5ejr7GNzwBNXDT7Pl6gT1bQv9LilyrCXjMeu+clau3W2j1poxzsucsty9bCczlrzNjrhwibVkPGbvKp3l1TnlfthX381vl7xC3ab7SWzc5nc5kTNnbjXNNwzRtHIV8y1gXGMhY85raqI6EZ4ZNl6eU+619pYOFq9aR/PqNfS9/yQ1yR/YlGaXOHHEcrkSA73c/pPPReI5m4+FTMQVu07nRPcovTuOMrhzK8dPdnpRWsW8PKfcL+0tHVzywTsYeE8jF3ecZOygdamWI5E8zu1vfIQTyeMTXWVHTv+SrgtO+FpXZMcTM9iYTJG8HEtx8r6Kub61YxY93EjDZhh86QVO60YGrzrIooBvqxHFUwQLKXcdRxSmFlfqgcPf5LUzm/n7f7+Xv9y7lP0Lt9K0sJXa963ybfFlbzIe44kWMkXycizFj3Gb1o5Z0PGfGHx6Fpd0/hZtgO62zkCvfo7iKYL5VHLWTNwnlCSSx3ky8QMU5WcznuXT17XT/o7309rxDl/r2nDokSnjiWHZLbwU1l3msaBvMyPzL6Ju5gK/yzB5VA+G8xgAPz1w+JuMjQf0KGM8kNzs6zgMpMZinj3+80iOJ2azlozHgv6ucqyxHQ7/xu8yTA7Hm07R8NZ++l7fzawVS/wuJxTSrZhhHQZgWEZ5fHQXtw2dpt3Huh7YsmEi+NKi2pqxkDEmBOrbFnL0nTDc9DoXbTpCovdW2tZc7XdZFfFinPOBw99kdGwUMg6EHxP4wVu/4p551zlyH+XYmtjJiE4dT3y1O3pT1a27zJgyeD31tL2lY2KX5r73n6T29HdDP6XZ7bHHns4+Xu/axIhkHWQ3Nur75JDH1n6LZ979I7bf8Qzb73iGP17y+wjCdXPD/cYhl8CEjIjMFpEnROSsiBwQkXV5rhMR+ZqI9Ix/fE1EJNe1TvJyLCUQ4zYnB7y7rxCqZOppJQHV3tJB7ftW0XJFO63N/SX/fFwkNm6j9qXv8z9GlvLT1o/w8//0zYkX9O13PBOoSSPZu1ZEbVwmSN1l3wKSwFzgWuBfRWSrqu7Iuu5O4EPANYACzwL7gAfcLM7LsRS/x22k/kLgqK81BFmlW9lkBlTZ/e/j+5qlzqO3YwDSejr7GGs6yVDix5xd1kPzkiWBX82fa9eKKI3LBKIlIyIzgQ8D96hqv6q+CDwFfCzH5bcD31DVQ6p6GPgG8HHPijWxV8lWNk69a91X382v575E7eYf0vv082XdRlTV1MDCq5tDsV1MlHetSAtKS+YKYERV92R8bSuwOse1y8a/l3ndslw3KiJ3kmr50NbWRldytzPVuiipQ77WOTJ3lD2z5jI240KGD9dx7lgy53XJIaVrV+7vBYnTdfYme3liz7MM6/kXhSc6n+WDjX/E7NrpWxT3vfV9RsfGp9OOjfG3z/8zd178X0uscRHVcxYxs+k0R5eco+rsCGeOvMxYYws1de6tpXL+ubko73fKvZ+RtiHGams4uOjdDCdrGQjwczQ5pNz3/PnnQ1r6eXH3pZ/0qTJnBSVkGoHTWV87Re4THhrHv5d5XaOIiKpq5oWquh5YD3DF5VfogtrgT/vsSu7Gzzp7DvSx8OCbDMzbxYlr5+ddsNa1K8mCpbUeV1c6p+t86OXHUBlLddSOU8Z4qv8H03ZxJAZ6ee43P5+YVTSiIzx3YiPrLvljrlk6t4xq5nD8ZCcD2zdzam+Cy15fyLmLVzL7D24s47am5/Rzs9BOBMXeT3rj0PT6oYbEfo7eupDRqk00XnHDtIuJnT6LqJTb69qVZO/wnimzzEZ0hL3Du0Px91WMoIRMP9Cc9bVmINc+GtnXNgP92QFjKtQU/hOcEgO9fG77/+S+hV9wbLuOSrayybfX2oZDj3DN2z9VVj3p45oPzN3I0ZZOGl5+iqEH93N21drAH9lcydhjev+xmlmv09gwCk2gTXUcXHICufAyLrsm57yhKRwZH6vg9oI0AcEtQQmZPUCNiHSoanp3xmuA7EF/xr92DfCbaa4zZTrRPcpQ8iiDyX0cb2sK9NYyhTywZQM7zrxR1gtIvneklbwo5AuonWd2lX2baYs61nC8bSEDrZvZt+UFFm46Qu9u91o1fkps3EZd16aJc2DGWudPrOBvBs4dy36/mud2HD6LyM42yi0QIaOqZ0XkceDLIvIJUrPL1gIrc1z+PeAzIvITUp0W/x34R69qjbr0Hmb9Gy9n1s9/TKLvGQ6sDP5mmdkCMQMsS76AcmrMaFKrZkEnbdufZujB/Qxe++5I7BAwcdBY/UucWlVF05WrcnbnduUZR8zm9KyuqM8SK1cgZpeN+zOgHjgOPAzcpao7ROS9IpK5IOD/Aj8GtgHbgX8d/5pxUNuaqxm66S7mbX8nF/17Hwc6N/pdUkmCMAPML4s61nD52k9y8iOLObTq9YkZaH2v76ans2/iI8gy6+zp7KP36eep23Q/3R3/xsmPLOaSD95R0QaXTs/qisMssXIFJmRUtVdVP6SqM1V1oapuGP/6C6ramHGdqupfqOrs8Y+/sPEYd7R2zGKgbTFzucjvUkpS6R+806dt+nUwVXqHgMRtg0jts8zpfJL219fT/vp6al/6fmBP20wvpEzX2v76eqqHnyZx2yANa2+puFWdGOjlj576fxjV7Fldo2X/ruNwtlG5AtFdZoyTCv3BFzMDLFdAVdK/7kbXW7Eyu9DS0zdregYY3r2TWYcOMvTg8sBMEujp7GPmiz+itmU7p1ZVcbp9zvgRBwAzWexQl+0DWzZwYnBq4I9o+dvNxO1so1JYyJiCRutb0TO/JUCN3mm5MQOs3IAIymDwpHf/HbD6v91Ez4kLUp8/ev5bsy/o52cf+adJP9s/93JHNuNMbNxGY/ebk74m586vRqgbPsKxFV3IymWujQGmfx8AF1TXsuH3/451//oZzo0muaC6lgc+8JWybjcOs8TKZSFjIifzD77UdTJOvyMN6mDwRMBk6T3XSM/v/Gri85G+M9S+8QJDD76dkQ//DpSxdCPdQhmu+i3dVw1SM2vy9PiR2edraVh+i6uzGbN/H3/xi78J5O8nSmITMooN25jpOfmO1I2uNy9c8sE7Jv59/GQnA5dtZs/eX3Phmbcx9MOHSr69uuEjHLqpm5rWZhqW3+jblPhcv4+9pw5OfD8sv5+wiU3ImMrU9AxACJbLOL2CuxJOd735IXNMZ/jUGMfW7i3rdpqXr/F9vVWu30e29OB/WH4/EKznfC4WMgEw+fCm8/s5OXl4UyWkoQkY8ruMojg1yO7EH26UBoMXdaxJdT1eX9xK+iDK9fvIVsngv1/8nFhSDAuZAHD78Ka4yDXIntrqrnRO/OHaYHCw5Pt9JAZ6+Y+P3VHx4L8fLQq3J5YkT/dUfBvhmTLkgGRP9h6cJkqcWt8S9sWYxWidc66kr0eZU8+bSg6yK5fTa7rcEKOQcf3wTOOjfIPsvcnSV7aH4Q+3Ur94dRPbD/x0yscvXt3kd2mTuL2Q1amV+n68MXF7lwEnWjEQq5Ax5Tp9dJDhoQF6Ol/zu5S8Cu1wXArbHiRY3G4dOLVS3483Jm7uMpAOGJ3TWvFtWciYgtrWXM2xme+l9SezOfPyixzo3Mjxk53T/6DHnNrh2LYHCQ4vWgdOTM7w642J2xNLnAgYiOHAf7LnNLWtxW0F7pVChzcFQduaq+npvIR5L/6Iwb5OTl3fzdisuZS1Ms8lTu1wHKUZYZmCPs01Fy8WsjoxOcOvqepuTSxxqpssLVYho61tSE/C7zKmyJym7PfJmPm0dsyir//dzD99IbMHD7PX/62uXBHVGWFBn+aaLUwLWaP0xsTJbrK0WIWMMXEUlP3TShGmhaxRe2PiZMCAjcmYEow1tsOZ/ukvjBi/tup3Shhny0WpdRAWydM9jgcMxLAlo61tJHsSgRuXMcEVtq6mTGHqdsoUtdZB0Dk9DpPJWjLGFBD2hZk2W85Mx41xmEwWMsYUEMaupkzW7WQKcTtgIIbdZcYUK6xdTZmi1u0UxqnYQedmwEBMWzKpcRnbx8wUZl1NwePH/mBR5dZAf7ZYhowxxbCupmAJ+/hYkLg50J/NusuMySNqXU1ecqNbK6hHWYeNF+MwmWLdkrEuMxNFQVjX43S3lm1c6gyvAwZiHDLa2uZ3Cca4otQXeKdDyY1uLRsfq5wfAQMxDhljwi5XOJTzAu90q8ONad82PlYZvwIGLGSMCa1c4VDqC7zTrQ63urUeW/sttt/xzJSPUsfNgtCV6DU/AwYsZGxcxkxR6guRHy9cucKhnBd4p1sdQe/WitsUaL8DBmIeMjYuY3Ip9YUoKGe7l/oC70aro9xuLS+COm5ToIMQMGBTmI2ZpNRt8f3YRj9fOCxomlfSC7wb2+mXO+3bi01I4zQFOigBAzFvyRiTrdjuo/Q7779/5TuBOdv9urlXlzRuEZTB9GJbGJW0duI0BTq9kj8IAQPWkjElqOo/7ncJriplr7IHtmzg1e7tbDn+BqPjL/he7W3mVDgEZbFpsS2MSlo7YToErVxBar1kin3I2PkyJWpqBE75XYUrin0hSocRMBEwha53WlDCwQnFBnul3ZJBabW5JagBAxYyxkwo9oUoVxgVut7kV2ywVzqeEqVgzhbkgIGAhIyIzAa+DfwH4ATweVXN2bktIvcC/x9wLuPLb1fVt9yuM+708FGGqrrYV5/wuxRXFPNClP3OG+CC6lp+dtuDtvV8GYoJdjeOXKh0b7WgHDkQ9ICBgIQM8C0gCcwFrgX+VUS2quqOPNc/oqof9aq4uOvp7KNh8/MMDb/AnpWjzJjbwYzROr/L8kUc+va9VEywu/GYVzqbze8jucMQLmm+zy4TkZnAh4F7VLVfVV8EngI+5mUdtigzt8TGbdS+9H32z3uasffU0Lx6DYs61vhdlm+i3rcfRE4/5pWul/F7vU2YAgZAVNXfAkRWAC+pakPG1z4LrFbVW3Ncfy/waWAUOArcp6r357ntO4E7Adra2q576J/yTy+VkWGkprqC/xNnJHWIWglOK2Hk9CAXVA8wWDdM7cyWia8nh5TaOvGvsCKFoc4w1Ai56+xN9vLVPV/n81d8jtm1s3yqbLLpHs/73rqfnx1/jhEdoUZquLn9A9x96SeLvv1Kf77YOrNpRtBqjXedUL//gbWvqur15f58ELrLGoHsZsQpoCnP9Y8C64Fu4J3AD0XkpKo+nH2hqq4fv5aOy6/QebXL8xYhZ4Ixw6wruZsFtUv8LmNCYus2rmjezfYlCeYvPZ/5XbuSLFha62NlxQlDnZXU6OXYQNeuJHUL+yfd30MvP8aOM2/wVP8PAtNdWOjxTAz08txvfs6Ipl6wR3SE505s5HM3frSox6/Sny+2zmxha71kcr27TESeFxHN8/Ei0A9kv7o3A2dy3Z6qvqGqR1R1VFVfBv4BuM3d/wtjgsnrLW0y78/vbqNyVLq3mtd7syVP94Q6YMCDkFHVG1VV8nysAvYANSLSkfFj1wD5Bv2n3AUQ/L6GENOBnHlvfOb1i3xvcvL9+bHbQaUqHd/xckwuM1zCGjAQgO4yVT0rIo8DXxaRT5CaXbYWWJnrehFZC/wSOAncAHwK+ELFddiizIJGWhumv8h42n3l9V5cGw49MnF/ozrG03s3MYa3ux1UqtL1Ml6st0mHC4S39ZLJ99ll4/4MqAeOAw8Dd6WnL4vIe0WkP+Pa/wy8Sao77XvA11T1ux7Xa0xOXnVfeb0XV2Kgl2eP/3zi/kbGRiYCJi0srZkgi0rrJVMgQkZVe1X1Q6o6U1UXZi7EVNUXVLUx4/P/oqqtqtqoqktV9Zv+VB0P1YM9019kAG+7r7weG3hgy4YpoZLNpnKXLz32EqVwSfO9u8wEnzQ1AEN+lxF4XnZf5RsbeLV7u2v3l55RlWnp7EsjvWWL26LWNZaLhYzJq6ezj5mJ/XRffBQIxhqIoHJj65NCsl/Yv/LyfTy6+ydcNzf/NP1K7y8M08HDInm6Bx2rB6IbLmmB6C4LElv5n5LYuI26Tfezb9ELHFmcpL5tod8lBZqfxw6HcSpxnE2Mu9TURD5gwEJmEjuOOdV6GdzwBEOJR0is6KLh5hUsXrWO9paO6X84xvzcbqbYg9aCzosjmP0U5XGXQqy7zEwxZ241DfOaOXHtlbR2vMPvckLBr3EJr7vp3OT3ppNuCftiykpZS8bkNdqWb2cfExR+dtM5KYpdfnFtuWSzkDEmxKKyK3RUuvzSwWLhcp51l5kpdDCaxytHURSmD0ehyy8OU5HLZS0Zk1uLbSNjvBHmLr/sDSwtYKayloyZpKr/uN8lmJgJY5dfXFou/UOV7/hhIZMl7htl6uGjDFX1sa8+QQO2Nsa4LyxdfnEJljQnAgYsZMy4ns4+Zr74I6pbtvPW6noalt9ga2OMId7hUnth5f+/FjKGns4+GjY/T/WlnZz83cUs7ljjd0nG+CpuwZKWDhgnwiXNQsYA0Nrcz+CcC237GBNbmcEC8Qw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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter5_129_6.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "from __future__ import division, print_function, unicode_literals\n", - "\n", - "import numpy as np\n", - "np.random.seed(42)\n", - "\n", - "import matplotlib\n", - "import matplotlib.pyplot as plt\n", - "plt.rcParams['axes.labelsize'] = 14\n", - "plt.rcParams['xtick.labelsize'] = 12\n", - "plt.rcParams['ytick.labelsize'] = 12\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "from sklearn import datasets\n", - "\n", - "\n", - "\n", - "from sklearn.pipeline import Pipeline\n", - "from sklearn.preprocessing import StandardScaler\n", - "from sklearn.svm import LinearSVC\n", - "\n", - "\n", - "from sklearn.datasets import make_moons\n", - "X, y = make_moons(n_samples=100, noise=0.15, random_state=42)\n", - "\n", - "def plot_dataset(X, y, axes):\n", - " plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"bs\")\n", - " plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"g^\")\n", - " plt.axis(axes)\n", - " plt.grid(True, which='both')\n", - " plt.xlabel(r\"$x_1$\", fontsize=20)\n", - " plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n", - "\n", - "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - "plt.show()\n", - "\n", - "from sklearn.datasets import make_moons\n", - "from sklearn.pipeline import Pipeline\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "\n", - "polynomial_svm_clf = Pipeline([\n", - " (\"poly_features\", PolynomialFeatures(degree=3)),\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", LinearSVC(C=10, loss=\"hinge\", random_state=42))\n", - " ])\n", - "\n", - "polynomial_svm_clf.fit(X, y)\n", - "\n", - "def plot_predictions(clf, axes):\n", - " x0s = np.linspace(axes[0], axes[1], 100)\n", - " x1s = np.linspace(axes[2], axes[3], 100)\n", - " x0, x1 = np.meshgrid(x0s, x1s)\n", - " X = np.c_[x0.ravel(), x1.ravel()]\n", - " y_pred = clf.predict(X).reshape(x0.shape)\n", - " y_decision = clf.decision_function(X).reshape(x0.shape)\n", - " plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)\n", - " plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)\n", - "\n", - "plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])\n", - "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - "\n", - "plt.show()\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "\n", - "poly_kernel_svm_clf = Pipeline([\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", SVC(kernel=\"poly\", degree=3, coef0=1, C=5))\n", - " ])\n", - "poly_kernel_svm_clf.fit(X, y)\n", - "\n", - "poly100_kernel_svm_clf = Pipeline([\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", SVC(kernel=\"poly\", degree=10, coef0=100, C=5))\n", - " ])\n", - "poly100_kernel_svm_clf.fit(X, y)\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "\n", - "plt.subplot(121)\n", - "plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\n", - "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - "plt.title(r\"$d=3, r=1, C=5$\", fontsize=18)\n", - "\n", - "plt.subplot(122)\n", - "plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\n", - "plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - "plt.title(r\"$d=10, r=100, C=5$\", fontsize=18)\n", - "\n", - "plt.show()\n", - "\n", - "def gaussian_rbf(x, landmark, gamma):\n", - " return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)\n", - "\n", - "gamma = 0.3\n", - "\n", - "x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)\n", - "x2s = gaussian_rbf(x1s, -2, gamma)\n", - "x3s = gaussian_rbf(x1s, 1, gamma)\n", - "\n", - "XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]\n", - "yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "\n", - "plt.subplot(121)\n", - "plt.grid(True, which='both')\n", - "plt.axhline(y=0, color='k')\n", - "plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c=\"red\")\n", - "plt.plot(X1D[:, 0][yk==0], np.zeros(4), \"bs\")\n", - "plt.plot(X1D[:, 0][yk==1], np.zeros(5), \"g^\")\n", - "plt.plot(x1s, x2s, \"g--\")\n", - "plt.plot(x1s, x3s, \"b:\")\n", - "plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])\n", - "plt.xlabel(r\"$x_1$\", fontsize=20)\n", - "plt.ylabel(r\"Similarity\", fontsize=14)\n", - "plt.annotate(r'$\\mathbf{x}$',\n", - " xy=(X1D[3, 0], 0),\n", - " xytext=(-0.5, 0.20),\n", - " ha=\"center\",\n", - " arrowprops=dict(facecolor='black', shrink=0.1),\n", - " fontsize=18,\n", - " )\n", - "plt.text(-2, 0.9, \"$x_2$\", ha=\"center\", fontsize=20)\n", - "plt.text(1, 0.9, \"$x_3$\", ha=\"center\", fontsize=20)\n", - "plt.axis([-4.5, 4.5, -0.1, 1.1])\n", - "\n", - "plt.subplot(122)\n", - "plt.grid(True, which='both')\n", - "plt.axhline(y=0, color='k')\n", - "plt.axvline(x=0, color='k')\n", - "plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], \"bs\")\n", - "plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], \"g^\")\n", - "plt.xlabel(r\"$x_2$\", fontsize=20)\n", - "plt.ylabel(r\"$x_3$ \", fontsize=20, rotation=0)\n", - "plt.annotate(r'$\\phi\\left(\\mathbf{x}\\right)$',\n", - " xy=(XK[3, 0], XK[3, 1]),\n", - " xytext=(0.65, 0.50),\n", - " ha=\"center\",\n", - " arrowprops=dict(facecolor='black', shrink=0.1),\n", - " fontsize=18,\n", - " )\n", - "plt.plot([-0.1, 1.1], [0.57, -0.1], \"r--\", linewidth=3)\n", - "plt.axis([-0.1, 1.1, -0.1, 1.1])\n", - " \n", - "plt.subplots_adjust(right=1)\n", - "\n", - "plt.show()\n", - "\n", - "\n", - "x1_example = X1D[3, 0]\n", - "for landmark in (-2, 1):\n", - " k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)\n", - " print(\"Phi({}, {}) = {}\".format(x1_example, landmark, k))\n", - "\n", - "rbf_kernel_svm_clf = Pipeline([\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", SVC(kernel=\"rbf\", gamma=5, C=0.001))\n", - " ])\n", - "rbf_kernel_svm_clf.fit(X, y)\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "\n", - "gamma1, gamma2 = 0.1, 5\n", - "C1, C2 = 0.001, 1000\n", - "hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)\n", - "\n", - "svm_clfs = []\n", - "for gamma, C in hyperparams:\n", - " rbf_kernel_svm_clf = Pipeline([\n", - " (\"scaler\", StandardScaler()),\n", - " (\"svm_clf\", SVC(kernel=\"rbf\", gamma=gamma, C=C))\n", - " ])\n", - " rbf_kernel_svm_clf.fit(X, y)\n", - " svm_clfs.append(rbf_kernel_svm_clf)\n", - "\n", - "plt.figure(figsize=(11, 7))\n", - "\n", - "for i, svm_clf in enumerate(svm_clfs):\n", - " plt.subplot(221 + i)\n", - " plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])\n", - " plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n", - " gamma, C = hyperparams[i]\n", - " plt.title(r\"$\\gamma = {}, C = {}$\".format(gamma, C), fontsize=16)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Mathematical optimization of convex functions\n", - "\n", - "A mathematical (quadratic) optimization problem, or just optimization problem, has the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " &\\mathrm{min}_{\\lambda}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{\\lambda}^T\\boldsymbol{P}\\boldsymbol{\\lambda}+\\boldsymbol{q}^T\\boldsymbol{\\lambda},\\\\ \\nonumber\n", - " &\\mathrm{subject\\hspace{0.1cm}to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{\\lambda} \\preceq \\boldsymbol{h} \\wedge \\boldsymbol{A}\\boldsymbol{\\lambda}=f.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to some constraints for say a selected set $i=1,2,\\dots, n$.\n", - "In our case we are optimizing with respect to the Lagrangian multipliers $\\lambda_i$, and the\n", - "vector $\\boldsymbol{\\lambda}=[\\lambda_1, \\lambda_2,\\dots, \\lambda_n]$ is the optimization variable we are dealing with.\n", - "\n", - "In our case we are particularly interested in a class of optimization problems called convex optmization problems. \n", - "In our discussion on gradient descent methods we discussed at length the definition of a convex function. \n", - "\n", - "Convex optimization problems play a central role in applied mathematics and we recommend strongly [Boyd and Vandenberghe's text on the topics](http://web.stanford.edu/~boyd/cvxbook/).\n", - "\n", - "\n", - "\n", - "\n", - "If we use Python as programming language and wish to venture beyond\n", - "**scikit-learn**, **tensorflow** and similar software which makes our\n", - "lives so much easier, we need to dive into the wonderful world of\n", - "quadratic programming. We can, if we wish, solve the minimization\n", - "problem using say standard gradient methods or conjugate gradient\n", - "methods. However, these methods tend to exhibit a rather slow\n", - "converge. So, welcome to the promised land of quadratic programming.\n", - "\n", - "The functions we need are contained in the quadratic programming package **CVXOPT** and we need to import it together with **numpy** as" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy\n", - "import cvxopt" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This will make our life much easier. You don't need t write your own optimizer.\n", - "\n", - "\n", - "\n", - "We remind ourselves about the general problem we want to solve" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " &\\mathrm{min}_{x}\\hspace{0.2cm} \\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{P}\\boldsymbol{x}+\\boldsymbol{q}^T\\boldsymbol{x},\\\\ \\nonumber\n", - " &\\mathrm{subject\\hspace{0.1cm} to} \\hspace{0.2cm} \\boldsymbol{G}\\boldsymbol{x} \\preceq \\boldsymbol{h} \\wedge \\boldsymbol{A}\\boldsymbol{x}=f.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " &\\mathrm{min}_{x}\\hspace{0.2cm} \\frac{1}{2}x^2+5x+3y \\\\ \\nonumber\n", - " &\\mathrm{subject to} \\\\ \\nonumber\n", - " &x, y \\geq 0 \\\\ \\nonumber\n", - " &x+3y \\geq 15 \\\\ \\nonumber\n", - " &2x+5y \\leq 100 \\\\ \\nonumber\n", - " &3x+4y \\leq 80. \\\\ \\nonumber\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The minimization problem can be rewritten in terms of vectors and matrices as (with $x$ and $y$ being the unknowns)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{2}\\begin{bmatrix} x\\\\ y \\end{bmatrix}^T \\begin{bmatrix} 1 & 0\\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} + \\begin{bmatrix}3\\\\ 4 \\end{bmatrix}^T \\begin{bmatrix}x \\\\ y \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Similarly, we can now set up the inequalities (we need to change $\\geq$ to $\\leq$ by multiplying with $-1$ on bot sides) as the following matrix-vector equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{bmatrix} -1 & 0 \\\\ 0 & -1 \\\\ -1 & -3 \\\\ 2 & 5 \\\\ 3 & 4\\end{bmatrix}\\begin{bmatrix} x \\\\ y\\end{bmatrix} \\preceq \\begin{bmatrix}0 \\\\ 0\\\\ -15 \\\\ 100 \\\\ 80\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We have collapsed all the inequalities into a single matrix $\\boldsymbol{G}$. We see also that our matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{P} =\\begin{bmatrix} 1 & 0\\\\ 0 & 0 \\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "is clearly positive semi-definite (all eigenvalues larger or equal zero). \n", - "Finally, the vector $\\boldsymbol{h}$ is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{h} = \\begin{bmatrix}0 \\\\ 0\\\\ -15 \\\\ 100 \\\\ 80\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Since we don't have any equalities the matrix $\\boldsymbol{A}$ is set to zero\n", - "The following code solves the equations for us" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "ename": "SyntaxError", - "evalue": "invalid character in identifier (, line 5)", - "output_type": "error", - "traceback": [ - "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m5\u001b[0m\n\u001b[0;31m P = matrix(numpy.diag([1,0]), tc=’d’)\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid character in identifier\n" - ] - } - ], - "source": [ - "# Import the necessary packages\n", - "import numpy\n", - "from cvxopt import matrix\n", - "from cvxopt import solvers\n", - "P = matrix(numpy.diag([1,0]), tc=’d’)\n", - "q = matrix(numpy.array([3,4]), tc=’d’)\n", - "G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)\n", - "h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)\n", - "# Construct the QP, invoke solver\n", - "sol = solvers.qp(P,q,G,h)\n", - "# Extract optimal value and solution\n", - "sol[’x’] \n", - "sol[’primal objective’]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the **slack** parameter $C$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{1}{2} \\boldsymbol{\\lambda}^T\\begin{bmatrix} y_1y_1K(\\boldsymbol{x}_1,\\boldsymbol{x}_1) & y_1y_2K(\\boldsymbol{x}_1,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_1,\\boldsymbol{x}_n) \\\\\n", - "y_2y_1K(\\boldsymbol{x}_2,\\boldsymbol{x}_1) & y_2y_2K(\\boldsymbol{x}_2,\\boldsymbol{x}_2) & \\dots & \\dots & y_1y_nK(\\boldsymbol{x}_2,\\boldsymbol{x}_n) \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "y_ny_1K(\\boldsymbol{x}_n,\\boldsymbol{x}_1) & y_ny_2K(\\boldsymbol{x}_n\\boldsymbol{x}_2) & \\dots & \\dots & y_ny_nK(\\boldsymbol{x}_n,\\boldsymbol{x}_n) \\\\\n", - "\\end{bmatrix}\\boldsymbol{\\lambda}-\\mathbb{I}\\boldsymbol{\\lambda},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "subject to $\\boldsymbol{y}^T\\boldsymbol{\\lambda}=0$. Here we defined the vectors $\\boldsymbol{\\lambda} =[\\lambda_1,\\lambda_2,\\dots,\\lambda_n]$ and \n", - "$\\boldsymbol{y}=[y_1,y_2,\\dots,y_n]$. \n", - "With the slack constants this leads to the additional constraint $0\\leq \\lambda_i \\leq C$.\n", - "\n", - "**code will be added**" - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter5.py b/doc/LectureNotes/_build/jupyter_execute/chapter5.py deleted file mode 100644 index 5555a53a4..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter5.py +++ /dev/null @@ -1,1079 +0,0 @@ -# Support Vector Machines, overarching aims - -A Support Vector Machine (SVM) is a very powerful and versatile -Machine Learning method, capable of performing linear or nonlinear -classification, regression, and even outlier detection. It is one of -the most popular models in Machine Learning, and anyone interested in -Machine Learning should have it in their toolbox. SVMs are -particularly well suited for classification of complex but small-sized or -medium-sized datasets. - -The case with two well-separated classes only can be understood in an -intuitive way in terms of lines in a two-dimensional space separating -the two classes (see figure below). - -The basic mathematics behind the SVM is however less familiar to most of us. -It relies on the definition of hyperplanes and the -definition of a **margin** which separates classes (in case of -classification problems) of variables. It is also used for regression -problems. - -With SVMs we distinguish between hard margin and soft margins. The -latter introduces a so-called softening parameter to be discussed -below. We distinguish also between linear and non-linear -approaches. The latter are the most frequent ones since it is rather -unlikely that we can separate classes easily by say straight lines. - - -## Hyperplanes and all that - -The theory behind support vector machines (SVM hereafter) is based on -the mathematical description of so-called hyperplanes. Let us start -with a two-dimensional case. This will also allow us to introduce our -first SVM examples. These will be tailored to the case of two specific -classes, as displayed in the figure here based on the usage of the petal data. - -We assume here that our data set can be well separated into two -domains, where a straight line does the job in the separating the two -classes. Here the two classes are represented by either squares or -circles. - -%matplotlib inline - -from sklearn import datasets -from sklearn.svm import SVC, LinearSVC -from sklearn.linear_model import SGDClassifier -from sklearn.preprocessing import StandardScaler -import matplotlib -import matplotlib.pyplot as plt -plt.rcParams['axes.labelsize'] = 14 -plt.rcParams['xtick.labelsize'] = 12 -plt.rcParams['ytick.labelsize'] = 12 - - -iris = datasets.load_iris() -X = iris["data"][:, (2, 3)] # petal length, petal width -y = iris["target"] - -setosa_or_versicolor = (y == 0) | (y == 1) -X = X[setosa_or_versicolor] -y = y[setosa_or_versicolor] - - - -C = 5 -alpha = 1 / (C * len(X)) - -lin_clf = LinearSVC(loss="hinge", C=C, random_state=42) -svm_clf = SVC(kernel="linear", C=C) -sgd_clf = SGDClassifier(loss="hinge", learning_rate="constant", eta0=0.001, alpha=alpha, - max_iter=100000, random_state=42) - -scaler = StandardScaler() -X_scaled = scaler.fit_transform(X) - -lin_clf.fit(X_scaled, y) -svm_clf.fit(X_scaled, y) -sgd_clf.fit(X_scaled, y) - -print("LinearSVC: ", lin_clf.intercept_, lin_clf.coef_) -print("SVC: ", svm_clf.intercept_, svm_clf.coef_) -print("SGDClassifier(alpha={:.5f}):".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_) - -# Compute the slope and bias of each decision boundary -w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1] -b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1] -w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1] -b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1] -w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1] -b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1] - -# Transform the decision boundary lines back to the original scale -line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]]) -line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]]) -line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]]) - -# Plot all three decision boundaries -plt.figure(figsize=(11, 4)) -plt.plot(line1[:, 0], line1[:, 1], "k:", label="LinearSVC") -plt.plot(line2[:, 0], line2[:, 1], "b--", linewidth=2, label="SVC") -plt.plot(line3[:, 0], line3[:, 1], "r-", label="SGDClassifier") -plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs") # label="Iris-Versicolor" -plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo") # label="Iris-Setosa" -plt.xlabel("Petal length", fontsize=14) -plt.ylabel("Petal width", fontsize=14) -plt.legend(loc="upper center", fontsize=14) -plt.axis([0, 5.5, 0, 2]) - -plt.show() - -The aim of the SVM algorithm is to find a hyperplane in a -$p$-dimensional space, where $p$ is the number of features that -distinctly classifies the data points. - -In a $p$-dimensional space, a hyperplane is what we call an affine subspace of dimension of $p-1$. -As an example, in two dimension, a hyperplane is simply as straight line while in three dimensions it is -a two-dimensional subspace, or stated simply, a plane. - -In two dimensions, with the variables $x_1$ and $x_2$, the hyperplane is defined as - -$$ -b+w_1x_1+w_2x_2=0, -$$ - -where $b$ is the intercept and $w_1$ and $w_2$ define the elements of a vector orthogonal to the line -$b+w_1x_1+w_2x_2=0$. -In two dimensions we define the vectors $\boldsymbol{x} =[x1,x2]$ and $\boldsymbol{w}=[w1,w2]$. -We can then rewrite the above equation as - -$$ -\boldsymbol{x}^T\boldsymbol{w}+b=0. -$$ - -We limit ourselves to two classes of outputs $y_i$ and assign these classes the values $y_i = \pm 1$. -In a $p$-dimensional space of say $p$ features we have a hyperplane defines as - -$$ -b+wx_1+w_2x_2+\dots +w_px_p=0. -$$ - -If we define a -matrix $\boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right]$ -of dimension $n\times p$, where $n$ represents the observations for each feature and each vector $x_i$ is a column vector of the matrix $\boldsymbol{X}$, - -$$ -\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}. -$$ - -If the above condition is not met for a given vector $\boldsymbol{x}_i$ we have - -$$ -b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0, -$$ - -if our output $y_i=1$. -In this case we say that $\boldsymbol{x}_i$ lies on one of the sides of the hyperplane and if - -$$ -b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0, -$$ - -for the class of observations $y_i=-1$, -then $\boldsymbol{x}_i$ lies on the other side. - -Equivalently, for the two classes of observations we have - -$$ -y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0. -$$ - -When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located. - - -### The two-dimensional case - -Let us try to develop our intuition about SVMs by limiting ourselves to a two-dimensional -plane. To separate the two classes of data points, there are many -possible lines (hyperplanes if you prefer a more strict naming) -that could be chosen. Our objective is to find a -plane that has the maximum margin, i.e the maximum distance between -data points of both classes. Maximizing the margin distance provides -some reinforcement so that future data points can be classified with -more confidence. - -What a linear classifier attempts to accomplish is to split the -feature space into two half spaces by placing a hyperplane between the -data points. This hyperplane will be our decision boundary. All -points on one side of the plane will belong to class one and all points -on the other side of the plane will belong to the second class two. - -Unfortunately there are many ways in which we can place a hyperplane -to divide the data. Below is an example of two candidate hyperplanes -for our data sample. - - -Let us define the function - -$$ -f(x) = \boldsymbol{w}^T\boldsymbol{x}+b = 0, -$$ - -as the function that determines the line $L$ that separates two classes (our two features), see the figure here. - - -Any point defined by $\boldsymbol{x}_i$ and $\boldsymbol{x}_2$ on the line $L$ will satisfy $\boldsymbol{w}^T(\boldsymbol{x}_1-\boldsymbol{x}_2)=0$. - -The signed distance $\delta$ from any point defined by a vector $\boldsymbol{x}$ and a point $\boldsymbol{x}_0$ on the line $L$ is then - -$$ -\delta = \frac{1}{\vert\vert \boldsymbol{w}\vert\vert}(\boldsymbol{w}^T\boldsymbol{x}+b). -$$ - -How do we find the parameter $b$ and the vector $\boldsymbol{w}$? What we could -do is to define a cost function which now contains the set of all -misclassified points $M$ and attempt to minimize this function - -$$ -C(\boldsymbol{w},b) = -\sum_{i\in M} y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b). -$$ - -We could now for example define all values $y_i =1$ as misclassified in case we have $\boldsymbol{w}^T\boldsymbol{x}_i+b < 0$ and the opposite if we have $y_i=-1$. Taking the derivatives gives us - -$$ -\frac{\partial C}{\partial b} = -\sum_{i\in M} y_i, -$$ - -and - -$$ -\frac{\partial C}{\partial \boldsymbol{w}} = -\sum_{i\in M} y_ix_i. -$$ - -We can now use the Newton-Raphson method or different variants of the gradient descent family (from plain gradient descent to various stochastic gradient descent approaches) to solve the equations - -$$ -b \leftarrow b +\eta \frac{\partial C}{\partial b}, -$$ - -and - -$$ -\boldsymbol{w} \leftarrow \boldsymbol{w} +\eta \frac{\partial C}{\partial \boldsymbol{w}}, -$$ - -where $\eta$ is our by now well-known learning rate. - - - -The equations we discussed above can be coded rather easily (the -framework is similar to what we developed for logistic -regression). We are going to set up a simple case with two classes only and we want to find a line which separates them the best possible way. - -There are however problems with this approach, although it looks -pretty straightforward to implement. When running the above code, we see that we can easily end up with many diffeent lines which separate the two classes. - - -For small -gaps between the entries, we may also end up needing many iterations -before the solutions converge and if the data cannot be separated -properly into two distinct classes, we may not experience a converge -at all. - - -### A better approach - -A better approach is rather to try to define a large margin between -the two classes (if they are well separated from the beginning). - -Thus, we wish to find a margin $M$ with $\boldsymbol{w}$ normalized to -$\vert\vert \boldsymbol{w}\vert\vert =1$ subject to the condition - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. -$$ - -All points are thus at a signed distance from the decision boundary defined by the line $L$. The parameters $b$ and $w_1$ and $w_2$ define this line. - -We seek thus the largest value $M$ defined by - -$$ -\frac{1}{\vert \vert \boldsymbol{w}\vert\vert}y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, -$$ - -or just - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq M\vert \vert \boldsymbol{w}\vert\vert \hspace{0.1cm}\forall i. -$$ - -If we scale the equation so that $\vert \vert \boldsymbol{w}\vert\vert = 1/M$, we have to find the minimum of -$\boldsymbol{w}^T\boldsymbol{w}=\vert \vert \boldsymbol{w}\vert\vert$ (the norm) subject to the condition - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. -$$ - -We have thus defined our margin as the invers of the norm of -$\boldsymbol{w}$. We want to minimize the norm in order to have a as large as -possible margin $M$. Before we proceed, we need to remind ourselves -about Lagrangian multipliers. - - -## A quick Reminder on Lagrangian Multipliers - -Consider a function of three independent variables $f(x,y,z)$ . For the function $f$ to be an -extreme we have - -$$ -df=0. -$$ - -A necessary and sufficient condition is - -$$ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, -$$ - -due to - -$$ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. -$$ - -In many problems the variables $x,y,z$ are often subject to constraints (such as those above for the margin) -so that they are no longer all independent. It is possible at least in principle to use each -constraint to eliminate one variable -and to proceed with a new and smaller set of independent varables. - -The use of so-called Lagrangian multipliers is an alternative technique when the elimination -of variables is incovenient or undesirable. Assume that we have an equation of constraint on -the variables $x,y,z$ - -$$ -\phi(x,y,z) = 0, -$$ - -resulting in - -$$ -d\phi = \frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy+\frac{\partial \phi}{\partial z}dz =0. -$$ - -Now we cannot set anymore - -$$ -\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}=0, -$$ - -if $df=0$ is wanted -because there are now only two independent variables! Assume $x$ and $y$ are the independent -variables. -Then $dz$ is no longer arbitrary. - - -However, we can add to - -$$ -df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz, -$$ - -a multiplum of $d\phi$, viz. $\lambda d\phi$, resulting in - -$$ -df+\lambda d\phi = (\frac{\partial f}{\partial z}+\lambda -\frac{\partial \phi}{\partial x})dx+(\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y})dy+ -(\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z})dz =0. -$$ - -Our multiplier is chosen so that - -$$ -\frac{\partial f}{\partial z}+\lambda\frac{\partial \phi}{\partial z} =0. -$$ - -We need to remember that we took $dx$ and $dy$ to be arbitrary and thus we must have - -$$ -\frac{\partial f}{\partial x}+\lambda\frac{\partial \phi}{\partial x} =0, -$$ - -and - -$$ -\frac{\partial f}{\partial y}+\lambda\frac{\partial \phi}{\partial y} =0. -$$ - -When all these equations are satisfied, $df=0$. We have four unknowns, $x,y,z$ and -$\lambda$. Actually we want only $x,y,z$, $\lambda$ needs not to be determined, -it is therefore often called -Lagrange's undetermined multiplier. -If we have a set of constraints $\phi_k$ we have the equations - -$$ -\frac{\partial f}{\partial x_i}+\sum_k\lambda_k\frac{\partial \phi_k}{\partial x_i} =0. -$$ - -In order to solve the above problem, we define the following Lagrangian function to be minimized - -$$ -\cal{L}(\lambda,b,\boldsymbol{w})=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-1\right], -$$ - -where $\lambda_i$ is a so-called Lagrange multiplier subject to the condition $\lambda_i \geq 0$. - -Taking the derivatives with respect to $b$ and $\boldsymbol{w}$ we obtain - -$$ -\frac{\partial \cal{L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, -$$ - -and - -$$ -\frac{\partial \cal{L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i. -$$ - -Inserting these constraints into the equation for $\cal{L}$ we obtain - -$$ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -$$ - -subject to the constraints $\lambda_i\geq 0$ and $\sum_i\lambda_iy_i=0$. -We must in addition satisfy the [Karush-Kuhn-Tucker](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions) (KKT) condition - -$$ -\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -1\right] \hspace{0.1cm}\forall i. -$$ - -1. If $\lambda_i > 0$, then $y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1$ and we say that $x_i$ is on the boundary. - -2. If $y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)> 1$, we say $x_i$ is not on the boundary and we set $\lambda_i=0$. - -When $\lambda_i > 0$, the vectors $\boldsymbol{x}_i$ are called support vectors. They are the vectors closest to the line (or hyperplane) and define the margin $M$. - - -We can rewrite - -$$ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -$$ - -and its constraints in terms of a matrix-vector problem where we minimize w.r.t. $\lambda$ the following problem - -$$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1\boldsymbol{x}_1^T\boldsymbol{x}_1 & y_1y_2\boldsymbol{x}_1^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_1^T\boldsymbol{x}_n \\ -y_2y_1\boldsymbol{x}_2^T\boldsymbol{x}_1 & y_2y_2\boldsymbol{x}_2^T\boldsymbol{x}_2 & \dots & \dots & y_1y_n\boldsymbol{x}_2^T\boldsymbol{x}_n \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1\boldsymbol{x}_n^T\boldsymbol{x}_1 & y_ny_2\boldsymbol{x}_n^T\boldsymbol{x}_2 & \dots & \dots & y_ny_n\boldsymbol{x}_n^T\boldsymbol{x}_n \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, -$$ - -subject to $\boldsymbol{y}^T\boldsymbol{\lambda}=0$. Here we defined the vectors $\boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n]$ and -$\boldsymbol{y}=[y_1,y_2,\dots,y_n]$. - - - -Solving the above problem, yields the values of $\lambda_i$. -To find the coefficients of your hyperplane we need simply to compute - -$$ -\boldsymbol{w}=\sum_{i} \lambda_iy_i\boldsymbol{x}_i. -$$ - -With our vector $\boldsymbol{w}$ we can in turn find the value of the intercept $b$ (here in two dimensions) via - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, -$$ - -resulting in - -$$ -b = \frac{1}{y_i}-\boldsymbol{w}^T\boldsymbol{x}_i, -$$ - -or if we write it out in terms of the support vectors only, with $N_s$ being their number, we have - -$$ -b = \frac{1}{N_s}\sum_{j\in N_s}\left(y_j-\sum_{i=1}^n\lambda_iy_i\boldsymbol{x}_i^T\boldsymbol{x}_j\right). -$$ - -With our hyperplane coefficients we can use our classifier to assign any observation by simply using - -$$ -y_i = \mathrm{sign}(\boldsymbol{w}^T\boldsymbol{x}_i+b). -$$ - -Below we discuss how to find the optimal values of $\lambda_i$. Before we proceed however, we discuss now the so-called soft classifier. - - -## A soft classifier - -Till now, the margin is strictly defined by the support vectors. This defines what is called a hard classifier, that is the margins are well defined. - -Suppose now that classes overlap in feature space, as shown in the -figure here. One way to deal with this problem before we define the -so-called **kernel approach**, is to allow a kind of slack in the sense -that we allow some points to be on the wrong side of the margin. - -We introduce thus the so-called **slack** variables $\boldsymbol{\xi} =[\xi_1,x_2,\dots,x_n]$ and -modify our previous equation - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1, -$$ - -to - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i, -$$ - -with the requirement $\xi_i\geq 0$. The total violation is now $\sum_i\xi$. -The value $\xi_i$ in the constraint the last constraint corresponds to the amount by which the prediction -$y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1$ is on the wrong side of its margin. Hence by bounding the sum $\sum_i \xi_i$, -we bound the total amount by which predictions fall on the wrong side of their margins. - -Misclassifications occur when $\xi_i > 1$. Thus bounding the total sum by some value $C$ bounds in turn the total number of -misclassifications. - - -This has in turn the consequences that we change our optmization problem to finding the minimum of - -$$ -\cal{L}=\frac{1}{2}\boldsymbol{w}^T\boldsymbol{w}-\sum_{i=1}^n\lambda_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)-(1-\xi_)\right]+C\sum_{i=1}^n\xi_i-\sum_{i=1}^n\gamma_i\xi_i, -$$ - -subject to - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, -$$ - -with the requirement $\xi_i\geq 0$. - -Taking the derivatives with respect to $b$ and $\boldsymbol{w}$ we obtain - -$$ -\frac{\partial \cal{L}}{\partial b} = -\sum_{i} \lambda_iy_i=0, -$$ - -and - -$$ -\frac{\partial \cal{L}}{\partial \boldsymbol{w}} = 0 = \boldsymbol{w}-\sum_{i} \lambda_iy_i\boldsymbol{x}_i, -$$ - -and - -$$ -\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. -$$ - -Inserting these constraints into the equation for $\cal{L}$ we obtain the same equation as before - -$$ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{x}_j, -$$ - -but now subject to the constraints $\lambda_i\geq 0$, $\sum_i\lambda_iy_i=0$ and $0\leq\lambda_i \leq C$. -We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads - -5 -0 - -< -< -< -! -! -M -A -T -H -_ -B -L -O -C -K - -$$ -\gamma_i\xi_i = 0, -$$ - -and - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. -$$ - -## Kernels and non-linearity - -The cases we have studied till now, were all characterized by two classes -with a close to linear separability. The classifiers we have described -so far find linear boundaries in our input feature space. It is -possible to make our procedure more flexible by exploring the feature -space using other basis expansions such as higher-order polynomials, -wavelets, splines etc. - -If our feature space is not easy to separate, as shown in the figure -here, we can achieve a better separation by introducing more complex -basis functions. The ideal would be, as shown in the next figure, to, via a specific transformation to -obtain a separation between the classes which is almost linear. - -The change of basis, from $x\rightarrow z=\phi(x)$ leads to the same type of equations to be solved, except that -we need to introduce for example a polynomial transformation to a two-dimensional training set. - -import numpy as np -import os - -np.random.seed(42) - -# To plot pretty figures -import matplotlib -import matplotlib.pyplot as plt -plt.rcParams['axes.labelsize'] = 14 -plt.rcParams['xtick.labelsize'] = 12 -plt.rcParams['ytick.labelsize'] = 12 - - -from sklearn.svm import SVC -from sklearn import datasets - - - -X1D = np.linspace(-4, 4, 9).reshape(-1, 1) -X2D = np.c_[X1D, X1D**2] -y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0]) - -plt.figure(figsize=(11, 4)) - -plt.subplot(121) -plt.grid(True, which='both') -plt.axhline(y=0, color='k') -plt.plot(X1D[:, 0][y==0], np.zeros(4), "bs") -plt.plot(X1D[:, 0][y==1], np.zeros(5), "g^") -plt.gca().get_yaxis().set_ticks([]) -plt.xlabel(r"$x_1$", fontsize=20) -plt.axis([-4.5, 4.5, -0.2, 0.2]) - -plt.subplot(122) -plt.grid(True, which='both') -plt.axhline(y=0, color='k') -plt.axvline(x=0, color='k') -plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], "bs") -plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], "g^") -plt.xlabel(r"$x_1$", fontsize=20) -plt.ylabel(r"$x_2$", fontsize=20, rotation=0) -plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16]) -plt.plot([-4.5, 4.5], [6.5, 6.5], "r--", linewidth=3) -plt.axis([-4.5, 4.5, -1, 17]) -plt.subplots_adjust(right=1) -plt.show() - -Suppose we define a polynomial transformation of degree two only (we continue to live in a plane with $x_i$ and $y_i$ as variables) - -$$ -z = \phi(x_i) =\left(x_i^2, y_i^2, \sqrt{2}x_iy_i\right). -$$ - -With our new basis, the equations we solved earlier are basically the same, that is we have now (without the slack option for simplicity) - -$$ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{z}_i^T\boldsymbol{z}_j, -$$ - -subject to the constraints $\lambda_i\geq 0$, $\sum_i\lambda_iy_i=0$, and for the support vectors - -$$ -y_i(\boldsymbol{w}^T\boldsymbol{z}_i+b)= 1 \hspace{0.1cm}\forall i, -$$ - -from which we also find $b$. -To compute $\boldsymbol{z}_i^T\boldsymbol{z}_j$ we define the kernel $K(\boldsymbol{x}_i,\boldsymbol{x}_j)$ as - -$$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\boldsymbol{z}_i^T\boldsymbol{z}_j= \phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). -$$ - -For the above example, the kernel reads - -$$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=[x_i^2, y_i^2, \sqrt{2}x_iy_i]^T\begin{bmatrix} x_j^2 \\ y_j^2 \\ \sqrt{2}x_jy_j \end{bmatrix}=x_i^2x_j^2+2x_ix_jy_iy_j+y_i^2y_j^2. -$$ - -We note that this is nothing but the dot product of the two original -vectors $(\boldsymbol{x}_i^T\boldsymbol{x}_j)^2$. Instead of thus computing the -product in the Lagrangian of $\boldsymbol{z}_i^T\boldsymbol{z}_j$ we simply compute -the dot product $(\boldsymbol{x}_i^T\boldsymbol{x}_j)^2$. - - -This leads to the so-called -kernel trick and the result leads to the same as if we went through -the trouble of performing the transformation -$\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j)$ during the SVM calculations. - - - -Using our definition of the kernel We can rewrite again the Lagrangian - -$$ -\cal{L}=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_j\boldsymbol{x}_i^T\boldsymbol{z}_j, -$$ - -subject to the constraints $\lambda_i\geq 0$, $\sum_i\lambda_iy_i=0$ in terms of a convex optimization problem - -$$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{1}\boldsymbol{\lambda}, -$$ - -subject to $\boldsymbol{y}^T\boldsymbol{\lambda}=0$. Here we defined the vectors $\boldsymbol{\lambda} =[\lambda_1,\lambda_2,\dots,\lambda_n]$ and -$\boldsymbol{y}=[y_1,y_2,\dots,y_n]$. -If we add the slack constants this leads to the additional constraint $0\leq \lambda_i \leq C$. - -We can rewrite this (see the solutions below) in terms of a convex optimization problem of the type - -$$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \hspace{0.2cm} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} -$$ - -Below we discuss how to solve these equations. Here we note that the matrix $\boldsymbol{P}$ has matrix elements $p_{ij}=y_iy_jK(\boldsymbol{x}_i,\boldsymbol{x}_j)$. -Given a kernel $K$ and the targets $y_i$ this matrix is easy to set up. The constraint $\boldsymbol{y}^T\boldsymbol{\lambda}=0$ leads to $f=0$ and $\boldsymbol{A}=\boldsymbol{y}$. How to set up the matrix $\boldsymbol{G}$ is discussed later. Here note that the inequalities $0\leq \lambda_i \leq C$ can be split up into -$0\leq \lambda_i$ and $\lambda_i \leq C$. These two inequalities define then the matrix $\boldsymbol{G}$ and the vector $\boldsymbol{h}$. - - - -## Different kernels and Mercer's theorem - -There are several popular kernels being used. These are -1. Linear: $K(\boldsymbol{x},\boldsymbol{y})=\boldsymbol{x}^T\boldsymbol{y}$, - -2. Polynomial: $K(\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{x}^T\boldsymbol{y}+\gamma)^d$, - -3. Gaussian Radial Basis Function: $K(\boldsymbol{x},\boldsymbol{y})=\exp{\left(-\gamma\vert\vert\boldsymbol{x}-\boldsymbol{y}\vert\vert^2\right)}$, - -4. Tanh: $K(\boldsymbol{x},\boldsymbol{y})=\tanh{(\boldsymbol{x}^T\boldsymbol{y}+\gamma)}$, - -and many other ones. - -An important theorem for us is [Mercer's -theorem](https://en.wikipedia.org/wiki/Mercer%27s_theorem). The -theorem states that if a kernel function $K$ is symmetric, continuous -and leads to a positive semi-definite matrix $\boldsymbol{P}$ then there -exists a function $\phi$ that maps $\boldsymbol{x}_i$ and $\boldsymbol{x}_j$ into -another space (possibly with much higher dimensions) such that - -$$ -K(\boldsymbol{x}_i,\boldsymbol{x}_j)=\phi(\boldsymbol{x}_i)^T\phi(\boldsymbol{x}_j). -$$ - -So you can use $K$ as a kernel since you know $\phi$ exists, even if -you don’t know what $\phi$ is. - -Note that some frequently used kernels (such as the Sigmoid kernel) -don’t respect all of Mercer’s conditions, yet they generally work well -in practice. - - -## The moons example - -from __future__ import division, print_function, unicode_literals - -import numpy as np -np.random.seed(42) - -import matplotlib -import matplotlib.pyplot as plt -plt.rcParams['axes.labelsize'] = 14 -plt.rcParams['xtick.labelsize'] = 12 -plt.rcParams['ytick.labelsize'] = 12 - - -from sklearn.svm import SVC -from sklearn import datasets - - - -from sklearn.pipeline import Pipeline -from sklearn.preprocessing import StandardScaler -from sklearn.svm import LinearSVC - - -from sklearn.datasets import make_moons -X, y = make_moons(n_samples=100, noise=0.15, random_state=42) - -def plot_dataset(X, y, axes): - plt.plot(X[:, 0][y==0], X[:, 1][y==0], "bs") - plt.plot(X[:, 0][y==1], X[:, 1][y==1], "g^") - plt.axis(axes) - plt.grid(True, which='both') - plt.xlabel(r"$x_1$", fontsize=20) - plt.ylabel(r"$x_2$", fontsize=20, rotation=0) - -plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) -plt.show() - -from sklearn.datasets import make_moons -from sklearn.pipeline import Pipeline -from sklearn.preprocessing import PolynomialFeatures - -polynomial_svm_clf = Pipeline([ - ("poly_features", PolynomialFeatures(degree=3)), - ("scaler", StandardScaler()), - ("svm_clf", LinearSVC(C=10, loss="hinge", random_state=42)) - ]) - -polynomial_svm_clf.fit(X, y) - -def plot_predictions(clf, axes): - x0s = np.linspace(axes[0], axes[1], 100) - x1s = np.linspace(axes[2], axes[3], 100) - x0, x1 = np.meshgrid(x0s, x1s) - X = np.c_[x0.ravel(), x1.ravel()] - y_pred = clf.predict(X).reshape(x0.shape) - y_decision = clf.decision_function(X).reshape(x0.shape) - plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2) - plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1) - -plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5]) -plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) - -plt.show() - - -from sklearn.svm import SVC - -poly_kernel_svm_clf = Pipeline([ - ("scaler", StandardScaler()), - ("svm_clf", SVC(kernel="poly", degree=3, coef0=1, C=5)) - ]) -poly_kernel_svm_clf.fit(X, y) - -poly100_kernel_svm_clf = Pipeline([ - ("scaler", StandardScaler()), - ("svm_clf", SVC(kernel="poly", degree=10, coef0=100, C=5)) - ]) -poly100_kernel_svm_clf.fit(X, y) - -plt.figure(figsize=(11, 4)) - -plt.subplot(121) -plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5]) -plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) -plt.title(r"$d=3, r=1, C=5$", fontsize=18) - -plt.subplot(122) -plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5]) -plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) -plt.title(r"$d=10, r=100, C=5$", fontsize=18) - -plt.show() - -def gaussian_rbf(x, landmark, gamma): - return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2) - -gamma = 0.3 - -x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1) -x2s = gaussian_rbf(x1s, -2, gamma) -x3s = gaussian_rbf(x1s, 1, gamma) - -XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)] -yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0]) - -plt.figure(figsize=(11, 4)) - -plt.subplot(121) -plt.grid(True, which='both') -plt.axhline(y=0, color='k') -plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c="red") -plt.plot(X1D[:, 0][yk==0], np.zeros(4), "bs") -plt.plot(X1D[:, 0][yk==1], np.zeros(5), "g^") -plt.plot(x1s, x2s, "g--") -plt.plot(x1s, x3s, "b:") -plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1]) -plt.xlabel(r"$x_1$", fontsize=20) -plt.ylabel(r"Similarity", fontsize=14) -plt.annotate(r'$\mathbf{x}$', - xy=(X1D[3, 0], 0), - xytext=(-0.5, 0.20), - ha="center", - arrowprops=dict(facecolor='black', shrink=0.1), - fontsize=18, - ) -plt.text(-2, 0.9, "$x_2$", ha="center", fontsize=20) -plt.text(1, 0.9, "$x_3$", ha="center", fontsize=20) -plt.axis([-4.5, 4.5, -0.1, 1.1]) - -plt.subplot(122) -plt.grid(True, which='both') -plt.axhline(y=0, color='k') -plt.axvline(x=0, color='k') -plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], "bs") -plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], "g^") -plt.xlabel(r"$x_2$", fontsize=20) -plt.ylabel(r"$x_3$ ", fontsize=20, rotation=0) -plt.annotate(r'$\phi\left(\mathbf{x}\right)$', - xy=(XK[3, 0], XK[3, 1]), - xytext=(0.65, 0.50), - ha="center", - arrowprops=dict(facecolor='black', shrink=0.1), - fontsize=18, - ) -plt.plot([-0.1, 1.1], [0.57, -0.1], "r--", linewidth=3) -plt.axis([-0.1, 1.1, -0.1, 1.1]) - -plt.subplots_adjust(right=1) - -plt.show() - - -x1_example = X1D[3, 0] -for landmark in (-2, 1): - k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma) - print("Phi({}, {}) = {}".format(x1_example, landmark, k)) - -rbf_kernel_svm_clf = Pipeline([ - ("scaler", StandardScaler()), - ("svm_clf", SVC(kernel="rbf", gamma=5, C=0.001)) - ]) -rbf_kernel_svm_clf.fit(X, y) - - -from sklearn.svm import SVC - -gamma1, gamma2 = 0.1, 5 -C1, C2 = 0.001, 1000 -hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2) - -svm_clfs = [] -for gamma, C in hyperparams: - rbf_kernel_svm_clf = Pipeline([ - ("scaler", StandardScaler()), - ("svm_clf", SVC(kernel="rbf", gamma=gamma, C=C)) - ]) - rbf_kernel_svm_clf.fit(X, y) - svm_clfs.append(rbf_kernel_svm_clf) - -plt.figure(figsize=(11, 7)) - -for i, svm_clf in enumerate(svm_clfs): - plt.subplot(221 + i) - plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5]) - plot_dataset(X, y, [-1.5, 2.5, -1, 1.5]) - gamma, C = hyperparams[i] - plt.title(r"$\gamma = {}, C = {}$".format(gamma, C), fontsize=16) - -plt.show() - -## Mathematical optimization of convex functions - -A mathematical (quadratic) optimization problem, or just optimization problem, has the form - -$$ -\begin{align*} - &\mathrm{min}_{\lambda}\hspace{0.2cm} \frac{1}{2}\boldsymbol{\lambda}^T\boldsymbol{P}\boldsymbol{\lambda}+\boldsymbol{q}^T\boldsymbol{\lambda},\\ \nonumber - &\mathrm{subject\hspace{0.1cm}to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{\lambda} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{\lambda}=f. -\end{align*} -$$ - -subject to some constraints for say a selected set $i=1,2,\dots, n$. -In our case we are optimizing with respect to the Lagrangian multipliers $\lambda_i$, and the -vector $\boldsymbol{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n]$ is the optimization variable we are dealing with. - -In our case we are particularly interested in a class of optimization problems called convex optmization problems. -In our discussion on gradient descent methods we discussed at length the definition of a convex function. - -Convex optimization problems play a central role in applied mathematics and we recommend strongly [Boyd and Vandenberghe's text on the topics](http://web.stanford.edu/~boyd/cvxbook/). - - - - -If we use Python as programming language and wish to venture beyond -**scikit-learn**, **tensorflow** and similar software which makes our -lives so much easier, we need to dive into the wonderful world of -quadratic programming. We can, if we wish, solve the minimization -problem using say standard gradient methods or conjugate gradient -methods. However, these methods tend to exhibit a rather slow -converge. So, welcome to the promised land of quadratic programming. - -The functions we need are contained in the quadratic programming package **CVXOPT** and we need to import it together with **numpy** as - -import numpy -import cvxopt - -This will make our life much easier. You don't need t write your own optimizer. - - - -We remind ourselves about the general problem we want to solve - -$$ -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}\boldsymbol{x}^T\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}^T\boldsymbol{x},\\ \nonumber - &\mathrm{subject\hspace{0.1cm} to} \hspace{0.2cm} \boldsymbol{G}\boldsymbol{x} \preceq \boldsymbol{h} \wedge \boldsymbol{A}\boldsymbol{x}=f. -\end{align*} -$$ - -Let us show how to perform the optmization using a simple case. Assume we want to optimize the following problem - -$$ -\begin{align*} - &\mathrm{min}_{x}\hspace{0.2cm} \frac{1}{2}x^2+5x+3y \\ \nonumber - &\mathrm{subject to} \\ \nonumber - &x, y \geq 0 \\ \nonumber - &x+3y \geq 15 \\ \nonumber - &2x+5y \leq 100 \\ \nonumber - &3x+4y \leq 80. \\ \nonumber -\end{align*} -$$ - -The minimization problem can be rewritten in terms of vectors and matrices as (with $x$ and $y$ being the unknowns) - -$$ -\frac{1}{2}\begin{bmatrix} x\\ y \end{bmatrix}^T \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix}3\\ 4 \end{bmatrix}^T \begin{bmatrix}x \\ y \end{bmatrix}. -$$ - -Similarly, we can now set up the inequalities (we need to change $\geq$ to $\leq$ by multiplying with $-1$ on bot sides) as the following matrix-vector equation - -$$ -\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ -1 & -3 \\ 2 & 5 \\ 3 & 4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix} \preceq \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -$$ - -We have collapsed all the inequalities into a single matrix $\boldsymbol{G}$. We see also that our matrix - -$$ -\boldsymbol{P} =\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} -$$ - -is clearly positive semi-definite (all eigenvalues larger or equal zero). -Finally, the vector $\boldsymbol{h}$ is defined as - -$$ -\boldsymbol{h} = \begin{bmatrix}0 \\ 0\\ -15 \\ 100 \\ 80\end{bmatrix}. -$$ - -Since we don't have any equalities the matrix $\boldsymbol{A}$ is set to zero -The following code solves the equations for us - -# Import the necessary packages -import numpy -from cvxopt import matrix -from cvxopt import solvers -P = matrix(numpy.diag([1,0]), tc=’d’) -q = matrix(numpy.array([3,4]), tc=’d’) -G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’) -h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’) -# Construct the QP, invoke solver -sol = solvers.qp(P,q,G,h) -# Extract optimal value and solution -sol[’x’] -sol[’primal objective’] - -We are now ready to return to our setup of the optmization problem for a more realistic case. Introducing the **slack** parameter $C$ we have - -$$ -\frac{1}{2} \boldsymbol{\lambda}^T\begin{bmatrix} y_1y_1K(\boldsymbol{x}_1,\boldsymbol{x}_1) & y_1y_2K(\boldsymbol{x}_1,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_1,\boldsymbol{x}_n) \\ -y_2y_1K(\boldsymbol{x}_2,\boldsymbol{x}_1) & y_2y_2K(\boldsymbol{x}_2,\boldsymbol{x}_2) & \dots & \dots & y_1y_nK(\boldsymbol{x}_2,\boldsymbol{x}_n) \\ -\dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots \\ -y_ny_1K(\boldsymbol{x}_n,\boldsymbol{x}_1) & y_ny_2K(\boldsymbol{x}_n\boldsymbol{x}_2) & \dots & \dots & y_ny_nK(\boldsymbol{x}_n,\boldsymbol{x}_n) \\ -\end{bmatrix}\boldsymbol{\lambda}-\mathbb{I}\boldsymbol{\lambda}, -$$ - -subject to $\boldsymbol{y}^T\boldsymbol{\lambda}=0$. 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With this basic algorithm we can in turn build more complex\n", - "networks, spanning from homogeneous and heterogenous forests (bagging,\n", - "random forests and more) to one of the most popular supervised\n", - "algorithms nowadays, the extreme gradient boosting, or just\n", - "XGBoost. But let us start with the simplest possible ingredient.\n", - "\n", - "Decision trees are supervised learning algorithms used for both,\n", - "classification and regression tasks.\n", - "\n", - "\n", - "The main idea of decision trees\n", - "is to find those descriptive features which contain the most\n", - "**information** regarding the target feature and then split the dataset\n", - "along the values of these features such that the target feature values\n", - "for the resulting underlying datasets are as pure as possible.\n", - "\n", - "The descriptive features which reproduce best the target/output features are normally said\n", - "to be the most informative ones. The process of finding the **most\n", - "informative** feature is done until we accomplish a stopping criteria\n", - "where we then finally end up in so called **leaf nodes**. \n", - "\n", - "## Basics of a tree\n", - "\n", - "A decision tree is typically divided into a **root node**, the **interior nodes**,\n", - "and the final **leaf nodes** or just **leaves**. These entities are then connected by so-called **branches**.\n", - "\n", - "The leaf nodes\n", - "contain the predictions we will make for new query instances presented\n", - "to our trained model. This is possible since the model has \n", - "learned the underlying structure of the training data and hence can,\n", - "given some assumptions, make predictions about the target feature value\n", - "(class) of unseen query instances.\n", - "\n", - "\n", - "## General Features\n", - "\n", - "The overarching approach to decision trees is a top-down approach.\n", - "\n", - "* A leaf provides the classification of a given instance.\n", - "\n", - "* A node specifies a test of some attribute of the instance.\n", - "\n", - "* A branch corresponds to a possible values of an attribute.\n", - "\n", - "* An instance is classified by starting at the root node of the tree, testing the attribute specified by this node, then moving down the tree branch corresponding to the value of the attribute in the given example.\n", - "\n", - "This process is then repeated for the subtree rooted at the new\n", - "node.\n", - "\n", - "\n", - "\n", - "In simplified terms, the process of training a decision tree and\n", - "predicting the target features of query instances is as follows:\n", - "\n", - "1. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature\n", - "\n", - "2. Train the decision tree model by continuously splitting the target feature along the values of the descriptive features using a measure of information gain during the training process\n", - "\n", - "3. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the *predictions* we want to make for new query instances\n", - "\n", - "4. Show query instances to the tree and run down the tree until we arrive at leaf nodes\n", - "\n", - "Then we are essentially done!" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "2nd degree coefficients:\n", - "zero power: 1.3961181500194275\n", - "first power: -0.0007183308296987448\n", - "second power: 0.0002882855342275337\n" - ] - }, - { - "data": { - "image/png": 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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter6_1_2.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.preprocessing import PolynomialFeatures\n", - "from sklearn.linear_model import LinearRegression\n", - "\n", - "steps=250\n", - "\n", - "distance=0\n", - "x=0\n", - "distance_list=[]\n", - "steps_list=[]\n", - "while x\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 7\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 8\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mIPython\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdisplay\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mImage\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 9\u001b[0;31m \u001b[0;32mfrom\u001b[0m \u001b[0mpydot\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mgraph_from_dot_data\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 10\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mpandas\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mpd\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 11\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mnumpy\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'pydot'" - ] - } - ], - "source": [ - "import os\n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.metrics import confusion_matrix\n", - "from sklearn.tree import export_graphviz\n", - "\n", - "from IPython.display import Image \n", - "from pydot import graph_from_dot_data\n", - "import pandas as pd\n", - "import numpy as np\n", - "\n", - "\n", - "cancer = load_breast_cancer()\n", - "X = pd.DataFrame(cancer.data, columns=cancer.feature_names)\n", - "print(X)\n", - "y = pd.Categorical.from_codes(cancer.target, cancer.target_names)\n", - "y = pd.get_dummies(y)\n", - "print(y)\n", - "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)\n", - "tree_clf = DecisionTreeClassifier(max_depth=5)\n", - "tree_clf.fit(X_train, y_train)\n", - "\n", - "export_graphviz(\n", - " tree_clf,\n", - " out_file=\"DataFiles/cancer.dot\",\n", - " feature_names=cancer.feature_names,\n", - " class_names=cancer.target_names,\n", - " rounded=True,\n", - " filled=True\n", - ")\n", - "cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'\n", - "os.system(cmd)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.datasets import make_moons\n", - "from sklearn.tree import export_graphviz\n", - "from pydot import graph_from_dot_data\n", - "import pandas as pd\n", - "import os\n", - "\n", - "np.random.seed(42)\n", - "X, y = make_moons(n_samples=100, noise=0.25, random_state=53)\n", - "X_train, X_test, y_train, y_test = train_test_split(X,y,random_state=0)\n", - "tree_clf = DecisionTreeClassifier(max_depth=5)\n", - "tree_clf.fit(X_train, y_train)\n", - "\n", - "export_graphviz(\n", - " tree_clf,\n", - " out_file=\"DataFiles/moons.dot\",\n", - " rounded=True,\n", - " filled=True\n", - ")\n", - "cmd = 'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'\n", - "os.system(cmd)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Other ways of visualizing the trees\n", - "\n", - "**Scikit-Learn** has also another way to visualize the trees which is very useful, here with the Iris data." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.datasets import load_iris\n", - "from sklearn import tree\n", - "X, y = load_iris(return_X_y=True)\n", - "tree_clf = tree.DecisionTreeClassifier()\n", - "tree_clf = tree_clf.fit(X, y)\n", - "# and then plot the tree\n", - "tree.plot_tree(tree_clf)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Alternatively, the tree can also be exported in textual format with the function exporttext.\n", - "This method doesn’t require the installation of external libraries and is more compact:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.datasets import load_iris\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.tree import export_text\n", - "iris = load_iris()\n", - "decision_tree = DecisionTreeClassifier(random_state=0, max_depth=2)\n", - "decision_tree = decision_tree.fit(iris.data, iris.target)\n", - "r = export_text(decision_tree, feature_names=iris['feature_names'])\n", - "print(r)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Algorithms for Setting up Decision Trees\n", - "\n", - "Two algorithms stand out in the set up of decision trees:\n", - "1. The CART (Classification And Regression Tree) algorithm for both classification and regression\n", - "\n", - "2. The ID3 algorithm based on the computation of the information gain for classification\n", - "\n", - "We discuss both algorithms with applications here. The popular library\n", - "**Scikit-Learn** uses the CART algorithm. For classification problems\n", - "you can use either the **gini** index or the **entropy** to split a tree\n", - "in two branches.\n", - "\n", - "### The CART algorithm for Classification\n", - "\n", - "For classification, the CART algorithm splits the data set in two subsets using a single feature $k$ and a threshold $t_k$.\n", - "This could be for example a threshold set by a number below a certain circumference of a malign tumor.\n", - "\n", - "How do we find these two quantities?\n", - "We search for the pair $(k,t_k)$ that produces the purest subset using for example the **gini** factor $G$.\n", - "The cost function it tries to minimize is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(k,t_k) = \\frac{m_{\\mathrm{left}}}{m}G_{\\mathrm{left}}+ \\frac{m_{\\mathrm{right}}}{m}G_{\\mathrm{right}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $G_{\\mathrm{left/right}}$ measures the impurity of the left/right subset and $m_{\\mathrm{left/right}}$\n", - " is the number of instances in the left/right subset\n", - "\n", - "Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets\n", - "and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the\n", - "$max\\_depth$ hyperparameter), or if it cannot find a split that will reduce impurity. A few other\n", - "hyperparameters control additional stopping conditions such as the $min\\_samples\\_split$,\n", - "$min\\_samples\\_leaf$, $min\\_weight\\_fraction\\_leaf$, and $max\\_leaf\\_nodes$.\n", - "\n", - "\n", - "### The CART algorithm for Regression\n", - "\n", - "The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the\n", - "training set in a way that minimizes say the **gini** or **entropy** impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(k,t_k) = \\frac{m_{\\mathrm{left}}}{m}\\mathrm{MSE}_{\\mathrm{left}}+ \\frac{m_{\\mathrm{right}}}{m}\\mathrm{MSE}_{\\mathrm{right}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here the MSE for a specific node is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{MSE}_{\\mathrm{node}}=\\frac{1}{m_\\mathrm{node}}\\sum_{i\\in \\mathrm{node}}(\\overline{y}_{\\mathrm{node}}-y_i)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\overline{y}_{\\mathrm{node}}=\\frac{1}{m_\\mathrm{node}}\\sum_{i\\in \\mathrm{node}}y_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "the mean value of all observations in a specific node.\n", - "\n", - "Without any regularization, the regression task for decision trees, \n", - "just like for classification tasks, is prone to overfitting.\n", - "\n", - "\n", - "\n", - "### Computing the Gini index\n", - "\n", - "The example we will look at is a classical one in many Machine\n", - "Learning applications. Based on various meteorological features, we\n", - "have several so-called attributes which decide whether we at the end\n", - "will do some outdoor activity like skiing, going for a bike ride etc\n", - "etc. The table here contains the feautures **outlook**, **temperature**,\n", - "**humidity** and **wind**. The target or output is whether we ride\n", - "(True=1) or whether we do something else that day (False=0). The\n", - "attributes for each feature are then sunny, overcast and rain for the\n", - "outlook, hot, cold and mild for temperature, high and normal for\n", - "humidity and weak and strong for wind.\n", - "\n", - "The table here summarizes the various attributes and\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
Day Outlook Temperature Humidity Wind Ride
1 Sunny Hot High Weak 0
2 Sunny Hot High Strong 1
3 Overcast Hot High Weak 1
4 Rain Mild High Weak 1
5 Rain Cool Normal Weak 1
6 Rain Cool Normal Strong 0
7 Overcast Cool Normal Strong 1
8 Sunny Mild High Weak 0
9 Sunny Cool Normal Weak 1
10 Rain Mild Normal Weak 1
11 Sunny Mild Normal Strong 1
12 Overcast Mild High Strong 1
13 Overcast Hot Normal Weak 1
14 Rain Mild High Strong 0
\n", - "\n", - "### Simple Python Code to read in Data and perform Classification" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.tree import export_graphviz\n", - "from sklearn.preprocessing import StandardScaler, OneHotEncoder\n", - "from sklearn.compose import ColumnTransformer\n", - "from IPython.display import Image \n", - "from pydot import graph_from_dot_data\n", - "import os\n", - "\n", - "# Where to save the figures and data files\n", - "PROJECT_ROOT_DIR = \"Results\"\n", - "FIGURE_ID = \"Results/FigureFiles\"\n", - "DATA_ID = \"DataFiles/\"\n", - "\n", - "if not os.path.exists(PROJECT_ROOT_DIR):\n", - " os.mkdir(PROJECT_ROOT_DIR)\n", - "\n", - "if not os.path.exists(FIGURE_ID):\n", - " os.makedirs(FIGURE_ID)\n", - "\n", - "if not os.path.exists(DATA_ID):\n", - " os.makedirs(DATA_ID)\n", - "\n", - "def image_path(fig_id):\n", - " return os.path.join(FIGURE_ID, fig_id)\n", - "\n", - "def data_path(dat_id):\n", - " return os.path.join(DATA_ID, dat_id)\n", - "\n", - "def save_fig(fig_id):\n", - " plt.savefig(image_path(fig_id) + \".png\", format='png')\n", - "\n", - "infile = open(data_path(\"rideclass.csv\"),'r')\n", - "\n", - "# Read the experimental data with Pandas\n", - "from IPython.display import display\n", - "ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))\n", - "ridedata = pd.DataFrame(ridedata)\n", - "\n", - "# Features and targets\n", - "X = ridedata.loc[:, ridedata.columns != 'Ride'].values\n", - "y = ridedata.loc[:, ridedata.columns == 'Ride'].values\n", - "\n", - "# Create the encoder.\n", - "encoder = OneHotEncoder(handle_unknown=\"ignore\")\n", - "# Assume for simplicity all features are categorical.\n", - "encoder.fit(X) \n", - "# Apply the encoder.\n", - "X = encoder.transform(X)\n", - "print(X)\n", - "# Then do a Classification tree\n", - "tree_clf = DecisionTreeClassifier(max_depth=2)\n", - "tree_clf.fit(X, y)\n", - "print(\"Train set accuracy with Decision Tree: {:.2f}\".format(tree_clf.score(X,y)))\n", - "#transfer to a decision tree graph\n", - "export_graphviz(\n", - " tree_clf,\n", - " out_file=\"DataFiles/ride.dot\",\n", - " rounded=True,\n", - " filled=True\n", - ")\n", - "cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'\n", - "os.system(cmd)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The above functions (gini, entropy and misclassification error) are\n", - "important components of the so-called CART algorithm. We will discuss\n", - "this algorithm below after we have discussed the information gain\n", - "algorithm ID3.\n", - "\n", - "In the example here we have converted all our attributes into numerical values $0,1,2$ etc." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Split a dataset based on an attribute and an attribute value\n", - "def test_split(index, value, dataset):\n", - "\tleft, right = list(), list()\n", - "\tfor row in dataset:\n", - "\t\tif row[index] < value:\n", - "\t\t\tleft.append(row)\n", - "\t\telse:\n", - "\t\t\tright.append(row)\n", - "\treturn left, right\n", - " \n", - "# Calculate the Gini index for a split dataset\n", - "def gini_index(groups, classes):\n", - "\t# count all samples at split point\n", - "\tn_instances = float(sum([len(group) for group in groups]))\n", - "\t# sum weighted Gini index for each group\n", - "\tgini = 0.0\n", - "\tfor group in groups:\n", - "\t\tsize = float(len(group))\n", - "\t\t# avoid divide by zero\n", - "\t\tif size == 0:\n", - "\t\t\tcontinue\n", - "\t\tscore = 0.0\n", - "\t\t# score the group based on the score for each class\n", - "\t\tfor class_val in classes:\n", - "\t\t\tp = [row[-1] for row in group].count(class_val) / size\n", - "\t\t\tscore += p * p\n", - "\t\t# weight the group score by its relative size\n", - "\t\tgini += (1.0 - score) * (size / n_instances)\n", - "\treturn gini\n", - "\n", - "# Select the best split point for a dataset\n", - "def get_split(dataset):\n", - "\tclass_values = list(set(row[-1] for row in dataset))\n", - "\tb_index, b_value, b_score, b_groups = 999, 999, 999, None\n", - "\tfor index in range(len(dataset[0])-1):\n", - "\t\tfor row in dataset:\n", - "\t\t\tgroups = test_split(index, row[index], dataset)\n", - "\t\t\tgini = gini_index(groups, class_values)\n", - "\t\t\tprint('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))\n", - "\t\t\tif gini < b_score:\n", - "\t\t\t\tb_index, b_value, b_score, b_groups = index, row[index], gini, groups\n", - "\treturn {'index':b_index, 'value':b_value, 'groups':b_groups}\n", - " \n", - "dataset = [[0,0,0,0,0],\n", - " [0,0,0,1,1],\n", - " [1,0,0,0,1],\n", - " [2,1,0,0,1],\n", - " [2,2,1,0,1],\n", - " [2,2,1,1,0],\n", - " [1,2,1,1,1],\n", - " [0,1,0,0,0],\n", - " [0,2,1,0,1],\n", - " [2,1,1,0,1],\n", - " [0,1,1,1,1],\n", - " [1,1,0,1,1],\n", - " [1,0,1,0,1],\n", - " [2,1,0,1,0]]\n", - "\n", - "split = get_split(dataset)\n", - "print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Entropy and the ID3 algorithm\n", - "\n", - "The ID3 algorithm learns decision trees by constructing\n", - "them in a top down way, beginning with the question **which attribute should be tested at the root of the tree**?\n", - "\n", - "1. Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.\n", - "\n", - "2. The best attribute is selected and used as the test at the root node of the tree.\n", - "\n", - "3. A descendant of the root node is then created for each possible value of this attribute.\n", - "\n", - "4. Training examples are sorted to the appropriate descendant node.\n", - "\n", - "5. The entire process is then repeated using the training examples associated with each descendant node to select the best attribute to test at that point in the tree.\n", - "\n", - "6. This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices. \n", - "\n", - "The ID3 algorithm selects which attribute to test at each node in the\n", - "tree.\n", - "\n", - "We would like to select the attribute that is most useful for classifying\n", - "examples.\n", - "\n", - "What is a good quantitative measure of the worth of an attribute?\n", - "\n", - "Information gain measures how well a given attribute separates the\n", - "training examples according to their target classification.\n", - "\n", - "The ID3 algorithm uses this information gain measure to select among the candidate\n", - "attributes at each step while growing the tree.\n", - "\n", - "\n", - "### Cancer Data again now with Decision Trees and other Methods" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.svm import SVC\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "# Logistic Regression\n", - "logreg = LogisticRegression(solver='lbfgs')\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n", - "# Support vector machine\n", - "svm = SVC(gamma='auto', C=100)\n", - "svm.fit(X_train, y_train)\n", - "print(\"Test set accuracy with SVM: {:.2f}\".format(svm.score(X_test,y_test)))\n", - "# Decision Trees\n", - "deep_tree_clf = DecisionTreeClassifier(max_depth=None)\n", - "deep_tree_clf.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Decision Trees: {:.2f}\".format(deep_tree_clf.score(X_test,y_test)))\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Logistic Regression\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "# Support Vector Machine\n", - "svm.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy SVM with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "# Decision Trees\n", - "deep_tree_clf.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy with Decision Trees and scaled data: {:.2f}\".format(deep_tree_clf.score(X_test_scaled,y_test)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Another example, the moons again" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from __future__ import division, print_function, unicode_literals\n", - "\n", - "# Common imports\n", - "import numpy as np\n", - "import os\n", - "\n", - "# to make this notebook's output stable across runs\n", - "np.random.seed(42)\n", - "\n", - "# To plot pretty figures\n", - "import matplotlib\n", - "import matplotlib.pyplot as plt\n", - "from matplotlib.colors import ListedColormap\n", - "plt.rcParams['axes.labelsize'] = 14\n", - "plt.rcParams['xtick.labelsize'] = 12\n", - "plt.rcParams['ytick.labelsize'] = 12\n", - "\n", - "\n", - "from sklearn.svm import SVC\n", - "from sklearn import datasets\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.datasets import make_moons\n", - "from sklearn.tree import export_graphviz\n", - "\n", - "Xm, ym = make_moons(n_samples=100, noise=0.25, random_state=53)\n", - "\n", - "deep_tree_clf1 = DecisionTreeClassifier(random_state=42)\n", - "deep_tree_clf2 = DecisionTreeClassifier(min_samples_leaf=4, random_state=42)\n", - "deep_tree_clf1.fit(Xm, ym)\n", - "deep_tree_clf2.fit(Xm, ym)\n", - "\n", - "\n", - "def plot_decision_boundary(clf, X, y, axes=[0, 7.5, 0, 3], iris=True, legend=False, plot_training=True):\n", - " x1s = np.linspace(axes[0], axes[1], 100)\n", - " x2s = np.linspace(axes[2], axes[3], 100)\n", - " x1, x2 = np.meshgrid(x1s, x2s)\n", - " X_new = np.c_[x1.ravel(), x2.ravel()]\n", - " y_pred = clf.predict(X_new).reshape(x1.shape)\n", - " custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])\n", - " plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)\n", - " if not iris:\n", - " custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])\n", - " plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)\n", - " if plot_training:\n", - " plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\", label=\"Iris-Setosa\")\n", - " plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\", label=\"Iris-Versicolor\")\n", - " plt.plot(X[:, 0][y==2], X[:, 1][y==2], \"g^\", label=\"Iris-Virginica\")\n", - " plt.axis(axes)\n", - " if iris:\n", - " plt.xlabel(\"Petal length\", fontsize=14)\n", - " plt.ylabel(\"Petal width\", fontsize=14)\n", - " else:\n", - " plt.xlabel(r\"$x_1$\", fontsize=18)\n", - " plt.ylabel(r\"$x_2$\", fontsize=18, rotation=0)\n", - " if legend:\n", - " plt.legend(loc=\"lower right\", fontsize=14)\n", - "plt.figure(figsize=(11, 4))\n", - "plt.subplot(121)\n", - "plot_decision_boundary(deep_tree_clf1, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)\n", - "plt.title(\"No restrictions\", fontsize=16)\n", - "plt.subplot(122)\n", - "plot_decision_boundary(deep_tree_clf2, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)\n", - "plt.title(\"min_samples_leaf = {}\".format(deep_tree_clf2.min_samples_leaf), fontsize=14)\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "np.random.seed(6)\n", - "Xs = np.random.rand(100, 2) - 0.5\n", - "ys = (Xs[:, 0] > 0).astype(np.float32) * 2\n", - "\n", - "angle = np.pi/4\n", - "rotation_matrix = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])\n", - "Xsr = Xs.dot(rotation_matrix)\n", - "\n", - "tree_clf_s = DecisionTreeClassifier(random_state=42)\n", - "tree_clf_s.fit(Xs, ys)\n", - "tree_clf_sr = DecisionTreeClassifier(random_state=42)\n", - "tree_clf_sr.fit(Xsr, ys)\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "plt.subplot(121)\n", - "plot_decision_boundary(tree_clf_s, Xs, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)\n", - "plt.subplot(122)\n", - "plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Quadratic training set + noise\n", - "np.random.seed(42)\n", - "m = 200\n", - "X = np.random.rand(m, 1)\n", - "y = 4 * (X - 0.5) ** 2\n", - "y = y + np.random.randn(m, 1) / 10" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.tree import DecisionTreeRegressor\n", - "\n", - "tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)\n", - "tree_reg.fit(X, y)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.tree import DecisionTreeRegressor\n", - "\n", - "tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)\n", - "tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)\n", - "tree_reg1.fit(X, y)\n", - "tree_reg2.fit(X, y)\n", - "\n", - "def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel=\"$y$\"):\n", - " x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)\n", - " y_pred = tree_reg.predict(x1)\n", - " plt.axis(axes)\n", - " plt.xlabel(\"$x_1$\", fontsize=18)\n", - " if ylabel:\n", - " plt.ylabel(ylabel, fontsize=18, rotation=0)\n", - " plt.plot(X, y, \"b.\")\n", - " plt.plot(x1, y_pred, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "plt.subplot(121)\n", - "plot_regression_predictions(tree_reg1, X, y)\n", - "for split, style in ((0.1973, \"k-\"), (0.0917, \"k--\"), (0.7718, \"k--\")):\n", - " plt.plot([split, split], [-0.2, 1], style, linewidth=2)\n", - "plt.text(0.21, 0.65, \"Depth=0\", fontsize=15)\n", - "plt.text(0.01, 0.2, \"Depth=1\", fontsize=13)\n", - "plt.text(0.65, 0.8, \"Depth=1\", fontsize=13)\n", - "plt.legend(loc=\"upper center\", fontsize=18)\n", - "plt.title(\"max_depth=2\", fontsize=14)\n", - "\n", - "plt.subplot(122)\n", - "plot_regression_predictions(tree_reg2, X, y, ylabel=None)\n", - "for split, style in ((0.1973, \"k-\"), (0.0917, \"k--\"), (0.7718, \"k--\")):\n", - " plt.plot([split, split], [-0.2, 1], style, linewidth=2)\n", - "for split in (0.0458, 0.1298, 0.2873, 0.9040):\n", - " plt.plot([split, split], [-0.2, 1], \"k:\", linewidth=1)\n", - "plt.text(0.3, 0.5, \"Depth=2\", fontsize=13)\n", - "plt.title(\"max_depth=3\", fontsize=14)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "tree_reg1 = DecisionTreeRegressor(random_state=42)\n", - "tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)\n", - "tree_reg1.fit(X, y)\n", - "tree_reg2.fit(X, y)\n", - "\n", - "x1 = np.linspace(0, 1, 500).reshape(-1, 1)\n", - "y_pred1 = tree_reg1.predict(x1)\n", - "y_pred2 = tree_reg2.predict(x1)\n", - "\n", - "plt.figure(figsize=(11, 4))\n", - "\n", - "plt.subplot(121)\n", - "plt.plot(X, y, \"b.\")\n", - "plt.plot(x1, y_pred1, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n", - "plt.axis([0, 1, -0.2, 1.1])\n", - "plt.xlabel(\"$x_1$\", fontsize=18)\n", - "plt.ylabel(\"$y$\", fontsize=18, rotation=0)\n", - "plt.legend(loc=\"upper center\", fontsize=18)\n", - "plt.title(\"No restrictions\", fontsize=14)\n", - "\n", - "plt.subplot(122)\n", - "plt.plot(X, y, \"b.\")\n", - "plt.plot(x1, y_pred2, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n", - "plt.axis([0, 1, -0.2, 1.1])\n", - "plt.xlabel(\"$x_1$\", fontsize=18)\n", - "plt.title(\"min_samples_leaf={}\".format(tree_reg2.min_samples_leaf), fontsize=14)\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Pros and cons of trees, pros\n", - "\n", - "* White box, easy to interpret model. Some people believe that decision trees more closely mirror human decision-making than do the regression and classification approaches discussed earlier (think of support vector machines)\n", - "\n", - "* Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!\n", - "\n", - "* No feature normalization needed\n", - "\n", - "* Tree models can handle both continuous and categorical data (Classification and Regression Trees)\n", - "\n", - "* Can model nonlinear relationships\n", - "\n", - "* Can model interactions between the different descriptive features\n", - "\n", - "* Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)\n", - "\n", - "### Disadvantages\n", - "\n", - "* Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches\n", - "\n", - "* If continuous features are used the tree may become quite large and hence less interpretable\n", - "\n", - "* Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented\n", - "\n", - "* Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests\n", - "\n", - "* Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones. \n", - "\n", - "* If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data\n", - "\n", - "* Features with many levels may be preferred over features with less levels since for them it is *more easy* to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain\n", - "\n", - "However, by aggregating many decision trees, using methods like\n", - "bagging, random forests, and boosting, the predictive performance of\n", - "trees can be substantially improved." - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter6.py b/doc/LectureNotes/_build/jupyter_execute/chapter6.py deleted file mode 100644 index 3cb2e71e3..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter6.py +++ /dev/null @@ -1,978 +0,0 @@ -# Decision trees, overarching aims - - -We start here with the most basic algorithm, the so-called decision -tree. With this basic algorithm we can in turn build more complex -networks, spanning from homogeneous and heterogenous forests (bagging, -random forests and more) to one of the most popular supervised -algorithms nowadays, the extreme gradient boosting, or just -XGBoost. But let us start with the simplest possible ingredient. - -Decision trees are supervised learning algorithms used for both, -classification and regression tasks. - - -The main idea of decision trees -is to find those descriptive features which contain the most -**information** regarding the target feature and then split the dataset -along the values of these features such that the target feature values -for the resulting underlying datasets are as pure as possible. - -The descriptive features which reproduce best the target/output features are normally said -to be the most informative ones. The process of finding the **most -informative** feature is done until we accomplish a stopping criteria -where we then finally end up in so called **leaf nodes**. - -## Basics of a tree - -A decision tree is typically divided into a **root node**, the **interior nodes**, -and the final **leaf nodes** or just **leaves**. These entities are then connected by so-called **branches**. - -The leaf nodes -contain the predictions we will make for new query instances presented -to our trained model. This is possible since the model has -learned the underlying structure of the training data and hence can, -given some assumptions, make predictions about the target feature value -(class) of unseen query instances. - - -## General Features - -The overarching approach to decision trees is a top-down approach. - -* A leaf provides the classification of a given instance. - -* A node specifies a test of some attribute of the instance. - -* A branch corresponds to a possible values of an attribute. - -* An instance is classified by starting at the root node of the tree, testing the attribute specified by this node, then moving down the tree branch corresponding to the value of the attribute in the given example. - -This process is then repeated for the subtree rooted at the new -node. - - - -In simplified terms, the process of training a decision tree and -predicting the target features of query instances is as follows: - -1. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature - -2. Train the decision tree model by continuously splitting the target feature along the values of the descriptive features using a measure of information gain during the training process - -3. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the *predictions* we want to make for new query instances - -4. Show query instances to the tree and run down the tree until we arrive at leaf nodes - -Then we are essentially done! - -%matplotlib inline - -import numpy as np -import matplotlib.pyplot as plt -from sklearn.preprocessing import PolynomialFeatures -from sklearn.linear_model import LinearRegression - -steps=250 - -distance=0 -x=0 -distance_list=[] -steps_list=[] -while x - -Day Outlook Temperature Humidity Wind Ride - - - 1 Sunny Hot High Weak 0 - 2 Sunny Hot High Strong 1 - 3 Overcast Hot High Weak 1 - 4 Rain Mild High Weak 1 - 5 Rain Cool Normal Weak 1 - 6 Rain Cool Normal Strong 0 - 7 Overcast Cool Normal Strong 1 - 8 Sunny Mild High Weak 0 - 9 Sunny Cool Normal Weak 1 - 10 Rain Mild Normal Weak 1 - 11 Sunny Mild Normal Strong 1 - 12 Overcast Mild High Strong 1 - 13 Overcast Hot Normal Weak 1 - 14 Rain Mild High Strong 0 - - - -### Simple Python Code to read in Data and perform Classification - -# Common imports -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from sklearn.tree import DecisionTreeClassifier -from sklearn.model_selection import train_test_split -from sklearn.tree import export_graphviz -from sklearn.preprocessing import StandardScaler, OneHotEncoder -from sklearn.compose import ColumnTransformer -from IPython.display import Image -from pydot import graph_from_dot_data -import os - -# Where to save the figures and data files -PROJECT_ROOT_DIR = "Results" -FIGURE_ID = "Results/FigureFiles" -DATA_ID = "DataFiles/" - -if not os.path.exists(PROJECT_ROOT_DIR): - os.mkdir(PROJECT_ROOT_DIR) - -if not os.path.exists(FIGURE_ID): - os.makedirs(FIGURE_ID) - -if not os.path.exists(DATA_ID): - os.makedirs(DATA_ID) - -def image_path(fig_id): - return os.path.join(FIGURE_ID, fig_id) - -def data_path(dat_id): - return os.path.join(DATA_ID, dat_id) - -def save_fig(fig_id): - plt.savefig(image_path(fig_id) + ".png", format='png') - -infile = open(data_path("rideclass.csv"),'r') - -# Read the experimental data with Pandas -from IPython.display import display -ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride')) -ridedata = pd.DataFrame(ridedata) - -# Features and targets -X = ridedata.loc[:, ridedata.columns != 'Ride'].values -y = ridedata.loc[:, ridedata.columns == 'Ride'].values - -# Create the encoder. -encoder = OneHotEncoder(handle_unknown="ignore") -# Assume for simplicity all features are categorical. -encoder.fit(X) -# Apply the encoder. -X = encoder.transform(X) -print(X) -# Then do a Classification tree -tree_clf = DecisionTreeClassifier(max_depth=2) -tree_clf.fit(X, y) -print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y))) -#transfer to a decision tree graph -export_graphviz( - tree_clf, - out_file="DataFiles/ride.dot", - rounded=True, - filled=True -) -cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png' -os.system(cmd) - -The above functions (gini, entropy and misclassification error) are -important components of the so-called CART algorithm. We will discuss -this algorithm below after we have discussed the information gain -algorithm ID3. - -In the example here we have converted all our attributes into numerical values $0,1,2$ etc. - -# Split a dataset based on an attribute and an attribute value -def test_split(index, value, dataset): - left, right = list(), list() - for row in dataset: - if row[index] < value: - left.append(row) - else: - right.append(row) - return left, right - -# Calculate the Gini index for a split dataset -def gini_index(groups, classes): - # count all samples at split point - n_instances = float(sum([len(group) for group in groups])) - # sum weighted Gini index for each group - gini = 0.0 - for group in groups: - size = float(len(group)) - # avoid divide by zero - if size == 0: - continue - score = 0.0 - # score the group based on the score for each class - for class_val in classes: - p = [row[-1] for row in group].count(class_val) / size - score += p * p - # weight the group score by its relative size - gini += (1.0 - score) * (size / n_instances) - return gini - -# Select the best split point for a dataset -def get_split(dataset): - class_values = list(set(row[-1] for row in dataset)) - b_index, b_value, b_score, b_groups = 999, 999, 999, None - for index in range(len(dataset[0])-1): - for row in dataset: - groups = test_split(index, row[index], dataset) - gini = gini_index(groups, class_values) - print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini)) - if gini < b_score: - b_index, b_value, b_score, b_groups = index, row[index], gini, groups - return {'index':b_index, 'value':b_value, 'groups':b_groups} - -dataset = [[0,0,0,0,0], - [0,0,0,1,1], - [1,0,0,0,1], - [2,1,0,0,1], - [2,2,1,0,1], - [2,2,1,1,0], - [1,2,1,1,1], - [0,1,0,0,0], - [0,2,1,0,1], - [2,1,1,0,1], - [0,1,1,1,1], - [1,1,0,1,1], - [1,0,1,0,1], - [2,1,0,1,0]] - -split = get_split(dataset) -print('Split: [X%d < %.3f]' % ((split['index']+1), split['value'])) - -## Entropy and the ID3 algorithm - -The ID3 algorithm learns decision trees by constructing -them in a top down way, beginning with the question **which attribute should be tested at the root of the tree**? - -1. Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples. - -2. The best attribute is selected and used as the test at the root node of the tree. - -3. A descendant of the root node is then created for each possible value of this attribute. - -4. Training examples are sorted to the appropriate descendant node. - -5. The entire process is then repeated using the training examples associated with each descendant node to select the best attribute to test at that point in the tree. - -6. This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices. - -The ID3 algorithm selects which attribute to test at each node in the -tree. - -We would like to select the attribute that is most useful for classifying -examples. - -What is a good quantitative measure of the worth of an attribute? - -Information gain measures how well a given attribute separates the -training examples according to their target classification. - -The ID3 algorithm uses this information gain measure to select among the candidate -attributes at each step while growing the tree. - - -### Cancer Data again now with Decision Trees and other Methods - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.datasets import load_breast_cancer -from sklearn.svm import SVC -from sklearn.linear_model import LogisticRegression -from sklearn.tree import DecisionTreeClassifier - -# Load the data -cancer = load_breast_cancer() - -X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) -print(X_train.shape) -print(X_test.shape) -# Logistic Regression -logreg = LogisticRegression(solver='lbfgs') -logreg.fit(X_train, y_train) -print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test))) -# Support vector machine -svm = SVC(gamma='auto', C=100) -svm.fit(X_train, y_train) -print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test))) -# Decision Trees -deep_tree_clf = DecisionTreeClassifier(max_depth=None) -deep_tree_clf.fit(X_train, y_train) -print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test))) -#now scale the data -from sklearn.preprocessing import StandardScaler -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) -# Logistic Regression -logreg.fit(X_train_scaled, y_train) -print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) -# Support Vector Machine -svm.fit(X_train_scaled, y_train) -print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) -# Decision Trees -deep_tree_clf.fit(X_train_scaled, y_train) -print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test))) - -### Another example, the moons again - -from __future__ import division, print_function, unicode_literals - -# Common imports -import numpy as np -import os - -# to make this notebook's output stable across runs -np.random.seed(42) - -# To plot pretty figures -import matplotlib -import matplotlib.pyplot as plt -from matplotlib.colors import ListedColormap -plt.rcParams['axes.labelsize'] = 14 -plt.rcParams['xtick.labelsize'] = 12 -plt.rcParams['ytick.labelsize'] = 12 - - -from sklearn.svm import SVC -from sklearn import datasets -from sklearn.tree import DecisionTreeClassifier -from sklearn.datasets import make_moons -from sklearn.tree import export_graphviz - -Xm, ym = make_moons(n_samples=100, noise=0.25, random_state=53) - -deep_tree_clf1 = DecisionTreeClassifier(random_state=42) -deep_tree_clf2 = DecisionTreeClassifier(min_samples_leaf=4, random_state=42) -deep_tree_clf1.fit(Xm, ym) -deep_tree_clf2.fit(Xm, ym) - - -def plot_decision_boundary(clf, X, y, axes=[0, 7.5, 0, 3], iris=True, legend=False, plot_training=True): - x1s = np.linspace(axes[0], axes[1], 100) - x2s = np.linspace(axes[2], axes[3], 100) - x1, x2 = np.meshgrid(x1s, x2s) - X_new = np.c_[x1.ravel(), x2.ravel()] - y_pred = clf.predict(X_new).reshape(x1.shape) - custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0']) - plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap) - if not iris: - custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50']) - plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8) - if plot_training: - plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", label="Iris-Setosa") - plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", label="Iris-Versicolor") - plt.plot(X[:, 0][y==2], X[:, 1][y==2], "g^", label="Iris-Virginica") - plt.axis(axes) - if iris: - plt.xlabel("Petal length", fontsize=14) - plt.ylabel("Petal width", fontsize=14) - else: - plt.xlabel(r"$x_1$", fontsize=18) - plt.ylabel(r"$x_2$", fontsize=18, rotation=0) - if legend: - plt.legend(loc="lower right", fontsize=14) -plt.figure(figsize=(11, 4)) -plt.subplot(121) -plot_decision_boundary(deep_tree_clf1, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False) -plt.title("No restrictions", fontsize=16) -plt.subplot(122) -plot_decision_boundary(deep_tree_clf2, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False) -plt.title("min_samples_leaf = {}".format(deep_tree_clf2.min_samples_leaf), fontsize=14) -plt.show() - -np.random.seed(6) -Xs = np.random.rand(100, 2) - 0.5 -ys = (Xs[:, 0] > 0).astype(np.float32) * 2 - -angle = np.pi/4 -rotation_matrix = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]]) -Xsr = Xs.dot(rotation_matrix) - -tree_clf_s = DecisionTreeClassifier(random_state=42) -tree_clf_s.fit(Xs, ys) -tree_clf_sr = DecisionTreeClassifier(random_state=42) -tree_clf_sr.fit(Xsr, ys) - -plt.figure(figsize=(11, 4)) -plt.subplot(121) -plot_decision_boundary(tree_clf_s, Xs, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False) -plt.subplot(122) -plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False) - -plt.show() - -# Quadratic training set + noise -np.random.seed(42) -m = 200 -X = np.random.rand(m, 1) -y = 4 * (X - 0.5) ** 2 -y = y + np.random.randn(m, 1) / 10 - -from sklearn.tree import DecisionTreeRegressor - -tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42) -tree_reg.fit(X, y) - -from sklearn.tree import DecisionTreeRegressor - -tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2) -tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3) -tree_reg1.fit(X, y) -tree_reg2.fit(X, y) - -def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel="$y$"): - x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1) - y_pred = tree_reg.predict(x1) - plt.axis(axes) - plt.xlabel("$x_1$", fontsize=18) - if ylabel: - plt.ylabel(ylabel, fontsize=18, rotation=0) - plt.plot(X, y, "b.") - plt.plot(x1, y_pred, "r.-", linewidth=2, label=r"$\hat{y}$") - -plt.figure(figsize=(11, 4)) -plt.subplot(121) -plot_regression_predictions(tree_reg1, X, y) -for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")): - plt.plot([split, split], [-0.2, 1], style, linewidth=2) -plt.text(0.21, 0.65, "Depth=0", fontsize=15) -plt.text(0.01, 0.2, "Depth=1", fontsize=13) -plt.text(0.65, 0.8, "Depth=1", fontsize=13) -plt.legend(loc="upper center", fontsize=18) -plt.title("max_depth=2", fontsize=14) - -plt.subplot(122) -plot_regression_predictions(tree_reg2, X, y, ylabel=None) -for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")): - plt.plot([split, split], [-0.2, 1], style, linewidth=2) -for split in (0.0458, 0.1298, 0.2873, 0.9040): - plt.plot([split, split], [-0.2, 1], "k:", linewidth=1) -plt.text(0.3, 0.5, "Depth=2", fontsize=13) -plt.title("max_depth=3", fontsize=14) - -plt.show() - -tree_reg1 = DecisionTreeRegressor(random_state=42) -tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10) -tree_reg1.fit(X, y) -tree_reg2.fit(X, y) - -x1 = np.linspace(0, 1, 500).reshape(-1, 1) -y_pred1 = tree_reg1.predict(x1) -y_pred2 = tree_reg2.predict(x1) - -plt.figure(figsize=(11, 4)) - -plt.subplot(121) -plt.plot(X, y, "b.") -plt.plot(x1, y_pred1, "r.-", linewidth=2, label=r"$\hat{y}$") -plt.axis([0, 1, -0.2, 1.1]) -plt.xlabel("$x_1$", fontsize=18) -plt.ylabel("$y$", fontsize=18, rotation=0) -plt.legend(loc="upper center", fontsize=18) -plt.title("No restrictions", fontsize=14) - -plt.subplot(122) -plt.plot(X, y, "b.") -plt.plot(x1, y_pred2, "r.-", linewidth=2, label=r"$\hat{y}$") -plt.axis([0, 1, -0.2, 1.1]) -plt.xlabel("$x_1$", fontsize=18) -plt.title("min_samples_leaf={}".format(tree_reg2.min_samples_leaf), fontsize=14) - -plt.show() - -## Pros and cons of trees, pros - -* White box, easy to interpret model. Some people believe that decision trees more closely mirror human decision-making than do the regression and classification approaches discussed earlier (think of support vector machines) - -* Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression! - -* No feature normalization needed - -* Tree models can handle both continuous and categorical data (Classification and Regression Trees) - -* Can model nonlinear relationships - -* Can model interactions between the different descriptive features - -* Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small) - -### Disadvantages - -* Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches - -* If continuous features are used the tree may become quite large and hence less interpretable - -* Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented - -* Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests - -* Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones. - -* If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data - -* Features with many levels may be preferred over features with less levels since for them it is *more easy* to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain - -However, by aggregating many decision trees, using methods like -bagging, random forests, and boosting, the predictive performance of -trees can be substantially improved. \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter6_1_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter6_1_1.png deleted file mode 100644 index 925a057adba2aeef6c9ae67eba98907eb353682a..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 26987 zcmZ^Kby!qixHX|Pg3^K@C|wHDJ%n^iH_{!_DGdSwBS?2icZZ~O=g{5Vc{jiNeD}Xk z;SqJ_%sG3%vDSLm4w0ArjEVLd4Gs#|HaiL zfs3HpimN-o!J*?keZ#{gBoV^FnNCZH2r0WH?I%0CD2oyVElM!^<8nU}ciMjIzaZkK z6dIftjDr6RPe>FO{kt|Ub|(T28igbtcBh<3f+LFd4HG;9Ji^RR9p`3&gM)I^Z7cHl zOV^XHm$7vw2UIOQ?i>AF5Xlfox;#zq3)u*ELGS}EW!NjP07yEGA=ilkj$mF=ci4=sG79;MKQ{MgSUQat^oSnKUlb09(ZEg=rA!b- 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b/doc/LectureNotes/_build/jupyter_execute/chapter7.ipynb deleted file mode 100644 index cc88d8ca2..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter7.ipynb +++ /dev/null @@ -1,1605 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods\n", - "\n", - "As stated previously and seen in many of the examples discussed in the previous chapter about\n", - "a single decision tree, we often end up overfitting our training\n", - "data. This normally means that we have a high variance. Can we reduce\n", - "the variance of a statistical learning method?\n", - "\n", - "This leads us to a set of different methods that can combine different\n", - "machine learning algorithms or just use one of them to construct\n", - "forests and jungles of trees, homogeneous ones or heterogenous\n", - "ones. These methods are recognized by different names which we will\n", - "try to explain here. These are\n", - "\n", - "1. Voting classifiers\n", - "\n", - "2. Bagging and Pasting\n", - "\n", - "3. Random forests\n", - "\n", - "4. Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)\n", - "\n", - "We discuss these methods here.\n", - "\n", - "### An Overview of Ensemble Methods\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Bagging\n", - "\n", - "The **plain** decision trees suffer from high\n", - "variance. This means that if we split the training data into two parts\n", - "at random, and fit a decision tree to both halves, the results that we\n", - "get could be quite different. In contrast, a procedure with low\n", - "variance will yield similar results if applied repeatedly to distinct\n", - "data sets; linear regression tends to have low variance, if the ratio\n", - "of $n$ to $p$ is moderately large. \n", - "\n", - "**Bootstrap aggregation**, or just **bagging**, is a\n", - "general-purpose procedure for reducing the variance of a statistical\n", - "learning method. \n", - "\n", - "\n", - "Bagging typically results in improved accuracy\n", - "over prediction using a single tree. Unfortunately, however, it can be\n", - "difficult to interpret the resulting model. Recall that one of the\n", - "advantages of decision trees is the attractive and easily interpreted\n", - "diagram that results.\n", - "\n", - "However, when we bag a large number of trees, it is no longer\n", - "possible to represent the resulting statistical learning procedure\n", - "using a single tree, and it is no longer clear which variables are\n", - "most important to the procedure. Thus, bagging improves prediction\n", - "accuracy at the expense of interpretability. Although the collection\n", - "of bagged trees is much more difficult to interpret than a single\n", - "tree, one can obtain an overall summary of the importance of each\n", - "predictor using the MSE (for bagging regression trees) or the Gini\n", - "index (for bagging classification trees). In the case of bagging\n", - "regression trees, we can record the total amount that the MSE is\n", - "decreased due to splits over a given predictor, averaged over all $B$ possible\n", - "trees. A large value indicates an important predictor. Similarly, in\n", - "the context of bagging classification trees, we can add up the total\n", - "amount that the Gini index is decreased by splits over a given\n", - "predictor, averaged over all $B$ trees." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "ename": "NameError", - "evalue": "name 'np' is not defined", - "output_type": "error", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0mheads_proba\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0.51\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mcoin_tosses\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mrand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m10000\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m10\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0mheads_proba\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mastype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mint32\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0mcumulative_heads_ratio\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcumsum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcoin_tosses\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0maxis\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m/\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0marange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m10001\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreshape\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfigure\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfigsize\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m8\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3.5\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcumulative_heads_ratio\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mNameError\u001b[0m: name 'np' is not defined" - ] - } - ], - "source": [ - "heads_proba = 0.51\n", - "coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)\n", - "cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)\n", - "plt.figure(figsize=(8,3.5))\n", - "plt.plot(cumulative_heads_ratio)\n", - "plt.plot([0, 10000], [0.51, 0.51], \"k--\", linewidth=2, label=\"51%\")\n", - "plt.plot([0, 10000], [0.5, 0.5], \"k-\", label=\"50%\")\n", - "plt.xlabel(\"Number of coin tosses\")\n", - "plt.ylabel(\"Heads ratio\")\n", - "plt.legend(loc=\"lower right\")\n", - "plt.axis([0, 10000, 0.42, 0.58])\n", - "save_fig(\"votingsimple\")\n", - "plt.show()" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.model_selection import train_test_split\n", - "from sklearn.datasets import make_moons\n", - "\n", - "X, y = make_moons(n_samples=500, noise=0.30, random_state=42)\n", - "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)\n", - "\n", - "from sklearn.ensemble import RandomForestClassifier\n", - "from sklearn.ensemble import VotingClassifier\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.svm import SVC\n", - "\n", - "log_clf = LogisticRegression(solver=\"liblinear\", random_state=42)\n", - "rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)\n", - "svm_clf = SVC(gamma=\"auto\", random_state=42)\n", - "\n", - "voting_clf = VotingClassifier(\n", - " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n", - " voting='hard')\n", - "\n", - "voting_clf.fit(X_train, y_train)\n", - "\n", - "from sklearn.metrics import accuracy_score\n", - "\n", - "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n", - " clf.fit(X_train, y_train)\n", - " y_pred = clf.predict(X_test)\n", - " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))\n", - "\n", - "log_clf = LogisticRegression(solver=\"liblinear\", random_state=42)\n", - "rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)\n", - "svm_clf = SVC(gamma=\"auto\", probability=True, random_state=42)\n", - "\n", - "voting_clf = VotingClassifier(\n", - " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n", - " voting='soft')\n", - "voting_clf.fit(X_train, y_train)\n", - "\n", - "from sklearn.metrics import accuracy_score\n", - "\n", - "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n", - " clf.fit(X_train, y_train)\n", - " y_pred = clf.predict(X_test)\n", - " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.model_selection import train_test_split\n", - "from sklearn.datasets import make_moons\n", - "\n", - "X, y = make_moons(n_samples=500, noise=0.30, random_state=42)\n", - "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)\n", - "from sklearn.ensemble import RandomForestClassifier\n", - "from sklearn.ensemble import VotingClassifier\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.svm import SVC\n", - "\n", - "log_clf = LogisticRegression(random_state=42)\n", - "rnd_clf = RandomForestClassifier(random_state=42)\n", - "svm_clf = SVC(random_state=42)\n", - "\n", - "voting_clf = VotingClassifier(\n", - " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n", - " voting='hard')\n", - "voting_clf.fit(X_train, y_train)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.metrics import accuracy_score\n", - "\n", - "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n", - " clf.fit(X_train, y_train)\n", - " y_pred = clf.predict(X_test)\n", - " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "log_clf = LogisticRegression(random_state=42)\n", - "rnd_clf = RandomForestClassifier(random_state=42)\n", - "svm_clf = SVC(probability=True, random_state=42)\n", - "\n", - "voting_clf = VotingClassifier(\n", - " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n", - " voting='soft')\n", - "voting_clf.fit(X_train, y_train)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.metrics import accuracy_score\n", - "\n", - "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n", - " clf.fit(X_train, y_train)\n", - " y_pred = clf.predict(X_test)\n", - " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Bagging Examples" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.ensemble import BaggingClassifier\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "\n", - "bag_clf = BaggingClassifier(\n", - " DecisionTreeClassifier(random_state=42), n_estimators=500,\n", - " max_samples=100, bootstrap=True, n_jobs=-1, random_state=42)\n", - "bag_clf.fit(X_train, y_train)\n", - "y_pred = bag_clf.predict(X_test)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.metrics import accuracy_score\n", - "print(accuracy_score(y_test, y_pred))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "tree_clf = DecisionTreeClassifier(random_state=42)\n", - "tree_clf.fit(X_train, y_train)\n", - "y_pred_tree = tree_clf.predict(X_test)\n", - "print(accuracy_score(y_test, y_pred_tree))" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "from matplotlib.colors import ListedColormap\n", - "\n", - "def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True):\n", - " x1s = np.linspace(axes[0], axes[1], 100)\n", - " x2s = np.linspace(axes[2], axes[3], 100)\n", - " x1, x2 = np.meshgrid(x1s, x2s)\n", - " X_new = np.c_[x1.ravel(), x2.ravel()]\n", - " y_pred = clf.predict(X_new).reshape(x1.shape)\n", - " custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])\n", - " plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)\n", - " if contour:\n", - " custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])\n", - " plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)\n", - " plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\", alpha=alpha)\n", - " plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\", alpha=alpha)\n", - " plt.axis(axes)\n", - " plt.xlabel(r\"$x_1$\", fontsize=18)\n", - " plt.ylabel(r\"$x_2$\", fontsize=18, rotation=0)\n", - "plt.figure(figsize=(11,4))\n", - "plt.subplot(121)\n", - "plot_decision_boundary(tree_clf, X, y)\n", - "plt.title(\"Decision Tree\", fontsize=14)\n", - "plt.subplot(122)\n", - "plot_decision_boundary(bag_clf, X, y)\n", - "plt.title(\"Decision Trees with Bagging\", fontsize=14)\n", - "save_fig(\"baggingtree\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Making your own Bootstrap: Changing the Level of the Decision Tree\n", - "\n", - "Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with\n", - "a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points $n$)." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.pipeline import make_pipeline\n", - "from sklearn.utils import resample\n", - "from sklearn.tree import DecisionTreeRegressor\n", - "\n", - "n = 100\n", - "n_boostraps = 100\n", - "maxdepth = 8\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "error = np.zeros(maxdepth)\n", - "bias = np.zeros(maxdepth)\n", - "variance = np.zeros(maxdepth)\n", - "polydegree = np.zeros(maxdepth)\n", - "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "# we produce a simple tree first as benchmark\n", - "simpletree = DecisionTreeRegressor(max_depth=3) \n", - "simpletree.fit(X_train_scaled, y_train)\n", - "simpleprediction = simpletree.predict(X_test_scaled)\n", - "for degree in range(1,maxdepth):\n", - " model = DecisionTreeRegressor(max_depth=degree) \n", - " y_pred = np.empty((y_test.shape[0], n_boostraps))\n", - " for i in range(n_boostraps):\n", - " x_, y_ = resample(X_train_scaled, y_train)\n", - " model.fit(x_, y_)\n", - " y_pred[:, i] = model.predict(X_test_scaled)#.ravel()\n", - "\n", - " polydegree[degree] = degree\n", - " error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n", - " bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n", - " variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n", - " print('Polynomial degree:', degree)\n", - " print('Error:', error[degree])\n", - " print('Bias^2:', bias[degree])\n", - " print('Var:', variance[degree])\n", - " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", - " \n", - "mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2)\n", - "print(mse_simpletree)\n", - "plt.xlim(1,maxdepth)\n", - "plt.plot(polydegree, error, label='MSE')\n", - "plt.plot(polydegree, bias, label='bias')\n", - "plt.plot(polydegree, variance, label='Variance')\n", - "plt.legend()\n", - "save_fig(\"baggingboot\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random forests\n", - "\n", - "Random forests provide an improvement over bagged trees by way of a\n", - "small tweak that decorrelates the trees. \n", - "\n", - "As in bagging, we build a\n", - "number of decision trees on bootstrapped training samples. But when\n", - "building these decision trees, each time a split in a tree is\n", - "considered, a random sample of $m$ predictors is chosen as split\n", - "candidates from the full set of $p$ predictors. The split is allowed to\n", - "use only one of those $m$ predictors. \n", - "\n", - "A fresh sample of $m$ predictors is\n", - "taken at each split, and typically we choose" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "m\\approx \\sqrt{p}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In building a random forest, at\n", - "each split in the tree, the algorithm is not even allowed to consider\n", - "a majority of the available predictors. \n", - "\n", - "The reason for this is rather clever. Suppose that there is one very\n", - "strong predictor in the data set, along with a number of other\n", - "moderately strong predictors. Then in the collection of bagged\n", - "variable importance random forest trees, most or all of the trees will\n", - "use this strong predictor in the top split. Consequently, all of the\n", - "bagged trees will look quite similar to each other. Hence the\n", - "predictions from the bagged trees will be highly correlated.\n", - "Unfortunately, averaging many highly correlated quantities does not\n", - "lead to as large of a reduction in variance as averaging many\n", - "uncorrelated quantities. In particular, this means that bagging will\n", - "not lead to a substantial reduction in variance over a single tree in\n", - "this setting.\n", - "\n", - "\n", - "The algorithm described here can be applied to both classification and regression problems.\n", - "\n", - "We will grow of forest of say $B$ trees.\n", - "1. For $b=1:B$\n", - "\n", - " * Draw a bootstrap sample from the training data organized in our $\\boldsymbol{X}$ matrix.\n", - "\n", - " * We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached\n", - "\n", - "1. we select $m \\le p$ variables at random from the $p$ predictors/features\n", - "\n", - "2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node\n", - "\n", - "3. split the node into daughter nodes\n", - "\n", - "\n", - "\n", - "4. Output then the ensemble of trees $\\{T_b\\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.svm import SVC\n", - "from sklearn.linear_model import LogisticRegression\n", - "from sklearn.tree import DecisionTreeClassifier\n", - "from sklearn.ensemble import BaggingClassifier\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "# Logistic Regression\n", - "logreg = LogisticRegression(solver='lbfgs')\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Logistic Regression: {:.2f}\".format(logreg.score(X_test,y_test)))\n", - "# Support vector machine\n", - "svm = SVC(gamma='auto', C=100)\n", - "svm.fit(X_train, y_train)\n", - "print(\"Test set accuracy with SVM: {:.2f}\".format(svm.score(X_test,y_test)))\n", - "# Decision Trees\n", - "deep_tree_clf = DecisionTreeClassifier(max_depth=None)\n", - "deep_tree_clf.fit(X_train, y_train)\n", - "print(\"Test set accuracy with Decision Trees: {:.2f}\".format(deep_tree_clf.score(X_test,y_test)))\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Logistic Regression\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy Logistic Regression with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "# Support Vector Machine\n", - "svm.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy SVM with scaled data: {:.2f}\".format(logreg.score(X_test_scaled,y_test)))\n", - "# Decision Trees\n", - "deep_tree_clf.fit(X_train_scaled, y_train)\n", - "print(\"Test set accuracy with Decision Trees and scaled data: {:.2f}\".format(deep_tree_clf.score(X_test_scaled,y_test)))\n", - "\n", - "\n", - "from sklearn.ensemble import RandomForestClassifier\n", - "from sklearn.preprocessing import LabelEncoder\n", - "from sklearn.model_selection import cross_validate\n", - "# Data set not specificied\n", - "#Instantiate the model with 500 trees and entropy as splitting criteria\n", - "Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion=\"entropy\")\n", - "Random_Forest_model.fit(X_train_scaled, y_train)\n", - "#Cross validation\n", - "accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']\n", - "print(accuracy)\n", - "print(\"Test set accuracy with Random Forests and scaled data: {:.2f}\".format(Random_Forest_model.score(X_test_scaled,y_test)))\n", - "\n", - "\n", - "import scikitplot as skplt\n", - "y_pred = Random_Forest_model.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "plt.show()\n", - "y_probas = Random_Forest_model.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Recall that the cumulative gains curve shows the percentage of the\n", - "overall number of cases in a given category *gained* by targeting a\n", - "percentage of the total number of cases.\n", - "\n", - "Similarly, the receiver operating characteristic curve, or ROC curve,\n", - "displays the diagnostic ability of a binary classifier system as its\n", - "discrimination threshold is varied. It plots the true positive rate against the false positive rate.\n", - "\n", - "\n", - "### Compare Bagging on Trees with Random Forests" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "bag_clf = BaggingClassifier(\n", - " DecisionTreeClassifier(splitter=\"random\", max_leaf_nodes=16, random_state=42),\n", - " n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "bag_clf.fit(X_train, y_train)\n", - "y_pred = bag_clf.predict(X_test)\n", - "from sklearn.ensemble import RandomForestClassifier\n", - "rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)\n", - "rnd_clf.fit(X_train, y_train)\n", - "y_pred_rf = rnd_clf.predict(X_test)\n", - "np.sum(y_pred == y_pred_rf) / len(y_pred)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Boosting, a Bird's Eye View\n", - "\n", - "The basic idea is to combine weak classifiers in order to create a good\n", - "classifier. With a weak classifier we often intend a classifier which\n", - "produces results which are only slightly better than we would get by\n", - "random guesses.\n", - "\n", - "This is done by applying in an iterative way a weak (or a standard\n", - "classifier like decision trees) to modify the data. In each iteration\n", - "we emphasize those observations which are misclassified by weighting\n", - "them with a factor.\n", - "\n", - "\n", - "\n", - "Boosting is a way of fitting an additive expansion in a set of\n", - "elementary basis functions like for example some simple polynomials.\n", - "Assume for example that we have a function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\beta_m$ are the expansion parameters to be determined in a\n", - "minimization process and $b(x;\\gamma_m)$ are some simple functions of\n", - "the multivariable parameter $x$ which is characterized by the\n", - "parameters $\\gamma_m$.\n", - "\n", - "As an example, consider the Sigmoid function we used in logistic\n", - "regression. In that case, we can translate the function\n", - "$b(x;\\gamma_m)$ into the Sigmoid function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma(t) = \\frac{1}{1+\\exp{(-t)}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $t=\\gamma_0+\\gamma_1 x$ and the parameters $\\gamma_0$ and\n", - "$\\gamma_1$ were determined by the Logistic Regression fitting\n", - "algorithm.\n", - "\n", - "As another example, consider the cost function we defined for linear regression" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case the function $f(x)$ was replaced by the design matrix\n", - "$\\boldsymbol{X}$ and the unknown linear regression parameters $\\boldsymbol{\\beta}$,\n", - "that is $\\boldsymbol{f}=\\boldsymbol{X}\\boldsymbol{\\beta}$. In linear regression we can \n", - "simply invert a matrix and obtain the parameters $\\beta$ by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\beta}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\\beta_m$ and $\\gamma_m$.\n", - "\n", - "\n", - "### Iterative Fitting, Regression and Squared-error Cost Function\n", - "\n", - "The way we proceed is as follows (here we specialize to the squared-error cost function)\n", - "\n", - "1. Establish a cost function, here $\\cal{C}(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ with $f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m)$.\n", - "\n", - "2. Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers.\n", - "\n", - "3. For $m=1:M$\n", - "\n", - "a. minimize $\\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta b(x;\\gamma))^2$ wrt $\\gamma$ and $\\beta$\n", - "\n", - "b. This gives the optimal values $\\beta_m$ and $\\gamma_m$\n", - "\n", - "c. Determine then the new values $f_m(x)=f_{m-1}(x) +\\beta_m b(x;\\gamma_m)$\n", - "\n", - "\n", - "We could use any of the algorithms we have discussed till now. If we\n", - "use trees, $\\gamma$ parameterizes the split variables and split points\n", - "at the internal nodes, and the predictions at the terminal nodes.\n", - "\n", - "\n", - "\n", - "To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.\n", - "\n", - "For simplicity we assume also that our functions $b(x;\\gamma)=1+\\gamma x$. \n", - "\n", - "This means that for every iteration $m$, we need to optimize" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\beta_m,\\gamma_m) = \\mathrm{argmin}_{\\beta,\\lambda}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta b(x;\\gamma))^2=\\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\\beta(1+\\gamma x_i))^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We start our iteration by simply setting $f_0(x)=0$. \n", - "Taking the derivatives with respect to $\\beta$ and $\\gamma$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{C}}{\\partial \\beta} = -2\\sum_{i}(1+\\gamma x_i)(y_i-\\beta(1+\\gamma x_i))=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial \\cal{C}}{\\partial \\gamma} =-2\\sum_{i}\\beta x_i(y_i-\\beta(1+\\gamma x_i))=0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can then rewrite these equations as (defining $\\boldsymbol{w}=\\boldsymbol{e}+\\gamma \\boldsymbol{x})$ with $\\boldsymbol{e}$ being the unit vector)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\gamma \\boldsymbol{w}^T(\\boldsymbol{y}-\\beta\\gamma \\boldsymbol{w})=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives us $\\beta = \\boldsymbol{w}^T\\boldsymbol{y}/(\\boldsymbol{w}^T\\boldsymbol{w})$. Similarly we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta\\gamma \\boldsymbol{x}^T(\\boldsymbol{y}-\\beta(1+\\gamma \\boldsymbol{x}))=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to $\\gamma =(\\boldsymbol{x}^T\\boldsymbol{y}-\\beta\\boldsymbol{x}^T\\boldsymbol{e})/(\\beta\\boldsymbol{x}^T\\boldsymbol{x})$. Inserting\n", - "for $\\beta$ gives us an equation for $\\gamma$. This is a non-linear equation in the unknown $\\gamma$ and has to be solved numerically. \n", - "\n", - "The solution to these two equations gives us in turn $\\beta_1$ and $\\gamma_1$ leading to the new expression for $f_1(x)$ as\n", - "$f_1(x) = \\beta_1(1+\\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$. \n", - "\n", - "\n", - "\n", - "### Iterative Fitting, Classification and AdaBoost\n", - "\n", - "Let us consider a binary classification problem with two outcomes $y_i \\in \\{-1,1\\}$ and $i=0,1,2,\\dots,n-1$ as our set of\n", - "observations. We define a classification function $G(x)$ which produces a prediction taking one or the other of the two values \n", - "$\\{-1,1\\}$.\n", - "\n", - "The error rate of the training sample is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{\\overline{err}}=\\frac{1}{n} \\sum_{i=0}^{n-1} I(y_i\\ne G(x_i)).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The iterative procedure starts with defining a weak classifier whose\n", - "error rate is barely better than random guessing. The iterative\n", - "procedure in boosting is to sequentially apply a weak\n", - "classification algorithm to repeatedly modified versions of the data\n", - "producing a sequence of weak classifiers $G_m(x)$.\n", - "\n", - "Here we will express our function $f(x)$ in terms of $G(x)$. That is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "will be a function of" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "G_M(x) = \\mathrm{sign} \\sum_{i=1}^M \\alpha_m G_m(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In our iterative procedure we define thus" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_m(x) = f_{m-1}(x)+\\beta_mG_m(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the\n", - "exponential cost/loss function defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{y},\\boldsymbol{f}) = \\sum_{i=0}^{n-1}\\exp{(-y_i(f_{m-1}(x_i)+\\beta G(x_i))}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We optimize $\\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case.\n", - "This is normally done in two steps. Let us however first rewrite the cost function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{y},\\boldsymbol{f}) = \\sum_{i=0}^{n-1}w_i^{m}\\exp{(-y_i\\beta G(x_i))},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined $w_i^m= \\exp{(-y_if_{m-1}(x_i))}$.\n", - "\n", - "\n", - "\n", - "First, for any $\\beta > 0$, we optimize $G$ by setting" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "G_m(x) = \\mathrm{sign} \\sum_{i=0}^{n-1} w_i^m I(y_i \\ne G_(x_i)),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is the classifier that minimizes the weighted error rate in predicting $y$.\n", - "\n", - "We can do this by rewriting" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\exp{-(\\beta)}\\sum_{y_i=G(x_i)}w_i^m+\\exp{(\\beta)}\\sum_{y_i\\ne G(x_i)}w_i^m,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which can be rewritten as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\exp{(\\beta)}-\\exp{-(\\beta)})\\sum_{i=0}^{n-1}w_i^mI(y_i\\ne G(x_i))+\\exp{(-\\beta)}\\sum_{i=0}^{n-1}w_i^m=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\beta_m = \\frac{1}{2}\\log{\\frac{1-\\mathrm{\\overline{err}}}{\\mathrm{\\overline{err}}}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have redefined the error as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{\\overline{err}}_m=\\frac{1}{n}\\frac{\\sum_{i=0}^{n-1}w_i^mI(y_i\\ne G(x_i)}{\\sum_{i=0}^{n-1}w_i^m},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which leads to an update of" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_m(x) = f_{m-1}(x) +\\beta_m G_m(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This leads to the new weights" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "w_i^{m+1} = w_i^m \\exp{(-y_i\\beta_m G_m(x_i))}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Adaptive boosting: AdaBoost, Basic Algorithm\n", - "\n", - "The algorithm here is rather straightforward. Assume that our weak\n", - "classifier is a decision tree and we consider a binary set of outputs\n", - "with $y_i \\in \\{-1,1\\}$ and $i=0,1,2,\\dots,n-1$ as our set of\n", - "observations. Our design matrix is given in terms of the\n", - "feature/predictor vectors\n", - "$\\boldsymbol{X}=[\\boldsymbol{x}_0\\boldsymbol{x}_1\\dots\\boldsymbol{x}_{p-1}]$. Finally, we define also a\n", - "classifier determined by our data via a function $G(x)$. This function tells us how well we are able to classify our outputs/targets $\\boldsymbol{y}$. \n", - "\n", - "We have already defined the misclassification error $\\mathrm{err}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{err}=\\frac{1}{n}\\sum_{i=0}^{n-1}I(y_i\\ne G(x_i)),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the function $I()$ is one if we misclassify and zero if we classify correctly. \n", - "\n", - "\n", - "With the above definitions we are now ready to set up the algorithm for AdaBoost.\n", - "The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.\n", - "1. We start by initializing all weights to $w_i = 1/n$, with $i=0,1,2,\\dots n-1$. It is easy to see that we must have $\\sum_{i=0}^{n-1}w_i = 1$.\n", - "\n", - "2. We rewrite the misclassification error as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{\\overline{err}}_m=\\frac{\\sum_{i=0}^{n-1}w_i^m I(y_i\\ne G(x_i))}{\\sum_{i=0}^{n-1}w_i},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "1. Then we start looping over all attempts at classifying, namely we start an iterative process for $m=1:M$, where $M$ is the final number of classifications. Our given classifier could for example be a plain decision tree.\n", - "\n", - "a. Fit then a given classifier to the training set using the weights $w_i$.\n", - "\n", - "b. Compute then $\\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly.\n", - "\n", - "c. Define a quantity $\\alpha_{m} = \\log{(1-\\mathrm{\\overline{err}}_m)/\\mathrm{\\overline{err}}_m}$\n", - "\n", - "d. Set the new weights to $w_i = w_i\\times \\exp{(\\alpha_m I(y_i\\ne G(x_i)}$.\n", - "\n", - "\n", - "5. Compute the new classifier $G(x)= \\sum_{i=0}^{n-1}\\alpha_m I(y_i\\ne G(x_i)$.\n", - "\n", - "For the iterations with $m \\le 2$ the weights are modified\n", - "individually at each steps. The observations which were misclassified\n", - "at iteration $m-1$ have a weight which is larger than those which were\n", - "classified properly. As this proceeds, the observations which were\n", - "difficult to classifiy correctly are given a larger influence. Each\n", - "new classification step $m$ is then forced to concentrate on those\n", - "observations that are missed in the previous iterations.\n", - "\n", - "\n", - "\n", - "\n", - "Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.ensemble import AdaBoostClassifier\n", - "\n", - "ada_clf = AdaBoostClassifier(\n", - " DecisionTreeClassifier(max_depth=1), n_estimators=200,\n", - " algorithm=\"SAMME.R\", learning_rate=0.5, random_state=42)\n", - "ada_clf.fit(X_train, y_train)\n", - "\n", - "from sklearn.ensemble import AdaBoostClassifier\n", - "\n", - "ada_clf = AdaBoostClassifier(\n", - " DecisionTreeClassifier(max_depth=1), n_estimators=200,\n", - " algorithm=\"SAMME.R\", learning_rate=0.5, random_state=42)\n", - "ada_clf.fit(X_train_scaled, y_train)\n", - "y_pred = ada_clf.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "plt.show()\n", - "y_probas = ada_clf.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent\n", - "\n", - "Gradient boosting is again a similar technique to Adaptive boosting,\n", - "it combines so-called weak classifiers or regressors into a strong\n", - "method via a series of iterations.\n", - "\n", - "In order to understand the method, let us illustrate its basics by\n", - "bringing back the essential steps in linear regression, where our cost\n", - "function was the least squares function.\n", - "\n", - "\n", - "We start again with our cost function $\\cal{C}(\\boldsymbol{y}m\\boldsymbol{f})=\\sum_{i=0}^{n-1}\\cal{L}(y_i, f(x_i))$ where we want to minimize\n", - "This means that for every iteration, we need to optimize" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\hat{\\boldsymbol{f}}) = \\mathrm{argmin}_{\\boldsymbol{f}}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_M(x) = \\sum_{m=0}^M h_m(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the steepest decent approach we approximate $h_m(x) = -\\rho_m g_m(x)$, where $\\rho_m$ is a scalar and $g_m(x)$ the gradient defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_m(x_i) = \\left[ \\frac{\\partial \\cal{L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{m-1}(x_i)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With the new gradient we can update $f_m(x) = f_{m-1}(x) -\\rho_m g_m(x)$. Using the above squared-error function we see that\n", - "the gradient is $g_m(x_i) = -2(y_i-f(x_i))$.\n", - "\n", - "Choosing $f_0(x)=0$ we obtain $g_m(x) = -2y_i$ and inserting this into the minimization problem for the cost function we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(\\rho_1) = \\mathrm{argmin}_{\\rho}\\hspace{0.1cm} \\sum_{i=0}^{n-1}(y_i+2\\rho y_i)^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Optimizing with respect to $\\rho$ we obtain (taking the derivative) that $\\rho_1 = -1/2$. We have then that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_1(x) = f_{0}(x) -\\rho_1 g_1(x)=-y_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can then proceed and compute" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_2(x_i) = \\left[ \\frac{\\partial \\cal{L}(y_i, f(x_i))}{\\partial f(x_i)}\\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and find a new value for $\\rho_2=-1/2$ and continue till we have reached $m=M$. We can modify the steepest descent method, or steepest boosting, by introducing what is called **gradient boosting**. \n", - "\n", - "\n", - "Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points,\n", - "so we do not learn a function that can generalize. However, we can modify the algorithm by\n", - "fitting a weak learner to approximate the negative gradient signal. \n", - "\n", - "Suppose we have a cost function $C(f)=\\sum_{i=0}^{n-1}L(y_i, f(x_i))$ where $y_i$ is our target and $f(x_i)$ the function which is meant to model $y_i$. The above cost function could be our standard squared-error function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{y},\\boldsymbol{f})=\\sum_{i=0}^{n-1}(y_i-f(x_i))^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The way we proceed in an iterative fashion is to\n", - "1. Initialize our estimate $f_0(x)$.\n", - "\n", - "2. For $m=1:M$, we\n", - "\n", - "a. compute the negative gradient vector $\\boldsymbol{u}_m = -\\partial C(\\boldsymbol{y},\\boldsymbol{f})/\\partial \\boldsymbol{f}(x)$ at $f(x) = f_{m-1}(x)$;\n", - "\n", - "b. fit the so-called base-learner to the negative gradient $h_m(u_m,x)$;\n", - "\n", - "c. update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$;\n", - "\n", - "\n", - "4. The final estimate is then $f_M(x) = \\sum_{m=1}^M h_m(u_m,x)$.\n", - "\n", - "## Gradient Boosting, Examples of Regression" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split\n", - "from sklearn.ensemble import GradientBoostingRegressor\n", - "from sklearn.preprocessing import StandardScaler\n", - "import scikitplot as skplt\n", - "from sklearn.metrics import mean_squared_error\n", - "\n", - "n = 100\n", - "maxdegree = 6\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "\n", - "error = np.zeros(maxdegree)\n", - "bias = np.zeros(maxdegree)\n", - "variance = np.zeros(maxdegree)\n", - "polydegree = np.zeros(maxdegree)\n", - "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "for degree in range(1,maxdegree):\n", - " model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0) \n", - " model.fit(X_train_scaled,y_train)\n", - " y_pred = model.predict(X_test_scaled)\n", - " polydegree[degree] = degree\n", - " error[degree] = np.mean( np.mean((y_test - y_pred)**2) )\n", - " bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )\n", - " variance[degree] = np.mean( np.var(y_pred) )\n", - " print('Max depth:', degree)\n", - " print('Error:', error[degree])\n", - " print('Bias^2:', bias[degree])\n", - " print('Var:', variance[degree])\n", - " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", - "\n", - "plt.xlim(1,maxdegree-1)\n", - "plt.plot(polydegree, error, label='Error')\n", - "plt.plot(polydegree, bias, label='bias')\n", - "plt.plot(polydegree, variance, label='Variance')\n", - "plt.legend()\n", - "save_fig(\"gdregression\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gradient Boosting, Classification Example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "import scikitplot as skplt\n", - "from sklearn.ensemble import GradientBoostingClassifier\n", - "from sklearn.model_selection import cross_validate\n", - "\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0) \n", - "gd_clf.fit(X_train_scaled, y_train)\n", - "#Cross validation\n", - "accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score']\n", - "print(accuracy)\n", - "print(\"Test set accuracy with Random Forests and scaled data: {:.2f}\".format(gd_clf.score(X_test_scaled,y_test)))\n", - "\n", - "import scikitplot as skplt\n", - "y_pred = gd_clf.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "save_fig(\"gdclassiffierconfusion\")\n", - "plt.show()\n", - "y_probas = gd_clf.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "save_fig(\"gdclassiffierroc\")\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "save_fig(\"gdclassiffiercgain\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## XGBoost: Extreme Gradient Boosting\n", - "\n", - "\n", - "[XGBoost](https://github.com/dmlc/xgboost) or Extreme Gradient\n", - "Boosting, is an optimized distributed gradient boosting library\n", - "designed to be highly efficient, flexible and portable. It implements\n", - "machine learning algorithms under the Gradient Boosting\n", - "framework. XGBoost provides a parallel tree boosting that solve many\n", - "data science problems in a fast and accurate way. See the [article by Chen and Guestrin](https://arxiv.org/abs/1603.02754).\n", - "\n", - "The authors design and build a highly scalable end-to-end tree\n", - "boosting system. It has a theoretically justified weighted quantile\n", - "sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.\n", - "\n", - "It is now the algorithm which wins essentially all ML competitions!!!\n", - "\n", - "## Regression Case" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split\n", - "import xgboost as xgb\n", - "from sklearn.preprocessing import StandardScaler\n", - "import scikitplot as skplt\n", - "from sklearn.metrics import mean_squared_error\n", - "\n", - "n = 100\n", - "maxdegree = 6\n", - "\n", - "# Make data set.\n", - "x = np.linspace(-3, 3, n).reshape(-1, 1)\n", - "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n", - "\n", - "error = np.zeros(maxdegree)\n", - "bias = np.zeros(maxdegree)\n", - "variance = np.zeros(maxdegree)\n", - "polydegree = np.zeros(maxdegree)\n", - "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "for degree in range(maxdegree):\n", - " model = xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200)\n", - "\n", - " model.fit(X_train_scaled,y_train)\n", - " y_pred = model.predict(X_test_scaled)\n", - " polydegree[degree] = degree\n", - " error[degree] = np.mean( np.mean((y_test - y_pred)**2) )\n", - " bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 )\n", - " variance[degree] = np.mean( np.var(y_pred) )\n", - " print('Max depth:', degree)\n", - " print('Error:', error[degree])\n", - " print('Bias^2:', bias[degree])\n", - " print('Var:', variance[degree])\n", - " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n", - "\n", - "plt.xlim(1,maxdegree-1)\n", - "plt.plot(polydegree, error, label='Error')\n", - "plt.plot(polydegree, bias, label='bias')\n", - "plt.plot(polydegree, variance, label='Variance')\n", - "plt.legend()\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "\n", - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.preprocessing import LabelEncoder\n", - "from sklearn.model_selection import cross_validate\n", - "import scikitplot as skplt\n", - "import xgboost as xgb\n", - "# Load the data\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "print(X_train.shape)\n", - "print(X_test.shape)\n", - "#now scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "\n", - "xg_clf = xgb.XGBClassifier()\n", - "xg_clf.fit(X_train_scaled,y_train)\n", - "\n", - "y_test = xg_clf.predict(X_test_scaled)\n", - "\n", - "print(\"Test set accuracy with Random Forests and scaled data: {:.2f}\".format(xg_clf.score(X_test_scaled,y_test)))\n", - "\n", - "import scikitplot as skplt\n", - "y_pred = xg_clf.predict(X_test_scaled)\n", - "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n", - "save_fig(\"xdclassiffierconfusion\")\n", - "plt.show()\n", - "y_probas = xg_clf.predict_proba(X_test_scaled)\n", - "skplt.metrics.plot_roc(y_test, y_probas)\n", - "save_fig(\"xdclassiffierroc\")\n", - "plt.show()\n", - "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n", - "save_fig(\"gdclassiffiercgain\")\n", - "plt.show()\n", - "\n", - "\n", - "xgb.plot_tree(xg_clf,num_trees=0)\n", - "plt.rcParams['figure.figsize'] = [50, 10]\n", - "save_fig(\"xgtree\")\n", - "plt.show()\n", - "\n", - "xgb.plot_importance(xg_clf)\n", - "plt.rcParams['figure.figsize'] = [5, 5]\n", - "save_fig(\"xgparams\")\n", - "plt.show()" - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter7.py b/doc/LectureNotes/_build/jupyter_execute/chapter7.py deleted file mode 100644 index 37183ca85..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter7.py +++ /dev/null @@ -1,1012 +0,0 @@ -# Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods - -As stated previously and seen in many of the examples discussed in the previous chapter about -a single decision tree, we often end up overfitting our training -data. This normally means that we have a high variance. Can we reduce -the variance of a statistical learning method? - -This leads us to a set of different methods that can combine different -machine learning algorithms or just use one of them to construct -forests and jungles of trees, homogeneous ones or heterogenous -ones. These methods are recognized by different names which we will -try to explain here. These are - -1. Voting classifiers - -2. Bagging and Pasting - -3. Random forests - -4. Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost) - -We discuss these methods here. - -### An Overview of Ensemble Methods - - - - - -## Bagging - -The **plain** decision trees suffer from high -variance. This means that if we split the training data into two parts -at random, and fit a decision tree to both halves, the results that we -get could be quite different. In contrast, a procedure with low -variance will yield similar results if applied repeatedly to distinct -data sets; linear regression tends to have low variance, if the ratio -of $n$ to $p$ is moderately large. - -**Bootstrap aggregation**, or just **bagging**, is a -general-purpose procedure for reducing the variance of a statistical -learning method. - - -Bagging typically results in improved accuracy -over prediction using a single tree. Unfortunately, however, it can be -difficult to interpret the resulting model. Recall that one of the -advantages of decision trees is the attractive and easily interpreted -diagram that results. - -However, when we bag a large number of trees, it is no longer -possible to represent the resulting statistical learning procedure -using a single tree, and it is no longer clear which variables are -most important to the procedure. Thus, bagging improves prediction -accuracy at the expense of interpretability. Although the collection -of bagged trees is much more difficult to interpret than a single -tree, one can obtain an overall summary of the importance of each -predictor using the MSE (for bagging regression trees) or the Gini -index (for bagging classification trees). In the case of bagging -regression trees, we can record the total amount that the MSE is -decreased due to splits over a given predictor, averaged over all $B$ possible -trees. A large value indicates an important predictor. Similarly, in -the context of bagging classification trees, we can add up the total -amount that the Gini index is decreased by splits over a given -predictor, averaged over all $B$ trees. - -heads_proba = 0.51 -coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32) -cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1) -plt.figure(figsize=(8,3.5)) -plt.plot(cumulative_heads_ratio) -plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%") -plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%") -plt.xlabel("Number of coin tosses") -plt.ylabel("Heads ratio") -plt.legend(loc="lower right") -plt.axis([0, 10000, 0.42, 0.58]) -save_fig("votingsimple") -plt.show() - -from sklearn.model_selection import train_test_split -from sklearn.datasets import make_moons - -X, y = make_moons(n_samples=500, noise=0.30, random_state=42) -X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42) - -from sklearn.ensemble import RandomForestClassifier -from sklearn.ensemble import VotingClassifier -from sklearn.linear_model import LogisticRegression -from sklearn.svm import SVC - -log_clf = LogisticRegression(solver="liblinear", random_state=42) -rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42) -svm_clf = SVC(gamma="auto", random_state=42) - -voting_clf = VotingClassifier( - estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)], - voting='hard') - -voting_clf.fit(X_train, y_train) - -from sklearn.metrics import accuracy_score - -for clf in (log_clf, rnd_clf, svm_clf, voting_clf): - clf.fit(X_train, y_train) - y_pred = clf.predict(X_test) - print(clf.__class__.__name__, accuracy_score(y_test, y_pred)) - -log_clf = LogisticRegression(solver="liblinear", random_state=42) -rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42) -svm_clf = SVC(gamma="auto", probability=True, random_state=42) - -voting_clf = VotingClassifier( - estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)], - voting='soft') -voting_clf.fit(X_train, y_train) - -from sklearn.metrics import accuracy_score - -for clf in (log_clf, rnd_clf, svm_clf, voting_clf): - clf.fit(X_train, y_train) - y_pred = clf.predict(X_test) - print(clf.__class__.__name__, accuracy_score(y_test, y_pred)) - -from sklearn.model_selection import train_test_split -from sklearn.datasets import make_moons - -X, y = make_moons(n_samples=500, noise=0.30, random_state=42) -X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42) -from sklearn.ensemble import RandomForestClassifier -from sklearn.ensemble import VotingClassifier -from sklearn.linear_model import LogisticRegression -from sklearn.svm import SVC - -log_clf = LogisticRegression(random_state=42) -rnd_clf = RandomForestClassifier(random_state=42) -svm_clf = SVC(random_state=42) - -voting_clf = VotingClassifier( - estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)], - voting='hard') -voting_clf.fit(X_train, y_train) - -from sklearn.metrics import accuracy_score - -for clf in (log_clf, rnd_clf, svm_clf, voting_clf): - clf.fit(X_train, y_train) - y_pred = clf.predict(X_test) - print(clf.__class__.__name__, accuracy_score(y_test, y_pred)) - -log_clf = LogisticRegression(random_state=42) -rnd_clf = RandomForestClassifier(random_state=42) -svm_clf = SVC(probability=True, random_state=42) - -voting_clf = VotingClassifier( - estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)], - voting='soft') -voting_clf.fit(X_train, y_train) - -from sklearn.metrics import accuracy_score - -for clf in (log_clf, rnd_clf, svm_clf, voting_clf): - clf.fit(X_train, y_train) - y_pred = clf.predict(X_test) - print(clf.__class__.__name__, accuracy_score(y_test, y_pred)) - -## Bagging Examples - -from sklearn.ensemble import BaggingClassifier -from sklearn.tree import DecisionTreeClassifier - -bag_clf = BaggingClassifier( - DecisionTreeClassifier(random_state=42), n_estimators=500, - max_samples=100, bootstrap=True, n_jobs=-1, random_state=42) -bag_clf.fit(X_train, y_train) -y_pred = bag_clf.predict(X_test) - -from sklearn.metrics import accuracy_score -print(accuracy_score(y_test, y_pred)) - -tree_clf = DecisionTreeClassifier(random_state=42) -tree_clf.fit(X_train, y_train) -y_pred_tree = tree_clf.predict(X_test) -print(accuracy_score(y_test, y_pred_tree)) - -%matplotlib inline - -from matplotlib.colors import ListedColormap - -def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True): - x1s = np.linspace(axes[0], axes[1], 100) - x2s = np.linspace(axes[2], axes[3], 100) - x1, x2 = np.meshgrid(x1s, x2s) - X_new = np.c_[x1.ravel(), x2.ravel()] - y_pred = clf.predict(X_new).reshape(x1.shape) - custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0']) - plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap) - if contour: - custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50']) - plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8) - plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", alpha=alpha) - plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", alpha=alpha) - plt.axis(axes) - plt.xlabel(r"$x_1$", fontsize=18) - plt.ylabel(r"$x_2$", fontsize=18, rotation=0) -plt.figure(figsize=(11,4)) -plt.subplot(121) -plot_decision_boundary(tree_clf, X, y) -plt.title("Decision Tree", fontsize=14) -plt.subplot(122) -plot_decision_boundary(bag_clf, X, y) -plt.title("Decision Trees with Bagging", fontsize=14) -save_fig("baggingtree") -plt.show() - -### Making your own Bootstrap: Changing the Level of the Decision Tree - -Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with -a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points $n$). - - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.pipeline import make_pipeline -from sklearn.utils import resample -from sklearn.tree import DecisionTreeRegressor - -n = 100 -n_boostraps = 100 -maxdepth = 8 - -# Make data set. -x = np.linspace(-3, 3, n).reshape(-1, 1) -y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape) -error = np.zeros(maxdepth) -bias = np.zeros(maxdepth) -variance = np.zeros(maxdepth) -polydegree = np.zeros(maxdepth) -X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2) - -from sklearn.preprocessing import StandardScaler -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) - -# we produce a simple tree first as benchmark -simpletree = DecisionTreeRegressor(max_depth=3) -simpletree.fit(X_train_scaled, y_train) -simpleprediction = simpletree.predict(X_test_scaled) -for degree in range(1,maxdepth): - model = DecisionTreeRegressor(max_depth=degree) - y_pred = np.empty((y_test.shape[0], n_boostraps)) - for i in range(n_boostraps): - x_, y_ = resample(X_train_scaled, y_train) - model.fit(x_, y_) - y_pred[:, i] = model.predict(X_test_scaled)#.ravel() - - polydegree[degree] = degree - error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) ) - bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 ) - variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) ) - print('Polynomial degree:', degree) - print('Error:', error[degree]) - print('Bias^2:', bias[degree]) - print('Var:', variance[degree]) - print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree])) - -mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2) -print(mse_simpletree) -plt.xlim(1,maxdepth) -plt.plot(polydegree, error, label='MSE') -plt.plot(polydegree, bias, label='bias') -plt.plot(polydegree, variance, label='Variance') -plt.legend() -save_fig("baggingboot") -plt.show() - -## Random forests - -Random forests provide an improvement over bagged trees by way of a -small tweak that decorrelates the trees. - -As in bagging, we build a -number of decision trees on bootstrapped training samples. But when -building these decision trees, each time a split in a tree is -considered, a random sample of $m$ predictors is chosen as split -candidates from the full set of $p$ predictors. The split is allowed to -use only one of those $m$ predictors. - -A fresh sample of $m$ predictors is -taken at each split, and typically we choose - -$$ -m\approx \sqrt{p}. -$$ - -In building a random forest, at -each split in the tree, the algorithm is not even allowed to consider -a majority of the available predictors. - -The reason for this is rather clever. Suppose that there is one very -strong predictor in the data set, along with a number of other -moderately strong predictors. Then in the collection of bagged -variable importance random forest trees, most or all of the trees will -use this strong predictor in the top split. Consequently, all of the -bagged trees will look quite similar to each other. Hence the -predictions from the bagged trees will be highly correlated. -Unfortunately, averaging many highly correlated quantities does not -lead to as large of a reduction in variance as averaging many -uncorrelated quantities. In particular, this means that bagging will -not lead to a substantial reduction in variance over a single tree in -this setting. - - -The algorithm described here can be applied to both classification and regression problems. - -We will grow of forest of say $B$ trees. -1. For $b=1:B$ - - * Draw a bootstrap sample from the training data organized in our $\boldsymbol{X}$ matrix. - - * We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached - -1. we select $m \le p$ variables at random from the $p$ predictors/features - -2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node - -3. split the node into daughter nodes - - - -4. Output then the ensemble of trees $\{T_b\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem. - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.datasets import load_breast_cancer -from sklearn.svm import SVC -from sklearn.linear_model import LogisticRegression -from sklearn.tree import DecisionTreeClassifier -from sklearn.ensemble import BaggingClassifier - -# Load the data -cancer = load_breast_cancer() - -X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) -print(X_train.shape) -print(X_test.shape) -# Logistic Regression -logreg = LogisticRegression(solver='lbfgs') -logreg.fit(X_train, y_train) -print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test))) -# Support vector machine -svm = SVC(gamma='auto', C=100) -svm.fit(X_train, y_train) -print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test))) -# Decision Trees -deep_tree_clf = DecisionTreeClassifier(max_depth=None) -deep_tree_clf.fit(X_train, y_train) -print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test))) -#now scale the data -from sklearn.preprocessing import StandardScaler -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) -# Logistic Regression -logreg.fit(X_train_scaled, y_train) -print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) -# Support Vector Machine -svm.fit(X_train_scaled, y_train) -print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test))) -# Decision Trees -deep_tree_clf.fit(X_train_scaled, y_train) -print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test))) - - -from sklearn.ensemble import RandomForestClassifier -from sklearn.preprocessing import LabelEncoder -from sklearn.model_selection import cross_validate -# Data set not specificied -#Instantiate the model with 500 trees and entropy as splitting criteria -Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy") -Random_Forest_model.fit(X_train_scaled, y_train) -#Cross validation -accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score'] -print(accuracy) -print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test))) - - -import scikitplot as skplt -y_pred = Random_Forest_model.predict(X_test_scaled) -skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) -plt.show() -y_probas = Random_Forest_model.predict_proba(X_test_scaled) -skplt.metrics.plot_roc(y_test, y_probas) -plt.show() -skplt.metrics.plot_cumulative_gain(y_test, y_probas) -plt.show() - -Recall that the cumulative gains curve shows the percentage of the -overall number of cases in a given category *gained* by targeting a -percentage of the total number of cases. - -Similarly, the receiver operating characteristic curve, or ROC curve, -displays the diagnostic ability of a binary classifier system as its -discrimination threshold is varied. It plots the true positive rate against the false positive rate. - - -### Compare Bagging on Trees with Random Forests - -bag_clf = BaggingClassifier( - DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42), - n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42) - -bag_clf.fit(X_train, y_train) -y_pred = bag_clf.predict(X_test) -from sklearn.ensemble import RandomForestClassifier -rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42) -rnd_clf.fit(X_train, y_train) -y_pred_rf = rnd_clf.predict(X_test) -np.sum(y_pred == y_pred_rf) / len(y_pred) - -## Boosting, a Bird's Eye View - -The basic idea is to combine weak classifiers in order to create a good -classifier. With a weak classifier we often intend a classifier which -produces results which are only slightly better than we would get by -random guesses. - -This is done by applying in an iterative way a weak (or a standard -classifier like decision trees) to modify the data. In each iteration -we emphasize those observations which are misclassified by weighting -them with a factor. - - - -Boosting is a way of fitting an additive expansion in a set of -elementary basis functions like for example some simple polynomials. -Assume for example that we have a function - -$$ -f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), -$$ - -where $\beta_m$ are the expansion parameters to be determined in a -minimization process and $b(x;\gamma_m)$ are some simple functions of -the multivariable parameter $x$ which is characterized by the -parameters $\gamma_m$. - -As an example, consider the Sigmoid function we used in logistic -regression. In that case, we can translate the function -$b(x;\gamma_m)$ into the Sigmoid function - -$$ -\sigma(t) = \frac{1}{1+\exp{(-t)}}, -$$ - -where $t=\gamma_0+\gamma_1 x$ and the parameters $\gamma_0$ and -$\gamma_1$ were determined by the Logistic Regression fitting -algorithm. - -As another example, consider the cost function we defined for linear regression - -$$ -C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. -$$ - -In this case the function $f(x)$ was replaced by the design matrix -$\boldsymbol{X}$ and the unknown linear regression parameters $\boldsymbol{\beta}$, -that is $\boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta}$. In linear regression we can -simply invert a matrix and obtain the parameters $\beta$ by - -$$ -\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. -$$ - -In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\beta_m$ and $\gamma_m$. - - -### Iterative Fitting, Regression and Squared-error Cost Function - -The way we proceed is as follows (here we specialize to the squared-error cost function) - -1. Establish a cost function, here $\cal{C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ with $f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)$. - -2. Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers. - -3. For $m=1:M$ - -a. minimize $\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2$ wrt $\gamma$ and $\beta$ - -b. This gives the optimal values $\beta_m$ and $\gamma_m$ - -c. Determine then the new values $f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)$ - - -We could use any of the algorithms we have discussed till now. If we -use trees, $\gamma$ parameterizes the split variables and split points -at the internal nodes, and the predictions at the terminal nodes. - - - -To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function. - -For simplicity we assume also that our functions $b(x;\gamma)=1+\gamma x$. - -This means that for every iteration $m$, we need to optimize - -$$ -(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2. -$$ - -We start our iteration by simply setting $f_0(x)=0$. -Taking the derivatives with respect to $\beta$ and $\gamma$ we obtain - -$$ -\frac{\partial \cal{C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0, -$$ - -and - -$$ -\frac{\partial \cal{C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0. -$$ - -We can then rewrite these equations as (defining $\boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x})$ with $\boldsymbol{e}$ being the unit vector) - -$$ -\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, -$$ - -which gives us $\beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w})$. Similarly we have - -$$ -\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0, -$$ - -which leads to $\gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x})$. Inserting -for $\beta$ gives us an equation for $\gamma$. This is a non-linear equation in the unknown $\gamma$ and has to be solved numerically. - -The solution to these two equations gives us in turn $\beta_1$ and $\gamma_1$ leading to the new expression for $f_1(x)$ as -$f_1(x) = \beta_1(1+\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$. - - - -### Iterative Fitting, Classification and AdaBoost - -Let us consider a binary classification problem with two outcomes $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of -observations. We define a classification function $G(x)$ which produces a prediction taking one or the other of the two values -$\{-1,1\}$. - -The error rate of the training sample is then - -$$ -\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)). -$$ - -The iterative procedure starts with defining a weak classifier whose -error rate is barely better than random guessing. The iterative -procedure in boosting is to sequentially apply a weak -classification algorithm to repeatedly modified versions of the data -producing a sequence of weak classifiers $G_m(x)$. - -Here we will express our function $f(x)$ in terms of $G(x)$. That is - -$$ -f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), -$$ - -will be a function of - -$$ -G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x). -$$ - -In our iterative procedure we define thus - -$$ -f_m(x) = f_{m-1}(x)+\beta_mG_m(x). -$$ - -The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the -exponential cost/loss function defined as - -$$ -C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}. -$$ - -We optimize $\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case. -This is normally done in two steps. Let us however first rewrite the cost function as - -$$ -C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))}, -$$ - -where we have defined $w_i^m= \exp{(-y_if_{m-1}(x_i))}$. - - - -First, for any $\beta > 0$, we optimize $G$ by setting - -$$ -G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)), -$$ - -which is the classifier that minimizes the weighted error rate in predicting $y$. - -We can do this by rewriting - -$$ -\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m, -$$ - -which can be rewritten as - -$$ -(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0, -$$ - -which leads to - -$$ -\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}}, -$$ - -where we have redefined the error as - -$$ -\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m}, -$$ - -which leads to an update of - -$$ -f_m(x) = f_{m-1}(x) +\beta_m G_m(x). -$$ - -This leads to the new weights - -$$ -w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))} -$$ - -### Adaptive boosting: AdaBoost, Basic Algorithm - -The algorithm here is rather straightforward. Assume that our weak -classifier is a decision tree and we consider a binary set of outputs -with $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of -observations. Our design matrix is given in terms of the -feature/predictor vectors -$\boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}]$. Finally, we define also a -classifier determined by our data via a function $G(x)$. This function tells us how well we are able to classify our outputs/targets $\boldsymbol{y}$. - -We have already defined the misclassification error $\mathrm{err}$ as - -$$ -\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)), -$$ - -where the function $I()$ is one if we misclassify and zero if we classify correctly. - - -With the above definitions we are now ready to set up the algorithm for AdaBoost. -The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases. -1. We start by initializing all weights to $w_i = 1/n$, with $i=0,1,2,\dots n-1$. It is easy to see that we must have $\sum_{i=0}^{n-1}w_i = 1$. - -2. We rewrite the misclassification error as - -$$ -\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i}, -$$ - -1. Then we start looping over all attempts at classifying, namely we start an iterative process for $m=1:M$, where $M$ is the final number of classifications. Our given classifier could for example be a plain decision tree. - -a. Fit then a given classifier to the training set using the weights $w_i$. - -b. Compute then $\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly. - -c. Define a quantity $\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}$ - -d. Set the new weights to $w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}$. - - -5. Compute the new classifier $G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)$. - -For the iterations with $m \le 2$ the weights are modified -individually at each steps. The observations which were misclassified -at iteration $m-1$ have a weight which is larger than those which were -classified properly. As this proceeds, the observations which were -difficult to classifiy correctly are given a larger influence. Each -new classification step $m$ is then forced to concentrate on those -observations that are missed in the previous iterations. - - - - -Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here. - -from sklearn.ensemble import AdaBoostClassifier - -ada_clf = AdaBoostClassifier( - DecisionTreeClassifier(max_depth=1), n_estimators=200, - algorithm="SAMME.R", learning_rate=0.5, random_state=42) -ada_clf.fit(X_train, y_train) - -from sklearn.ensemble import AdaBoostClassifier - -ada_clf = AdaBoostClassifier( - DecisionTreeClassifier(max_depth=1), n_estimators=200, - algorithm="SAMME.R", learning_rate=0.5, random_state=42) -ada_clf.fit(X_train_scaled, y_train) -y_pred = ada_clf.predict(X_test_scaled) -skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) -plt.show() -y_probas = ada_clf.predict_proba(X_test_scaled) -skplt.metrics.plot_roc(y_test, y_probas) -plt.show() -skplt.metrics.plot_cumulative_gain(y_test, y_probas) -plt.show() - -## Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent - -Gradient boosting is again a similar technique to Adaptive boosting, -it combines so-called weak classifiers or regressors into a strong -method via a series of iterations. - -In order to understand the method, let us illustrate its basics by -bringing back the essential steps in linear regression, where our cost -function was the least squares function. - - -We start again with our cost function $\cal{C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}\cal{L}(y_i, f(x_i))$ where we want to minimize -This means that for every iteration, we need to optimize - -$$ -(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. -$$ - -We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as - -$$ -f_M(x) = \sum_{m=0}^M h_m(x). -$$ - -In the steepest decent approach we approximate $h_m(x) = -\rho_m g_m(x)$, where $\rho_m$ is a scalar and $g_m(x)$ the gradient defined as - -$$ -g_m(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. -$$ - -With the new gradient we can update $f_m(x) = f_{m-1}(x) -\rho_m g_m(x)$. Using the above squared-error function we see that -the gradient is $g_m(x_i) = -2(y_i-f(x_i))$. - -Choosing $f_0(x)=0$ we obtain $g_m(x) = -2y_i$ and inserting this into the minimization problem for the cost function we have - -$$ -(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. -$$ - -Optimizing with respect to $\rho$ we obtain (taking the derivative) that $\rho_1 = -1/2$. We have then that - -$$ -f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i. -$$ - -We can then proceed and compute - -$$ -g_2(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i, -$$ - -and find a new value for $\rho_2=-1/2$ and continue till we have reached $m=M$. We can modify the steepest descent method, or steepest boosting, by introducing what is called **gradient boosting**. - - -Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points, -so we do not learn a function that can generalize. However, we can modify the algorithm by -fitting a weak learner to approximate the negative gradient signal. - -Suppose we have a cost function $C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i))$ where $y_i$ is our target and $f(x_i)$ the function which is meant to model $y_i$. The above cost function could be our standard squared-error function - -$$ -C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. -$$ - -The way we proceed in an iterative fashion is to -1. Initialize our estimate $f_0(x)$. - -2. For $m=1:M$, we - -a. compute the negative gradient vector $\boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x)$ at $f(x) = f_{m-1}(x)$; - -b. fit the so-called base-learner to the negative gradient $h_m(u_m,x)$; - -c. update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$; - - -4. The final estimate is then $f_M(x) = \sum_{m=1}^M h_m(u_m,x)$. - -## Gradient Boosting, Examples of Regression - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.ensemble import GradientBoostingRegressor -from sklearn.preprocessing import StandardScaler -import scikitplot as skplt -from sklearn.metrics import mean_squared_error - -n = 100 -maxdegree = 6 - -# Make data set. -x = np.linspace(-3, 3, n).reshape(-1, 1) -y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape) - -error = np.zeros(maxdegree) -bias = np.zeros(maxdegree) -variance = np.zeros(maxdegree) -polydegree = np.zeros(maxdegree) -X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2) -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) - -for degree in range(1,maxdegree): - model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0) - model.fit(X_train_scaled,y_train) - y_pred = model.predict(X_test_scaled) - polydegree[degree] = degree - error[degree] = np.mean( np.mean((y_test - y_pred)**2) ) - bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 ) - variance[degree] = np.mean( np.var(y_pred) ) - print('Max depth:', degree) - print('Error:', error[degree]) - print('Bias^2:', bias[degree]) - print('Var:', variance[degree]) - print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree])) - -plt.xlim(1,maxdegree-1) -plt.plot(polydegree, error, label='Error') -plt.plot(polydegree, bias, label='bias') -plt.plot(polydegree, variance, label='Variance') -plt.legend() -save_fig("gdregression") -plt.show() - -## Gradient Boosting, Classification Example - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.datasets import load_breast_cancer -import scikitplot as skplt -from sklearn.ensemble import GradientBoostingClassifier -from sklearn.model_selection import cross_validate - -# Load the data -cancer = load_breast_cancer() - -X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) -print(X_train.shape) -print(X_test.shape) -#now scale the data -from sklearn.preprocessing import StandardScaler -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) - -gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0) -gd_clf.fit(X_train_scaled, y_train) -#Cross validation -accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score'] -print(accuracy) -print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test))) - -import scikitplot as skplt -y_pred = gd_clf.predict(X_test_scaled) -skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) -save_fig("gdclassiffierconfusion") -plt.show() -y_probas = gd_clf.predict_proba(X_test_scaled) -skplt.metrics.plot_roc(y_test, y_probas) -save_fig("gdclassiffierroc") -plt.show() -skplt.metrics.plot_cumulative_gain(y_test, y_probas) -save_fig("gdclassiffiercgain") -plt.show() - -## XGBoost: Extreme Gradient Boosting - - -[XGBoost](https://github.com/dmlc/xgboost) or Extreme Gradient -Boosting, is an optimized distributed gradient boosting library -designed to be highly efficient, flexible and portable. It implements -machine learning algorithms under the Gradient Boosting -framework. XGBoost provides a parallel tree boosting that solve many -data science problems in a fast and accurate way. See the [article by Chen and Guestrin](https://arxiv.org/abs/1603.02754). - -The authors design and build a highly scalable end-to-end tree -boosting system. It has a theoretically justified weighted quantile -sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning. - -It is now the algorithm which wins essentially all ML competitions!!! - -## Regression Case - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -import xgboost as xgb -from sklearn.preprocessing import StandardScaler -import scikitplot as skplt -from sklearn.metrics import mean_squared_error - -n = 100 -maxdegree = 6 - -# Make data set. -x = np.linspace(-3, 3, n).reshape(-1, 1) -y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape) - -error = np.zeros(maxdegree) -bias = np.zeros(maxdegree) -variance = np.zeros(maxdegree) -polydegree = np.zeros(maxdegree) -X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2) -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) - -for degree in range(maxdegree): - model = xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200) - - model.fit(X_train_scaled,y_train) - y_pred = model.predict(X_test_scaled) - polydegree[degree] = degree - error[degree] = np.mean( np.mean((y_test - y_pred)**2) ) - bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 ) - variance[degree] = np.mean( np.var(y_pred) ) - print('Max depth:', degree) - print('Error:', error[degree]) - print('Bias^2:', bias[degree]) - print('Var:', variance[degree]) - print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree])) - -plt.xlim(1,maxdegree-1) -plt.plot(polydegree, error, label='Error') -plt.plot(polydegree, bias, label='bias') -plt.plot(polydegree, variance, label='Variance') -plt.legend() -plt.show() - -As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now. - - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.datasets import load_breast_cancer -from sklearn.preprocessing import LabelEncoder -from sklearn.model_selection import cross_validate -import scikitplot as skplt -import xgboost as xgb -# Load the data -cancer = load_breast_cancer() - -X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) -print(X_train.shape) -print(X_test.shape) -#now scale the data -from sklearn.preprocessing import StandardScaler -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) - -xg_clf = xgb.XGBClassifier() -xg_clf.fit(X_train_scaled,y_train) - -y_test = xg_clf.predict(X_test_scaled) - -print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test))) - -import scikitplot as skplt -y_pred = xg_clf.predict(X_test_scaled) -skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) -save_fig("xdclassiffierconfusion") -plt.show() -y_probas = xg_clf.predict_proba(X_test_scaled) -skplt.metrics.plot_roc(y_test, y_probas) -save_fig("xdclassiffierroc") -plt.show() -skplt.metrics.plot_cumulative_gain(y_test, y_probas) -save_fig("gdclassiffiercgain") -plt.show() - - -xgb.plot_tree(xg_clf,num_trees=0) -plt.rcParams['figure.figsize'] = [50, 10] -save_fig("xgtree") -plt.show() - -xgb.plot_importance(xg_clf) -plt.rcParams['figure.figsize'] = [5, 5] -save_fig("xgparams") -plt.show() \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter8.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter8.ipynb deleted file mode 100644 index ba922a06b..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter8.ipynb +++ /dev/null @@ -1,1734 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Basic ideas of the Principal Component Analysis (PCA)\n", - "\n", - "The principal component analysis deals with the problem of fitting a\n", - "low-dimensional affine subspace $S$ of dimension $d$ much smaller than\n", - "the total dimension $D$ of the problem at hand (our data\n", - "set). Mathematically it can be formulated as a statistical problem or\n", - "a geometric problem. In our discussion of the theorem for the\n", - "classical PCA, we will stay with a statistical approach. \n", - "Historically, the PCA was first formulated in a statistical setting in order to estimate the principal component of a multivariate random variable.\n", - "\n", - "We have a data set defined by a design/feature matrix $\\boldsymbol{X}$ (see below for its definition) \n", - "* Each data point is determined by $p$ extrinsic (measurement) variables\n", - "\n", - "* We may want to ask the following question: Are there fewer intrinsic variables (say $d << p$) that still approximately describe the data?\n", - "\n", - "* If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do. \n", - "\n", - "A good read is for example [Vidal, Ma and Sastry](https://www.springer.com/gp/book/9780387878102).\n", - "\n", - "\n", - "\n", - "## Introducing the Covariance and Correlation functions\n", - "\n", - "Before we discuss the PCA theorem, we need to remind ourselves about\n", - "the definition of the covariance and the correlation function. These are quantities \n", - "\n", - "Suppose we have defined two vectors\n", - "$\\hat{x}$ and $\\hat{y}$ with $n$ elements each. The covariance matrix $\\boldsymbol{C}$ is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{y},\\boldsymbol{y}] \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where for example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With this definition and recalling that the variance is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{var}[\\boldsymbol{x}]=\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "we can rewrite the covariance matrix as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}] & \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}] & \\mathrm{var}[\\boldsymbol{y}] \\\\\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The covariance takes values between zero and infinity and may thus\n", - "lead to problems with loss of numerical precision for particularly\n", - "large values. It is common to scale the covariance matrix by\n", - "introducing instead the correlation matrix defined via the so-called\n", - "correlation function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]=\\frac{\\mathrm{cov}[\\boldsymbol{x},\\boldsymbol{y}]}{\\sqrt{\\mathrm{var}[\\boldsymbol{x}] \\mathrm{var}[\\boldsymbol{y}]}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The correlation function is then given by values $\\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}]\n", - "\\in [-1,1]$. This avoids eventual problems with too large values. We\n", - "can then define the correlation matrix for the two vectors $\\boldsymbol{x}$\n", - "and $\\boldsymbol{y}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{K}[\\boldsymbol{x},\\boldsymbol{y}] = \\begin{bmatrix} 1 & \\mathrm{corr}[\\boldsymbol{x},\\boldsymbol{y}] \\\\\n", - " \\mathrm{corr}[\\boldsymbol{y},\\boldsymbol{x}] & 1 \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the above example this is the function we constructed using **pandas**.\n", - "\n", - "\n", - "In our derivation of the various regression algorithms like **Ordinary Least Squares** or **Ridge regression**\n", - "we defined the design/feature matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n", - "x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n", - "x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n", - "\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n", - "x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n", - "x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$ refering to the column numbers and the\n", - "entries $n$ being the row elements.\n", - "We can rewrite the design/feature matrix in terms of its column vectors as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix} \\boldsymbol{x}_0 & \\boldsymbol{x}_1 & \\boldsymbol{x}_2 & \\dots & \\dots & \\boldsymbol{x}_{p-1}\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with a given vector" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{x}_i^T = \\begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \\dots & \\dots x_{n-1,i}\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With these definitions, we can now rewrite our $2\\times 2$\n", - "correaltion/covariance matrix in terms of a moe general design/feature\n", - "matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$. This leads to a $p\\times p$\n", - "covariance matrix for the vectors $\\boldsymbol{x}_i$ with $i=0,1,\\dots,p-1$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\begin{bmatrix}\n", - "\\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & \\mathrm{var}[\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{cov}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{cov}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & \\mathrm{var}[\\boldsymbol{x}_{p-1}]\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the correlation matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{K}[\\boldsymbol{x}] = \\begin{bmatrix}\n", - "1 & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_0,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & 1 & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_2] & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_1,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_1] & 1 & \\dots & \\dots & \\mathrm{corr}[\\boldsymbol{x}_2,\\boldsymbol{x}_{p-1}]\\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\dots & \\dots & \\dots & \\dots & \\dots & \\dots \\\\\n", - "\\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_0] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_1] & \\mathrm{corr}[\\boldsymbol{x}_{p-1},\\boldsymbol{x}_{2}] & \\dots & \\dots & 1\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The Numpy function **np.cov** calculates the covariance elements using\n", - "the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have\n", - "the exact mean values. The following simple function uses the\n", - "**np.vstack** function which takes each vector of dimension $1\\times n$\n", - "and produces a $2\\times n$ matrix $\\boldsymbol{W}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{W} = \\begin{bmatrix} x_0 & y_0 \\\\\n", - " x_1 & y_1 \\\\\n", - " x_2 & y_2\\\\\n", - " \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2}\\\\\n", - " x_{n-1} & y_{n-1} & \n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in turn is converted into into the $2\\times 2$ covariance matrix\n", - "$\\boldsymbol{C}$ via the Numpy function **np.cov()**. We note that we can also calculate\n", - "the mean value of each set of samples $\\boldsymbol{x}$ etc using the Numpy\n", - "function **np.mean(x)**. We can also extract the eigenvalues of the\n", - "covariance matrix through the **np.linalg.eig()** function." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "-0.2310524795427768\n", - "3.1831903689627863\n", - "[[0.86117632 2.59603122]\n", - " [2.59603122 9.06044826]]\n" - ] - } - ], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "W = np.vstack((x, y))\n", - "C = np.cov(W)\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Correlation Matrix\n", - "\n", - "The previous example can be converted into the correlation matrix by\n", - "simply scaling the matrix elements with the variances. We should also\n", - "subtract the mean values for each column. This leads to the following\n", - "code which sets up the correlations matrix for the previous example in\n", - "a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\\times 2$ correlation matrix (since we have only two vectors)." - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0.08745913868064381\n", - "2.1128123336365077\n", - "[[1. 0.6798478]\n", - " [0.6798478 1. ]]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "n = 100\n", - "# define two vectors \n", - "x = np.random.random(size=n)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "#scaling the x and y vectors \n", - "x = x - np.mean(x)\n", - "y = y - np.mean(y)\n", - "variance_x = np.sum(x@x)/n\n", - "variance_y = np.sum(y@y)/n\n", - "print(variance_x)\n", - "print(variance_y)\n", - "cov_xy = np.sum(x@y)/n\n", - "cov_xx = np.sum(x@x)/n\n", - "cov_yy = np.sum(y@y)/n\n", - "C = np.zeros((2,2))\n", - "C[0,0]= cov_xx/variance_x\n", - "C[1,1]= cov_yy/variance_y\n", - "C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)\n", - "C[1,0]= C[0,1]\n", - "print(C)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see that the matrix elements along the diagonal are one as they\n", - "should be and that the matrix is symmetric. Furthermore, diagonalizing\n", - "this matrix we easily see that it is a positive definite matrix.\n", - "\n", - "The above procedure with **numpy** can be made more compact if we use **pandas**.\n", - "\n", - "\n", - "We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[ 1.21030078 4.50152769]\n", - " [ 0.85231126 2.49259646]\n", - " [-0.27754082 -0.99161035]\n", - " [-0.05499028 -0.95681341]\n", - " [ 0.65405197 0.57844946]\n", - " [ 0.12802926 0.88441436]\n", - " [ 0.43424077 1.60051748]\n", - " [-1.57205214 -3.89834457]\n", - " [-0.77949144 -1.89899948]\n", - " [-0.59485937 -2.31173763]]\n", - " 0 1\n", - "0 1.210301 4.501528\n", - "1 0.852311 2.492596\n", - "2 -0.277541 -0.991610\n", - "3 -0.054990 -0.956813\n", - "4 0.654052 0.578449\n", - "5 0.128029 0.884414\n", - "6 0.434241 1.600517\n", - "7 -1.572052 -3.898345\n", - "8 -0.779491 -1.898999\n", - "9 -0.594859 -2.311738\n", - " 0 1\n", - "0 1.000000 0.957565\n", - "1 0.957565 1.000000\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "import pandas as pd\n", - "n = 10\n", - "x = np.random.normal(size=n)\n", - "x = x - np.mean(x)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "y = y - np.mean(y)\n", - "X = (np.vstack((x, y))).T\n", - "print(X)\n", - "Xpd = pd.DataFrame(X)\n", - "print(Xpd)\n", - "correlation_matrix = Xpd.corr()\n", - "print(correlation_matrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We expand this model to the Franke function discussed above." - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - " 0 1 2 3 4 5 6 7 \\\n", - "0 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 \n", - "1 0.0 0.091583 0.077830 0.092209 0.086333 0.080182 0.084446 0.080360 \n", - "2 0.0 0.077830 0.067325 0.077735 0.073465 0.068967 0.071391 0.068389 \n", - "3 0.0 0.092209 0.077735 0.099350 0.092714 0.085784 0.094735 0.090145 \n", - "4 0.0 0.086333 0.073465 0.092714 0.086992 0.080974 0.088533 0.084586 \n", - "5 0.0 0.080182 0.068967 0.085784 0.080974 0.075877 0.082067 0.078752 \n", - "6 0.0 0.084446 0.071391 0.094735 0.088533 0.082067 0.092739 0.088427 \n", - "7 0.0 0.080360 0.068389 0.090145 0.084586 0.078752 0.088427 0.084585 \n", - "8 0.0 0.076446 0.065511 0.085713 0.080765 0.075532 0.084250 0.080847 \n", - "9 0.0 0.072627 0.062704 0.081361 0.076998 0.072349 0.080136 0.077151 \n", - "10 0.0 0.076483 0.065094 0.088105 0.082612 0.076881 0.087850 0.084014 \n", - "11 0.0 0.073208 0.062649 0.084453 0.079459 0.074213 0.084413 0.080946 \n", - "12 0.0 0.070145 0.060358 0.081016 0.076483 0.071690 0.081166 0.078040 \n", - "13 0.0 0.067263 0.058198 0.077760 0.073657 0.069286 0.078077 0.075268 \n", - "14 0.0 0.064527 0.056149 0.074647 0.070949 0.066978 0.075114 0.072600 \n", - "\n", - " 8 9 10 11 12 13 14 \n", - "0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 \n", - "1 0.076446 0.072627 0.076483 0.073208 0.070145 0.067263 0.064527 \n", - "2 0.065511 0.062704 0.065094 0.062649 0.060358 0.058198 0.056149 \n", - "3 0.085713 0.081361 0.088105 0.084453 0.081016 0.077760 0.074647 \n", - "4 0.080765 0.076998 0.082612 0.079459 0.076483 0.073657 0.070949 \n", - "5 0.075532 0.072349 0.076881 0.074213 0.071690 0.069286 0.066978 \n", - "6 0.084250 0.080136 0.087850 0.084413 0.081166 0.078077 0.075114 \n", - "7 0.080847 0.077151 0.084014 0.080946 0.078040 0.075268 0.072600 \n", - "8 0.077525 0.074225 0.080285 0.077563 0.074977 0.072501 0.070112 \n", - "9 0.074225 0.071304 0.076603 0.074208 0.071924 0.069731 0.067608 \n", - "10 0.080285 0.076603 0.084360 0.081287 0.078374 0.075595 0.072921 \n", - "11 0.077563 0.074208 0.081287 0.078509 0.075868 0.073341 0.070901 \n", - "12 0.074977 0.071924 0.078374 0.075868 0.073479 0.071184 0.068960 \n", - "13 0.072501 0.069731 0.075595 0.073341 0.071184 0.069105 0.067084 \n", - "14 0.070112 0.067608 0.072921 0.070901 0.068960 0.067084 0.065252 \n" - ] - } - ], - "source": [ - "# Common imports\n", - "import numpy as np\n", - "import pandas as pd\n", - "\n", - "\n", - "def FrankeFunction(x,y):\n", - "\tterm1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))\n", - "\tterm2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))\n", - "\tterm3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))\n", - "\tterm4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)\n", - "\treturn term1 + term2 + term3 + term4\n", - "\n", - "\n", - "def create_X(x, y, n ):\n", - "\tif len(x.shape) > 1:\n", - "\t\tx = np.ravel(x)\n", - "\t\ty = np.ravel(y)\n", - "\n", - "\tN = len(x)\n", - "\tl = int((n+1)*(n+2)/2)\t\t# Number of elements in beta\n", - "\tX = np.ones((N,l))\n", - "\n", - "\tfor i in range(1,n+1):\n", - "\t\tq = int((i)*(i+1)/2)\n", - "\t\tfor k in range(i+1):\n", - "\t\t\tX[:,q+k] = (x**(i-k))*(y**k)\n", - "\n", - "\treturn X\n", - "\n", - "\n", - "# Making meshgrid of datapoints and compute Franke's function\n", - "n = 4\n", - "N = 100\n", - "x = np.sort(np.random.uniform(0, 1, N))\n", - "y = np.sort(np.random.uniform(0, 1, N))\n", - "z = FrankeFunction(x, y)\n", - "X = create_X(x, y, n=n) \n", - "\n", - "Xpd = pd.DataFrame(X)\n", - "# subtract the mean values and set up the covariance matrix\n", - "Xpd = Xpd - Xpd.mean()\n", - "covariance_matrix = Xpd.cov()\n", - "print(covariance_matrix)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note here that the covariance is zero for the first rows and\n", - "columns since all matrix elements in the design matrix were set to one\n", - "(we are fitting the function in terms of a polynomial of degree $n$). We would however not include the intercept\n", - "and wee can simply\n", - "drop these elements and construct a correlation\n", - "matrix without them. \n", - "\n", - "\n", - "\n", - "We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\\boldsymbol{X}$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}= \\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}].\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To see this let us simply look at a design matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{2\\times 2}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{00} & x_{01}\\\\\n", - "x_{10} & x_{11}\\\\\n", - "\\end{bmatrix}=\\begin{bmatrix}\n", - "\\boldsymbol{x}_{0} & \\boldsymbol{x}_{1}\\\\\n", - "\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we then compute the expectation value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\\\\n", - "x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is just" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]=\\begin{bmatrix} \\mathrm{var}[\\boldsymbol{x}_0] & \\mathrm{cov}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] \\\\\n", - " \\mathrm{cov}[\\boldsymbol{x}_1,\\boldsymbol{x}_0] & \\mathrm{var}[\\boldsymbol{x}_1] \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we wrote $$\\boldsymbol{C}[\\boldsymbol{x}_0,\\boldsymbol{x}_1] = \\boldsymbol{C}[\\boldsymbol{x}]$$ to indicate that this the covariance of the vectors $\\boldsymbol{x}$ of the design/feature matrix $\\boldsymbol{X}$.\n", - "\n", - "It is easy to generalize this to a matrix $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$.\n", - "\n", - "\n", - "\n", - "## Towards the PCA theorem\n", - "\n", - "We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}^T\\boldsymbol{X}= \\mathbb{E}[\\boldsymbol{X}^T\\boldsymbol{X}].\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices $\\boldsymbol{S}$.\n", - "These matrices are defined as $\\boldsymbol{S}\\in {\\mathbb{R}}^{p\\times p}$ and obey the orthogonality requirements $\\boldsymbol{S}\\boldsymbol{S}^T=\\boldsymbol{S}^T\\boldsymbol{S}=\\boldsymbol{I}$. The matrix can be written out in terms of the column vectors $\\boldsymbol{s}_i$ as $\\boldsymbol{S}=[\\boldsymbol{s}_0,\\boldsymbol{s}_1,\\dots,\\boldsymbol{s}_{p-1}]$ and $\\boldsymbol{s}_i \\in {\\mathbb{R}}^{p}$.\n", - "\n", - "Assume also that there is a transformation $\\boldsymbol{S}^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S}=\\boldsymbol{C}[\\boldsymbol{y}]$ such that the new matrix $\\boldsymbol{C}[\\boldsymbol{y}]$ is diagonal with elements $[\\lambda_0,\\lambda_1,\\lambda_2,\\dots,\\lambda_{p-1}]$. \n", - "\n", - "That is we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{y}] = \\mathbb{E}[\\boldsymbol{S}^T\\boldsymbol{X}^T\\boldsymbol{X}T\\boldsymbol{S}]=\\boldsymbol{S}^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "since the matrix $\\boldsymbol{S}$ is not a data dependent matrix. Multiplying with $\\boldsymbol{S}$ from the left we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{S}\\boldsymbol{C}[\\boldsymbol{y}] = \\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and since $\\boldsymbol{C}[\\boldsymbol{y}]$ is diagonal we have for a given eigenvalue $i$ of the covariance matrix that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{S}_i\\lambda_i = \\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S}_i.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is\n", - "$\\lambda_0 > \\lambda_1 > \\dots > \\lambda_{p-1}$. \n", - "\n", - "\n", - "The eigenvalues tell us then how much we need to stretch the\n", - "corresponding eigenvectors. Dimensions with large eigenvalues have\n", - "thus large variations (large variance) and define therefore useful\n", - "dimensions. The data points are more spread out in the direction of\n", - "these eigenvectors. Smaller eigenvalues mean on the other hand that\n", - "the corresponding eigenvectors are shrunk accordingly and the data\n", - "points are tightly bunched together and there is not much variation in\n", - "these specific directions. Hopefully then we could leave it out\n", - "dimensions where the eigenvalues are very small. If $p$ is very large,\n", - "we could then aim at reducing $p$ to $l << p$ and handle only $l$\n", - "features/predictors.\n", - "\n", - "### The Algorithm before theorem\n", - "\n", - "Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. \n", - "* Set up the datapoints for the design/feature matrix $\\boldsymbol{X}$ with $\\boldsymbol{X}\\in {\\mathbb{R}}^{n\\times p}$, with the predictors/features $p$ referring to the column numbers and the entries $n$ being the row elements." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{X}=\\begin{bmatrix}\n", - "x_{0,0} & x_{0,1} & x_{0,2}& \\dots & \\dots x_{0,p-1}\\\\\n", - "x_{1,0} & x_{1,1} & x_{1,2}& \\dots & \\dots x_{1,p-1}\\\\\n", - "x_{2,0} & x_{2,1} & x_{2,2}& \\dots & \\dots x_{2,p-1}\\\\\n", - "\\dots & \\dots & \\dots & \\dots \\dots & \\dots \\\\\n", - "x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \\dots & \\dots x_{n-2,p-1}\\\\\n", - "x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \\dots & \\dots x_{n-1,p-1}\\\\\n", - "\\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "* Center the data by subtracting the mean value for each column. This leads to a new matrix $\\boldsymbol{X}\\rightarrow \\overline{\\boldsymbol{X}}$.\n", - "\n", - "* Compute then the covariance/correlation matrix $\\mathbb{E}[\\overline{\\boldsymbol{X}}^T\\overline{\\boldsymbol{X}}]$.\n", - "\n", - "* Find the eigenpairs of $\\boldsymbol{C}$ with eigenvalues $[\\lambda_0,\\lambda_1,\\dots,\\lambda_{p-1}]$ and eigenvectors $[\\boldsymbol{s}_0,\\boldsymbol{s}_1,\\dots,\\boldsymbol{s}_{p-1}]$.\n", - "\n", - "* Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.\n", - "\n", - "* Keep only those $l$ eigenvalues larger than a selected threshold value, discarding thus $p-l$ features since we expect small variations in the data here.\n", - "\n", - "### Writing our own PCA code\n", - "\n", - "We will use a simple example first with two-dimensional data\n", - "drawn from a multivariate normal distribution with the following mean and covariance matrix (we have fixed these quantities but will play around with them below):" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = (-1,2) \\qquad \\Sigma = \\begin{bmatrix} 4 & 2 \\\\\n", - "2 & 2\n", - "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that the mean refers to each column of data. \n", - "We will generate $n = 10000$ points $X = \\{ x_1, \\ldots, x_N \\}$ from\n", - "this distribution, and store them in the $1000 \\times 2$ matrix $\\boldsymbol{X}$. This is our design matrix where we have forced the covariance and mean values to take specific values.\n", - "\n", - "The following Python code aids in setting up the data and writing out the design matrix.\n", - "Note that the function **multivariate** returns also the covariance discussed above and that it is defined by dividing by $n-1$ instead of $n$." - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "import pandas as pd\n", - "import matplotlib.pyplot as plt\n", - "from IPython.display import display\n", - "n = 10000\n", - "mean = (-1, 2)\n", - "cov = [[4, 2], [2, 2]]\n", - "X = np.random.multivariate_normal(mean, cov, n)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now we are going to implement the PCA algorithm. We will break it down into various substeps.\n", - "\n", - "\n", - "The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu_n = \\frac{1}{n} \\sum_{i=1}^n x_i\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the mean-centered data $\\bar{X} = \\{ \\bar{x}_1, \\ldots, \\bar{x}_n \\}$ takes the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\bar{x}_i = x_i - \\mu_n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "When you are done with these steps, print out $\\mu_n$ to verify it is\n", - "close to $\\mu$ and plot your mean centered data to verify it is\n", - "centered at the origin! \n", - "The following code elements perform these operations using **pandas** or using our own functionality for doing so. The latter, using **numpy** is rather simple through the **mean()** function." - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "df = pd.DataFrame(X)\n", - "# Pandas does the centering for us\n", - "df = df -df.mean()\n", - "# we center it ourselves\n", - "X_centered = X - X.mean(axis=0)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Alternatively, we could use the functions we discussed\n", - "earlier for scaling the data set. That is, we could have used the\n", - "**StandardScaler** function in **Scikit-Learn**, a function which ensures\n", - "that for each feature/predictor we study the mean value is zero and\n", - "the variance is one (every column in the design/feature matrix). You\n", - "would then not get the same results, since we divide by the\n", - "variance. The diagonal covariance matrix elements will then be one,\n", - "while the non-diagonal ones need to be divided by $2\\sqrt{2}$ for our\n", - "specific case.\n", - "\n", - "\n", - "Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\Sigma_n = \\frac{1}{n-1} \\sum_{i=1}^n \\bar{x}_i^T \\bar{x}_i = \\frac{1}{n-1} \\sum_{i=1}^n (x_i - \\mu_n)^T (x_i - \\mu_n)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the data points $x_i \\in \\mathbb{R}^p$ (here in this example $p = 2$) are column vectors and $x^T$ is the transpose of $x$.\n", - "We can write our own code or simply use either the functionaly of **numpy** or that of **pandas**, as follows" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - " 0 1\n", - "0 3.923640 1.961854\n", - "1 1.961854 1.947452\n", - "[[3.92363958 1.96185372]\n", - " [1.96185372 1.94745179]]\n" - ] - } - ], - "source": [ - "print(df.cov())\n", - "print(np.cov(X_centered.T))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Note that the way we define the covariance matrix here has a factor $n-1$ instead of $n$. This is included in the **cov()** function by **numpy** and **pandas**. \n", - "Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific $2\\times 2$ covariance matrix." - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Centered covariance using own code\n", - "[[3.92363958 1.96185372]\n", - " [1.96185372 1.94745179]]\n" - ] - }, - { - "data": { - "image/png": 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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter8_65_1.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "# extract the relevant columns from the centered design matrix of dim n x 2\n", - "x = X_centered[:,0]\n", - "y = X_centered[:,1]\n", - "Cov = np.zeros((2,2))\n", - "Cov[0,1] = np.sum(x.T@y)/(n-1.0)\n", - "Cov[0,0] = np.sum(x.T@x)/(n-1.0)\n", - "Cov[1,1] = np.sum(y.T@y)/(n-1.0)\n", - "Cov[1,0]= Cov[0,1]\n", - "print(\"Centered covariance using own code\")\n", - "print(Cov)\n", - "plt.plot(x, y, 'x')\n", - "plt.axis('equal')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Depending on the number of points $n$, we will get results that are close to the covariance values defined above.\n", - "The plot shows how the data are clustered around a line with slope close to one. Is this expected? Try to change the covariance and the mean values. For example, try to make the variance of the first element much larger than that of the second diagonal element. Try also to shrink the covariance (the non-diagonal elements) and see how the data points are distributed. \n", - "\n", - "### Diagonalize the sample covariance matrix to obtain the principal components\n", - "\n", - "Now we are ready to solve for the principal components! To do so we\n", - "diagonalize the sample covariance matrix $\\Sigma$. We can use the\n", - "function **np.linalg.eig** to do so. It will return the eigenvalues and\n", - "eigenvectors of $\\Sigma$. Once we have these we can perform the \n", - "following tasks:\n", - "\n", - "* We compute the percentage of the total variance captured by the first principal component\n", - "\n", - "* We plot the mean centered data and lines along the first and second principal components\n", - "\n", - "* Then we project the mean centered data onto the first and second principal components, and plot the projected data. \n", - "\n", - "* Finally, we approximate the data as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "x_i \\approx \\tilde{x}_i = \\mu_n + \\langle x_i, v_0 \\rangle v_0\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $v_0$ is the first principal component. \n", - "\n", - "Collecting all these steps we can write our own PCA function and\n", - "compare this with the functionality included in **Scikit-Learn**. \n", - "\n", - "The code here outlines some of the elements we could include in the\n", - "analysis. Feel free to extend upon this in order to address the above\n", - "questions." - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Eigenvalues of Covariance matrix\n", - "5.132179379442221\n", - "0.7389119925478163\n", - "First eigenvector\n", - "[0.85141702 0.52448933]\n", - "Second eigenvector\n", - "[-0.52448933 0.85141702]\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Eigenvector of largest eigenvalue\n", - "[0.85141702 0.52448933]\n" - ] - } - ], - "source": [ - "# diagonalize and obtain eigenvalues, not necessarily sorted\n", - "EigValues, EigVectors = np.linalg.eig(Cov)\n", - "# sort eigenvectors and eigenvalues\n", - "#permute = EigValues.argsort()\n", - "#EigValues = EigValues[permute]\n", - "#EigVectors = EigVectors[:,permute]\n", - "print(\"Eigenvalues of Covariance matrix\")\n", - "for i in range(2):\n", - " print(EigValues[i])\n", - "FirstEigvector = EigVectors[:,0]\n", - "SecondEigvector = EigVectors[:,1]\n", - "print(\"First eigenvector\")\n", - "print(FirstEigvector)\n", - "print(\"Second eigenvector\")\n", - "print(SecondEigvector)\n", - "#thereafter we do a PCA with Scikit-learn\n", - "from sklearn.decomposition import PCA\n", - "pca = PCA(n_components = 2)\n", - "X2Dsl = pca.fit_transform(X)\n", - "print(\"Eigenvector of largest eigenvalue\")\n", - "print(pca.components_.T[:, 0])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This code does not contain all the above elements, but it shows how we can use **Scikit-Learn** to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then? \n", - "\n", - "\n", - "## Classical PCA Theorem\n", - "\n", - "We assume now that we have a design matrix $\\boldsymbol{X}$ which has been\n", - "centered as discussed above. For the sake of simplicity we skip the\n", - "overline symbol. The matrix is defined in terms of the various column\n", - "vectors $[\\boldsymbol{x}_0,\\boldsymbol{x}_1,\\dots, \\boldsymbol{x}_{p-1}]$ each with dimension\n", - "$\\boldsymbol{x}\\in {\\mathbb{R}}^{n}$.\n", - "\n", - "\n", - "\n", - "The PCA theorem states that minimizing the above reconstruction error\n", - "corresponds to setting $\\boldsymbol{W}=\\boldsymbol{S}$, the orthogonal matrix which\n", - "diagonalizes the empirical covariance(correlation) matrix. The optimal\n", - "low-dimensional encoding of the data is then given by a set of vectors\n", - "$\\boldsymbol{z}_i$ with at most $l$ vectors, with $l << p$, defined by the\n", - "orthogonal projection of the data onto the columns spanned by the\n", - "eigenvectors of the covariance(correlations matrix).\n", - "\n", - "\n", - "\n", - "\n", - "To show the PCA theorem let us start with the assumption that there is one vector $\\boldsymbol{s}_0$ which corresponds to a solution which minimized the reconstruction error $J$. This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of $\\boldsymbol{w}_0$ and $\\boldsymbol{z}_0$ as\n", - "\n", - "\n", - "\n", - "We are almost there, we have obtained a relation between minimizing\n", - "the reconstruction error and the variance and the covariance\n", - "matrix. Minimizing the error is equivalent to maximizing the variance\n", - "of the projected data.\n", - "\n", - "\n", - "We could trivially maximize the variance of the projection (and\n", - "thereby minimize the error in the reconstruction function) by letting\n", - "the norm-2 of $\\boldsymbol{w}_0$ go to infinity. However, this norm since we\n", - "want the matrix $\\boldsymbol{W}$ to be an orthogonal matrix, is constrained by\n", - "$\\vert\\vert \\boldsymbol{w}_0 \\vert\\vert_2^2=1$. Imposing this condition via a\n", - "Lagrange multiplier we can then in turn maximize" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "J(\\boldsymbol{w}_0)= \\boldsymbol{w}_0^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0+\\lambda_0(1-\\boldsymbol{w}_0^T\\boldsymbol{w}_0).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Taking the derivative with respect to $\\boldsymbol{w}_0$ we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial J(\\boldsymbol{w}_0)}{\\partial \\boldsymbol{w}_0}= 2\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0-2\\lambda_0\\boldsymbol{w}_0=0,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "meaning that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0=\\lambda_0\\boldsymbol{w}_0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix**! If we left multiply with $\\boldsymbol{w}_0^T$ we have the variance of the projected data is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{w}_0^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0=\\lambda_0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we want to maximize the variance (minimize the construction error)\n", - "we simply pick the eigenvector of the covariance matrix with the\n", - "largest eigenvalue. This establishes the link between the minimization\n", - "of the reconstruction function $J$ in terms of an orthogonal matrix\n", - "and the maximization of the variance and thereby the covariance of our\n", - "observations encoded in the design/feature matrix $\\boldsymbol{X}$.\n", - "\n", - "The proof\n", - "for the other eigenvectors $\\boldsymbol{w}_1,\\boldsymbol{w}_2,\\dots$ can be\n", - "established by applying the above arguments and using the fact that\n", - "our basis of eigenvectors is orthogonal, see [Murphy chapter\n", - "12.2](https://mitpress.mit.edu/books/machine-learning-1). The\n", - "discussion in chapter 12.2 of Murphy's text has also a nice link with\n", - "the Singular Value Decomposition theorem. For categorical data, see\n", - "chapter 12.4 and discussion therein.\n", - "\n", - "For more details, see for example [Vidal, Ma and Sastry, chapter 2](https://www.springer.com/gp/book/9780387878102).\n", - "\n", - "\n", - "## Geometric Interpretation and link with Singular Value Decomposition\n", - "\n", - "For a detailed demonstration of the geometric interpretation, see [Vidal, Ma and Sastry, section 2.1.2](https://www.springer.com/gp/book/9780387878102).\n", - "\n", - "\n", - "Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.\n", - "First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.\n", - "\n", - "The following Python code uses NumPy’s **svd()** function to obtain all the principal components of the\n", - "training set, then extracts the first two principal components. First we center the data using either **pandas** or our own code" - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "data": { - "text/html": [ - "
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Scikit-Learn’s PCA classes take care of centering\n", - "the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t\n", - "forget to center the data first.\n", - "\n", - "Once you have identified all the principal components, you can reduce the dimensionality of the dataset\n", - "down to $d$ dimensions by projecting it onto the hyperplane defined by the first $d$ principal components.\n", - "Selecting this hyperplane ensures that the projection will preserve as much variance as possible." - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "W2 = V.T[:, :2]\n", - "X2D = X_centered.dot(W2)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## PCA and scikit-learn\n", - "\n", - "Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The\n", - "following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note\n", - "that it automatically takes care of centering the data):" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[ 1.5378811 -0.94639099]\n", - " [-0.86145244 0.89288636]\n", - " [ 0.00445655 0.81633628]\n", - " [-0.07145103 -1.00433417]\n", - " [-2.03707133 -0.48476997]\n", - " [-0.72174172 -1.4557763 ]\n", - " [ 0.55854694 1.60673226]\n", - " [-1.6999536 0.43766686]\n", - " [ 1.10405456 0.31718909]\n", - " [ 2.18673098 -0.17953942]]\n" - ] - } - ], - "source": [ - "#thereafter we do a PCA with Scikit-learn\n", - "from sklearn.decomposition import PCA\n", - "pca = PCA(n_components = 2)\n", - "X2D = pca.fit_transform(X)\n", - "print(X2D)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "After fitting the PCA transformer to the dataset, you can access the principal components using the\n", - "components variable (note that it contains the PCs as horizontal vectors, so, for example, the first\n", - "principal component is equal to" - ] - }, - { - "cell_type": "code", - "execution_count": 13, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "data": { - "text/plain": [ - "array([-0.62373464, -0.5303329 , 0.317367 , 0.01873344, 0.47815203])" - ] - }, - "execution_count": 13, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "pca.components_.T[:, 0]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Another very useful piece of information is the explained variance ratio of each principal component,\n", - "available via the $explained\\_variance\\_ratio$ variable. It indicates the proportion of the dataset’s\n", - "variance that lies along the axis of each principal component. \n", - "\n", - "## Back to the Cancer Data\n", - "We can now repeat the above but applied to real data, in this case our breast cancer data.\n", - "Here we compute performance scores on the training data using logistic regression." - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Train set accuracy from Logistic Regression: 0.95\n", - "Train set accuracy scaled data: 0.99\n", - "Train set accuracy scaled and PCA data: 0.96\n" - ] - }, - { - "name": "stderr", - "output_type": "stream", - "text": [ - "/Users/mhjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/linear_model/_logistic.py:762: ConvergenceWarning: lbfgs failed to converge (status=1):\n", - "STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.\n", - "\n", - "Increase the number of iterations (max_iter) or scale the data as shown in:\n", - " https://scikit-learn.org/stable/modules/preprocessing.html\n", - "Please also refer to the documentation for alternative solver options:\n", - " https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression\n", - " n_iter_i = _check_optimize_result(\n" - ] - } - ], - "source": [ - "import matplotlib.pyplot as plt\n", - "import numpy as np\n", - "from sklearn.model_selection import train_test_split \n", - "from sklearn.datasets import load_breast_cancer\n", - "from sklearn.linear_model import LogisticRegression\n", - "cancer = load_breast_cancer()\n", - "\n", - "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n", - "\n", - "logreg = LogisticRegression()\n", - "logreg.fit(X_train, y_train)\n", - "print(\"Train set accuracy from Logistic Regression: {:.2f}\".format(logreg.score(X_train,y_train)))\n", - "# We scale the data\n", - "from sklearn.preprocessing import StandardScaler\n", - "scaler = StandardScaler()\n", - "scaler.fit(X_train)\n", - "X_train_scaled = scaler.transform(X_train)\n", - "X_test_scaled = scaler.transform(X_test)\n", - "# Then perform again a log reg fit\n", - "logreg.fit(X_train_scaled, y_train)\n", - "print(\"Train set accuracy scaled data: {:.2f}\".format(logreg.score(X_train_scaled,y_train)))\n", - "#thereafter we do a PCA with Scikit-learn\n", - "from sklearn.decomposition import PCA\n", - "pca = PCA(n_components = 2)\n", - "X2D_train = pca.fit_transform(X_train_scaled)\n", - "# and finally compute the log reg fit and the score on the training data\t\n", - "logreg.fit(X2D_train,y_train)\n", - "print(\"Train set accuracy scaled and PCA data: {:.2f}\".format(logreg.score(X2D_train,y_train)))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We see that our training data after the PCA decomposition has a performance similar to the non-scaled data. \n", - "\n", - "\n", - "Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to\n", - "choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).\n", - "Unless, of course, you are reducing dimensionality for data visualization — in that case you will\n", - "generally want to reduce the dimensionality down to 2 or 3.\n", - "The following code computes PCA without reducing dimensionality, then computes the minimum number\n", - "of dimensions required to preserve 95% of the training set’s variance:" - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pca = PCA()\n", - "pca.fit(X)\n", - "cumsum = np.cumsum(pca.explained_variance_ratio_)\n", - "d = np.argmax(cumsum >= 0.95) + 1" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "You could then set $n\\_components=d$ and run PCA again. However, there is a much better option: instead\n", - "of specifying the number of principal components you want to preserve, you can set $n\\_components$ to be\n", - "a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:" - ] - }, - { - "cell_type": "code", - "execution_count": 16, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "pca = PCA(n_components=0.95)\n", - "X_reduced = pca.fit_transform(X)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Incremental PCA\n", - "\n", - "One problem with the preceding implementation of PCA is that it requires the whole training set to fit in\n", - "memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have\n", - "been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch\n", - "at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new\n", - "instances arrive).\n", - "\n", - "\n", - "### Randomized PCA\n", - "\n", - "Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic\n", - "algorithm that quickly finds an approximation of the first d principal components. Its computational\n", - "complexity is $O(m \\times d^2)+O(d^3)$, instead of $O(m \\times n^2) + O(n^3)$, so it is dramatically faster than the\n", - "previous algorithms when $d$ is much smaller than $n$.\n", - "\n", - "\n", - "### Kernel PCA\n", - "\n", - "The kernel trick is a mathematical technique that implicitly maps instances into a\n", - "very high-dimensional space (called the feature space), enabling nonlinear classification and regression\n", - "with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature\n", - "space corresponds to a complex nonlinear decision boundary in the original space.\n", - "It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear\n", - "projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at\n", - "preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a\n", - "twisted manifold.\n", - "For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an" - ] - }, - { - "cell_type": "code", - "execution_count": 17, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from sklearn.decomposition import KernelPCA\n", - "rbf_pca = KernelPCA(n_components = 2, kernel=\"rbf\", gamma=0.04)\n", - "X_reduced = rbf_pca.fit_transform(X)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Other techniques\n", - "\n", - "\n", - "There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.\n", - "\n", - "Here are some of the most popular:\n", - "* **Multidimensional Scaling (MDS)** reduces dimensionality while trying to preserve the distances between the instances.\n", - "\n", - "* **Isomap** creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.\n", - "\n", - "* **t-Distributed Stochastic Neighbor Embedding** (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).\n", - "\n", - "* Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures." - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter8.py b/doc/LectureNotes/_build/jupyter_execute/chapter8.py deleted file mode 100644 index a61a1ae60..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter8.py +++ /dev/null @@ -1,791 +0,0 @@ -# Basic ideas of the Principal Component Analysis (PCA) - -The principal component analysis deals with the problem of fitting a -low-dimensional affine subspace $S$ of dimension $d$ much smaller than -the total dimension $D$ of the problem at hand (our data -set). Mathematically it can be formulated as a statistical problem or -a geometric problem. In our discussion of the theorem for the -classical PCA, we will stay with a statistical approach. -Historically, the PCA was first formulated in a statistical setting in order to estimate the principal component of a multivariate random variable. - -We have a data set defined by a design/feature matrix $\boldsymbol{X}$ (see below for its definition) -* Each data point is determined by $p$ extrinsic (measurement) variables - -* We may want to ask the following question: Are there fewer intrinsic variables (say $d << p$) that still approximately describe the data? - -* If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do. - -A good read is for example [Vidal, Ma and Sastry](https://www.springer.com/gp/book/9780387878102). - - - -## Introducing the Covariance and Correlation functions - -Before we discuss the PCA theorem, we need to remind ourselves about -the definition of the covariance and the correlation function. These are quantities - -Suppose we have defined two vectors -$\hat{x}$ and $\hat{y}$ with $n$ elements each. The covariance matrix $\boldsymbol{C}$ is defined as - -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\ - \end{bmatrix}, -$$ - -where for example - -$$ -\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -$$ - -With this definition and recalling that the variance is defined as - -$$ -\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2, -$$ - -we can rewrite the covariance matrix as - -$$ -\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] & \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] & \mathrm{var}[\boldsymbol{y}] \\ - \end{bmatrix}. -$$ - -The covariance takes values between zero and infinity and may thus -lead to problems with loss of numerical precision for particularly -large values. It is common to scale the covariance matrix by -introducing instead the correlation matrix defined via the so-called -correlation function - -$$ -\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}. -$$ - -The correlation function is then given by values $\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] -\in [-1,1]$. This avoids eventual problems with too large values. We -can then define the correlation matrix for the two vectors $\boldsymbol{x}$ -and $\boldsymbol{y}$ as - -$$ -\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 & \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\ - \mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] & 1 \\ - \end{bmatrix}, -$$ - -In the above example this is the function we constructed using **pandas**. - - -In our derivation of the various regression algorithms like **Ordinary Least Squares** or **Ridge regression** -we defined the design/feature matrix $\boldsymbol{X}$ as - -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -$$ - -with $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$, with the predictors/features $p$ refering to the column numbers and the -entries $n$ being the row elements. -We can rewrite the design/feature matrix in terms of its column vectors as - -$$ -\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 & \boldsymbol{x}_1 & \boldsymbol{x}_2 & \dots & \dots & \boldsymbol{x}_{p-1}\end{bmatrix}, -$$ - -with a given vector - -$$ -\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} & x_{1,i} & x_{2,i}& \dots & \dots x_{n-1,i}\end{bmatrix}. -$$ - -With these definitions, we can now rewrite our $2\times 2$ -correaltion/covariance matrix in terms of a moe general design/feature -matrix $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$. This leads to a $p\times p$ -covariance matrix for the vectors $\boldsymbol{x}_i$ with $i=0,1,\dots,p-1$ - -$$ -\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix} -\mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] & \mathrm{var}[\boldsymbol{x}_2] & \dots & \dots & \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & \mathrm{var}[\boldsymbol{x}_{p-1}]\\ -\end{bmatrix}, -$$ - -and the correlation matrix - -$$ -\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix} -1 & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] & 1 & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\ -\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] & 1 & \dots & \dots & \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\dots & \dots & \dots & \dots & \dots & \dots \\ -\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] & \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] & \dots & \dots & 1\\ -\end{bmatrix}, -$$ - -The Numpy function **np.cov** calculates the covariance elements using -the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have -the exact mean values. The following simple function uses the -**np.vstack** function which takes each vector of dimension $1\times n$ -and produces a $2\times n$ matrix $\boldsymbol{W}$ - -$$ -\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\ - x_1 & y_1 \\ - x_2 & y_2\\ - \dots & \dots \\ - x_{n-2} & y_{n-2}\\ - x_{n-1} & y_{n-1} & - \end{bmatrix}, -$$ - -which in turn is converted into into the $2\times 2$ covariance matrix -$\boldsymbol{C}$ via the Numpy function **np.cov()**. We note that we can also calculate -the mean value of each set of samples $\boldsymbol{x}$ etc using the Numpy -function **np.mean(x)**. We can also extract the eigenvalues of the -covariance matrix through the **np.linalg.eig()** function. - -# Importing various packages -import numpy as np -n = 100 -x = np.random.normal(size=n) -print(np.mean(x)) -y = 4+3*x+np.random.normal(size=n) -print(np.mean(y)) -W = np.vstack((x, y)) -C = np.cov(W) -print(C) - -## Correlation Matrix - -The previous example can be converted into the correlation matrix by -simply scaling the matrix elements with the variances. We should also -subtract the mean values for each column. This leads to the following -code which sets up the correlations matrix for the previous example in -a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the $2\times 2$ correlation matrix (since we have only two vectors). - -import numpy as np -n = 100 -# define two vectors -x = np.random.random(size=n) -y = 4+3*x+np.random.normal(size=n) -#scaling the x and y vectors -x = x - np.mean(x) -y = y - np.mean(y) -variance_x = np.sum(x@x)/n -variance_y = np.sum(y@y)/n -print(variance_x) -print(variance_y) -cov_xy = np.sum(x@y)/n -cov_xx = np.sum(x@x)/n -cov_yy = np.sum(y@y)/n -C = np.zeros((2,2)) -C[0,0]= cov_xx/variance_x -C[1,1]= cov_yy/variance_y -C[0,1]= cov_xy/np.sqrt(variance_y*variance_x) -C[1,0]= C[0,1] -print(C) - -We see that the matrix elements along the diagonal are one as they -should be and that the matrix is symmetric. Furthermore, diagonalizing -this matrix we easily see that it is a positive definite matrix. - -The above procedure with **numpy** can be made more compact if we use **pandas**. - - -We whow here how we can set up the correlation matrix using **pandas**, as done in this simple code - -import numpy as np -import pandas as pd -n = 10 -x = np.random.normal(size=n) -x = x - np.mean(x) -y = 4+3*x+np.random.normal(size=n) -y = y - np.mean(y) -X = (np.vstack((x, y))).T -print(X) -Xpd = pd.DataFrame(X) -print(Xpd) -correlation_matrix = Xpd.corr() -print(correlation_matrix) - -We expand this model to the Franke function discussed above. - -# Common imports -import numpy as np -import pandas as pd - - -def FrankeFunction(x,y): - term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2)) - term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1)) - term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2)) - term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2) - return term1 + term2 + term3 + term4 - - -def create_X(x, y, n ): - if len(x.shape) > 1: - x = np.ravel(x) - y = np.ravel(y) - - N = len(x) - l = int((n+1)*(n+2)/2) # Number of elements in beta - X = np.ones((N,l)) - - for i in range(1,n+1): - q = int((i)*(i+1)/2) - for k in range(i+1): - X[:,q+k] = (x**(i-k))*(y**k) - - return X - - -# Making meshgrid of datapoints and compute Franke's function -n = 4 -N = 100 -x = np.sort(np.random.uniform(0, 1, N)) -y = np.sort(np.random.uniform(0, 1, N)) -z = FrankeFunction(x, y) -X = create_X(x, y, n=n) - -Xpd = pd.DataFrame(X) -# subtract the mean values and set up the covariance matrix -Xpd = Xpd - Xpd.mean() -covariance_matrix = Xpd.cov() -print(covariance_matrix) - -We note here that the covariance is zero for the first rows and -columns since all matrix elements in the design matrix were set to one -(we are fitting the function in terms of a polynomial of degree $n$). We would however not include the intercept -and wee can simply -drop these elements and construct a correlation -matrix without them. - - - -We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix $\boldsymbol{X}$ as - -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}]. -$$ - -To see this let us simply look at a design matrix $\boldsymbol{X}\in {\mathbb{R}}^{2\times 2}$ - -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{00} & x_{01}\\ -x_{10} & x_{11}\\ -\end{bmatrix}=\begin{bmatrix} -\boldsymbol{x}_{0} & \boldsymbol{x}_{1}\\ -\end{bmatrix}. -$$ - -If we then compute the expectation value - -$$ -\mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}=\begin{bmatrix} -x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\ -x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\ -\end{bmatrix}, -$$ - -which is just - -$$ -\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] & \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\ - \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] & \mathrm{var}[\boldsymbol{x}_1] \\ - \end{bmatrix}, -$$ - -where we wrote $$\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]$$ to indicate that this the covariance of the vectors $\boldsymbol{x}$ of the design/feature matrix $\boldsymbol{X}$. - -It is easy to generalize this to a matrix $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$. - - - -## Towards the PCA theorem - -We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as - -$$ -\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}^T\boldsymbol{X}= \mathbb{E}[\boldsymbol{X}^T\boldsymbol{X}]. -$$ - -Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices $\boldsymbol{S}$. -These matrices are defined as $\boldsymbol{S}\in {\mathbb{R}}^{p\times p}$ and obey the orthogonality requirements $\boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I}$. The matrix can be written out in terms of the column vectors $\boldsymbol{s}_i$ as $\boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}]$ and $\boldsymbol{s}_i \in {\mathbb{R}}^{p}$. - -Assume also that there is a transformation $\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}=\boldsymbol{C}[\boldsymbol{y}]$ such that the new matrix $\boldsymbol{C}[\boldsymbol{y}]$ is diagonal with elements $[\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}]$. - -That is we have - -$$ -\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}^T\boldsymbol{X}^T\boldsymbol{X}T\boldsymbol{S}]=\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}, -$$ - -since the matrix $\boldsymbol{S}$ is not a data dependent matrix. Multiplying with $\boldsymbol{S}$ from the left we have - -$$ -\boldsymbol{S}\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}, -$$ - -and since $\boldsymbol{C}[\boldsymbol{y}]$ is diagonal we have for a given eigenvalue $i$ of the covariance matrix that - -$$ -\boldsymbol{S}_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}_i. -$$ - -In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is -$\lambda_0 > \lambda_1 > \dots > \lambda_{p-1}$. - - -The eigenvalues tell us then how much we need to stretch the -corresponding eigenvectors. Dimensions with large eigenvalues have -thus large variations (large variance) and define therefore useful -dimensions. The data points are more spread out in the direction of -these eigenvectors. Smaller eigenvalues mean on the other hand that -the corresponding eigenvectors are shrunk accordingly and the data -points are tightly bunched together and there is not much variation in -these specific directions. Hopefully then we could leave it out -dimensions where the eigenvalues are very small. If $p$ is very large, -we could then aim at reducing $p$ to $l << p$ and handle only $l$ -features/predictors. - -### The Algorithm before theorem - -Here's how we would proceed in setting up the algorithm for the PCA, see also discussion below here. -* Set up the datapoints for the design/feature matrix $\boldsymbol{X}$ with $\boldsymbol{X}\in {\mathbb{R}}^{n\times p}$, with the predictors/features $p$ referring to the column numbers and the entries $n$ being the row elements. - -$$ -\boldsymbol{X}=\begin{bmatrix} -x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\ -x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\ -x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\ -\dots & \dots & \dots & \dots \dots & \dots \\ -x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\ -x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\ -\end{bmatrix}, -$$ - -* Center the data by subtracting the mean value for each column. This leads to a new matrix $\boldsymbol{X}\rightarrow \overline{\boldsymbol{X}}$. - -* Compute then the covariance/correlation matrix $\mathbb{E}[\overline{\boldsymbol{X}}^T\overline{\boldsymbol{X}}]$. - -* Find the eigenpairs of $\boldsymbol{C}$ with eigenvalues $[\lambda_0,\lambda_1,\dots,\lambda_{p-1}]$ and eigenvectors $[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}]$. - -* Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues. - -* Keep only those $l$ eigenvalues larger than a selected threshold value, discarding thus $p-l$ features since we expect small variations in the data here. - -### Writing our own PCA code - -We will use a simple example first with two-dimensional data -drawn from a multivariate normal distribution with the following mean and covariance matrix (we have fixed these quantities but will play around with them below): - -$$ -\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 & 2 \\ -2 & 2 -\end{bmatrix} -$$ - -Note that the mean refers to each column of data. -We will generate $n = 10000$ points $X = \{ x_1, \ldots, x_N \}$ from -this distribution, and store them in the $1000 \times 2$ matrix $\boldsymbol{X}$. This is our design matrix where we have forced the covariance and mean values to take specific values. - -The following Python code aids in setting up the data and writing out the design matrix. -Note that the function **multivariate** returns also the covariance discussed above and that it is defined by dividing by $n-1$ instead of $n$. - -%matplotlib inline - -import numpy as np -import pandas as pd -import matplotlib.pyplot as plt -from IPython.display import display -n = 10000 -mean = (-1, 2) -cov = [[4, 2], [2, 2]] -X = np.random.multivariate_normal(mean, cov, n) - -Now we are going to implement the PCA algorithm. We will break it down into various substeps. - - -The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is - -$$ -\mu_n = \frac{1}{n} \sum_{i=1}^n x_i -$$ - -and the mean-centered data $\bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \}$ takes the form - -$$ -\bar{x}_i = x_i - \mu_n. -$$ - -When you are done with these steps, print out $\mu_n$ to verify it is -close to $\mu$ and plot your mean centered data to verify it is -centered at the origin! -The following code elements perform these operations using **pandas** or using our own functionality for doing so. The latter, using **numpy** is rather simple through the **mean()** function. - -df = pd.DataFrame(X) -# Pandas does the centering for us -df = df -df.mean() -# we center it ourselves -X_centered = X - X.mean(axis=0) - -Alternatively, we could use the functions we discussed -earlier for scaling the data set. That is, we could have used the -**StandardScaler** function in **Scikit-Learn**, a function which ensures -that for each feature/predictor we study the mean value is zero and -the variance is one (every column in the design/feature matrix). You -would then not get the same results, since we divide by the -variance. The diagonal covariance matrix elements will then be one, -while the non-diagonal ones need to be divided by $2\sqrt{2}$ for our -specific case. - - -Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation - -$$ -\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n) -$$ - -where the data points $x_i \in \mathbb{R}^p$ (here in this example $p = 2$) are column vectors and $x^T$ is the transpose of $x$. -We can write our own code or simply use either the functionaly of **numpy** or that of **pandas**, as follows - -print(df.cov()) -print(np.cov(X_centered.T)) - -Note that the way we define the covariance matrix here has a factor $n-1$ instead of $n$. This is included in the **cov()** function by **numpy** and **pandas**. -Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific $2\times 2$ covariance matrix. - -# extract the relevant columns from the centered design matrix of dim n x 2 -x = X_centered[:,0] -y = X_centered[:,1] -Cov = np.zeros((2,2)) -Cov[0,1] = np.sum(x.T@y)/(n-1.0) -Cov[0,0] = np.sum(x.T@x)/(n-1.0) -Cov[1,1] = np.sum(y.T@y)/(n-1.0) -Cov[1,0]= Cov[0,1] -print("Centered covariance using own code") -print(Cov) -plt.plot(x, y, 'x') -plt.axis('equal') -plt.show() - -Depending on the number of points $n$, we will get results that are close to the covariance values defined above. -The plot shows how the data are clustered around a line with slope close to one. Is this expected? Try to change the covariance and the mean values. For example, try to make the variance of the first element much larger than that of the second diagonal element. Try also to shrink the covariance (the non-diagonal elements) and see how the data points are distributed. - -### Diagonalize the sample covariance matrix to obtain the principal components - -Now we are ready to solve for the principal components! To do so we -diagonalize the sample covariance matrix $\Sigma$. We can use the -function **np.linalg.eig** to do so. It will return the eigenvalues and -eigenvectors of $\Sigma$. Once we have these we can perform the -following tasks: - -* We compute the percentage of the total variance captured by the first principal component - -* We plot the mean centered data and lines along the first and second principal components - -* Then we project the mean centered data onto the first and second principal components, and plot the projected data. - -* Finally, we approximate the data as - -$$ -x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0 -$$ - -where $v_0$ is the first principal component. - -Collecting all these steps we can write our own PCA function and -compare this with the functionality included in **Scikit-Learn**. - -The code here outlines some of the elements we could include in the -analysis. Feel free to extend upon this in order to address the above -questions. - -# diagonalize and obtain eigenvalues, not necessarily sorted -EigValues, EigVectors = np.linalg.eig(Cov) -# sort eigenvectors and eigenvalues -#permute = EigValues.argsort() -#EigValues = EigValues[permute] -#EigVectors = EigVectors[:,permute] -print("Eigenvalues of Covariance matrix") -for i in range(2): - print(EigValues[i]) -FirstEigvector = EigVectors[:,0] -SecondEigvector = EigVectors[:,1] -print("First eigenvector") -print(FirstEigvector) -print("Second eigenvector") -print(SecondEigvector) -#thereafter we do a PCA with Scikit-learn -from sklearn.decomposition import PCA -pca = PCA(n_components = 2) -X2Dsl = pca.fit_transform(X) -print("Eigenvector of largest eigenvalue") -print(pca.components_.T[:, 0]) - -This code does not contain all the above elements, but it shows how we can use **Scikit-Learn** to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then? - - -## Classical PCA Theorem - -We assume now that we have a design matrix $\boldsymbol{X}$ which has been -centered as discussed above. For the sake of simplicity we skip the -overline symbol. The matrix is defined in terms of the various column -vectors $[\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}]$ each with dimension -$\boldsymbol{x}\in {\mathbb{R}}^{n}$. - - - -The PCA theorem states that minimizing the above reconstruction error -corresponds to setting $\boldsymbol{W}=\boldsymbol{S}$, the orthogonal matrix which -diagonalizes the empirical covariance(correlation) matrix. The optimal -low-dimensional encoding of the data is then given by a set of vectors -$\boldsymbol{z}_i$ with at most $l$ vectors, with $l << p$, defined by the -orthogonal projection of the data onto the columns spanned by the -eigenvectors of the covariance(correlations matrix). - - - - -To show the PCA theorem let us start with the assumption that there is one vector $\boldsymbol{s}_0$ which corresponds to a solution which minimized the reconstruction error $J$. This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of $\boldsymbol{w}_0$ and $\boldsymbol{z}_0$ as - - - -We are almost there, we have obtained a relation between minimizing -the reconstruction error and the variance and the covariance -matrix. Minimizing the error is equivalent to maximizing the variance -of the projected data. - - -We could trivially maximize the variance of the projection (and -thereby minimize the error in the reconstruction function) by letting -the norm-2 of $\boldsymbol{w}_0$ go to infinity. However, this norm since we -want the matrix $\boldsymbol{W}$ to be an orthogonal matrix, is constrained by -$\vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1$. Imposing this condition via a -Lagrange multiplier we can then in turn maximize - -$$ -J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0). -$$ - -Taking the derivative with respect to $\boldsymbol{w}_0$ we obtain - -$$ -\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0, -$$ - -meaning that - -$$ -\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0. -$$ - -**The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix**! If we left multiply with $\boldsymbol{w}_0^T$ we have the variance of the projected data is - -$$ -\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. -$$ - -If we want to maximize the variance (minimize the construction error) -we simply pick the eigenvector of the covariance matrix with the -largest eigenvalue. This establishes the link between the minimization -of the reconstruction function $J$ in terms of an orthogonal matrix -and the maximization of the variance and thereby the covariance of our -observations encoded in the design/feature matrix $\boldsymbol{X}$. - -The proof -for the other eigenvectors $\boldsymbol{w}_1,\boldsymbol{w}_2,\dots$ can be -established by applying the above arguments and using the fact that -our basis of eigenvectors is orthogonal, see [Murphy chapter -12.2](https://mitpress.mit.edu/books/machine-learning-1). The -discussion in chapter 12.2 of Murphy's text has also a nice link with -the Singular Value Decomposition theorem. For categorical data, see -chapter 12.4 and discussion therein. - -For more details, see for example [Vidal, Ma and Sastry, chapter 2](https://www.springer.com/gp/book/9780387878102). - - -## Geometric Interpretation and link with Singular Value Decomposition - -For a detailed demonstration of the geometric interpretation, see [Vidal, Ma and Sastry, section 2.1.2](https://www.springer.com/gp/book/9780387878102). - - -Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm. -First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it. - -The following Python code uses NumPy’s **svd()** function to obtain all the principal components of the -training set, then extracts the first two principal components. First we center the data using either **pandas** or our own code - -import numpy as np -import pandas as pd -from IPython.display import display -np.random.seed(100) -# setting up a 10 x 5 vanilla matrix -rows = 10 -cols = 5 -X = np.random.randn(rows,cols) -df = pd.DataFrame(X) -# Pandas does the centering for us -df = df -df.mean() -display(df) - -# we center it ourselves -X_centered = X - X.mean(axis=0) -# Then check the difference between pandas and our own set up -print(X_centered-df) -#Now we do an SVD -U, s, V = np.linalg.svd(X_centered) -c1 = V.T[:, 0] -c2 = V.T[:, 1] -W2 = V.T[:, :2] -X2D = X_centered.dot(W2) -print(X2D) - -PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering -the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t -forget to center the data first. - -Once you have identified all the principal components, you can reduce the dimensionality of the dataset -down to $d$ dimensions by projecting it onto the hyperplane defined by the first $d$ principal components. -Selecting this hyperplane ensures that the projection will preserve as much variance as possible. - -W2 = V.T[:, :2] -X2D = X_centered.dot(W2) - -## PCA and scikit-learn - -Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The -following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note -that it automatically takes care of centering the data): - -#thereafter we do a PCA with Scikit-learn -from sklearn.decomposition import PCA -pca = PCA(n_components = 2) -X2D = pca.fit_transform(X) -print(X2D) - -After fitting the PCA transformer to the dataset, you can access the principal components using the -components variable (note that it contains the PCs as horizontal vectors, so, for example, the first -principal component is equal to - -pca.components_.T[:, 0] - -Another very useful piece of information is the explained variance ratio of each principal component, -available via the $explained\_variance\_ratio$ variable. It indicates the proportion of the dataset’s -variance that lies along the axis of each principal component. - -## Back to the Cancer Data -We can now repeat the above but applied to real data, in this case our breast cancer data. -Here we compute performance scores on the training data using logistic regression. - -import matplotlib.pyplot as plt -import numpy as np -from sklearn.model_selection import train_test_split -from sklearn.datasets import load_breast_cancer -from sklearn.linear_model import LogisticRegression -cancer = load_breast_cancer() - -X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0) - -logreg = LogisticRegression() -logreg.fit(X_train, y_train) -print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train))) -# We scale the data -from sklearn.preprocessing import StandardScaler -scaler = StandardScaler() -scaler.fit(X_train) -X_train_scaled = scaler.transform(X_train) -X_test_scaled = scaler.transform(X_test) -# Then perform again a log reg fit -logreg.fit(X_train_scaled, y_train) -print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train))) -#thereafter we do a PCA with Scikit-learn -from sklearn.decomposition import PCA -pca = PCA(n_components = 2) -X2D_train = pca.fit_transform(X_train_scaled) -# and finally compute the log reg fit and the score on the training data -logreg.fit(X2D_train,y_train) -print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train))) - -We see that our training data after the PCA decomposition has a performance similar to the non-scaled data. - - -Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to -choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%). -Unless, of course, you are reducing dimensionality for data visualization — in that case you will -generally want to reduce the dimensionality down to 2 or 3. -The following code computes PCA without reducing dimensionality, then computes the minimum number -of dimensions required to preserve 95% of the training set’s variance: - -pca = PCA() -pca.fit(X) -cumsum = np.cumsum(pca.explained_variance_ratio_) -d = np.argmax(cumsum >= 0.95) + 1 - -You could then set $n\_components=d$ and run PCA again. However, there is a much better option: instead -of specifying the number of principal components you want to preserve, you can set $n\_components$ to be -a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve: - -pca = PCA(n_components=0.95) -X_reduced = pca.fit_transform(X) - -### Incremental PCA - -One problem with the preceding implementation of PCA is that it requires the whole training set to fit in -memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have -been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch -at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new -instances arrive). - - -### Randomized PCA - -Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic -algorithm that quickly finds an approximation of the first d principal components. Its computational -complexity is $O(m \times d^2)+O(d^3)$, instead of $O(m \times n^2) + O(n^3)$, so it is dramatically faster than the -previous algorithms when $d$ is much smaller than $n$. - - -### Kernel PCA - -The kernel trick is a mathematical technique that implicitly maps instances into a -very high-dimensional space (called the feature space), enabling nonlinear classification and regression -with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature -space corresponds to a complex nonlinear decision boundary in the original space. -It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear -projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at -preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a -twisted manifold. -For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an - -from sklearn.decomposition import KernelPCA -rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04) -X_reduced = rbf_pca.fit_transform(X) - -## Other techniques - - -There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn. - -Here are some of the most popular: -* **Multidimensional Scaling (MDS)** reduces dimensionality while trying to preserve the distances between the instances. - -* **Isomap** creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances. - -* **t-Distributed Stochastic Neighbor Embedding** (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D). - -* Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. 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\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y = f\\left(\\sum_{i=1}^n w_ix_i\\right) = f(u)\n", - "\\label{artificialNeuron} \\tag{1}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here, the output $y$ of the neuron is the value of its activation function, which have as input\n", - "a weighted sum of signals $x_i, \\dots ,x_n$ received by $n$ other neurons.\n", - "\n", - "Conceptually, it is helpful to divide neural networks into four\n", - "categories:\n", - "1. general purpose neural networks for supervised learning,\n", - "\n", - "2. neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs),\n", - "\n", - "3. neural networks for sequential data such as Recurrent Neural Networks (RNNs), and\n", - "\n", - "4. neural networks for unsupervised learning such as Deep Boltzmann Machines.\n", - "\n", - "In natural science, DNNs and CNNs have already found numerous\n", - "applications. In statistical physics, they have been applied to detect\n", - "phase transitions in 2D Ising and Potts models, lattice gauge\n", - "theories, and different phases of polymers, or solving the\n", - "Navier-Stokes equation in weather forecasting. Deep learning has also\n", - "found interesting applications in quantum physics. Various quantum\n", - "phase transitions can be detected and studied using DNNs and CNNs,\n", - "topological phases, and even non-equilibrium many-body\n", - "localization. Representing quantum states as DNNs quantum state\n", - "tomography are among some of the impressive achievements to reveal the\n", - "potential of DNNs to facilitate the study of quantum systems.\n", - "\n", - "In quantum information theory, it has been shown that one can perform\n", - "gate decompositions with the help of neural. \n", - "\n", - "The applications are not limited to the natural sciences. There is a\n", - "plethora of applications in essentially all disciplines, from the\n", - "humanities to life science and medicine.\n", - "\n", - "\n", - "An artificial neural network (ANN), is a computational model that\n", - "consists of layers of connected neurons, or nodes or units. We will\n", - "refer to these interchangeably as units or nodes, and sometimes as\n", - "neurons.\n", - "\n", - "It is supposed to mimic a biological nervous system by letting each\n", - "neuron interact with other neurons by sending signals in the form of\n", - "mathematical functions between layers. A wide variety of different\n", - "ANNs have been developed, but most of them consist of an input layer,\n", - "an output layer and eventual layers in-between, called *hidden\n", - "layers*. All layers can contain an arbitrary number of nodes, and each\n", - "connection between two nodes is associated with a weight variable.\n", - "\n", - "Neural networks (also called neural nets) are neural-inspired\n", - "nonlinear models for supervised learning. As we will see, neural nets\n", - "can be viewed as natural, more powerful extensions of supervised\n", - "learning methods such as linear and logistic regression and soft-max\n", - "methods we discussed earlier.\n", - "\n", - "\n", - "### Feed-forward neural networks\n", - "\n", - "The feed-forward neural network (FFNN) was the first and simplest type\n", - "of ANNs that were devised. In this network, the information moves in\n", - "only one direction: forward through the layers.\n", - "\n", - "Nodes are represented by circles, while the arrows display the\n", - "connections between the nodes, including the direction of information\n", - "flow. Additionally, each arrow corresponds to a weight variable\n", - "(figure to come). We observe that each node in a layer is connected\n", - "to *all* nodes in the subsequent layer, making this a so-called\n", - "*fully-connected* FFNN.\n", - "\n", - "\n", - "\n", - "### Convolutional Neural Network\n", - "\n", - "A different variant of FFNNs are *convolutional neural networks*\n", - "(CNNs), which have a connectivity pattern inspired by the animal\n", - "visual cortex. Individual neurons in the visual cortex only respond to\n", - "stimuli from small sub-regions of the visual field, called a receptive\n", - "field. This makes the neurons well-suited to exploit the strong\n", - "spatially local correlation present in natural images. The response of\n", - "each neuron can be approximated mathematically as a convolution\n", - "operation. (figure to come)\n", - "\n", - "Convolutional neural networks emulate the behaviour of neurons in the\n", - "visual cortex by enforcing a *local* connectivity pattern between\n", - "nodes of adjacent layers: Each node in a convolutional layer is\n", - "connected only to a subset of the nodes in the previous layer, in\n", - "contrast to the fully-connected FFNN. Often, CNNs consist of several\n", - "convolutional layers that learn local features of the input, with a\n", - "fully-connected layer at the end, which gathers all the local data and\n", - "produces the outputs. They have wide applications in image and video\n", - "recognition.\n", - "\n", - "### Recurrent neural networks\n", - "\n", - "So far we have only mentioned ANNs where information flows in one\n", - "direction: forward. *Recurrent neural networks* on the other hand,\n", - "have connections between nodes that form directed *cycles*. This\n", - "creates a form of internal memory which are able to capture\n", - "information on what has been calculated before; the output is\n", - "dependent on the previous computations. Recurrent NNs make use of\n", - "sequential information by performing the same task for every element\n", - "in a sequence, where each element depends on previous elements. An\n", - "example of such information is sentences, making recurrent NNs\n", - "especially well-suited for handwriting and speech recognition.\n", - "\n", - "### Other types of networks\n", - "\n", - "There are many other kinds of ANNs that have been developed. One type\n", - "that is specifically designed for interpolation in multidimensional\n", - "space is the radial basis function (RBF) network. RBFs are typically\n", - "made up of three layers: an input layer, a hidden layer with\n", - "non-linear radial symmetric activation functions and a linear output\n", - "layer (''linear'' here means that each node in the output layer has a\n", - "linear activation function). The layers are normally fully-connected\n", - "and there are no cycles, thus RBFs can be viewed as a type of\n", - "fully-connected FFNN. They are however usually treated as a separate\n", - "type of NN due the unusual activation functions.\n", - "\n", - "\n", - "## Multilayer perceptrons\n", - "\n", - "One uses often so-called fully-connected feed-forward neural networks\n", - "with three or more layers (an input layer, one or more hidden layers\n", - "and an output layer) consisting of neurons that have non-linear\n", - "activation functions.\n", - "\n", - "Such networks are often called *multilayer perceptrons* (MLPs).\n", - "\n", - "\n", - "According to the *Universal approximation theorem*, a feed-forward\n", - "neural network with just a single hidden layer containing a finite\n", - "number of neurons can approximate a continuous multidimensional\n", - "function to arbitrary accuracy, assuming the activation function for\n", - "the hidden layer is a **non-constant, bounded and\n", - "monotonically-increasing continuous function**.\n", - "\n", - "Note that the requirements on the activation function only applies to\n", - "the hidden layer, the output nodes are always assumed to be linear, so\n", - "as to not restrict the range of output values.\n", - "\n", - "\n", - "\n", - "The output $y$ is produced via the activation function $f$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y = f\\left(\\sum_{i=1}^n w_ix_i + b_i\\right) = f(z),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This function receives $x_i$ as inputs.\n", - "Here the activation $z=(\\sum_{i=1}^n w_ix_i+b_i)$. \n", - "In an FFNN of such neurons, the *inputs* $x_i$ are the *outputs* of\n", - "the neurons in the preceding layer. Furthermore, an MLP is\n", - "fully-connected, which means that each neuron receives a weighted sum\n", - "of the outputs of *all* neurons in the previous layer.\n", - "\n", - "\n", - "First, for each node $i$ in the first hidden layer, we calculate a weighted sum $z_i^1$ of the input coordinates $x_j$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} z_i^1 = \\sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1\n", - "\\label{_auto1} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here $b_i$ is the so-called bias which is normally needed in\n", - "case of zero activation weights or inputs. How to fix the biases and\n", - "the weights will be discussed below. The value of $z_i^1$ is the\n", - "argument to the activation function $f_i$ of each node $i$, The\n", - "variable $M$ stands for all possible inputs to a given node $i$ in the\n", - "first layer. We define the output $y_i^1$ of all neurons in layer 1 as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y_i^1 = f(z_i^1) = f\\left(\\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\\right)\n", - "\\label{outputLayer1} \\tag{3}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we assume that all nodes in the same layer have identical\n", - "activation functions, hence the notation $f$. In general, we could assume in the more general case that different layers have different activation functions.\n", - "In this case we would identify these functions with a superscript $l$ for the $l$-th layer," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y_i^l = f^l(u_i^l) = f^l\\left(\\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\\right)\n", - "\\label{generalLayer} \\tag{4}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $N_l$ is the number of nodes in layer $l$. When the output of\n", - "all the nodes in the first hidden layer are computed, the values of\n", - "the subsequent layer can be calculated and so forth until the output\n", - "is obtained.\n", - "\n", - "\n", - "\n", - "\n", - "The output of neuron $i$ in layer 2 is thus," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y_i^2 = f^2\\left(\\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\\right) \n", - "\\label{_auto2} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \n", - " = f^2\\left[\\sum_{j=1}^N w_{ij}^2f^1\\left(\\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\\right) + b_i^2\\right]\n", - "\\label{outputLayer2} \\tag{6}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have substituted $y_k^1$ with the inputs $x_k$. Finally, the ANN output reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y_i^3 = f^3\\left(\\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\\right) \n", - "\\label{_auto3} \\tag{7}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \n", - " = f_3\\left[\\sum_{j} w_{ij}^3 f^2\\left(\\sum_{k} w_{jk}^2 f^1\\left(\\sum_{m} w_{km}^1 x_m + b_k^1\\right) + b_j^2\\right)\n", - " + b_1^3\\right]\n", - "\\label{_auto4} \\tag{8}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can generalize this expression to an MLP with $l$ hidden\n", - "layers. The complete functional form is," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "y^{l+1}_i = f^{l+1}\\left[\\!\\sum_{j=1}^{N_l} w_{ij}^3 f^l\\left(\\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\\left(\\dots f^1\\left(\\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\\right)\\dots\\right)+b_k^2\\right)+b_1^3\\right] \n", - "\\label{completeNN} \\tag{9}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which illustrates a basic property of MLPs: The only independent\n", - "variables are the input values $x_n$.\n", - "\n", - "\n", - "This confirms that an MLP, despite its quite convoluted mathematical\n", - "form, is nothing more than an analytic function, specifically a\n", - "mapping of real-valued vectors $\\hat{x} \\in \\mathbb{R}^n \\rightarrow\n", - "\\hat{y} \\in \\mathbb{R}^m$.\n", - "\n", - "Furthermore, the flexibility and universality of an MLP can be\n", - "illustrated by realizing that the expression is essentially a nested\n", - "sum of scaled activation functions of the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " f(x) = c_1 f(c_2 x + c_3) + c_4\n", - "\\label{_auto5} \\tag{10}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the parameters $c_i$ are weights and biases. By adjusting these\n", - "parameters, the activation functions can be shifted up and down or\n", - "left and right, change slope or be rescaled which is the key to the\n", - "flexibility of a neural network.\n", - "\n", - "\n", - "We can introduce a more convenient notation for the activations in an A NN. \n", - "\n", - "Additionally, we can represent the biases and activations\n", - "as layer-wise column vectors $\\hat{b}_l$ and $\\hat{y}_l$, so that the $i$-th element of each vector \n", - "is the bias $b_i^l$ and activation $y_i^l$ of node $i$ in layer $l$ respectively. \n", - "\n", - "We have that $\\mathrm{W}_l$ is an $N_{l-1} \\times N_l$ matrix, while $\\hat{b}_l$ and $\\hat{y}_l$ are $N_l \\times 1$ column vectors. \n", - "With this notation, the sum becomes a matrix-vector multiplication, and we can write\n", - "the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\hat{y}_2 = f_2(\\mathrm{W}_2 \\hat{y}_{1} + \\hat{b}_{2}) = \n", - " f_2\\left(\\left[\\begin{array}{ccc}\n", - " w^2_{11} &w^2_{12} &w^2_{13} \\\\\n", - " w^2_{21} &w^2_{22} &w^2_{23} \\\\\n", - " w^2_{31} &w^2_{32} &w^2_{33} \\\\\n", - " \\end{array} \\right] \\cdot\n", - " \\left[\\begin{array}{c}\n", - " y^1_1 \\\\\n", - " y^1_2 \\\\\n", - " y^1_3 \\\\\n", - " \\end{array}\\right] + \n", - " \\left[\\begin{array}{c}\n", - " b^2_1 \\\\\n", - " b^2_2 \\\\\n", - " b^2_3 \\\\\n", - " \\end{array}\\right]\\right).\n", - "\\label{_auto6} \\tag{11}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Matrix-vector notation and activation\n", - "\n", - "The activation of node $i$ in layer 2 is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " y^2_i = f_2\\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\\Bigr) = \n", - " f_2\\left(\\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\\right).\n", - "\\label{_auto7} \\tag{12}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This is not just a convenient and compact notation, but also a useful\n", - "and intuitive way to think about MLPs: The output is calculated by a\n", - "series of matrix-vector multiplications and vector additions that are\n", - "used as input to the activation functions. For each operation\n", - "$\\mathrm{W}_l \\hat{y}_{l-1}$ we move forward one layer.\n", - "\n", - "\n", - "\n", - "### Activation functions\n", - "\n", - "A property that characterizes a neural network, other than its\n", - "connectivity, is the choice of activation function(s). As described\n", - "in, the following restrictions are imposed on an activation function\n", - "for a FFNN to fulfill the universal approximation theorem\n", - "\n", - " * Non-constant\n", - "\n", - " * Bounded\n", - "\n", - " * Monotonically-increasing\n", - "\n", - " * Continuous\n", - "\n", - "The second requirement excludes all linear functions. Furthermore, in\n", - "a MLP with only linear activation functions, each layer simply\n", - "performs a linear transformation of its inputs.\n", - "\n", - "Regardless of the number of layers, the output of the NN will be\n", - "nothing but a linear function of the inputs. Thus we need to introduce\n", - "some kind of non-linearity to the NN to be able to fit non-linear\n", - "functions Typical examples are the logistic *Sigmoid*" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(x) = \\frac{1}{1 + e^{-x}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the *hyperbolic tangent* function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(x) = \\tanh(x)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The *sigmoid* function are more biologically plausible because the\n", - "output of inactive neurons are zero. Such activation function are\n", - "called *one-sided*. However, it has been shown that the hyperbolic\n", - "tangent performs better than the sigmoid for training MLPs. has\n", - "become the most popular for *deep neural networks*" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "data": { - "image/png": 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uen05p+f34zenjNIJQ9IiBbpIlPvDS0u56/XlnDmhP7/++kiFueySJuBEotidry3jzteLOHNCP4W5tEqBLhKl7n1zBb9/aSmnjO3Lr7+uaRZpnQJdJAo9+t5qfvX8Ik4a1ZtbTx2tMJc2UaCLRJnnPtrA9U9/zHHDenLb6WN0hyFpM31TRKLIG0tLuOrxeRw6sBt3nT2O5ET9ikrb6dsiEiXmrtnC9x4u5KCemdx7fr7OAJU9pkAXiQJFxZVc+OBsemal8NC3DyUrNSnokiQGKdBFAlZcUcMFD7xPYifj79+eoAttyV7TiUUiAaqqbeCiB+dQWlnH4987nAHddf9P2XsaoYsEpKGxiUse+4AFG8q56+yxjM7rEnRJEuM0QhcJgLtzw8wFFCwp4eZvjOLYYblBlyRxQCN0kQDc99ZKHntvDRd/6UDOOqx/0OVInFCgi7SzVxZ+wq9nLWLyiF78+CtDgy5H4ogCXaQdLdhQzuUz5jKqbza3nT5Gp/RLRCnQRdpJSUUt33loDtlpSdx7Xj5pyTpxSCJLO0VF2kFtQyMXP1LI1up6nrz4CHpm6VhziTwFush+5u5c/6+PKVy9hbvOGsfIvtlBlyRxSlMuIvvZA2+v4snCdVx+3GBOGt076HIkjinQRfajt5Zt5lfPL+QrI3K58rjBQZcjcU6BLrKfrC2r5tLpH3BQzwz+eJqOaJH9T4Eush9sr2vkew8X0tjkTDs3n/QU7a6S/U/fMpEIc3eufeojFm3axn3n5zOwhy64Je1DI3SRCHvg7VU8PW8DV08aomu0SLtSoItE0OxVZdw8axHHD8/lkmMOCroc6WDaFOhmNtnMlphZkZn9dBdtTjOzhWa2wMwei2yZItGvuKKGSx79gLyuafzhtEO0E1TaXatz6GaWANwFHA+sA2ab2Ux3X9iszWDgWuAod99iZj33V8Ei0aihsYnLHpvLtpp6Hvr2BN1CTgLRlhH6BKDI3Ve4ex0wA5iyU5vvAne5+xYAdy+ObJki0e3WF5fw3soyfnPKKA7unRV0OdJBteUol77A2mbP1wGH7dRmCICZvQ0kAD939xd2fiMzmwpMBcjNzaWgoGAvSo6sysrKqKgjGqgvQva0Hwo/aWDa3FqO7Z9I1/IiCgqK9l9x7UzfiR1ioS8iddhiIjAYmAjkAW+Y2Sh339q8kbtPA6YB5Ofn+8SJEyO0+b1XUFBANNQRDdQXIXvSD2tKq7ms4E0Oycvm7qlHkJIYX1dQ1Hdih1joi7ZMuawH+jV7nhde1tw6YKa717v7SmApoYAXiVs19Y18/9FCOplx51nj4i7MJfa0JdBnA4PNbJCZJQNnADN3avM0odE5ZtaD0BTMisiVKRJ9bnp2IQs2bOOPpx1Cv26dgy5HpPVAd/cG4FLgRWAR8IS7LzCzX5jZ18LNXgRKzWwh8DrwI3cv3V9FiwTtmXnrmf5+6J6gxx2sk4ckOrRpDt3dZwGzdlp2Q7OfHbg6/BCJa0XFlVz71HwOHdiVH355SNDliPyPzhQV2QPb6xq55NEPSE1K4E9njiUxQb9CEj10cS6RPXDTswtY8kkFD154KL2z04IuR+QzNLwQaaOn565nxuy1/GDigUwcqpOhJfoo0EXaYHlJJT/7V2je/OrjNW8u0UmBLtKKmvrQvHlKYifNm0tU0xy6SCt++dxCFm+q4IELNG8u0U1DDZHdeO6jDTz63hq+d/QBHDNM8+YS3RToIruwurSKa/85n7H9u/DDrwwNuhyRVinQRVpQ19DEZdPnYgZ/OmMsSZo3lxigOXSRFtzywmI+WlfOPeeM03VaJGYo0EV2Mre4gfs+WMl5Rwxg8sjeQZcj0mb6O1KkmY3l27l3fi3De2fxsxMPDrockT2iQBcJa2hs4vLpc2logjvPGktqkq5vLrFFgS4Sdsery5i9agvnj0jhgJyMoMsR2WMKdBHgrWWbufP1Ir41Po8j+2jXksQmBbp0eCUVtVz5+DwOzMngpikjgi5HZK9pKCIdWlOTc/UT86ioqeeR70ygc7J+JSR2aYQuHdrd/1nOm8s2c+PJIxjWKyvockT2iQJdOqzZq8r448tLOfmQPpw5oV/Q5YjsMwW6dEhlVXVc9thc+nVN4+ZvjMTMgi5JZJ9pwlA6nKYm50dPfkhZVR1P/eBIMlOTgi5JJCI0QpcO5963VvDq4mKuO+lgRvbNDrockYhRoEuHUri6jFteWMKJo3px3hEDgi5HJKIU6NJhbAnPm/ftksZvvzla8+YSdzSHLh1CU5NzzZMfsrmyjn9+/0iyNG8ucUgjdOkQ/vrGCl5bXMz1Xz2YUXmaN5f4pECXuPfeilJ+/9ISThrdm3MP17y5xK82BbqZTTazJWZWZGY/3U27b5qZm1l+5EoU2XslFbVcNn0uA7p15hbNm0ucazXQzSwBuAs4ARgOnGlmw1tolwlcAbwX6SJF9kZjk3PFjLlsq6nnL+eMIyNFu4wkvrVlhD4BKHL3Fe5eB8wAprTQ7pfALUBNBOsT2Wu3v7KUd5aX8sspI3WdFukQ2jJk6QusbfZ8HXBY8wZmNg7o5+7Pm9mPdvVGZjYVmAqQm5tLQUHBHhccaZWVlVFRRzSIp76YV9zAnz+o5Yt9E8mpXE5BwfI2vzae+mFfqS92iIW+2Oe/Qc2sE/BH4ILW2rr7NGAaQH5+vk+cOHFfN7/PCgoKiIY6okG89MWa0mou//ObjOiTxd8uPnKPbyUXL/0QCeqLHWKhL9oy5bIeaH4purzwsk9lAiOBAjNbBRwOzNSOUQlCTX0jFz9SiJlxzznjdV9Q6VDaEuizgcFmNsjMkoEzgJmfrnT3cnfv4e4D3X0g8C7wNXefs18qFtkFd+f6pz9m0aZt3H76GPp16xx0SSLtqtVAd/cG4FLgRWAR8IS7LzCzX5jZ1/Z3gSJt9ch7a/hH4TouO3YwxwzrGXQ5Iu2uTXPo7j4LmLXTsht20XbivpclsmdmryrjppkLOGZoDlceNzjockQCoTNFJeZtKq/h+498QF7XNG4/YyydOunkIemYdKaFxLTahka+/2gh1XUNPPbdw8hO00W3pONSoEvMcndueHoBc9ds5S9nj2NIbmbQJYkESlMuErMeemcVj89Zy2XHHsSJo3oHXY5I4BToEpPeWraZXz6/iOOH53LVpCFBlyMSFRToEnNWba7iksc+4KCcDG47fYx2goqEKdAlppRvr+eih2bTyeDe8/N1BUWRZvTbIDGjvrGJHzxayJqyah6+6DCdCSqyEwW6xAR354ZnFvB2USm//9YhHH5A96BLEok6mnKRmHDfWyuZ/v4afjDxQE4dnxd0OSJRSYEuUe+Fjzfy61mLOGFkL3745aFBlyMStRToEtUKV5dxxYx5jO3XRUe0iLRCgS5Ra3lJJRc9NIc+XdK49/xDdW1zkVYo0CUqlVTUcsED75NgxoMXHkq39OSgSxKJejrKRaLOtpp6LnjgfUoqapkx9QgGdE8PuiSRmKARukSVmvpGvvvQHJZsquDuc8Yzpl+XoEsSiRkaoUvUaGhs4vLpc3lvZRl3nDGGY4bqrkMie0IjdIkKTU3OtU/N56WFn/Dzk4czZUzfoEsSiTkKdAmcu3PjzAU8WbiOy48bzAVHDQq6JJGYpECXQLk7N89axMPvrmbq0Qdw1STdD1RkbynQJVC3vbyUv725kvOPGMC1JwzDTCcOiewt7RSVQLg7t7+yjD+9VsRp+XncePIIhbnIPlKgS7tzd/748lL+/FoR3xqfx29OGa1T+kUiQIEu7crd+d2LS/hLwXJOz+/Hb04ZpTAXiRAFurSbT3eA/u3NlZw5oT+//vpIhblIBCnQpV00Njk/e2o+j89Zy3lHDODnJ49QmItEmAJd9ru6hiauenwez8/fyGXHHsTVxw/RDlCR/aBNhy2a2WQzW2JmRWb20xbWX21mC83sIzN71cwGRL5UiUVVtQ185+9zeH7+Rn524jCu+fJQhbnIftJqoJtZAnAXcAIwHDjTzIbv1GwukO/uo4F/ALdGulCJPcUVNZw+7b+8XbSZW745iqlHHxh0SSJxrS0j9AlAkbuvcPc6YAYwpXkDd3/d3avDT98FdNPHDq6ouJJT/vIOy4uruPe8fE4/tH/QJYnEPXP33TcwOxWY7O7fCT8/FzjM3S/dRfs7gU3u/qsW1k0FpgLk5uaOnzFjxj6Wv+8qKyvJyMgIuoyoEKm+WFTayJ3zakgwuGp8KoOyY+tOQ/pO7KC+2CFa+uKYY44pdPf8ltZFdKeomZ0D5ANfamm9u08DpgHk5+f7xIkTI7n5vVJQUEA01BENItEX099fwx9e+pgB3dN54IIJ9O/eOTLFtSN9J3ZQX+wQC33RlkBfD/Rr9jwvvOwzzGwScB3wJXevjUx5EisaGpu4edZi7n97JUcPyeHOs8aSlZoUdFkiHUpbAn02MNjMBhEK8jOAs5o3MLOxwF8JTc0UR7xKiWqllbVcMWMebxVt5sKjBnLdiQeTmKDrvom0t1YD3d0bzOxS4EUgAbjf3ReY2S+AOe4+E/gdkAE8GT4kbY27f20/1i1R4sO1W/n+I4Vsrqrj1m+O5rRD+7X+IhHZL9o0h+7us4BZOy27odnPkyJcl0Q5d+ex99dw08yF5GSm8M+Lj2RUXnbQZYl0aDpTVPZY+fZ6fvbUfJ6fv5Gjh+Rwx+lj6JqeHHRZIh2eAl32yAdrtnD59LlsLK/hJ5OH8b2jD9A1WUSihAJd2qS+sYk7XyvizteL6J2dypMXH8G4/l2DLktEmlGgS6uWfVLB1U98yPz15Xx9TB9umjKS7DQdkigSbRToskv1jU3c++ZKbntlKRkpidx99jhOGNU76LJEZBcU6NKij9Zt5Sf/nM+ijdv4yohcfvX1UeRkpgRdlojshgJdPmNbTT23v7yMB99ZSU5mCvecM57JI3sFXZaItIECXQBoanLeXFfPD39fQGlVHWcf1p8fTx6m0/dFYogCXShcvYVfPb+QuWvqGNe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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter9_29_3.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "%matplotlib inline\n", - "\n", - "\"\"\"The sigmoid function (or the logistic curve) is a \n", - "function that takes any real number, z, and outputs a number (0,1).\n", - "It is useful in neural networks for assigning weights on a relative scale.\n", - "The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n", - "\n", - "import numpy\n", - "import matplotlib.pyplot as plt\n", - "import math as mt\n", - "\n", - "z = numpy.arange(-5, 5, .1)\n", - "sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n", - "sigma = sigma_fn(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, sigma)\n", - "ax.set_ylim([-0.1, 1.1])\n", - "ax.set_xlim([-5,5])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('sigmoid function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"Step Function\"\"\"\n", - "z = numpy.arange(-5, 5, .02)\n", - "step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n", - "step = step_fn(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, step)\n", - "ax.set_ylim([-0.5, 1.5])\n", - "ax.set_xlim([-5,5])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('step function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"Sine Function\"\"\"\n", - "z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n", - "t = numpy.sin(z)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, t)\n", - "ax.set_ylim([-1.0, 1.0])\n", - "ax.set_xlim([-2*mt.pi,2*mt.pi])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('sine function')\n", - "\n", - "plt.show()\n", - "\n", - "\"\"\"Plots a graph of the squashing function used by a rectified linear\n", - "unit\"\"\"\n", - "z = numpy.arange(-2, 2, .1)\n", - "zero = numpy.zeros(len(z))\n", - "y = numpy.max([zero, z], axis=0)\n", - "\n", - "fig = plt.figure()\n", - "ax = fig.add_subplot(111)\n", - "ax.plot(z, y)\n", - "ax.set_ylim([-2.0, 2.0])\n", - "ax.set_xlim([-2.0, 2.0])\n", - "ax.grid(True)\n", - "ax.set_xlabel('z')\n", - "ax.set_title('Rectified linear unit')\n", - "\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The multilayer perceptron (MLP)\n", - "\n", - "The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of\n", - "1. A neural network with one or more layers of nodes between the input and the output nodes.\n", - "\n", - "2. The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer.\n", - "\n", - "3. The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer.\n", - "\n", - "As a convention it is normal to call a network with one layer of input units, one layer of hidden\n", - "units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc.\n", - "\n", - "For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units.\n", - "Hereafter we will call the various entities of a layer for nodes.\n", - "There are also no connections within a single layer.\n", - "\n", - "The number of input nodes does not need to equal the number of output\n", - "nodes. This applies also to the hidden layers. Each layer may have its\n", - "own number of nodes and activation functions.\n", - "\n", - "The hidden layers have their name from the fact that they are not\n", - "linked to observables and as we will see below when we define the\n", - "so-called activation $\\hat{z}$, we can think of this as a basis\n", - "expansion of the original inputs $\\hat{x}$. The difference however\n", - "between neural networks and say linear regression is that now these\n", - "basis functions (which will correspond to the weights in the network)\n", - "are learned from data. This results in an important difference between\n", - "neural networks and deep learning approaches on one side and methods\n", - "like logistic regression or linear regression and their modifications on the other side.\n", - "\n", - "\n", - "### From one to many layers, the universal approximation theorem\n", - "\n", - "A neural network with only one layer, what we called the simple\n", - "perceptron, is best suited if we have a standard binary model with\n", - "clear (linear) boundaries between the outcomes. As such it could\n", - "equally well be replaced by standard linear regression or logistic\n", - "regression. Networks with one or more hidden layers approximate\n", - "systems with more complex boundaries.\n", - "\n", - "As stated earlier, \n", - "an important theorem in studies of neural networks, restated without\n", - "proof here, is the [universal approximation\n", - "theorem](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf).\n", - "\n", - "It states that a feed-forward network with a single hidden layer\n", - "containing a finite number of neurons can approximate continuous\n", - "functions on compact subsets of real functions. The theorem thus\n", - "states that simple neural networks can represent a wide variety of\n", - "interesting functions when given appropriate parameters. It is the\n", - "multilayer feedforward architecture itself which gives neural networks\n", - "the potential of being universal approximators.\n", - "\n", - "\n", - "\n", - "## Deriving the back propagation code for a multilayer perceptron model\n", - "\n", - "\n", - "\n", - "As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications.\n", - "The unknowwn quantities are our weights $w_{ij}$ and we need to find an algorithm for changing them so that our errors are as small as possible.\n", - "This leads us to the famous [back propagation algorithm](https://www.nature.com/articles/323533a0).\n", - "\n", - "The questions we want to ask are how do changes in the biases and the\n", - "weights in our network change the cost function and how can we use the\n", - "final output to modify the weights?\n", - "\n", - "To derive these equations let us start with a plain regression problem\n", - "and define our cost function as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\cal C}(\\hat{W}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the $t_i$s are our $n$ targets (the values we want to\n", - "reproduce), while the outputs of the network after having propagated\n", - "all inputs $\\hat{x}$ are given by $y_i$. Below we will demonstrate\n", - "how the basic equations arising from the back propagation algorithm\n", - "can be modified in order to study classification problems with $K$\n", - "classes.\n", - "\n", - "\n", - "With our definition of the targets $\\hat{t}$, the outputs of the\n", - "network $\\hat{y}$ and the inputs $\\hat{x}$ we\n", - "define now the activation $z_j^l$ of node/neuron/unit $j$ of the\n", - "$l$-th layer as a function of the bias, the weights which add up from\n", - "the previous layer $l-1$ and the forward passes/outputs\n", - "$\\hat{a}^{l-1}$ from the previous layer as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "z_j^l = \\sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $b_k^l$ are the biases from layer $l$. Here $M_{l-1}$\n", - "represents the total number of nodes/neurons/units of layer $l-1$. The\n", - "figure here illustrates this equation. We can rewrite this in a more\n", - "compact form as the matrix-vector products we discussed earlier," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{z}^l = \\left(\\hat{W}^l\\right)^T\\hat{a}^{l-1}+\\hat{b}^l.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With the activation values $\\hat{z}^l$ we can in turn define the\n", - "output of layer $l$ as $\\hat{a}^l = f(\\hat{z}^l)$ where $f$ is our\n", - "activation function. In the examples here we will use the sigmoid\n", - "function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers\n", - "and their nodes. It means we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_j^l = f(z_j^l) = \\frac{1}{1+\\exp{-(z_j^l)}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Derivatives and the chain rule\n", - "\n", - "From the definition of the activation $z_j^l$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial z_j^l}{\\partial w_{ij}^l} = a_i^{l-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial z_j^l}{\\partial a_i^{l-1}} = w_{ji}^l.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With our definition of the activation function we have that (note that this function depends only on $z_j^l$)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial a_j^l}{\\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With these definitions we can now compute the derivative of the cost function in terms of the weights.\n", - "\n", - "Let us specialize to the output layer $l=L$. Our cost function is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "{\\cal C}(\\hat{W^L}) = \\frac{1}{2}\\sum_{i=1}^n\\left(y_i - t_i\\right)^2=\\frac{1}{2}\\sum_{i=1}^n\\left(a_i^L - t_i\\right)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The derivative of this function with respect to the weights is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\left(a_j^L - t_j\\right)\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The last partial derivative can easily be computed and reads (by applying the chain rule)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial a_j^L}{\\partial w_{jk}^{L}} = \\frac{\\partial a_j^L}{\\partial z_{j}^{L}}\\frac{\\partial z_j^L}{\\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Bringing it together, first back propagation equation\n", - "\n", - "We have thus" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\left(a_j^L - t_j\\right)a_j^L(1-a_j^L)a_k^{L-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Defining" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^L = a_j^L(1-a_j^L)\\left(a_j^L - t_j\\right) = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and using the Hadamard product of two vectors we can write this as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{\\delta}^L = f'(\\hat{z}^L)\\circ\\frac{\\partial {\\cal C}}{\\partial (\\hat{a}^L)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This is an important expression. The second term on the right handside\n", - "measures how fast the cost function is changing as a function of the $j$th\n", - "output activation. If, for example, the cost function doesn't depend\n", - "much on a particular output node $j$, then $\\delta_j^L$ will be small,\n", - "which is what we would expect. The first term on the right, measures\n", - "how fast the activation function $f$ is changing at a given activation\n", - "value $z_j^L$.\n", - "\n", - "Notice that everything in the above equations is easily computed. In\n", - "particular, we compute $z_j^L$ while computing the behaviour of the\n", - "network, and it is only a small additional overhead to compute\n", - "$f'(z^L_j)$. The exact form of the derivative with respect to the\n", - "output depends on the form of the cost function.\n", - "However, provided the cost function is known there should be little\n", - "trouble in calculating" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "With the definition of $\\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is also easy to see that our previous equation can be written as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^L =\\frac{\\partial {\\cal C}}{\\partial z_j^L}= \\frac{\\partial {\\cal C}}{\\partial a_j^L}\\frac{\\partial a_j^L}{\\partial z_j^L},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L}\\frac{\\partial b_j^L}{\\partial z_j^L}=\\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "That is, the error $\\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias. \n", - "\n", - "We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are\n", - "\n", - "**The starting equations.**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\frac{\\partial{\\cal C}(\\hat{W^L})}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1},\n", - "\\label{_auto8} \\tag{13}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", - "\\label{_auto9} \\tag{14}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", - "\\label{_auto10} \\tag{15}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "An interesting consequence of the above equations is that when the\n", - "activation $a_k^{L-1}$ is small, the gradient term, that is the\n", - "derivative of the cost function with respect to the weights, will also\n", - "tend to be small. We say then that the weight learns slowly, meaning\n", - "that it changes slowly when we minimize the weights via say gradient\n", - "descent. In this case we say the system learns slowly.\n", - "\n", - "Another interesting feature is that is when the activation function,\n", - "represented by the sigmoid function here, is rather flat when we move towards\n", - "its end values $0$ and $1$ (see the above Python codes). In these\n", - "cases, the derivatives of the activation function will also be close\n", - "to zero, meaning again that the gradients will be small and the\n", - "network learns slowly again.\n", - "\n", - "\n", - "\n", - "We need a fourth equation and we are set. We are going to propagate\n", - "backwards in order to the determine the weights and biases. In order\n", - "to do so we need to represent the error in the layer before the final\n", - "one $L-1$ in terms of the errors in the final output layer.\n", - "\n", - "### Final back propagating equation\n", - "\n", - "We have that (replacing $L$ with a general layer $l$)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^l =\\frac{\\partial {\\cal C}}{\\partial z_j^l}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^l =\\sum_k \\frac{\\partial {\\cal C}}{\\partial z_k^{l+1}}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}}=\\sum_k \\delta_k^{l+1}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and recalling that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "z_j^{l+1} = \\sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $M_l$ being the number of nodes in layer $l$, we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^l =\\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This is our final equation.\n", - "\n", - "We are now ready to set up the algorithm for back propagation and learning the weights and biases.\n", - "\n", - "\n", - "### Setting up the Back propagation algorithm\n", - "\n", - "The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.\n", - "\n", - "First, we set up the input data $\\hat{x}$ and the activations\n", - "$\\hat{z}_1$ of the input layer and compute the activation function and\n", - "the pertinent outputs $\\hat{a}^1$.\n", - "\n", - "\n", - "\n", - "Secondly, we perform then the feed forward till we reach the output\n", - "layer and compute all $\\hat{z}_l$ of the input layer and compute the\n", - "activation function and the pertinent outputs $\\hat{a}^l$ for\n", - "$l=2,3,\\dots,L$.\n", - "\n", - "\n", - "\n", - "Thereafter we compute the ouput error $\\hat{\\delta}^L$ by computing all" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^L = f'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Then we compute the back propagate error for each $l=L-1,L-2,\\dots,2$ as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,2$ and update the weights and biases according to the rules" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "w_{jk}^l\\leftarrow = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The parameter $\\eta$ is the learning parameter discussed in connection with the gradient descent methods.\n", - "Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training." - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter9.py b/doc/LectureNotes/_build/jupyter_execute/chapter9.py deleted file mode 100644 index 8972a6556..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/chapter9.py +++ /dev/null @@ -1,840 +0,0 @@ -# Neural networks - -Artificial neural networks are computational systems that can learn to -perform tasks by considering examples, generally without being -programmed with any task-specific rules. It is supposed to mimic a -biological system, wherein neurons interact by sending signals in the -form of mathematical functions between layers. All layers can contain -an arbitrary number of neurons, and each connection is represented by -a weight variable. - - -The field of artificial neural networks has a long history of -development, and is closely connected with the advancement of computer -science and computers in general. A model of artificial neurons was -first developed by McCulloch and Pitts in 1943 to study signal -processing in the brain and has later been refined by others. The -general idea is to mimic neural networks in the human brain, which is -composed of billions of neurons that communicate with each other by -sending electrical signals. Each neuron accumulates its incoming -signals, which must exceed an activation threshold to yield an -output. If the threshold is not overcome, the neuron remains inactive, -i.e. has zero output. - -This behaviour has inspired a simple mathematical model for an artificial neuron. - - -
- -$$ -\begin{equation} - y = f\left(\sum_{i=1}^n w_ix_i\right) = f(u) -\label{artificialNeuron} \tag{1} -\end{equation} -$$ - -Here, the output $y$ of the neuron is the value of its activation function, which have as input -a weighted sum of signals $x_i, \dots ,x_n$ received by $n$ other neurons. - -Conceptually, it is helpful to divide neural networks into four -categories: -1. general purpose neural networks for supervised learning, - -2. neural networks designed specifically for image processing, the most prominent example of this class being Convolutional Neural Networks (CNNs), - -3. neural networks for sequential data such as Recurrent Neural Networks (RNNs), and - -4. neural networks for unsupervised learning such as Deep Boltzmann Machines. - -In natural science, DNNs and CNNs have already found numerous -applications. In statistical physics, they have been applied to detect -phase transitions in 2D Ising and Potts models, lattice gauge -theories, and different phases of polymers, or solving the -Navier-Stokes equation in weather forecasting. Deep learning has also -found interesting applications in quantum physics. Various quantum -phase transitions can be detected and studied using DNNs and CNNs, -topological phases, and even non-equilibrium many-body -localization. Representing quantum states as DNNs quantum state -tomography are among some of the impressive achievements to reveal the -potential of DNNs to facilitate the study of quantum systems. - -In quantum information theory, it has been shown that one can perform -gate decompositions with the help of neural. - -The applications are not limited to the natural sciences. There is a -plethora of applications in essentially all disciplines, from the -humanities to life science and medicine. - - -An artificial neural network (ANN), is a computational model that -consists of layers of connected neurons, or nodes or units. We will -refer to these interchangeably as units or nodes, and sometimes as -neurons. - -It is supposed to mimic a biological nervous system by letting each -neuron interact with other neurons by sending signals in the form of -mathematical functions between layers. A wide variety of different -ANNs have been developed, but most of them consist of an input layer, -an output layer and eventual layers in-between, called *hidden -layers*. All layers can contain an arbitrary number of nodes, and each -connection between two nodes is associated with a weight variable. - -Neural networks (also called neural nets) are neural-inspired -nonlinear models for supervised learning. As we will see, neural nets -can be viewed as natural, more powerful extensions of supervised -learning methods such as linear and logistic regression and soft-max -methods we discussed earlier. - - -### Feed-forward neural networks - -The feed-forward neural network (FFNN) was the first and simplest type -of ANNs that were devised. In this network, the information moves in -only one direction: forward through the layers. - -Nodes are represented by circles, while the arrows display the -connections between the nodes, including the direction of information -flow. Additionally, each arrow corresponds to a weight variable -(figure to come). We observe that each node in a layer is connected -to *all* nodes in the subsequent layer, making this a so-called -*fully-connected* FFNN. - - - -### Convolutional Neural Network - -A different variant of FFNNs are *convolutional neural networks* -(CNNs), which have a connectivity pattern inspired by the animal -visual cortex. Individual neurons in the visual cortex only respond to -stimuli from small sub-regions of the visual field, called a receptive -field. This makes the neurons well-suited to exploit the strong -spatially local correlation present in natural images. The response of -each neuron can be approximated mathematically as a convolution -operation. (figure to come) - -Convolutional neural networks emulate the behaviour of neurons in the -visual cortex by enforcing a *local* connectivity pattern between -nodes of adjacent layers: Each node in a convolutional layer is -connected only to a subset of the nodes in the previous layer, in -contrast to the fully-connected FFNN. Often, CNNs consist of several -convolutional layers that learn local features of the input, with a -fully-connected layer at the end, which gathers all the local data and -produces the outputs. They have wide applications in image and video -recognition. - -### Recurrent neural networks - -So far we have only mentioned ANNs where information flows in one -direction: forward. *Recurrent neural networks* on the other hand, -have connections between nodes that form directed *cycles*. This -creates a form of internal memory which are able to capture -information on what has been calculated before; the output is -dependent on the previous computations. Recurrent NNs make use of -sequential information by performing the same task for every element -in a sequence, where each element depends on previous elements. An -example of such information is sentences, making recurrent NNs -especially well-suited for handwriting and speech recognition. - -### Other types of networks - -There are many other kinds of ANNs that have been developed. One type -that is specifically designed for interpolation in multidimensional -space is the radial basis function (RBF) network. RBFs are typically -made up of three layers: an input layer, a hidden layer with -non-linear radial symmetric activation functions and a linear output -layer (''linear'' here means that each node in the output layer has a -linear activation function). The layers are normally fully-connected -and there are no cycles, thus RBFs can be viewed as a type of -fully-connected FFNN. They are however usually treated as a separate -type of NN due the unusual activation functions. - - -## Multilayer perceptrons - -One uses often so-called fully-connected feed-forward neural networks -with three or more layers (an input layer, one or more hidden layers -and an output layer) consisting of neurons that have non-linear -activation functions. - -Such networks are often called *multilayer perceptrons* (MLPs). - - -According to the *Universal approximation theorem*, a feed-forward -neural network with just a single hidden layer containing a finite -number of neurons can approximate a continuous multidimensional -function to arbitrary accuracy, assuming the activation function for -the hidden layer is a **non-constant, bounded and -monotonically-increasing continuous function**. - -Note that the requirements on the activation function only applies to -the hidden layer, the output nodes are always assumed to be linear, so -as to not restrict the range of output values. - - - -The output $y$ is produced via the activation function $f$ - -$$ -y = f\left(\sum_{i=1}^n w_ix_i + b_i\right) = f(z), -$$ - -This function receives $x_i$ as inputs. -Here the activation $z=(\sum_{i=1}^n w_ix_i+b_i)$. -In an FFNN of such neurons, the *inputs* $x_i$ are the *outputs* of -the neurons in the preceding layer. Furthermore, an MLP is -fully-connected, which means that each neuron receives a weighted sum -of the outputs of *all* neurons in the previous layer. - - -First, for each node $i$ in the first hidden layer, we calculate a weighted sum $z_i^1$ of the input coordinates $x_j$, - - -
- -$$ -\begin{equation} z_i^1 = \sum_{j=1}^{M} w_{ij}^1 x_j + b_i^1 -\label{_auto1} \tag{2} -\end{equation} -$$ - -Here $b_i$ is the so-called bias which is normally needed in -case of zero activation weights or inputs. How to fix the biases and -the weights will be discussed below. The value of $z_i^1$ is the -argument to the activation function $f_i$ of each node $i$, The -variable $M$ stands for all possible inputs to a given node $i$ in the -first layer. We define the output $y_i^1$ of all neurons in layer 1 as - - -
- -$$ -\begin{equation} - y_i^1 = f(z_i^1) = f\left(\sum_{j=1}^M w_{ij}^1 x_j + b_i^1\right) -\label{outputLayer1} \tag{3} -\end{equation} -$$ - -where we assume that all nodes in the same layer have identical -activation functions, hence the notation $f$. In general, we could assume in the more general case that different layers have different activation functions. -In this case we would identify these functions with a superscript $l$ for the $l$-th layer, - - -
- -$$ -\begin{equation} - y_i^l = f^l(u_i^l) = f^l\left(\sum_{j=1}^{N_{l-1}} w_{ij}^l y_j^{l-1} + b_i^l\right) -\label{generalLayer} \tag{4} -\end{equation} -$$ - -where $N_l$ is the number of nodes in layer $l$. When the output of -all the nodes in the first hidden layer are computed, the values of -the subsequent layer can be calculated and so forth until the output -is obtained. - - - - -The output of neuron $i$ in layer 2 is thus, - - -
- -$$ -\begin{equation} - y_i^2 = f^2\left(\sum_{j=1}^N w_{ij}^2 y_j^1 + b_i^2\right) -\label{_auto2} \tag{5} -\end{equation} -$$ - - -
- -$$ -\begin{equation} - = f^2\left[\sum_{j=1}^N w_{ij}^2f^1\left(\sum_{k=1}^M w_{jk}^1 x_k + b_j^1\right) + b_i^2\right] -\label{outputLayer2} \tag{6} -\end{equation} -$$ - -where we have substituted $y_k^1$ with the inputs $x_k$. Finally, the ANN output reads - - -
- -$$ -\begin{equation} - y_i^3 = f^3\left(\sum_{j=1}^N w_{ij}^3 y_j^2 + b_i^3\right) -\label{_auto3} \tag{7} -\end{equation} -$$ - - -
- -$$ -\begin{equation} - = f_3\left[\sum_{j} w_{ij}^3 f^2\left(\sum_{k} w_{jk}^2 f^1\left(\sum_{m} w_{km}^1 x_m + b_k^1\right) + b_j^2\right) - + b_1^3\right] -\label{_auto4} \tag{8} -\end{equation} -$$ - -We can generalize this expression to an MLP with $l$ hidden -layers. The complete functional form is, - - -
- -$$ -\begin{equation} -y^{l+1}_i = f^{l+1}\left[\!\sum_{j=1}^{N_l} w_{ij}^3 f^l\left(\sum_{k=1}^{N_{l-1}}w_{jk}^{l-1}\left(\dots f^1\left(\sum_{n=1}^{N_0} w_{mn}^1 x_n+ b_m^1\right)\dots\right)+b_k^2\right)+b_1^3\right] -\label{completeNN} \tag{9} -\end{equation} -$$ - -which illustrates a basic property of MLPs: The only independent -variables are the input values $x_n$. - - -This confirms that an MLP, despite its quite convoluted mathematical -form, is nothing more than an analytic function, specifically a -mapping of real-valued vectors $\hat{x} \in \mathbb{R}^n \rightarrow -\hat{y} \in \mathbb{R}^m$. - -Furthermore, the flexibility and universality of an MLP can be -illustrated by realizing that the expression is essentially a nested -sum of scaled activation functions of the form - - -
- -$$ -\begin{equation} - f(x) = c_1 f(c_2 x + c_3) + c_4 -\label{_auto5} \tag{10} -\end{equation} -$$ - -where the parameters $c_i$ are weights and biases. By adjusting these -parameters, the activation functions can be shifted up and down or -left and right, change slope or be rescaled which is the key to the -flexibility of a neural network. - - -We can introduce a more convenient notation for the activations in an A NN. - -Additionally, we can represent the biases and activations -as layer-wise column vectors $\hat{b}_l$ and $\hat{y}_l$, so that the $i$-th element of each vector -is the bias $b_i^l$ and activation $y_i^l$ of node $i$ in layer $l$ respectively. - -We have that $\mathrm{W}_l$ is an $N_{l-1} \times N_l$ matrix, while $\hat{b}_l$ and $\hat{y}_l$ are $N_l \times 1$ column vectors. -With this notation, the sum becomes a matrix-vector multiplication, and we can write -the equation for the activations of hidden layer 2 (assuming three nodes for simplicity) as - - -
- -$$ -\begin{equation} - \hat{y}_2 = f_2(\mathrm{W}_2 \hat{y}_{1} + \hat{b}_{2}) = - f_2\left(\left[\begin{array}{ccc} - w^2_{11} &w^2_{12} &w^2_{13} \\ - w^2_{21} &w^2_{22} &w^2_{23} \\ - w^2_{31} &w^2_{32} &w^2_{33} \\ - \end{array} \right] \cdot - \left[\begin{array}{c} - y^1_1 \\ - y^1_2 \\ - y^1_3 \\ - \end{array}\right] + - \left[\begin{array}{c} - b^2_1 \\ - b^2_2 \\ - b^2_3 \\ - \end{array}\right]\right). -\label{_auto6} \tag{11} -\end{equation} -$$ - -### Matrix-vector notation and activation - -The activation of node $i$ in layer 2 is - - -
- -$$ -\begin{equation} - y^2_i = f_2\Bigr(w^2_{i1}y^1_1 + w^2_{i2}y^1_2 + w^2_{i3}y^1_3 + b^2_i\Bigr) = - f_2\left(\sum_{j=1}^3 w^2_{ij} y_j^1 + b^2_i\right). -\label{_auto7} \tag{12} -\end{equation} -$$ - -This is not just a convenient and compact notation, but also a useful -and intuitive way to think about MLPs: The output is calculated by a -series of matrix-vector multiplications and vector additions that are -used as input to the activation functions. For each operation -$\mathrm{W}_l \hat{y}_{l-1}$ we move forward one layer. - - - -### Activation functions - -A property that characterizes a neural network, other than its -connectivity, is the choice of activation function(s). As described -in, the following restrictions are imposed on an activation function -for a FFNN to fulfill the universal approximation theorem - - * Non-constant - - * Bounded - - * Monotonically-increasing - - * Continuous - -The second requirement excludes all linear functions. Furthermore, in -a MLP with only linear activation functions, each layer simply -performs a linear transformation of its inputs. - -Regardless of the number of layers, the output of the NN will be -nothing but a linear function of the inputs. Thus we need to introduce -some kind of non-linearity to the NN to be able to fit non-linear -functions Typical examples are the logistic *Sigmoid* - -$$ -f(x) = \frac{1}{1 + e^{-x}}, -$$ - -and the *hyperbolic tangent* function - -$$ -f(x) = \tanh(x) -$$ - -The *sigmoid* function are more biologically plausible because the -output of inactive neurons are zero. Such activation function are -called *one-sided*. However, it has been shown that the hyperbolic -tangent performs better than the sigmoid for training MLPs. has -become the most popular for *deep neural networks* - -%matplotlib inline - -"""The sigmoid function (or the logistic curve) is a -function that takes any real number, z, and outputs a number (0,1). -It is useful in neural networks for assigning weights on a relative scale. -The value z is the weighted sum of parameters involved in the learning algorithm.""" - -import numpy -import matplotlib.pyplot as plt -import math as mt - -z = numpy.arange(-5, 5, .1) -sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z))) -sigma = sigma_fn(z) - -fig = plt.figure() -ax = fig.add_subplot(111) -ax.plot(z, sigma) -ax.set_ylim([-0.1, 1.1]) -ax.set_xlim([-5,5]) -ax.grid(True) -ax.set_xlabel('z') -ax.set_title('sigmoid function') - -plt.show() - -"""Step Function""" -z = numpy.arange(-5, 5, .02) -step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0) -step = step_fn(z) - -fig = plt.figure() -ax = fig.add_subplot(111) -ax.plot(z, step) -ax.set_ylim([-0.5, 1.5]) -ax.set_xlim([-5,5]) -ax.grid(True) -ax.set_xlabel('z') -ax.set_title('step function') - -plt.show() - -"""Sine Function""" -z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1) -t = numpy.sin(z) - -fig = plt.figure() -ax = fig.add_subplot(111) -ax.plot(z, t) -ax.set_ylim([-1.0, 1.0]) -ax.set_xlim([-2*mt.pi,2*mt.pi]) -ax.grid(True) -ax.set_xlabel('z') -ax.set_title('sine function') - -plt.show() - -"""Plots a graph of the squashing function used by a rectified linear -unit""" -z = numpy.arange(-2, 2, .1) -zero = numpy.zeros(len(z)) -y = numpy.max([zero, z], axis=0) - -fig = plt.figure() -ax = fig.add_subplot(111) -ax.plot(z, y) -ax.set_ylim([-2.0, 2.0]) -ax.set_xlim([-2.0, 2.0]) -ax.grid(True) -ax.set_xlabel('z') -ax.set_title('Rectified linear unit') - -plt.show() - -## The multilayer perceptron (MLP) - -The multilayer perceptron is a very popular, and easy to implement approach, to deep learning. It consists of -1. A neural network with one or more layers of nodes between the input and the output nodes. - -2. The multilayer network structure, or architecture, or topology, consists of an input layer, one or more hidden layers, and one output layer. - -3. The input nodes pass values to the first hidden layer, its nodes pass the information on to the second and so on till we reach the output layer. - -As a convention it is normal to call a network with one layer of input units, one layer of hidden -units and one layer of output units as a two-layer network. A network with two layers of hidden units is called a three-layer network etc etc. - -For an MLP network there is no direct connection between the output nodes/neurons/units and the input nodes/neurons/units. -Hereafter we will call the various entities of a layer for nodes. -There are also no connections within a single layer. - -The number of input nodes does not need to equal the number of output -nodes. This applies also to the hidden layers. Each layer may have its -own number of nodes and activation functions. - -The hidden layers have their name from the fact that they are not -linked to observables and as we will see below when we define the -so-called activation $\hat{z}$, we can think of this as a basis -expansion of the original inputs $\hat{x}$. The difference however -between neural networks and say linear regression is that now these -basis functions (which will correspond to the weights in the network) -are learned from data. This results in an important difference between -neural networks and deep learning approaches on one side and methods -like logistic regression or linear regression and their modifications on the other side. - - -### From one to many layers, the universal approximation theorem - -A neural network with only one layer, what we called the simple -perceptron, is best suited if we have a standard binary model with -clear (linear) boundaries between the outcomes. As such it could -equally well be replaced by standard linear regression or logistic -regression. Networks with one or more hidden layers approximate -systems with more complex boundaries. - -As stated earlier, -an important theorem in studies of neural networks, restated without -proof here, is the [universal approximation -theorem](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.441.7873&rep=rep1&type=pdf). - -It states that a feed-forward network with a single hidden layer -containing a finite number of neurons can approximate continuous -functions on compact subsets of real functions. The theorem thus -states that simple neural networks can represent a wide variety of -interesting functions when given appropriate parameters. It is the -multilayer feedforward architecture itself which gives neural networks -the potential of being universal approximators. - - - -## Deriving the back propagation code for a multilayer perceptron model - - - -As we have seen now in a feed forward network, we can express the final output of our network in terms of basic matrix-vector multiplications. -The unknowwn quantities are our weights $w_{ij}$ and we need to find an algorithm for changing them so that our errors are as small as possible. -This leads us to the famous [back propagation algorithm](https://www.nature.com/articles/323533a0). - -The questions we want to ask are how do changes in the biases and the -weights in our network change the cost function and how can we use the -final output to modify the weights? - -To derive these equations let us start with a plain regression problem -and define our cost function as - -$$ -{\cal C}(\hat{W}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2, -$$ - -where the $t_i$s are our $n$ targets (the values we want to -reproduce), while the outputs of the network after having propagated -all inputs $\hat{x}$ are given by $y_i$. Below we will demonstrate -how the basic equations arising from the back propagation algorithm -can be modified in order to study classification problems with $K$ -classes. - - -With our definition of the targets $\hat{t}$, the outputs of the -network $\hat{y}$ and the inputs $\hat{x}$ we -define now the activation $z_j^l$ of node/neuron/unit $j$ of the -$l$-th layer as a function of the bias, the weights which add up from -the previous layer $l-1$ and the forward passes/outputs -$\hat{a}^{l-1}$ from the previous layer as - -$$ -z_j^l = \sum_{i=1}^{M_{l-1}}w_{ij}^la_i^{l-1}+b_j^l, -$$ - -where $b_k^l$ are the biases from layer $l$. Here $M_{l-1}$ -represents the total number of nodes/neurons/units of layer $l-1$. The -figure here illustrates this equation. We can rewrite this in a more -compact form as the matrix-vector products we discussed earlier, - -$$ -\hat{z}^l = \left(\hat{W}^l\right)^T\hat{a}^{l-1}+\hat{b}^l. -$$ - -With the activation values $\hat{z}^l$ we can in turn define the -output of layer $l$ as $\hat{a}^l = f(\hat{z}^l)$ where $f$ is our -activation function. In the examples here we will use the sigmoid -function discussed in our logistic regression lectures. We will also use the same activation function $f$ for all layers -and their nodes. It means we have - -$$ -a_j^l = f(z_j^l) = \frac{1}{1+\exp{-(z_j^l)}}. -$$ - -### Derivatives and the chain rule - -From the definition of the activation $z_j^l$ we have - -$$ -\frac{\partial z_j^l}{\partial w_{ij}^l} = a_i^{l-1}, -$$ - -and - -$$ -\frac{\partial z_j^l}{\partial a_i^{l-1}} = w_{ji}^l. -$$ - -With our definition of the activation function we have that (note that this function depends only on $z_j^l$) - -$$ -\frac{\partial a_j^l}{\partial z_j^{l}} = a_j^l(1-a_j^l)=f(z_j^l)(1-f(z_j^l)). -$$ - -With these definitions we can now compute the derivative of the cost function in terms of the weights. - -Let us specialize to the output layer $l=L$. Our cost function is - -$$ -{\cal C}(\hat{W^L}) = \frac{1}{2}\sum_{i=1}^n\left(y_i - t_i\right)^2=\frac{1}{2}\sum_{i=1}^n\left(a_i^L - t_i\right)^2, -$$ - -The derivative of this function with respect to the weights is - -$$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)\frac{\partial a_j^L}{\partial w_{jk}^{L}}, -$$ - -The last partial derivative can easily be computed and reads (by applying the chain rule) - -$$ -\frac{\partial a_j^L}{\partial w_{jk}^{L}} = \frac{\partial a_j^L}{\partial z_{j}^{L}}\frac{\partial z_j^L}{\partial w_{jk}^{L}}=a_j^L(1-a_j^L)a_k^{L-1}, -$$ - -### Bringing it together, first back propagation equation - -We have thus - -$$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \left(a_j^L - t_j\right)a_j^L(1-a_j^L)a_k^{L-1}, -$$ - -Defining - -$$ -\delta_j^L = a_j^L(1-a_j^L)\left(a_j^L - t_j\right) = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, -$$ - -and using the Hadamard product of two vectors we can write this as - -$$ -\hat{\delta}^L = f'(\hat{z}^L)\circ\frac{\partial {\cal C}}{\partial (\hat{a}^L)}. -$$ - -This is an important expression. The second term on the right handside -measures how fast the cost function is changing as a function of the $j$th -output activation. If, for example, the cost function doesn't depend -much on a particular output node $j$, then $\delta_j^L$ will be small, -which is what we would expect. The first term on the right, measures -how fast the activation function $f$ is changing at a given activation -value $z_j^L$. - -Notice that everything in the above equations is easily computed. In -particular, we compute $z_j^L$ while computing the behaviour of the -network, and it is only a small additional overhead to compute -$f'(z^L_j)$. The exact form of the derivative with respect to the -output depends on the form of the cost function. -However, provided the cost function is known there should be little -trouble in calculating - -$$ -\frac{\partial {\cal C}}{\partial (a_j^L)} -$$ - -With the definition of $\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely - -$$ -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}. -$$ - -It is also easy to see that our previous equation can be written as - -$$ -\delta_j^L =\frac{\partial {\cal C}}{\partial z_j^L}= \frac{\partial {\cal C}}{\partial a_j^L}\frac{\partial a_j^L}{\partial z_j^L}, -$$ - -which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely - -$$ -\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}\frac{\partial b_j^L}{\partial z_j^L}=\frac{\partial {\cal C}}{\partial b_j^L}, -$$ - -That is, the error $\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias. - -We have now three equations that are essential for the computations of the derivatives of the cost function at the output layer. These equations are needed to start the algorithm and they are - -**The starting equations.** - - -
- -$$ -\begin{equation} -\frac{\partial{\cal C}(\hat{W^L})}{\partial w_{jk}^L} = \delta_j^La_k^{L-1}, -\label{_auto8} \tag{13} -\end{equation} -$$ - -and - - -
- -$$ -\begin{equation} -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}, -\label{_auto9} \tag{14} -\end{equation} -$$ - -and - - -
- -$$ -\begin{equation} -\delta_j^L = \frac{\partial {\cal C}}{\partial b_j^L}, -\label{_auto10} \tag{15} -\end{equation} -$$ - -An interesting consequence of the above equations is that when the -activation $a_k^{L-1}$ is small, the gradient term, that is the -derivative of the cost function with respect to the weights, will also -tend to be small. We say then that the weight learns slowly, meaning -that it changes slowly when we minimize the weights via say gradient -descent. In this case we say the system learns slowly. - -Another interesting feature is that is when the activation function, -represented by the sigmoid function here, is rather flat when we move towards -its end values $0$ and $1$ (see the above Python codes). In these -cases, the derivatives of the activation function will also be close -to zero, meaning again that the gradients will be small and the -network learns slowly again. - - - -We need a fourth equation and we are set. We are going to propagate -backwards in order to the determine the weights and biases. In order -to do so we need to represent the error in the layer before the final -one $L-1$ in terms of the errors in the final output layer. - -### Final back propagating equation - -We have that (replacing $L$ with a general layer $l$) - -$$ -\delta_j^l =\frac{\partial {\cal C}}{\partial z_j^l}. -$$ - -We want to express this in terms of the equations for layer $l+1$. Using the chain rule and summing over all $k$ entries we have - -$$ -\delta_j^l =\sum_k \frac{\partial {\cal C}}{\partial z_k^{l+1}}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}=\sum_k \delta_k^{l+1}\frac{\partial z_k^{l+1}}{\partial z_j^{l}}, -$$ - -and recalling that - -$$ -z_j^{l+1} = \sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1}, -$$ - -with $M_l$ being the number of nodes in layer $l$, we obtain - -$$ -\delta_j^l =\sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l), -$$ - -This is our final equation. - -We are now ready to set up the algorithm for back propagation and learning the weights and biases. - - -### Setting up the Back propagation algorithm - -The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm. - -First, we set up the input data $\hat{x}$ and the activations -$\hat{z}_1$ of the input layer and compute the activation function and -the pertinent outputs $\hat{a}^1$. - - - -Secondly, we perform then the feed forward till we reach the output -layer and compute all $\hat{z}_l$ of the input layer and compute the -activation function and the pertinent outputs $\hat{a}^l$ for -$l=2,3,\dots,L$. - - - -Thereafter we compute the ouput error $\hat{\delta}^L$ by computing all - -$$ -\delta_j^L = f'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}. -$$ - -Then we compute the back propagate error for each $l=L-1,L-2,\dots,2$ as - -$$ -\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}f'(z_j^l). -$$ - -Finally, we update the weights and the biases using gradient descent for each $l=L-1,L-2,\dots,2$ and update the weights and biases according to the rules - -$$ -w_{jk}^l\leftarrow = w_{jk}^l- \eta \delta_j^la_k^{l-1}, -$$ - -$$ -b_j^l \leftarrow 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z-ch_^+-#b4A{1&aJ@M@JnLNFncy7e-dom1K|l9kx777;G*H32El!%dxVgj1ynK{&xkcL<$Lb zj8@#2KufAj$j3$NFqOucYsw^{sZibcQ;YYjEHqI`%-Z6McZ=pZ#pr*Pr|gUv>xd)m zwPzT$KR2)_xh$urd|{D+pK70FqMD{dLx@AtjCtbtV7U=jGy}XG4q)b7BCMSJmx#8U mKe7H> it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available.\n", - "\n", - " * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from .\n", - "\n", - "When dealing with matrices and vectors a central issue is memory\n", - "handling and allocation. If our code is written in Python the way we\n", - "declare these objects and the way they are handled, interpreted and\n", - "used by say a linear algebra library, requires codes that interface\n", - "our Python program with such libraries. For Python programmers,\n", - "**Numpy** is by now the standard Python package for numerical arrays in\n", - "Python as well as the source of functions which act on these\n", - "arrays. These functions span from eigenvalue solvers to functions that\n", - "compute the mean value, variance or the covariance matrix. If you are\n", - "not familiar with how arrays are handled in say Python or compiled\n", - "languages like C++ and Fortran, the sections in this chapter may be\n", - "useful. For C++ programmer, **Armadillo** is widely used library for\n", - "linear algebra and eigenvalue problems. In addition it offers a\n", - "convenient way to handle and organize arrays. We discuss this library\n", - "as well. Before we proceed we believe it may be convenient to repeat some basic features of \n", - " matrices and vectors.\n", - "\n", - "\n", - "## Basic Matrix Features\n", - "\n", - "Matrix properties reminder" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A} =\n", - " \\begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", - " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", - " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", - " a_{41} & a_{42} & a_{43} & a_{44}\n", - " \\end{bmatrix}\\qquad\n", - "\\mathbf{I} =\n", - " \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\\n", - " 0 & 1 & 0 & 0 \\\\\n", - " 0 & 0 & 1 & 0 \\\\\n", - " 0 & 0 & 0 & 1\n", - " \\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The inverse of a matrix is defined by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}^{-1} \\cdot \\mathbf{A} = I\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \\left (A^{T} \\right )^{-1}$ real orthogonal $\\sum_k a_{ik} a_{jk} = \\sum_k a_{ki} a_{kj} = \\delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \\left (A^{\\dagger} \\right )^{-1}$ unitary $\\sum_k a_{ik} a_{jk}^{ * } = \\sum_k a_{ki}^{ * } a_{kj} = \\delta_{ij}$
\n", - "### Some famous Matrices\n", - "\n", - " * Diagonal if $a_{ij}=0$ for $i\\ne j$\n", - "\n", - " * Upper triangular if $a_{ij}=0$ for $i > j$\n", - "\n", - " * Lower triangular if $a_{ij}=0$ for $i < j$\n", - "\n", - " * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$\n", - "\n", - " * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$\n", - "\n", - " * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$\n", - "\n", - " * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$\n", - "\n", - " * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$\n", - "\n", - " * Banded, block upper triangular, block lower triangular....\n", - "\n", - "Some Equivalent Statements. For an $N\\times N$ matrix $\\mathbf{A}$ the following properties are all equivalent\n", - "\n", - " * If the inverse of $\\mathbf{A}$ exists, $\\mathbf{A}$ is nonsingular.\n", - "\n", - " * The equation $\\mathbf{Ax}=0$ implies $\\mathbf{x}=0$.\n", - "\n", - " * The rows of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * The columns of $\\mathbf{A}$ form a basis of $R^N$.\n", - "\n", - " * $\\mathbf{A}$ is a product of elementary matrices.\n", - "\n", - " * $0$ is not eigenvalue of $\\mathbf{A}$.\n", - "\n", - "## Numpy and arrays\n", - "[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[ 0.44960994 -0.31524949 -0.60668732 -1.03920139 -0.23088568 -0.05148059\n", - " -1.4727093 0.29019465 0.82846181 -0.09720925]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "n = 10\n", - "x = np.random.normal(size=n)\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$.\n", - "Another alternative is to declare a vector as follows" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1 2 3]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "x = np.array([1, 2, 3])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++\n", - "start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \\dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1.38629436 1.94591015 2.07944154]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4, 7, 8]))\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Here we have used Numpy's unary function $np.log$. This function is\n", - "highly tuned to compute array elements since the code is vectorized\n", - "and does not require looping. We normaly recommend that you use the\n", - "Numpy intrinsic functions instead of the corresponding **log** function\n", - "from Python's **math** module. The looping is done explicitely by the\n", - "**np.log** function. The alternative, and slower way to compute the\n", - "logarithms of a vector would be to write" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1 1 2]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "from math import log\n", - "x = np.array([4, 7, 8])\n", - "for i in range(0, len(x)):\n", - " x[i] = log(x[i])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We note that our code is much longer already and we need to import the **log** function from the **math** module. \n", - "The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automacally our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1.38629436 1.94591015 2.07944154]\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4, 7, 8], dtype = np.float64))\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is" - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "ename": "SyntaxError", - "evalue": "invalid syntax (, line 3)", - "output_type": "error", - "traceback": [ - "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m3\u001b[0m\n\u001b[0;31m print(x)\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" - ] - } - ], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", - "print(x)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "x = np.log(np.array([4.0, 7.0, 8.0])\n", - "print(x.itemsize)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Having defined vectors, we are now ready to try out matrices. We can define a $3 \\times 3 $ real matrix $\\hat{A}$\n", - "as (recall that we user lowercase letters for vectors and uppercase letters for matrices)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "# print the first column, row-major order and elements start with 0\n", - "print(A[:,0])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can continue this was by printing out other columns or rows. The example here prints out the second column" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ]))\n", - "# print the first column, row-major order and elements start with 0\n", - "print(A[1,:])" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to zero\n", - "A = np.zeros( (n, n) )\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or initializing all elements to" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to one\n", - "A = np.ones( (n, n) )\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or as unitarily distributed random numbers (see the material on random number generators in the statistics part)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import numpy as np\n", - "n = 10\n", - "# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \\in [0, 1]\n", - "A = np.random.rand(n, n)\n", - "print(A)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n", - "As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n", - "$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n", - " \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n", - " \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where for example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma_{xy} =\\frac{1}{n} \\sum_{i=0}^{n-1}(x_i- \\overline{x})(y_i- \\overline{y}).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. For a more in-depth discussion of the covariance and covariance matrix and its meaning, we refer you to the lectures on statistics. \n", - "The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $ 3\\times n$ matrix $\\hat{W}$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n", - " x_1 & y_1 & z_1 \\\\\n", - " x_2 & y_2 & z_2 \\\\\n", - " \\dots & \\dots & \\dots \\\\\n", - " x_{n-2} & y_{n-2} & z_{n-2} \\\\\n", - " x_{n-1} & y_{n-1} & z_{n-1}\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which in turn is converted into into the $3 times 3$ covariance matrix\n", - "$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. In our review of\n", - "statistical functions and quantities we will discuss more about the\n", - "meaning of the covariance matrix. Here we note that we can calculate\n", - "the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n", - "function **np.mean(x)**. We can also extract the eigenvalues of the\n", - "covariance matrix through the **np.linalg.eig()** function." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Importing various packages\n", - "import numpy as np\n", - "\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "z = x**3+np.random.normal(size=n)\n", - "print(np.mean(z))\n", - "W = np.vstack((x, y, z))\n", - "Sigma = np.cov(W)\n", - "print(Sigma)\n", - "Eigvals, Eigvecs = np.linalg.eig(Sigma)\n", - "print(Eigvals)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from scipy import sparse\n", - "eye = np.eye(4)\n", - "print(eye)\n", - "sparse_mtx = sparse.csr_matrix(eye)\n", - "print(sparse_mtx)\n", - "x = np.linspace(-10,10,100)\n", - "y = np.sin(x)\n", - "plt.plot(x,y,marker='x')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Gaussian Elimination\n", - "\n", - "We start with the linear set of equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}\\mathbf{x} = \\mathbf{w}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We assume also that the matrix $\\mathbf{A}$ is non-singular and that the\n", - "matrix elements along the diagonal satisfy $a_{ii} \\ne 0$. Simple $4\\times 4 $ example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{bmatrix}\n", - " a_{11}& a_{12} &a_{13}& a_{14}\\\\\n", - " a_{21}& a_{22} &a_{23}& a_{24}\\\\\n", - " a_{31}& a_{32} &a_{33}& a_{34}\\\\\n", - " a_{41}& a_{42} &a_{43}& a_{44}\\\\\n", - " \\end{bmatrix} \\begin{bmatrix}\n", - " x_1\\\\\n", - " x_2\\\\\n", - " x_3 \\\\\n", - " x_4 \\\\\n", - " \\end{bmatrix}\n", - " =\\begin{bmatrix}\n", - " w_1\\\\\n", - " w_2\\\\\n", - " w_3 \\\\\n", - " w_4\\\\\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown $x_1$\n", - "from the remaining $n-1$ equations. Then we use the new second equation to eliminate the second unknown\n", - "$x_2$ from the remaining $n-2$ equations. With $n-1$ such eliminations\n", - "we obtain a so-called upper triangular set of equations of the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=y_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_{22}x_2 + b_{23}x_3 + b_{24}x_4=y_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_{33}x_3 + b_{34}x_4=y_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "b_{44}x_4=y_4. \\nonumber\n", - "\\label{eq:gaussbacksub} \\tag{1}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can solve this system of equations recursively starting from $x_n$ (in our case $x_4$) and proceed with\n", - "what is called a backward substitution. \n", - "\n", - "\n", - "This process can be expressed mathematically as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " x_m = \\frac{1}{b_{mm}}\\left(y_m-\\sum_{k=m+1}^nb_{mk}x_k\\right)\\quad m=n-1,n-2,\\dots,1.\n", - "\\label{_auto1} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To arrive at such an upper triangular system of equations, we start by eliminating\n", - "the unknown $x_1$ for $j=2,n$. We achieve this by multiplying the first equation by $a_{j1}/a_{11}$ and then subtract\n", - "the result from the $j$th equation. We assume obviously that $a_{11}\\ne 0$ and that\n", - "$\\mathbf{A}$ is not singular.\n", - "\n", - "\n", - "Our actual $4\\times 4$ example reads after the first operation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{bmatrix}\n", - " a_{11}& a_{12} &a_{13}& a_{14}\\\\\n", - " 0& (a_{22}-\\frac{a_{21}a_{12}}{a_{11}}) &(a_{23}-\\frac{a_{21}a_{13}}{a_{11}}) & (a_{24}-\\frac{a_{21}a_{14}}{a_{11}})\\\\\n", - "0& (a_{32}-\\frac{a_{31}a_{12}}{a_{11}})& (a_{33}-\\frac{a_{31}a_{13}}{a_{11}})& (a_{34}-\\frac{a_{31}a_{14}}{a_{11}})\\\\\n", - "0&(a_{42}-\\frac{a_{41}a_{12}}{a_{11}}) &(a_{43}-\\frac{a_{41}a_{13}}{a_{11}}) & (a_{44}-\\frac{a_{41}a_{14}}{a_{11}}) \\\\\n", - " \\end{bmatrix} \\begin{bmatrix}\n", - " x_1\\\\\n", - " x_2\\\\\n", - " x_3 \\\\\n", - " x_4 \\\\\n", - " \\end{bmatrix} \n", - " =\\begin{bmatrix}\n", - " y_1\\\\\n", - " w_2^{(2)}\\\\\n", - " w_3^{(2)} \\\\\n", - " w_4^{(2)}\\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=y_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a^{(2)}_{22}x_2 + a^{(2)}_{23}x_3 + a^{(2)}_{24}x_4=w^{(2)}_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a^{(2)}_{32}x_2 + a^{(2)}_{33}x_3 + a^{(2)}_{34}x_4=w^{(2)}_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a^{(2)}_{42}x_2 + a^{(2)}_{43}x_3 + a^{(2)}_{44}x_4=w^{(2)}_4, \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \n", - "\\label{_auto2} \\tag{3}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The new coefficients are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " b_{1k} = a_{1k}^{(1)} \\quad k=1,\\dots,n,\n", - "\\label{_auto3} \\tag{4}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where each $a_{1k}^{(1)}$ is equal to the original $a_{1k}$ element. The other coefficients are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "a_{jk}^{(2)} = a_{jk}^{(1)}-\\frac{a_{j1}^{(1)}a_{1k}^{(1)}}{a_{11}^{(1)}} \\quad j,k=2,\\dots,n,\n", - "\\label{_auto4} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with a new right-hand side given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "y_{1}=w_1^{(1)}, \\quad w_j^{(2)} =w_j^{(1)}-\\frac{a_{j1}^{(1)}w_1^{(1)}}{a_{11}^{(1)}} \\quad j=2,\\dots,n.\n", - "\\label{_auto5} \\tag{6}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We have also set $w_1^{(1)}=w_1$, the original vector element.\n", - "We see that the system of unknowns $x_1,\\dots,x_n$ is transformed into an $(n-1)\\times (n-1)$ problem.\n", - "\n", - "\n", - "\n", - "This step is called forward substitution.\n", - "Proceeding with these substitutions, we obtain the\n", - "general expressions for the new coefficients" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " a_{jk}^{(m+1)} = a_{jk}^{(m)}-\\frac{a_{jm}^{(m)}a_{mk}^{(m)}}{a_{mm}^{(m)}} \\quad j,k=m+1,\\dots,n,\n", - "\\label{_auto6} \\tag{7}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $m=1,\\dots,n-1$ and a\n", - "right-hand side given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " w_j^{(m+1)} =w_j^{(m)}-\\frac{a_{jm}^{(m)}w_m^{(m)}}{a_{mm}^{(m)}}\\quad j=m+1,\\dots,n.\n", - "\\label{_auto7} \\tag{8}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This set of $n-1$ elimations leads us to an equations which is solved by back substitution.\n", - "If the arithmetics is exact and the matrix $\\mathbf{A}$ is not singular, then the computed answer will be exact.\n", - "\n", - "Even though the matrix elements along the diagonal are not zero,\n", - "numerically small numbers may appear and subsequent divisions may lead to large numbers, which, if added\n", - "to a small number may yield losses of precision. Suppose for example that our first division in $(a_{22}-a_{21}a_{12}/a_{11})$\n", - "results in $-10^{-7}$ and that $a_{22}$ is one.\n", - "one. We are then\n", - "adding $10^7+1$. With single precision this results in $10^7$.\n", - "\n", - "\n", - "\n", - "\n", - " * Gaussian elimination, $O(2/3n^3)$ flops, general matrix\n", - "\n", - " * LU decomposition, upper triangular and lower tridiagonal matrices, $O(2/3n^3)$ flops, general matrix. Get easily the inverse, determinant and can solve linear equations with back-substitution only, $O(n^2)$ flops\n", - "\n", - " * Cholesky decomposition. Real symmetric or hermitian positive definite matrix, $O(1/3n^3)$ flops.\n", - "\n", - " * Tridiagonal linear systems, important for differential equations. Normally positive definite and non-singular. $O(8n)$ flops for symmetric. Special case of banded matrices.\n", - "\n", - " * Singular value decomposition\n", - "\n", - " * the QR method will be discussed in chapter 7 in connection with eigenvalue systems. $O(4/3n^3)$ flops.\n", - "\n", - "The LU decomposition method means that we can rewrite\n", - "this matrix as the product of two matrices $\\mathbf{L}$ and $\\mathbf{U}$\n", - "where" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{bmatrix}\n", - " a_{11} & a_{12} & a_{13} & a_{14} \\\\\n", - " a_{21} & a_{22} & a_{23} & a_{24} \\\\\n", - " a_{31} & a_{32} & a_{33} & a_{34} \\\\\n", - " a_{41} & a_{42} & a_{43} & a_{44}\n", - " \\end{bmatrix}\n", - " = \\begin{bmatrix}\n", - " 1 & 0 & 0 & 0 \\\\\n", - " l_{21} & 1 & 0 & 0 \\\\\n", - " l_{31} & l_{32} & 1 & 0 \\\\\n", - " l_{41} & l_{42} & l_{43} & 1\n", - " \\end{bmatrix}\n", - " \\begin{bmatrix}\n", - " u_{11} & u_{12} & u_{13} & u_{14} \\\\\n", - " 0 & u_{22} & u_{23} & u_{24} \\\\\n", - " 0 & 0 & u_{33} & u_{34} \\\\\n", - " 0 & 0 & 0 & u_{44}\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "LU decomposition forms the backbone of other algorithms in linear algebra, such as the\n", - "solution of linear equations given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The above set of equations is conveniently solved by using LU decomposition as an intermediate step.\n", - "\n", - "The matrix $\\mathbf{A}\\in \\mathbb{R}^{n\\times n}$ has an LU factorization if the determinant\n", - "is different from zero. If the LU factorization exists and $\\mathbf{A}$ is non-singular, then the LU factorization\n", - "is unique and the determinant is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "det\\{\\mathbf{A}\\}=det\\{\\mathbf{LU}\\}= det\\{\\mathbf{L}\\}det\\{\\mathbf{U}\\}=u_{11}u_{22}\\dots u_{nn}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "There are at least three main advantages with LU decomposition compared with standard Gaussian elimination:\n", - "\n", - " * It is straightforward to compute the determinant of a matrix\n", - "\n", - " * If we have to solve sets of linear equations with the same matrix but with different vectors $\\mathbf{y}$, the number of FLOPS is of the order $n^3$.\n", - "\n", - " * The inverse is such an operation \n", - "\n", - "With the LU decomposition it is rather\n", - "simple to solve a system of linear equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This can be written in matrix form as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{Ax}=\\mathbf{w}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\mathbf{A}$ and $\\mathbf{w}$ are known and we have to solve for\n", - "$\\mathbf{x}$. Using the LU dcomposition we write" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A} \\mathbf{x} \\equiv \\mathbf{L} \\mathbf{U} \\mathbf{x} =\\mathbf{w}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The previous equation can be calculated in two steps" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{L} \\mathbf{y} = \\mathbf{w};\\qquad \\mathbf{Ux}=\\mathbf{y}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To show that this is correct we use to the LU decomposition\n", - "to rewrite our system of linear equations as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{LUx}=\\mathbf{w},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and since the determinant of $\\mathbf{L}$ is equal to 1 (by construction\n", - "since the diagonals of $\\mathbf{L}$ equal 1) we can use the inverse of\n", - "$\\mathbf{L}$ to obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{Ux}=\\mathbf{L^{-1}w}=\\mathbf{y},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which yields the intermediate step" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{L^{-1}w}=\\mathbf{y}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and as soon as we have $\\mathbf{y}$ we can obtain $\\mathbf{x}$\n", - "through $\\mathbf{Ux}=\\mathbf{y}$.\n", - "\n", - "\n", - "For our four-dimentional example this takes the form" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "y_1=w_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "l_{21}y_1 + y_2=w_2\\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "l_{31}y_1 + l_{32}y_2 + y_3 =w_3\\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "l_{41}y_1 + l_{42}y_2 + l_{43}y_3 + y_4=w_4. \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u_{11}x_1 +u_{12}x_2 +u_{13}x_3 + u_{14}x_4=y_1 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u_{22}x_2 + u_{23}x_3 + u_{24}x_4=y_2\\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u_{33}x_3 + u_{34}x_4=y_3\\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "u_{44}x_4=y_4 \\nonumber\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This example shows the basis for the algorithm\n", - "needed to solve the set of $n$ linear equations.\n", - "\n", - "\n", - "\n", - "The algorithm goes as follows\n", - "\n", - " * Set up the matrix $\\bf A$ and the vector $\\bf w$ with their correct dimensions. This determines the dimensionality of the unknown vector $\\bf x$.\n", - "\n", - " * Then LU decompose the matrix $\\bf A$ through a call to the function `ludcmp(double a, int n, int indx, double &d)`. This functions returns the LU decomposed matrix $\\bf A$, its determinant and the vector indx which keeps track of the number of interchanges of rows. If the determinant is zero, the solution is malconditioned.\n", - "\n", - " * Thereafter you call the function `lubksb(double a, int n, int indx, double w)` which uses the LU decomposed matrix $\\bf A$ and the vector $\\bf w$ and returns $\\bf x$ in the same place as $\\bf w$. Upon exit the original content in $\\bf w$ is destroyed. If you wish to keep this information, you should make a backup of it in your calling function.\n", - "\n", - "### LU Decomposition, the inverse of a matrix\n", - "\n", - "If the inverse exists then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}^{-1}\\mathbf{A}=\\mathbf{I},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "the identity matrix. With an LU decomposed matrix we can rewrite the last equation as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{LU}\\mathbf{A}^{-1}=\\mathbf{I}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we assume that the first column (that is column 1) of the inverse matrix\n", - "can be written as a vector with unknown entries" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{A}_1^{-1}= \\begin{bmatrix}\n", - " a_{11}^{-1} \\\\\n", - " a_{21}^{-1} \\\\\n", - " \\dots \\\\\n", - " a_{n1}^{-1} \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "then we have a linear set of equations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{LU}\\begin{bmatrix}\n", - " a_{11}^{-1} \\\\\n", - " a_{21}^{-1} \\\\\n", - " \\dots \\\\\n", - " a_{n1}^{-1} \\\\\n", - " \\end{bmatrix} =\\begin{bmatrix}\n", - " 1 \\\\\n", - " 0 \\\\\n", - " \\dots \\\\\n", - " 0 \\\\\n", - " \\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In a similar way we can compute the unknow entries of the second column," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathbf{LU}\\begin{bmatrix}\n", - " a_{12}^{-1} \\\\\n", - " a_{22}^{-1} \\\\\n", - " \\dots \\\\\n", - " a_{n2}^{-1} \\\\\n", - " \\end{bmatrix}=\\begin{bmatrix}\n", - " 0 \\\\\n", - " 1 \\\\\n", - " \\dots \\\\\n", - " 0 \\\\\n", - " \\end{bmatrix},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and continue till we have solved all $n$ sets of linear equations." - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/linalg.py b/doc/LectureNotes/_build/jupyter_execute/linalg.py deleted file mode 100644 index 729fa16cd..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/linalg.py +++ /dev/null @@ -1,759 +0,0 @@ -# Linear Algebra, Handling of Arrays and more Python Features - -## Introduction - -The aim of this set of lectures is to review some central linear algebra algorithms that we will need in our -data analysis part and in the construction of Machine Learning algorithms (ML). -This will allow us to introduce some central programming features of high-level languages like Python and -compiled languages like C++ and/or Fortran. - -As discussed in the introductory notes, these series of lectures focuses both on using -central Python packages like **tensorflow** and **scikit-learn** as well -as writing your own codes for some central ML algorithms. The -latter can be written in a language of your choice, be it Python, Julia, R, -Rust, C++, Fortran etc. In order to avoid confusion however, in these lectures we will limit our -attention to Python, C++ and Fortran. - - -## Important Matrix and vector handling packages - -There are several central software packages for linear algebra and eigenvalue problems. Several of the more -popular ones have been wrapped into ofter software packages like those from the widely used text **Numerical Recipes**. The original source codes in many of the available packages are often taken from the widely used -software package LAPACK, which follows two other popular packages -developed in the 1970s, namely EISPACK and LINPACK. We describe them shortly here. - - * LINPACK: package for linear equations and least square problems. - - * LAPACK:package for solving symmetric, unsymmetric and generalized eigenvalue problems. From LAPACK's website it is possible to download for free all source codes from this library. Both C/C++ and Fortran versions are available. - - * BLAS (I, II and III): (Basic Linear Algebra Subprograms) are routines that provide standard building blocks for performing basic vector and matrix operations. Blas I is vector operations, II vector-matrix operations and III matrix-matrix operations. Highly parallelized and efficient codes, all available for download from . - -When dealing with matrices and vectors a central issue is memory -handling and allocation. If our code is written in Python the way we -declare these objects and the way they are handled, interpreted and -used by say a linear algebra library, requires codes that interface -our Python program with such libraries. For Python programmers, -**Numpy** is by now the standard Python package for numerical arrays in -Python as well as the source of functions which act on these -arrays. These functions span from eigenvalue solvers to functions that -compute the mean value, variance or the covariance matrix. If you are -not familiar with how arrays are handled in say Python or compiled -languages like C++ and Fortran, the sections in this chapter may be -useful. For C++ programmer, **Armadillo** is widely used library for -linear algebra and eigenvalue problems. In addition it offers a -convenient way to handle and organize arrays. We discuss this library -as well. Before we proceed we believe it may be convenient to repeat some basic features of - matrices and vectors. - - -## Basic Matrix Features - -Matrix properties reminder - -$$ -\mathbf{A} = - \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix}\qquad -\mathbf{I} = - \begin{bmatrix} 1 & 0 & 0 & 0 \\ - 0 & 1 & 0 & 0 \\ - 0 & 0 & 1 & 0 \\ - 0 & 0 & 0 & 1 - \end{bmatrix} -$$ - -The inverse of a matrix is defined by - -$$ -\mathbf{A}^{-1} \cdot \mathbf{A} = I -$$ - - - - - - - - - - - - -
Relations Name matrix elements
$A = A^{T}$ symmetric $a_{ij} = a_{ji}$
$A = \left (A^{T} \right )^{-1}$ real orthogonal $\sum_k a_{ik} a_{jk} = \sum_k a_{ki} a_{kj} = \delta_{ij}$
$A = A^{ * }$ real matrix $a_{ij} = a_{ij}^{ * }$
$A = A^{\dagger}$ hermitian $a_{ij} = a_{ji}^{ * }$
$A = \left (A^{\dagger} \right )^{-1}$ unitary $\sum_k a_{ik} a_{jk}^{ * } = \sum_k a_{ki}^{ * } a_{kj} = \delta_{ij}$
-### Some famous Matrices - - * Diagonal if $a_{ij}=0$ for $i\ne j$ - - * Upper triangular if $a_{ij}=0$ for $i > j$ - - * Lower triangular if $a_{ij}=0$ for $i < j$ - - * Upper Hessenberg if $a_{ij}=0$ for $i > j+1$ - - * Lower Hessenberg if $a_{ij}=0$ for $i < j+1$ - - * Tridiagonal if $a_{ij}=0$ for $|i -j| > 1$ - - * Lower banded with bandwidth $p$: $a_{ij}=0$ for $i > j+p$ - - * Upper banded with bandwidth $p$: $a_{ij}=0$ for $i < j+p$ - - * Banded, block upper triangular, block lower triangular.... - -Some Equivalent Statements. For an $N\times N$ matrix $\mathbf{A}$ the following properties are all equivalent - - * If the inverse of $\mathbf{A}$ exists, $\mathbf{A}$ is nonsingular. - - * The equation $\mathbf{Ax}=0$ implies $\mathbf{x}=0$. - - * The rows of $\mathbf{A}$ form a basis of $R^N$. - - * The columns of $\mathbf{A}$ form a basis of $R^N$. - - * $\mathbf{A}$ is a product of elementary matrices. - - * $0$ is not eigenvalue of $\mathbf{A}$. - -## Numpy and arrays -[Numpy](http://www.numpy.org/) provides an easy way to handle arrays in Python. The standard way to import this library is as - -import numpy as np -n = 10 -x = np.random.normal(size=n) -print(x) - -Here we have defined a vector $x$ with $n=10$ elements with its values given by the Normal distribution $N(0,1)$. -Another alternative is to declare a vector as follows - -import numpy as np -x = np.array([1, 2, 3]) -print(x) - -Here we have defined a vector with three elements, with $x_0=1$, $x_1=2$ and $x_2=3$. Note that both Python and C++ -start numbering array elements from $0$ and on. This means that a vector with $n$ elements has a sequence of entities $x_0, x_1, x_2, \dots, x_{n-1}$. We could also let (recommended) Numpy to compute the logarithms of a specific array as - -import numpy as np -x = np.log(np.array([4, 7, 8])) -print(x) - -Here we have used Numpy's unary function $np.log$. This function is -highly tuned to compute array elements since the code is vectorized -and does not require looping. We normaly recommend that you use the -Numpy intrinsic functions instead of the corresponding **log** function -from Python's **math** module. The looping is done explicitely by the -**np.log** function. The alternative, and slower way to compute the -logarithms of a vector would be to write - -import numpy as np -from math import log -x = np.array([4, 7, 8]) -for i in range(0, len(x)): - x[i] = log(x[i]) -print(x) - -We note that our code is much longer already and we need to import the **log** function from the **math** module. -The attentive reader will also notice that the output is $[1, 1, 2]$. Python interprets automacally our numbers as integers (like the **automatic** keyword in C++). To change this we could define our array elements to be double precision numbers as - -import numpy as np -x = np.log(np.array([4, 7, 8], dtype = np.float64)) -print(x) - -or simply write them as double precision numbers (Python uses 64 bits as default for floating point type variables), that is - -import numpy as np -x = np.log(np.array([4.0, 7.0, 8.0]) -print(x) - -To check the number of bytes (remember that one byte contains eight bits for double precision variables), you can use simple use the **itemsize** functionality (the array $x$ is actually an object which inherits the functionalities defined in Numpy) as - -import numpy as np -x = np.log(np.array([4.0, 7.0, 8.0]) -print(x.itemsize) - -Having defined vectors, we are now ready to try out matrices. We can define a $3 \times 3 $ real matrix $\hat{A}$ -as (recall that we user lowercase letters for vectors and uppercase letters for matrices) - -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -print(A) - -If we use the **shape** function we would get $(3, 3)$ as output, that is verifying that our matrix is a $3\times 3$ matrix. We can slice the matrix and print for example the first column (Python organized matrix elements in a row-major order, see below) as - -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -# print the first column, row-major order and elements start with 0 -print(A[:,0]) - -We can continue this was by printing out other columns or rows. The example here prints out the second column - -import numpy as np -A = np.log(np.array([ [4.0, 7.0, 8.0], [3.0, 10.0, 11.0], [4.0, 5.0, 7.0] ])) -# print the first column, row-major order and elements start with 0 -print(A[1,:]) - -Numpy contains many other functionalities that allow us to slice, subdivide etc etc arrays. We strongly recommend that you look up the [Numpy website for more details](http://www.numpy.org/). Useful functions when defining a matrix are the **np.zeros** function which declares a matrix of a given dimension and sets all elements to zero - -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to zero -A = np.zeros( (n, n) ) -print(A) - -or initializing all elements to - -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to one -A = np.ones( (n, n) ) -print(A) - -or as unitarily distributed random numbers (see the material on random number generators in the statistics part) - -import numpy as np -n = 10 -# define a matrix of dimension 10 x 10 and set all elements to random numbers with x \in [0, 1] -A = np.random.rand(n, n) -print(A) - -As we will see throughout these lectures, there are several extremely useful functionalities in Numpy. -As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors -$\hat{x}, \hat{y}, \hat{z}$ with $n$ elements each. The covariance matrix is defined as - -$$ -\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ - \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ - \sigma_{zx} & \sigma_{zy} & \sigma_{zz} - \end{bmatrix}, -$$ - -where for example - -$$ -\sigma_{xy} =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}). -$$ - -The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. For a more in-depth discussion of the covariance and covariance matrix and its meaning, we refer you to the lectures on statistics. -The following simple function uses the **np.vstack** function which takes each vector of dimension $1\times n$ and produces a $ 3\times n$ matrix $\hat{W}$ - -$$ -\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\ - x_1 & y_1 & z_1 \\ - x_2 & y_2 & z_2 \\ - \dots & \dots & \dots \\ - x_{n-2} & y_{n-2} & z_{n-2} \\ - x_{n-1} & y_{n-1} & z_{n-1} - \end{bmatrix}, -$$ - -which in turn is converted into into the $3 times 3$ covariance matrix -$\hat{\Sigma}$ via the Numpy function **np.cov()**. In our review of -statistical functions and quantities we will discuss more about the -meaning of the covariance matrix. Here we note that we can calculate -the mean value of each set of samples $\hat{x}$ etc using the Numpy -function **np.mean(x)**. We can also extract the eigenvalues of the -covariance matrix through the **np.linalg.eig()** function. - -# Importing various packages -import numpy as np - -n = 100 -x = np.random.normal(size=n) -print(np.mean(x)) -y = 4+3*x+np.random.normal(size=n) -print(np.mean(y)) -z = x**3+np.random.normal(size=n) -print(np.mean(z)) -W = np.vstack((x, y, z)) -Sigma = np.cov(W) -print(Sigma) -Eigvals, Eigvecs = np.linalg.eig(Sigma) -print(Eigvals) - -%matplotlib inline - -import numpy as np -import matplotlib.pyplot as plt -from scipy import sparse -eye = np.eye(4) -print(eye) -sparse_mtx = sparse.csr_matrix(eye) -print(sparse_mtx) -x = np.linspace(-10,10,100) -y = np.sin(x) -plt.plot(x,y,marker='x') -plt.show() - -## Gaussian Elimination - -We start with the linear set of equations - -$$ -\mathbf{A}\mathbf{x} = \mathbf{w}. -$$ - -We assume also that the matrix $\mathbf{A}$ is non-singular and that the -matrix elements along the diagonal satisfy $a_{ii} \ne 0$. Simple $4\times 4 $ example - -$$ -\begin{bmatrix} - a_{11}& a_{12} &a_{13}& a_{14}\\ - a_{21}& a_{22} &a_{23}& a_{24}\\ - a_{31}& a_{32} &a_{33}& a_{34}\\ - a_{41}& a_{42} &a_{43}& a_{44}\\ - \end{bmatrix} \begin{bmatrix} - x_1\\ - x_2\\ - x_3 \\ - x_4 \\ - \end{bmatrix} - =\begin{bmatrix} - w_1\\ - w_2\\ - w_3 \\ - w_4\\ - \end{bmatrix}. -$$ - -or - -$$ -a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \nonumber -$$ - -$$ -a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \nonumber -$$ - -$$ -a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \nonumber -$$ - -$$ -a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \nonumber -$$ - -The basic idea of Gaussian elimination is to use the first equation to eliminate the first unknown $x_1$ -from the remaining $n-1$ equations. Then we use the new second equation to eliminate the second unknown -$x_2$ from the remaining $n-2$ equations. With $n-1$ such eliminations -we obtain a so-called upper triangular set of equations of the form - -$$ -b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=y_1 \nonumber -$$ - -$$ -b_{22}x_2 + b_{23}x_3 + b_{24}x_4=y_2 \nonumber -$$ - -$$ -b_{33}x_3 + b_{34}x_4=y_3 \nonumber -$$ - - -
- -$$ -b_{44}x_4=y_4. \nonumber -\label{eq:gaussbacksub} \tag{1} -$$ - -We can solve this system of equations recursively starting from $x_n$ (in our case $x_4$) and proceed with -what is called a backward substitution. - - -This process can be expressed mathematically as - - -
- -$$ -\begin{equation} - x_m = \frac{1}{b_{mm}}\left(y_m-\sum_{k=m+1}^nb_{mk}x_k\right)\quad m=n-1,n-2,\dots,1. -\label{_auto1} \tag{2} -\end{equation} -$$ - -To arrive at such an upper triangular system of equations, we start by eliminating -the unknown $x_1$ for $j=2,n$. We achieve this by multiplying the first equation by $a_{j1}/a_{11}$ and then subtract -the result from the $j$th equation. We assume obviously that $a_{11}\ne 0$ and that -$\mathbf{A}$ is not singular. - - -Our actual $4\times 4$ example reads after the first operation - -$$ -\begin{bmatrix} - a_{11}& a_{12} &a_{13}& a_{14}\\ - 0& (a_{22}-\frac{a_{21}a_{12}}{a_{11}}) &(a_{23}-\frac{a_{21}a_{13}}{a_{11}}) & (a_{24}-\frac{a_{21}a_{14}}{a_{11}})\\ -0& (a_{32}-\frac{a_{31}a_{12}}{a_{11}})& (a_{33}-\frac{a_{31}a_{13}}{a_{11}})& (a_{34}-\frac{a_{31}a_{14}}{a_{11}})\\ -0&(a_{42}-\frac{a_{41}a_{12}}{a_{11}}) &(a_{43}-\frac{a_{41}a_{13}}{a_{11}}) & (a_{44}-\frac{a_{41}a_{14}}{a_{11}}) \\ - \end{bmatrix} \begin{bmatrix} - x_1\\ - x_2\\ - x_3 \\ - x_4 \\ - \end{bmatrix} - =\begin{bmatrix} - y_1\\ - w_2^{(2)}\\ - w_3^{(2)} \\ - w_4^{(2)}\\ - \end{bmatrix}, -$$ - -or - -$$ -b_{11}x_1 +b_{12}x_2 +b_{13}x_3 + b_{14}x_4=y_1 \nonumber -$$ - -$$ -a^{(2)}_{22}x_2 + a^{(2)}_{23}x_3 + a^{(2)}_{24}x_4=w^{(2)}_2 \nonumber -$$ - -$$ -a^{(2)}_{32}x_2 + a^{(2)}_{33}x_3 + a^{(2)}_{34}x_4=w^{(2)}_3 \nonumber -$$ - -$$ -a^{(2)}_{42}x_2 + a^{(2)}_{43}x_3 + a^{(2)}_{44}x_4=w^{(2)}_4, \nonumber -$$ - - -
- -$$ -\begin{equation} -\label{_auto2} \tag{3} -\end{equation} -$$ - -The new coefficients are - - -
- -$$ -\begin{equation} - b_{1k} = a_{1k}^{(1)} \quad k=1,\dots,n, -\label{_auto3} \tag{4} -\end{equation} -$$ - -where each $a_{1k}^{(1)}$ is equal to the original $a_{1k}$ element. The other coefficients are - - -
- -$$ -\begin{equation} -a_{jk}^{(2)} = a_{jk}^{(1)}-\frac{a_{j1}^{(1)}a_{1k}^{(1)}}{a_{11}^{(1)}} \quad j,k=2,\dots,n, -\label{_auto4} \tag{5} -\end{equation} -$$ - -with a new right-hand side given by - - -
- -$$ -\begin{equation} -y_{1}=w_1^{(1)}, \quad w_j^{(2)} =w_j^{(1)}-\frac{a_{j1}^{(1)}w_1^{(1)}}{a_{11}^{(1)}} \quad j=2,\dots,n. -\label{_auto5} \tag{6} -\end{equation} -$$ - -We have also set $w_1^{(1)}=w_1$, the original vector element. -We see that the system of unknowns $x_1,\dots,x_n$ is transformed into an $(n-1)\times (n-1)$ problem. - - - -This step is called forward substitution. -Proceeding with these substitutions, we obtain the -general expressions for the new coefficients - - -
- -$$ -\begin{equation} - a_{jk}^{(m+1)} = a_{jk}^{(m)}-\frac{a_{jm}^{(m)}a_{mk}^{(m)}}{a_{mm}^{(m)}} \quad j,k=m+1,\dots,n, -\label{_auto6} \tag{7} -\end{equation} -$$ - -with $m=1,\dots,n-1$ and a -right-hand side given by - - -
- -$$ -\begin{equation} - w_j^{(m+1)} =w_j^{(m)}-\frac{a_{jm}^{(m)}w_m^{(m)}}{a_{mm}^{(m)}}\quad j=m+1,\dots,n. -\label{_auto7} \tag{8} -\end{equation} -$$ - -This set of $n-1$ elimations leads us to an equations which is solved by back substitution. -If the arithmetics is exact and the matrix $\mathbf{A}$ is not singular, then the computed answer will be exact. - -Even though the matrix elements along the diagonal are not zero, -numerically small numbers may appear and subsequent divisions may lead to large numbers, which, if added -to a small number may yield losses of precision. Suppose for example that our first division in $(a_{22}-a_{21}a_{12}/a_{11})$ -results in $-10^{-7}$ and that $a_{22}$ is one. -one. We are then -adding $10^7+1$. With single precision this results in $10^7$. - - - - - * Gaussian elimination, $O(2/3n^3)$ flops, general matrix - - * LU decomposition, upper triangular and lower tridiagonal matrices, $O(2/3n^3)$ flops, general matrix. Get easily the inverse, determinant and can solve linear equations with back-substitution only, $O(n^2)$ flops - - * Cholesky decomposition. Real symmetric or hermitian positive definite matrix, $O(1/3n^3)$ flops. - - * Tridiagonal linear systems, important for differential equations. Normally positive definite and non-singular. $O(8n)$ flops for symmetric. Special case of banded matrices. - - * Singular value decomposition - - * the QR method will be discussed in chapter 7 in connection with eigenvalue systems. $O(4/3n^3)$ flops. - -The LU decomposition method means that we can rewrite -this matrix as the product of two matrices $\mathbf{L}$ and $\mathbf{U}$ -where - -$$ -\begin{bmatrix} - a_{11} & a_{12} & a_{13} & a_{14} \\ - a_{21} & a_{22} & a_{23} & a_{24} \\ - a_{31} & a_{32} & a_{33} & a_{34} \\ - a_{41} & a_{42} & a_{43} & a_{44} - \end{bmatrix} - = \begin{bmatrix} - 1 & 0 & 0 & 0 \\ - l_{21} & 1 & 0 & 0 \\ - l_{31} & l_{32} & 1 & 0 \\ - l_{41} & l_{42} & l_{43} & 1 - \end{bmatrix} - \begin{bmatrix} - u_{11} & u_{12} & u_{13} & u_{14} \\ - 0 & u_{22} & u_{23} & u_{24} \\ - 0 & 0 & u_{33} & u_{34} \\ - 0 & 0 & 0 & u_{44} - \end{bmatrix}. -$$ - -LU decomposition forms the backbone of other algorithms in linear algebra, such as the -solution of linear equations given by - -$$ -a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \nonumber -$$ - -$$ -a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \nonumber -$$ - -$$ -a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \nonumber -$$ - -$$ -a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \nonumber -$$ - -The above set of equations is conveniently solved by using LU decomposition as an intermediate step. - -The matrix $\mathbf{A}\in \mathbb{R}^{n\times n}$ has an LU factorization if the determinant -is different from zero. If the LU factorization exists and $\mathbf{A}$ is non-singular, then the LU factorization -is unique and the determinant is given by - -$$ -det\{\mathbf{A}\}=det\{\mathbf{LU}\}= det\{\mathbf{L}\}det\{\mathbf{U}\}=u_{11}u_{22}\dots u_{nn}. -$$ - -There are at least three main advantages with LU decomposition compared with standard Gaussian elimination: - - * It is straightforward to compute the determinant of a matrix - - * If we have to solve sets of linear equations with the same matrix but with different vectors $\mathbf{y}$, the number of FLOPS is of the order $n^3$. - - * The inverse is such an operation - -With the LU decomposition it is rather -simple to solve a system of linear equations - -$$ -a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=w_1 \nonumber -$$ - -$$ -a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=w_2 \nonumber -$$ - -$$ -a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=w_3 \nonumber -$$ - -$$ -a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=w_4. \nonumber -$$ - -This can be written in matrix form as - -$$ -\mathbf{Ax}=\mathbf{w}. -$$ - -where $\mathbf{A}$ and $\mathbf{w}$ are known and we have to solve for -$\mathbf{x}$. Using the LU dcomposition we write - -$$ -\mathbf{A} \mathbf{x} \equiv \mathbf{L} \mathbf{U} \mathbf{x} =\mathbf{w}. -$$ - -The previous equation can be calculated in two steps - -$$ -\mathbf{L} \mathbf{y} = \mathbf{w};\qquad \mathbf{Ux}=\mathbf{y}. -$$ - -To show that this is correct we use to the LU decomposition -to rewrite our system of linear equations as - -$$ -\mathbf{LUx}=\mathbf{w}, -$$ - -and since the determinant of $\mathbf{L}$ is equal to 1 (by construction -since the diagonals of $\mathbf{L}$ equal 1) we can use the inverse of -$\mathbf{L}$ to obtain - -$$ -\mathbf{Ux}=\mathbf{L^{-1}w}=\mathbf{y}, -$$ - -which yields the intermediate step - -$$ -\mathbf{L^{-1}w}=\mathbf{y} -$$ - -and as soon as we have $\mathbf{y}$ we can obtain $\mathbf{x}$ -through $\mathbf{Ux}=\mathbf{y}$. - - -For our four-dimentional example this takes the form - -$$ -y_1=w_1 \nonumber -$$ - -$$ -l_{21}y_1 + y_2=w_2\nonumber -$$ - -$$ -l_{31}y_1 + l_{32}y_2 + y_3 =w_3\nonumber -$$ - -$$ -l_{41}y_1 + l_{42}y_2 + l_{43}y_3 + y_4=w_4. \nonumber -$$ - -and - -$$ -u_{11}x_1 +u_{12}x_2 +u_{13}x_3 + u_{14}x_4=y_1 \nonumber -$$ - -$$ -u_{22}x_2 + u_{23}x_3 + u_{24}x_4=y_2\nonumber -$$ - -$$ -u_{33}x_3 + u_{34}x_4=y_3\nonumber -$$ - -$$ -u_{44}x_4=y_4 \nonumber -$$ - -This example shows the basis for the algorithm -needed to solve the set of $n$ linear equations. - - - -The algorithm goes as follows - - * Set up the matrix $\bf A$ and the vector $\bf w$ with their correct dimensions. This determines the dimensionality of the unknown vector $\bf x$. - - * Then LU decompose the matrix $\bf A$ through a call to the function `ludcmp(double a, int n, int indx, double &d)`. This functions returns the LU decomposed matrix $\bf A$, its determinant and the vector indx which keeps track of the number of interchanges of rows. If the determinant is zero, the solution is malconditioned. - - * Thereafter you call the function `lubksb(double a, int n, int indx, double w)` which uses the LU decomposed matrix $\bf A$ and the vector $\bf w$ and returns $\bf x$ in the same place as $\bf w$. Upon exit the original content in $\bf w$ is destroyed. If you wish to keep this information, you should make a backup of it in your calling function. - -### LU Decomposition, the inverse of a matrix - -If the inverse exists then - -$$ -\mathbf{A}^{-1}\mathbf{A}=\mathbf{I}, -$$ - -the identity matrix. With an LU decomposed matrix we can rewrite the last equation as - -$$ -\mathbf{LU}\mathbf{A}^{-1}=\mathbf{I}. -$$ - -If we assume that the first column (that is column 1) of the inverse matrix -can be written as a vector with unknown entries - -$$ -\mathbf{A}_1^{-1}= \begin{bmatrix} - a_{11}^{-1} \\ - a_{21}^{-1} \\ - \dots \\ - a_{n1}^{-1} \\ - \end{bmatrix}, -$$ - -then we have a linear set of equations - -$$ -\mathbf{LU}\begin{bmatrix} - a_{11}^{-1} \\ - a_{21}^{-1} \\ - \dots \\ - a_{n1}^{-1} \\ - \end{bmatrix} =\begin{bmatrix} - 1 \\ - 0 \\ - \dots \\ - 0 \\ - \end{bmatrix}. -$$ - -In a similar way we can compute the unknow entries of the second column, - -$$ -\mathbf{LU}\begin{bmatrix} - a_{12}^{-1} \\ - a_{22}^{-1} \\ - \dots \\ - a_{n2}^{-1} \\ - \end{bmatrix}=\begin{bmatrix} - 0 \\ - 1 \\ - \dots \\ - 0 \\ - \end{bmatrix}, -$$ - -and continue till we have solved all $n$ sets of linear equations. \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/statistics.ipynb b/doc/LectureNotes/_build/jupyter_execute/statistics.ipynb deleted file mode 100644 index bb37fef09..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/statistics.ipynb +++ /dev/null @@ -1,3003 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Elements of Probability Theory and Statistical Data Analysis\n", - "\n", - "\n", - "## Domains and probabilities\n", - "Consider the following simple example, namely the tossing of two dice, resulting in the following possible values" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{2,3,4,5,6,7,8,9,10,11,12\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These values are called the *domain*. \n", - "To this domain we have the corresponding *probabilities*" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Tossing the dice\n", - "The numbers in the domain are the outcomes of the physical process of tossing say two dice.\n", - "We cannot tell beforehand whether the outcome is 3 or 5 or any other number in this domain.\n", - "This defines the randomness of the outcome, or unexpectedness or any other synonimous word which\n", - "encompasses the uncertitude of the final outcome. \n", - "\n", - "The only thing we can tell beforehand\n", - "is that say the outcome 2 has a certain probability. \n", - "If our favorite hobby is to spend an hour every evening throwing dice and \n", - "registering the sequence of outcomes, we will note that the numbers in the above domain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{2,3,4,5,6,7,8,9,10,11,12\\},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "appear in a random order. After 11 throws the results may look like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{10,8,6,3,6,9,11,8,12,4,5\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Stochastic variables\n", - "\n", - "**Random variables are characterized by a domain which contains all possible values that the random value may take. This domain has a corresponding probability distribution function(PDF)**.\n", - "\n", - "\n", - "\n", - "\n", - "## Stochastic variables and the main concepts, the discrete case\n", - "There are two main concepts associated with a stochastic variable. The\n", - "*domain* is the set $\\mathbb D = \\{x\\}$ of all accessible values\n", - "the variable can assume, so that $X \\in \\mathbb D$. An example of a\n", - "discrete domain is the set of six different numbers that we may get by\n", - "throwing of a dice, $x\\in\\{1,\\,2,\\,3,\\,4,\\,5,\\,6\\}$.\n", - "\n", - "The *probability distribution function (PDF)* is a function\n", - "$p(x)$ on the domain which, in the discrete case, gives us the\n", - "probability or relative frequency with which these values of $X$\n", - "occur" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\mathrm{Prob}(X=x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Stochastic variables and the main concepts, the continuous case\n", - "In the continuous case, the PDF does not directly depict the\n", - "actual probability. Instead we define the probability for the\n", - "stochastic variable to assume any value on an infinitesimal interval\n", - "around $x$ to be $p(x)dx$. The continuous function $p(x)$ then gives us\n", - "the *density* of the probability rather than the probability\n", - "itself. The probability for a stochastic variable to assume any value\n", - "on a non-infinitesimal interval $[a,\\,b]$ is then just the integral" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{Prob}(a\\leq X\\leq b) = \\int_a^b p(x)dx.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Qualitatively speaking, a stochastic variable represents the values of\n", - "numbers chosen as if by chance from some specified PDF so that the\n", - "selection of a large set of these numbers reproduces this PDF.\n", - "\n", - "\n", - "\n", - "\n", - "## The cumulative probability\n", - "Of interest to us is the *cumulative probability\n", - "distribution function* (**CDF**), $P(x)$, which is just the probability\n", - "for a stochastic variable $X$ to assume any value less than $x$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(x)=\\mathrm{Prob(}X\\leq x\\mathrm{)} =\n", - "\\int_{-\\infty}^x p(x^{\\prime})dx^{\\prime}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The relation between a CDF and its corresponding PDF is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\frac{d}{dx}P(x).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Properties of PDFs\n", - "\n", - "There are two properties that all PDFs must satisfy. The first one is\n", - "positivity (assuming that the PDF is normalized)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "0 \\leq p(x) \\leq 1.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Naturally, it would be nonsensical for any of the values of the domain\n", - "to occur with a probability greater than $1$ or less than $0$. Also,\n", - "the PDF must be normalized. That is, all the probabilities must add up\n", - "to unity. The probability of \"anything\" to happen is always unity. For\n", - "both discrete and continuous PDFs, this condition is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\sum_{x_i\\in\\mathbb D} p(x_i) & = 1,\\\\\n", - "\\int_{x\\in\\mathbb D} p(x)\\,dx & = 1.\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Important distributions, the uniform distribution\n", - "The first one\n", - "is the most basic PDF; namely the uniform distribution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "p(x) = \\frac{1}{b-a}\\theta(x-a)\\theta(b-x).\n", - "\\label{eq:unifromPDF} \\tag{1}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "For $a=0$ and $b=1$ we have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{array}{ll}\n", - "p(x)dx = dx & \\in [0,1].\n", - "\\end{array}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The latter distribution is used to generate random numbers. For other PDFs, one needs normally a mapping from this distribution to say for example the exponential distribution.\n", - "\n", - "\n", - "\n", - "\n", - "## Gaussian distribution\n", - "The second one is the Gaussian Distribution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} \\exp{(-\\frac{(x-\\mu)^2}{2\\sigma^2})},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with mean value $\\mu$ and standard deviation $\\sigma$. If $\\mu=0$ and $\\sigma=1$, it is normally called the **standard normal distribution**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\frac{1}{\\sqrt{2\\pi}} \\exp{(-\\frac{x^2}{2})},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The following simple Python code plots the above distribution for different values of $\\mu$ and $\\sigma$." - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "data": { - "image/png": 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\n", - "text/plain": [ - "
" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/statistics_29_0.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "%matplotlib inline\n", - "\n", - "import numpy as np\n", - "from math import acos, exp, sqrt\n", - "from matplotlib import pyplot as plt\n", - "from matplotlib import rc, rcParams\n", - "import matplotlib.units as units\n", - "import matplotlib.ticker as ticker\n", - "rc('text',usetex=True)\n", - "rc('font',**{'family':'serif','serif':['Gaussian distribution']})\n", - "font = {'family' : 'serif',\n", - " 'color' : 'darkred',\n", - " 'weight' : 'normal',\n", - " 'size' : 16,\n", - " }\n", - "pi = acos(-1.0)\n", - "mu0 = 0.0\n", - "sigma0 = 1.0\n", - "mu1= 1.0\n", - "sigma1 = 2.0\n", - "mu2 = 2.0\n", - "sigma2 = 4.0\n", - "\n", - "x = np.linspace(-20.0, 20.0)\n", - "v0 = np.exp(-(x*x-2*x*mu0+mu0*mu0)/(2*sigma0*sigma0))/sqrt(2*pi*sigma0*sigma0)\n", - "v1 = np.exp(-(x*x-2*x*mu1+mu1*mu1)/(2*sigma1*sigma1))/sqrt(2*pi*sigma1*sigma1)\n", - "v2 = np.exp(-(x*x-2*x*mu2+mu2*mu2)/(2*sigma2*sigma2))/sqrt(2*pi*sigma2*sigma2)\n", - "plt.plot(x, v0, 'b-', x, v1, 'r-', x, v2, 'g-')\n", - "plt.title(r'{\\bf Gaussian distributions}', fontsize=20)\n", - "plt.text(-19, 0.3, r'Parameters: $\\mu = 0$, $\\sigma = 1$', fontdict=font)\n", - "plt.text(-19, 0.18, r'Parameters: $\\mu = 1$, $\\sigma = 2$', fontdict=font)\n", - "plt.text(-19, 0.08, r'Parameters: $\\mu = 2$, $\\sigma = 4$', fontdict=font)\n", - "plt.xlabel(r'$x$',fontsize=20)\n", - "plt.ylabel(r'$p(x)$ [MeV]',fontsize=20)\n", - "\n", - "# Tweak spacing to prevent clipping of ylabel \n", - "plt.subplots_adjust(left=0.15)\n", - "plt.savefig('gaussian.pdf', format='pdf')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Exponential distribution\n", - "Another important distribution in science is the exponential distribution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\alpha\\exp{-(\\alpha x)}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Expectation values\n", - "Let $h(x)$ be an arbitrary continuous function on the domain of the stochastic\n", - "variable $X$ whose PDF is $p(x)$. We define the *expectation value*\n", - "of $h$ with respect to $p$ as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\langle h \\rangle_X \\equiv \\int\\! h(x)p(x)\\,dx\n", - "\\label{eq:expectation_value_of_h_wrt_p} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Whenever the PDF is known implicitly, like in this case, we will drop\n", - "the index $X$ for clarity. \n", - "A particularly useful class of special expectation values are the\n", - "*moments*. The $n$-th moment of the PDF $p$ is defined as\n", - "follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x^n \\rangle \\equiv \\int\\! x^n p(x)\\,dx\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Stochastic variables and the main concepts, mean values\n", - "The zero-th moment $\\langle 1\\rangle$ is just the normalization condition of\n", - "$p$. The first moment, $\\langle x\\rangle$, is called the *mean* of $p$\n", - "and often denoted by the letter $\\mu$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x\\rangle = \\mu \\equiv \\int x p(x)dx,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for a continuous distribution and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x\\rangle = \\mu \\equiv \\sum_{i=1}^N x_i p(x_i),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for a discrete distribution. \n", - "Qualitatively it represents the centroid or the average value of the\n", - "PDF and is therefore simply called the expectation value of $p(x)$.\n", - "\n", - "\n", - "\n", - "\n", - "## Stochastic variables and the main concepts, central moments, the variance\n", - "\n", - "A special version of the moments is the set of *central moments*, the n-th central moment defined as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle (x-\\langle x\\rangle )^n\\rangle \\equiv \\int\\! (x-\\langle x\\rangle)^n p(x)\\,dx\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The zero-th and first central moments are both trivial, equal $1$ and\n", - "$0$, respectively. But the second central moment, known as the\n", - "*variance* of $p$, is of particular interest. For the stochastic\n", - "variable $X$, the variance is denoted as $\\sigma^2_X$ or $\\mathrm{Var}(X)$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\\sigma^2_X &=\\mathrm{Var}(X) = \\langle (x-\\langle x\\rangle)^2\\rangle =\n", - "\\int (x-\\langle x\\rangle)^2 p(x)dx\\\\\n", - "& = \\int\\left(x^2 - 2 x \\langle x\\rangle^{2} +\\langle x\\rangle^2\\right)p(x)dx\\\\\n", - "& = \\langle x^2\\rangle - 2 \\langle x\\rangle\\langle x\\rangle + \\langle x\\rangle^2\\\\\n", - "& = \\langle x^2 \\rangle - \\langle x\\rangle^2\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The square root of the variance, $\\sigma =\\sqrt{\\langle (x-\\langle x\\rangle)^2\\rangle}$ is called the \n", - "**standard deviation** of $p$. It is the RMS (root-mean-square)\n", - "value of the deviation of the PDF from its mean value, interpreted\n", - "qualitatively as the \"spread\" of $p$ around its mean.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions\n", - "\n", - "The following table collects properties of probability distribution functions.\n", - "In our notation we reserve the label $p(x)$ for the probability of a certain event,\n", - "while $P(x)$ is the cumulative probability. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
Discrete PDF Continuous PDF
Domain $\\left\\{x_1, x_2, x_3, \\dots, x_N\\right\\}$ $[a,b]$
Probability $p(x_i)$ $p(x)dx$
Cumulative $P_i=\\sum_{l=1}^ip(x_l)$ $P(x)=\\int_a^xp(t)dt$
Positivity $0 \\le p(x_i) \\le 1$ $p(x) \\ge 0$
Positivity $0 \\le P_i \\le 1$ $0 \\le P(x) \\le 1$
Monotonic $P_i \\ge P_j$ if $x_i \\ge x_j$ $P(x_i) \\ge P(x_j)$ if $x_i \\ge x_j$
Normalization $P_N=1$ $P(b)=1$
\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions\n", - "With a PDF we can compute expectation values of selected quantities such as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x^k\\rangle=\\sum_{i=1}^{N}x_i^kp(x_i),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "if we have a discrete PDF or" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x^k\\rangle=\\int_a^b x^kp(x)dx,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "in the case of a continuous PDF. We have already defined the mean value $\\mu$\n", - "and the variance $\\sigma^2$.\n", - "\n", - "\n", - "\n", - "\n", - "## The three famous Probability Distribution Functions\n", - "\n", - "There are at least three PDFs which one may encounter. These are the\n", - "\n", - "**Uniform distribution**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x)=\\frac{1}{b-a}\\Theta(x-a)\\Theta(b-x),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "yielding probabilities different from zero in the interval $[a,b]$.\n", - "\n", - "**The exponential distribution**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x)=\\alpha \\exp{(-\\alpha x)},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "yielding probabilities different from zero in the interval $[0,\\infty)$ and with mean value" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \\int_0^{\\infty}xp(x)dx=\\int_0^{\\infty}x\\alpha \\exp{(-\\alpha x)}dx=\\frac{1}{\\alpha},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with variance" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2=\\int_0^{\\infty}x^2p(x)dx-\\mu^2 = \\frac{1}{\\alpha^2}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Probability Distribution Functions, the normal distribution\n", - "Finally, we have the so-called univariate normal distribution, or just the **normal distribution**" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x)=\\frac{1}{b\\sqrt{2\\pi}}\\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with probabilities different from zero in the interval $(-\\infty,\\infty)$.\n", - "The integral $\\int_{-\\infty}^{\\infty}\\exp{\\left(-(x^2\\right)}dx$ appears in many calculations, its value\n", - "is $\\sqrt{\\pi}$, a result we will need when we compute the mean value and the variance.\n", - "The mean value is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \\int_0^{\\infty}xp(x)dx=\\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}x \\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}dx,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which becomes with a suitable change of variables" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu =\\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}b\\sqrt{2}(a+b\\sqrt{2}y)\\exp{-y^2}dy=a.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Probability Distribution Functions, the normal distribution\n", - "Similarly, the variance becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2 = \\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}(x-\\mu)^2 \\exp{\\left(-\\frac{(x-a)^2}{2b^2}\\right)}dx,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and inserting the mean value and performing a variable change we obtain" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2 = \\frac{1}{b\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty}b\\sqrt{2}(b\\sqrt{2}y)^2\\exp{\\left(-y^2\\right)}dy=\n", - "\\frac{2b^2}{\\sqrt{\\pi}}\\int_{-\\infty}^{\\infty}y^2\\exp{\\left(-y^2\\right)}dy,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and performing a final integration by parts we obtain the well-known result $\\sigma^2=b^2$.\n", - "It is useful to introduce the standard normal distribution as well, defined by $\\mu=a=0$, viz. a distribution\n", - "centered around zero and with a variance $\\sigma^2=1$, leading to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " p(x)=\\frac{1}{\\sqrt{2\\pi}}\\exp{\\left(-\\frac{x^2}{2}\\right)}.\n", - "\\label{_auto1} \\tag{3}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Probability Distribution Functions, the cumulative distribution\n", - "\n", - "The exponential and uniform distributions have simple cumulative functions,\n", - "whereas the normal distribution does not, being proportional to the so-called\n", - "error function $erf(x)$, given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "P(x) = \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^x\\exp{\\left(-\\frac{t^2}{2}\\right)}dt,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which is difficult to evaluate in a quick way.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions, other important distribution\n", - "\n", - "Some other PDFs which one encounters often in the natural sciences are the binomial distribution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} \\hspace{0.5cm}x=0,1,\\dots,n,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $y$ is the probability for a specific event, such as the tossing of a coin or moving left or right\n", - "in case of a random walker. Note that $x$ is a discrete stochastic variable. \n", - "\n", - "The sequence of binomial trials is characterized by the following definitions\n", - "\n", - " * Every experiment is thought to consist of $N$ independent trials.\n", - "\n", - " * In every independent trial one registers if a specific situation happens or not, such as the jump to the left or right of a random walker.\n", - "\n", - " * The probability for every outcome in a single trial has the same value, for example the outcome of tossing (either heads or tails) a coin is always $1/2$.\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions, the binomial distribution\n", - "\n", - "In order to compute the mean and variance we need to recall Newton's binomial\n", - "formula" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(a+b)^m=\\sum_{n=0}^m \\left(\\begin{array}{c} m \\\\ n\\end{array}\\right)a^nb^{m-n},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which can be used to show that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sum_{x=0}^n\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "the PDF is normalized to one. \n", - "The mean value is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \\sum_{x=0}^n x\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} =\n", - "\\sum_{x=0}^n x\\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "resulting in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \n", - "\\sum_{x=0}^n x\\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which we rewrite as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu=ny\\sum_{\\nu=0}^n\\left(\\begin{array}{c} n-1 \\\\ \\nu\\end{array}\\right)y^{\\nu}(1-y)^{n-1-\\nu} =ny(y+1-y)^{n-1}=ny.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The variance is slightly trickier to get. It reads $\\sigma^2=ny(1-y)$. \n", - "\n", - "\n", - "## Probability Distribution Functions, Poisson's distribution\n", - "\n", - "Another important distribution with discrete stochastic variables $x$ is \n", - "the Poisson model, which resembles the exponential distribution and reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "p(x) = \\frac{\\lambda^x}{x!} e^{-\\lambda} \\hspace{0.5cm}x=0,1,\\dots,;\\lambda > 0.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case both the mean value and the variance are easier to calculate," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu = \\sum_{x=0}^{\\infty} x \\frac{\\lambda^x}{x!} e^{-\\lambda} = \\lambda e^{-\\lambda}\\sum_{x=1}^{\\infty}\n", - "\\frac{\\lambda^{x-1}}{(x-1)!}=\\lambda,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and the variance is $\\sigma^2=\\lambda$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Probability Distribution Functions, Poisson's distribution\n", - "An example of applications of the Poisson distribution could be the counting\n", - "of the number of $\\alpha$-particles emitted from a radioactive source in a given time interval.\n", - "In the limit of $n\\rightarrow \\infty$ and for small probabilities $y$, the binomial distribution\n", - "approaches the Poisson distribution. Setting $\\lambda = ny$, with $y$ the probability for an event in\n", - "the binomial distribution we can show that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\lim_{n\\rightarrow \\infty}\\left(\\begin{array}{c} n \\\\ x\\end{array}\\right)y^x(1-y)^{n-x} e^{-\\lambda}=\\sum_{x=1}^{\\infty}\\frac{\\lambda^x}{x!} e^{-\\lambda}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Meet the covariance!\n", - "An important quantity in a statistical analysis is the so-called covariance. \n", - "\n", - "Consider the set $\\{X_i\\}$ of $n$\n", - "stochastic variables (not necessarily uncorrelated) with the\n", - "multivariate PDF $P(x_1,\\dots,x_n)$. The *covariance* of two\n", - "of the stochastic variables, $X_i$ and $X_j$, is defined as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\mathrm{Cov}(X_i,\\,X_j) = \\langle (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle \n", - "\\label{_auto2} \\tag{4}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \n", - "=\\int\\cdots\\int (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)P(x_1,\\dots,x_n)\\,dx_1\\dots dx_n,\n", - "\\label{eq:def_covariance} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x_i\\rangle =\n", - "\\int\\cdots\\int x_i P(x_1,\\dots,x_n)\\,dx_1\\dots dx_n.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Meet the covariance in matrix disguise\n", - "If we consider the above covariance as a matrix" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C_{ij} =\\mathrm{Cov}(X_i,\\,X_j),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "then the diagonal elements are just the familiar\n", - "variances, $C_{ii} = \\mathrm{Cov}(X_i,\\,X_i) = \\mathrm{Var}(X_i)$. It turns out that\n", - "all the off-diagonal elements are zero if the stochastic variables are\n", - "uncorrelated.\n", - "\n", - "\n", - "\n", - "\n", - "## Covariance" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "4.752993882123058\n", - "[[ 4.70339379 7.80064137 18.20965525 1.99515112 14.01360746 10.49675434\n", - " 5.81431951 6.88529462 8.48275939 8.04252289]\n", - " [ 7.80064137 12.93746782 30.20095624 3.30898476 23.24175499 17.40900715\n", - " 9.64312649 11.41935301 14.06876966 13.33863154]\n", - " [18.20965525 30.20095624 70.50048516 7.72442532 54.25507027 40.63922482\n", - " 22.51071426 26.65710056 32.84184377 31.13742451]\n", - " [ 1.99515112 3.30898476 7.72442532 0.84633101 5.9444873 4.45265953\n", - " 2.46639907 2.92070022 3.59833509 3.41158944]\n", - " [14.01360746 23.24175499 54.25507027 5.9444873 41.75308359 31.27473511\n", - " 17.32357417 20.51450938 25.27410326 23.96243304]\n", - " [10.49675434 17.40900715 40.63922482 4.45265953 31.27473511 23.42603161\n", - " 12.97605223 15.36619075 18.93131756 17.94882393]\n", - " [ 5.81431951 9.64312649 22.51071426 2.46639907 17.32357417 12.97605223\n", - " 7.18764212 8.51157793 10.48635849 9.94213961]\n", - " [ 6.88529462 11.41935301 26.65710056 2.92070022 20.51450938 15.36619075\n", - " 8.51157793 10.07937759 12.41790507 11.77344318]\n", - " [ 8.48275939 14.06876966 32.84184377 3.59833509 25.27410326 18.93131756\n", - " 10.48635849 12.41790507 15.29899688 14.50501268]\n", - " [ 8.04252289 13.33863154 31.13742451 3.41158944 23.96243304 17.94882393\n", - " 9.94213961 11.77344318 14.50501268 13.75223451]]\n" - ] - } - ], - "source": [ - "# Importing various packages\n", - "from math import exp, sqrt\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "\n", - "def covariance(x, y, n):\n", - " sum = 0.0\n", - " mean_x = np.mean(x)\n", - " mean_y = np.mean(y)\n", - " for i in range(0, n):\n", - " sum += (x[(i)]-mean_x)*(y[i]-mean_y)\n", - " return sum/n\n", - "\n", - "n = 10\n", - "\n", - "x=np.random.normal(size=n)\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "covxy = covariance(x,y,n)\n", - "print(covxy)\n", - "z = np.vstack((x, y))\n", - "c = np.cov(z.T)\n", - "\n", - "print(c)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Meet the covariance, uncorrelated events\n", - "\n", - "Consider the stochastic variables $X_i$ and $X_j$, ($i\\neq j$). We have" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "Cov(X_i,\\,X_j) &= \\langle (x_i-\\langle x_i\\rangle)(x_j-\\langle x_j\\rangle)\\rangle\\\\\n", - "&=\\langle x_i x_j - x_i\\langle x_j\\rangle - \\langle x_i\\rangle x_j + \\langle x_i\\rangle\\langle x_j\\rangle\\rangle\\\\\n", - "&=\\langle x_i x_j\\rangle - \\langle x_i\\langle x_j\\rangle\\rangle - \\langle \\langle x_i\\rangle x_j \\rangle +\n", - "\\langle \\langle x_i\\rangle\\langle x_j\\rangle\\rangle \\\\\n", - "&=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle +\n", - "\\langle x_i\\rangle\\langle x_j\\rangle \\\\\n", - "&=\\langle x_i x_j\\rangle - \\langle x_i\\rangle\\langle x_j\\rangle\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If $X_i$ and $X_j$ are independent (assuming $i \\neq j$), we have that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x_i x_j\\rangle = \\langle x_i\\rangle\\langle x_j\\rangle,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "leading to" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "Cov(X_i, X_j) = 0 \\hspace{0.1cm} (i\\neq j).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Numerical experiments and the covariance\n", - "\n", - "Now that we have constructed an idealized mathematical framework, let\n", - "us try to apply it to empirical observations. Examples of relevant\n", - "physical phenomena may be spontaneous decays of nuclei, or a purely\n", - "mathematical set of numbers produced by some deterministic\n", - "mechanism. It is the latter we will deal with, using so-called pseudo-random\n", - "number generators. In general our observations will contain only a limited set of\n", - "observables. We remind the reader that\n", - "a *stochastic process* is a process that produces sequentially a\n", - "chain of values" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\{x_1, x_2,\\dots\\,x_k,\\dots\\}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Numerical experiments and the covariance\n", - "We will call these\n", - "values our *measurements* and the entire set as our measured\n", - "*sample*. The action of measuring all the elements of a sample\n", - "we will call a stochastic *experiment* (since, operationally,\n", - "they are often associated with results of empirical observation of\n", - "some physical or mathematical phenomena; precisely an experiment). We\n", - "assume that these values are distributed according to some \n", - "PDF $p_X^{\\phantom X}(x)$, where $X$ is just the formal symbol for the\n", - "stochastic variable whose PDF is $p_X^{\\phantom X}(x)$. Instead of\n", - "trying to determine the full distribution $p$ we are often only\n", - "interested in finding the few lowest moments, like the mean\n", - "$\\mu_X^{\\phantom X}$ and the variance $\\sigma_X^{\\phantom X}$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Numerical experiments and the covariance, actual situations\n", - "In practical situations however, a sample is always of finite size. Let that\n", - "size be $n$. The expectation value of a sample $\\alpha$, the **sample mean**, is then defined as follows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\langle x_{\\alpha} \\rangle \\equiv \\frac{1}{n}\\sum_{k=1}^n x_{\\alpha,k}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The *sample variance* is:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mathrm{Var}(x) \\equiv \\frac{1}{n}\\sum_{k=1}^n (x_{\\alpha,k} - \\langle x_{\\alpha} \\rangle)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with its square root being the *standard deviation of the sample*.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Numerical experiments and the covariance, our observables\n", - "You can think of the above observables as a set of quantities which define\n", - "a given experiment. This experiment is then repeated several times, say $m$ times.\n", - "The total average is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\langle X_m \\rangle= \\frac{1}{m}\\sum_{\\alpha=1}^mx_{\\alpha}=\\frac{1}{mn}\\sum_{\\alpha, k} x_{\\alpha,k},\n", - "\\label{eq:exptmean} \\tag{6}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the last sums end at $m$ and $n$.\n", - "The total variance is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2_m= \\frac{1}{mn^2}\\sum_{\\alpha=1}^m(\\langle x_{\\alpha} \\rangle-\\langle X_m \\rangle)^2,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which we rewrite as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\sigma^2_m=\\frac{1}{m}\\sum_{\\alpha=1}^m\\sum_{kl=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,l}-\\langle X_m \\rangle).\n", - "\\label{eq:exptvariance} \\tag{7}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Numerical experiments and the covariance, the sample variance\n", - "\n", - "We define also the sample variance $\\sigma^2$ of all $mn$ individual experiments as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\sigma^2=\\frac{1}{mn}\\sum_{\\alpha=1}^m\\sum_{k=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)^2.\n", - "\\label{eq:sampleexptvariance} \\tag{8}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These quantities, being known experimental values or the results from our calculations, \n", - "may differ, in some cases\n", - "significantly, from the similarly named\n", - "exact values for the mean value $\\mu_X$, the variance $\\mathrm{Var}(X)$\n", - "and the covariance $\\mathrm{Cov}(X,Y)$.\n", - "\n", - "\n", - "\n", - "\n", - "## Numerical experiments and the covariance, central limit theorem\n", - "\n", - "The central limit theorem states that the PDF $\\tilde{p}(z)$ of\n", - "the average of $m$ random values corresponding to a PDF $p(x)$ \n", - "is a normal distribution whose mean is the \n", - "mean value of the PDF $p(x)$ and whose variance is the variance\n", - "of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$.\n", - "\n", - "The central limit theorem leads then to the well-known expression for the\n", - "standard deviation, given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma_m=\n", - "\\frac{\\sigma}{\\sqrt{m}}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic. We need therefore a more precise defintion of the error and the variance in our results.\n", - "\n", - "\n", - "\n", - "\n", - "## Definition of Correlation Functions and Standard Deviation\n", - "Our estimate of the true average $\\mu_{X}$ is the sample mean $\\langle X_m \\rangle$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu_{X}^{\\phantom X} \\approx X_m=\\frac{1}{mn}\\sum_{\\alpha=1}^m\\sum_{k=1}^n x_{\\alpha,k}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "We can then use Eq. ([7](#eq:exptvariance))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2_m=\\frac{1}{mn^2}\\sum_{\\alpha=1}^m\\sum_{kl=1}^n (x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,l}-\\langle X_m \\rangle),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and rewrite it as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma^2_m=\\frac{\\sigma^2}{n}+\\frac{2}{mn^2}\\sum_{\\alpha=1}^m\\sum_{k\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\kappa_d = \\frac{f_d}{\\sigma^2}\n", - "\\label{eq:autocorrelformal} \\tag{9}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which gives us a useful measure of the correlation pair correlation\n", - "starting always at $1$ for $d=0$.\n", - "\n", - "\n", - "\n", - "\n", - "## Definition of Correlation Functions and Standard Deviation, sample variance\n", - "\n", - "The sample variance of the $mn$ experiments can now be\n", - "written in terms of the autocorrelation function" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\sigma_m^2=\\frac{\\sigma^2}{n}+\\frac{2}{n}\\cdot\\sigma^2\\sum_{d=1}^{n-1}\n", - "\\frac{f_d}{\\sigma^2}=\\left(1+2\\sum_{d=1}^{n-1}\\kappa_d\\right)\\frac{1}{n}\\sigma^2=\\frac{\\tau}{n}\\cdot\\sigma^2\n", - "\\label{eq:error_estimate_corr_time} \\tag{10}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and we see that $\\sigma_m$ can be expressed in terms of the\n", - "uncorrelated sample variance times a correction factor $\\tau$ which\n", - "accounts for the correlation between measurements. We call this\n", - "correction factor the *autocorrelation time*" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\\tau = 1+2\\sum_{d=1}^{n-1}\\kappa_d\n", - "\\label{eq:autocorrelation_time} \\tag{11}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "\n", - "For a correlation free experiment, $\\tau$\n", - "equals 1.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Definition of Correlation Functions and Standard Deviation\n", - "From the point of view of\n", - "Eq. ([10](#eq:error_estimate_corr_time)) we can interpret a sequential\n", - "correlation as an effective reduction of the number of measurements by\n", - "a factor $\\tau$. The effective number of measurements becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "n_\\mathrm{eff} = \\frac{n}{\\tau}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To neglect the autocorrelation time $\\tau$ will always cause our\n", - "simple uncorrelated estimate of $\\sigma_m^2\\approx \\sigma^2/n$ to\n", - "be less than the true sample error. The estimate of the error will be\n", - "too \"good\". On the other hand, the calculation of the full\n", - "autocorrelation time poses an efficiency problem if the set of\n", - "measurements is very large. The solution to this problem is given by \n", - "more practically oriented methods like the blocking technique.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Code to compute the Covariance matrix and the Covariance" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "-0.027477551848353866\n", - "3.7802249527556957\n", - "-0.4029931539585397\n", - "0.8404538734963756 8.457307018163291 11.102672153764402\n", - "2.530192962097468 2.272095736442208 6.697993128260892\n", - "[[ 0.84045387 2.53019296 2.27209574]\n", - " [ 2.53019296 8.45730702 6.69799313]\n", - " [ 2.27209574 6.69799313 11.10267215]]\n", - "[17.29496013 0.06631095 3.03916197]\n" - ] - } - ], - "source": [ - "# Importing various packages\n", - "from math import exp, sqrt\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "\n", - "# Sample covariance, note the factor 1/(n-1)\n", - "def covariance(x, y, n):\n", - " sum = 0.0\n", - " mean_x = np.mean(x)\n", - " mean_y = np.mean(y)\n", - " for i in range(0, n):\n", - " sum += (x[(i)]-mean_x)*(y[i]-mean_y)\n", - " return sum/(n-1.)\n", - "\n", - "n = 100\n", - "x = np.random.normal(size=n)\n", - "print(np.mean(x))\n", - "y = 4+3*x+np.random.normal(size=n)\n", - "print(np.mean(y))\n", - "z = x**3+np.random.normal(size=n)\n", - "print(np.mean(z))\n", - "covxx = covariance(x,x,n)\n", - "covyy = covariance(y,y,n)\n", - "covzz = covariance(z,z,n)\n", - "covxy = covariance(x,y,n)\n", - "covxz = covariance(x,z,n)\n", - "covyz = covariance(y,z,n)\n", - "print(covxx,covyy, covzz)\n", - "print(covxy,covxz, covyz)\n", - "w = np.vstack((x, y, z))\n", - "#print(w)\n", - "c = np.cov(w)\n", - "print(c)\n", - "#eigen = np.zeros(n)\n", - "Eigvals, Eigvecs = np.linalg.eig(c)\n", - "print(Eigvals)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random Numbers\n", - "\n", - "Uniform deviates are just random numbers that lie within a specified range\n", - "(typically 0 to 1), with any one number in the range just as likely as any other. They\n", - "are, in other words, what you probably think random numbers are. However,\n", - "we want to distinguish uniform deviates from other sorts of random numbers, for\n", - "example numbers drawn from a normal (Gaussian) distribution of specified mean\n", - "and standard deviation. These other sorts of deviates are almost always generated by\n", - "performing appropriate operations on one or more uniform deviates, as we will see\n", - "in subsequent sections. So, a reliable source of random uniform deviates, the subject\n", - "of this section, is an essential building block for any sort of stochastic modeling\n", - "or Monte Carlo computer work.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random Numbers, better name: pseudo random numbers\n", - "\n", - "A disclaimer is however appropriate. It should be fairly obvious that \n", - "something as deterministic as a computer cannot generate purely random numbers.\n", - "\n", - "Numbers generated by any of the standard algorithms are in reality pseudo random\n", - "numbers, hopefully abiding to the following criteria:\n", - "\n", - " * they produce a uniform distribution in the interval [0,1].\n", - "\n", - " * correlations between random numbers are negligible\n", - "\n", - " * the period before the same sequence of random numbers is repeated is as large as possible and finally\n", - "\n", - " * the algorithm should be fast.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG\n", - " The most common random number generators are based on so-called\n", - "Linear congruential relations of the type" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_i=(aN_{i-1}+c) \\mathrm{MOD} (M),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which yield a number in the interval [0,1] through" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "x_i=N_i/M\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The number \n", - "$M$ is called the period and it should be as large as possible \n", - " and \n", - "$N_0$ is the starting value, or seed. The function $\\mathrm{MOD}$ means the remainder,\n", - "that is if we were to evaluate $(13)\\mathrm{MOD}(9)$, the outcome is the remainder\n", - "of the division $13/9$, namely $4$.\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG and periodic outputs\n", - "\n", - "The problem with such generators is that their outputs are periodic;\n", - "they \n", - "will start to repeat themselves with a period that is at most $M$. If however\n", - "the parameters $a$ and $c$ are badly chosen, the period may be even shorter.\n", - "\n", - "Consider the following example" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_i=(6N_{i-1}+7) \\mathrm{MOD} (5),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with a seed $N_0=2$. This generator produces the sequence\n", - "$4,1,3,0,2,4,1,3,0,2,...\\dots$, i.e., a sequence with period $5$.\n", - "However, increasing $M$ may not guarantee a larger period as the following\n", - "example shows" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_i=(27N_{i-1}+11) \\mathrm{MOD} (54),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which still, with $N_0=2$, results in $11,38,11,38,11,38,\\dots$, a period of\n", - "just $2$.\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG and its period\n", - "Typical periods for the random generators provided in the program library \n", - "are of the order of $\\sim 10^9$ or larger. Other random number generators which have\n", - "become increasingly popular are so-called shift-register generators.\n", - "In these generators each successive number depends on many preceding\n", - "values (rather than the last values as in the linear congruential\n", - "generator).\n", - "For example, you could make a shift register generator whose $l$th \n", - "number is the sum of the $l-i$th and $l-j$th values with modulo $M$," - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_l=(aN_{l-i}+cN_{l-j})\\mathrm{MOD}(M).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random number generator RNG, other examples\n", - "Such a generator again produces a sequence of pseudorandom numbers\n", - "but this time with a period much larger than $M$.\n", - "It is also possible to construct more elaborate algorithms by including\n", - "more than two past terms in the sum of each iteration.\n", - "One example is the generator of [Marsaglia and Zaman](http://dl.acm.org/citation.cfm?id=187154)\n", - "which consists of two congruential relations" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " N_l=(N_{l-3}-N_{l-1})\\mathrm{MOD}(2^{31}-69),\n", - "\\label{eq:mz1} \\tag{12}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "followed by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " N_l=(69069N_{l-1}+1013904243)\\mathrm{MOD}(2^{32}),\n", - "\\label{eq:mz2} \\tag{13}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which according to the authors has a period larger than $2^{94}$.\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, other examples\n", - "Instead of using modular addition, we could use the bitwise\n", - "exclusive-OR ($\\oplus$) operation so that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_l=(N_{l-i})\\oplus (N_{l-j})\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the bitwise action of $\\oplus$ means that if $N_{l-i}=N_{l-j}$ the result is\n", - "$0$ whereas if $N_{l-i}\\ne N_{l-j}$ the result is\n", - "$1$. As an example, consider the case where $N_{l-i}=6$ and $N_{l-j}=11$. The first\n", - "one has a bit representation (using 4 bits only) which reads $0110$ whereas the \n", - "second number is $1011$. Employing the $\\oplus$ operator yields \n", - "$1101$, or $2^3+2^2+2^0=13$.\n", - "\n", - "In Fortran90, the bitwise $\\oplus$ operation is coded through the intrinsic\n", - "function $\\mathrm{IEOR}(m,n)$ where $m$ and $n$ are the input numbers, while in $C$\n", - "it is given by $m\\wedge n$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, RAN0\n", - "\n", - "We show here how the linear congruential algorithm can be implemented, namely" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "N_i=(aN_{i-1}) \\mathrm{MOD} (M).\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "However, since $a$ and $N_{i-1}$ are integers and their multiplication \n", - "could become greater than the standard 32 bit integer, there is a trick via \n", - "Schrage's algorithm which approximates the multiplication\n", - "of large integers through the factorization" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "M=aq+r,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where we have defined" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "q=[M/a],\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "r = M\\hspace{0.1cm}\\mathrm{MOD} \\hspace{0.1cm}a.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where the brackets denote integer division. In the code below the numbers \n", - "$q$ and $r$ are chosen so that $r < q$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, RAN0\n", - "\n", - "To see how this works we note first that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "(aN_{i-1}) \\mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\\mathrm{MOD} (M),\n", - "\\label{eq:rntrick1} \\tag{14}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "since we can add or subtract any integer multiple of $M$ from $aN_{i-1}$.\n", - "The last term $[N_{i-1}/q]M\\mathrm{MOD}(M)$ is zero since the integer division \n", - "$[N_{i-1}/q]$ just yields a constant which is multiplied with $M$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, RAN0\n", - "We can now rewrite Eq. ([14](#eq:rntrick1)) as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "(aN_{i-1}) \\mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\\mathrm{MOD} (M),\n", - "\\label{eq:rntrick2} \\tag{15}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which results\n", - "in" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "(aN_{i-1}) \\mathrm{MOD} (M)= \\left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\\right)\\mathrm{MOD} (M),\n", - "\\label{eq:rntrick3} \\tag{16}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "yielding" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "(aN_{i-1}) \\mathrm{MOD} (M)= \\left(a(N_{i-1}\\mathrm{MOD} (q)) -[N_{i-1}/q]r)\\right)\\mathrm{MOD} (M).\n", - "\\label{eq:rntrick4} \\tag{17}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Random number generator RNG, RAN0\n", - "The term $[N_{i-1}/q]r$ is always smaller or equal $N_{i-1}(r/q)$ and with $r < q$ we obtain always a \n", - "number smaller than $N_{i-1}$, which is smaller than $M$. \n", - "And since the number $N_{i-1}\\mathrm{MOD} (q)$ is between zero and $q-1$ then\n", - "$a(N_{i-1}\\mathrm{MOD} (q))< aq$. Combined with our definition of $q=[M/a]$ ensures that \n", - "this term is also smaller than $M$ meaning that both terms fit into a\n", - "32-bit signed integer. None of these two terms can be negative, but their difference could.\n", - "The algorithm below adds $M$ if their difference is negative.\n", - "Note that the program uses the bitwise $\\oplus$ operator to generate\n", - "the starting point for each generation of a random number. The period\n", - "of $ran0$ is $\\sim 2.1\\times 10^{9}$. A special feature of this\n", - "algorithm is that is should never be called with the initial seed \n", - "set to $0$.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Random number generator RNG, RAN0 code" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " /*\n", - " ** The function\n", - " ** ran0()\n", - " ** is an \"Minimal\" random number generator of Park and Miller\n", - " ** Set or reset the input value\n", - " ** idum to any integer value (except the unlikely value MASK)\n", - " ** to initialize the sequence; idum must not be altered between\n", - " ** calls for sucessive deviates in a sequence.\n", - " ** The function returns a uniform deviate between 0.0 and 1.0.\n", - " */\n", - " double ran0(long &idum)\n", - " {\n", - " const int a = 16807, m = 2147483647, q = 127773;\n", - " const int r = 2836, MASK = 123459876;\n", - " const double am = 1./m;\n", - " long k;\n", - " double ans;\n", - " idum ^= MASK;\n", - " k = (*idum)/q;\n", - " idum = a*(idum - k*q) - r*k;\n", - " // add m if negative difference\n", - " if(idum < 0) idum += m;\n", - " ans=am*(idum);\n", - " idum ^= MASK;\n", - " return ans;\n", - " } // End: function ran0() \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Properties of Selected Random Number Generators\n", - "\n", - "As mentioned previously, the underlying PDF for the generation of\n", - "random numbers is the uniform distribution, meaning that the \n", - "probability for finding a number $x$ in the interval [0,1] is $p(x)=1$.\n", - "\n", - "A random number generator should produce numbers which are uniformly distributed\n", - "in this interval. The table shows the distribution of $N=10000$ random\n", - "numbers generated by the functions in the program library.\n", - "We note in this table that the number of points in the various\n", - "intervals $0.0-0.1$, $0.1-0.2$ etc are fairly close to $1000$, with some minor\n", - "deviations. \n", - "\n", - "Two additional measures are the standard deviation $\\sigma$ and the mean\n", - "$\\mu=\\langle x\\rangle$.\n", - "\n", - "\n", - "\n", - "\n", - "## Properties of Selected Random Number Generators\n", - "For the uniform distribution, the mean value $\\mu$ is then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\mu=\\langle x\\rangle=\\frac{1}{2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "while the standard deviation is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma=\\sqrt{\\langle x^2\\rangle-\\mu^2}=\\frac{1}{\\sqrt{12}}=0.2886.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Properties of Selected Random Number Generators\n", - "The various random number generators produce results which agree rather well with\n", - "these limiting values. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "
$x$-bin ran0 ran1 ran2 ran3
0.0-0.1 1013 991 938 1047
0.1-0.2 1002 1009 1040 1030
0.2-0.3 989 999 1030 993
0.3-0.4 939 960 1023 937
0.4-0.5 1038 1001 1002 992
0.5-0.6 1037 1047 1009 1009
0.6-0.7 1005 989 1003 989
0.7-0.8 986 962 985 954
0.8-0.9 1000 1027 1009 1023
0.9-1.0 991 1015 961 1026
$\\mu$ 0.4997 0.5018 0.4992 0.4990
$\\sigma$ 0.2882 0.2892 0.2861 0.2915
\n", - "\n", - "\n", - "\n", - "\n", - "## Simple demonstration of RNGs using python\n", - "The following simple Python code plots the distribution of the produced random numbers using the linear congruential RNG employed by Python. The trend displayed in the previous table is seen rather clearly." - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "data": { - "image/png": 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/statistics_178_0.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "#!/usr/bin/env python\n", - "import numpy as np\n", - "import matplotlib.mlab as mlab\n", - "import matplotlib.pyplot as plt\n", - "import random\n", - "\n", - "# initialize the rng with a seed\n", - "random.seed() \n", - "counts = 10000\n", - "values = np.zeros(counts) \n", - "for i in range (1, counts, 1):\n", - " values[i] = random.random()\n", - "\n", - "# the histogram of the data\n", - "n, bins, patches = plt.hist(values, 10, facecolor='green')\n", - "\n", - "plt.xlabel('$x$')\n", - "plt.ylabel('Number of counts')\n", - "plt.title(r'Test of uniform distribution')\n", - "plt.axis([0, 1, 0, 1100])\n", - "plt.grid(True)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Properties of Selected Random Number Generators\n", - "Since our random numbers, which are typically generated via a linear congruential algorithm,\n", - "are never fully independent, we can then define \n", - "an important test which measures the degree of correlation, namely the so-called \n", - "auto-correlation function defined previously, see again Eq. ([9](#eq:autocorrelformal)).\n", - "We rewrite it here as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C_k=\\frac{f_d}\n", - " {\\sigma^2},\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $C_0=1$. Recall that \n", - "$\\sigma^2=\\langle x_i^2\\rangle-\\langle x_i\\rangle^2$ and that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f_d = \\frac{1}{nm}\\sum_{\\alpha=1}^m\\sum_{k=1}^{n-d}(x_{\\alpha,k}-\\langle X_m \\rangle)(x_{\\alpha,k+d}-\\langle X_m \\rangle),\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The non-vanishing of $C_k$ for $k\\ne 0$ means that the random\n", - "numbers are not independent. The independence of the random numbers is crucial \n", - "in the evaluation of other expectation values. If they are not independent, our\n", - "assumption for approximating $\\sigma_N$ is no longer valid.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Autocorrelation function\n", - "This program computes the autocorrelation function as discussed in the equation on the previous slide for random numbers generated with the normal distribution $N(0,1)$." - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0.07147629989718486 0.9457130616859395\n" - ] - }, - { - "data": { - "image/png": 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\n", 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" - ] - }, - "metadata": { - "filenames": { - "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/statistics_184_1.png" - }, - "needs_background": "light" - }, - "output_type": "display_data" - } - ], - "source": [ - "# Importing various packages\n", - "from math import exp, sqrt\n", - "from random import random, seed\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "\n", - "def autocovariance(x, n, k, mean_x):\n", - " sum = 0.0\n", - " for i in range(0, n-k):\n", - " sum += (x[(i+k)]-mean_x)*(x[i]-mean_x)\n", - " return sum/n\n", - "\n", - "n = 1000\n", - "x=np.random.normal(size=n)\n", - "autocor = np.zeros(n)\n", - "figaxis = np.zeros(n)\n", - "mean_x=np.mean(x)\n", - "var_x = np.var(x)\n", - "print(mean_x, var_x)\n", - "for i in range (0, n):\n", - " figaxis[i] = i\n", - " autocor[i]=(autocovariance(x, n, i, mean_x))/var_x \n", - "\n", - "plt.plot(figaxis, autocor, \"r-\")\n", - "plt.axis([0,n,-0.1, 1.0])\n", - "plt.xlabel(r'$i$')\n", - "plt.ylabel(r'$\\gamma_i$')\n", - "plt.title(r'Autocorrelation function')\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As can be seen from the plot, the first point gives back the variance and a value of one. \n", - "For the remaining values we notice that there are still non-zero values for the auto-correlation function.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Correlation function and which random number generators should I use\n", - "The program here computes the correlation function for one of the standard functions included with the c++ compiler." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " // This function computes the autocorrelation function for \n", - " // the standard c++ random number generator\n", - " \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " using namespace std;\n", - " // output file as global variable\n", - " ofstream ofile; \n", - " \n", - " // Main function begins here \n", - " int main(int argc, char* argv[])\n", - " {\n", - " int n;\n", - " char *outfilename;\n", - " \n", - " cin >> n;\n", - " double MCint = 0.; double MCintsqr2=0.;\n", - " double invers_period = 1./RAND_MAX; // initialise the random number generator\n", - " srand(time(NULL)); // This produces the so-called seed in MC jargon\n", - " // Compute the variance and the mean value of the uniform distribution\n", - " // Compute also the specific values x for each cycle in order to be able to\n", - " // the covariance and the correlation function \n", - " // Read in output file, abort if there are too few command-line arguments\n", - " if( argc <= 2 ){\n", - " cout << \"Bad Usage: \" << argv[0] << \n", - " \t \" read also output file and number of cycles on same line\" << endl;\n", - " exit(1);\n", - " }\n", - " else{\n", - " outfilename=argv[1];\n", - " }\n", - " ofile.open(outfilename); \n", - " // Get the number of Monte-Carlo samples\n", - " n = atoi(argv[2]);\n", - " double *X; \n", - " X = new double[n];\n", - " for (int i = 0; i < n; i++){\n", - " double x = double(rand())*invers_period; \n", - " X[i] = x;\n", - " MCint += x;\n", - " MCintsqr2 += x*x;\n", - " }\n", - " double Mean = MCint/((double) n );\n", - " MCintsqr2 = MCintsqr2/((double) n );\n", - " double STDev = sqrt(MCintsqr2-Mean*Mean);\n", - " double Variance = MCintsqr2-Mean*Mean;\n", - " // Write mean value and standard deviation \n", - " cout << \" Standard deviation= \" << STDev << \" Integral = \" << Mean << endl;\n", - " \n", - " // Now we compute the autocorrelation function\n", - " double *autocor; autocor = new double[n];\n", - " for (int j = 0; j < n; j++){\n", - " double sum = 0.0;\n", - " for (int k = 0; k < (n-j); k++){\n", - " \t sum += (X[k]-Mean)*(X[k+j]-Mean); \n", - " }\n", - " autocor[j] = sum/Variance/((double) n );\n", - " ofile << setiosflags(ios::showpoint | ios::uppercase);\n", - " ofile << setw(15) << setprecision(8) << j;\n", - " ofile << setw(15) << setprecision(8) << autocor[j] << endl;\n", - " }\n", - " ofile.close(); // close output file\n", - " return 0;\n", - " } // end of main program \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Which RNG should I use?\n", - "* C++ has a class called **random**. The [random class](http://www.cplusplus.com/reference/random/) contains a large selection of RNGs and is highly recommended. Some of these RNGs have very large periods making it thereby very safe to use these RNGs in case one is performing large calculations. In particular, the [Mersenne twister random number engine](http://www.cplusplus.com/reference/random/mersenne_twister_engine/) has a period of $2^{19937}$. \n", - "\n", - "* Add RNGs in Python\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## How to use the Mersenne generator\n", - "The following part of a c++ code (from project 4) sets up the uniform distribution for $x\\in [0,1]$." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " /*\n", - " \n", - " // You need this \n", - " #include \n", - " \n", - " // Initialize the seed and call the Mersienne algo\n", - " std::random_device rd;\n", - " std::mt19937_64 gen(rd());\n", - " // Set up the uniform distribution for x \\in [[0, 1]\n", - " std::uniform_real_distribution RandomNumberGenerator(0.0,1.0);\n", - " \n", - " // Now use the RNG\n", - " int ix = (int) (RandomNumberGenerator(gen)*NSpins);\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Why blocking?\n", - "**Statistical analysis.**\n", - "\n", - " * Monte Carlo simulations can be treated as *computer experiments*\n", - "\n", - " * The results can be analysed with the same statistical tools as we would use analysing experimental data.\n", - "\n", - " * As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.\n", - "\n", - "A very good article which explains blocking is H. Flyvbjerg and H. G. Petersen, *Error estimates on averages of correlated data*, [Journal of Chemical Physics 91, 461-466 (1989)](http://scitation.aip.org/content/aip/journal/jcp/91/1/10.1063/1.457480).\n", - "\n", - " \n", - "\n", - "\n", - "\n", - "\n", - "## Why blocking?\n", - "**Statistical analysis.**\n", - "\n", - " * As in other experiments, Monte Carlo experiments have two classes of errors:\n", - "\n", - " * Statistical errors\n", - "\n", - " * Systematical errors\n", - "\n", - "\n", - " * Statistical errors can be estimated using standard tools from statistics\n", - "\n", - " * Systematical errors are method specific and must be treated differently from case to case. (In VMC a common source is the step length or time step in importance sampling)\n", - "\n", - " \n", - "\n", - "\n", - "\n", - "## Code to demonstrate the calculation of the autocorrelation function\n", - "The following code computes the autocorrelation function, the covariance and the standard deviation\n", - "for standard RNG. \n", - "The [following file](https://github.com/CompPhysics/ComputationalPhysics2/tree/gh-pages/doc/Programs/LecturePrograms/programs/Blocking/autocorrelation.cpp) gives the code." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " // This function computes the autocorrelation function for \n", - " // the Mersenne random number generator with a uniform distribution\n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " #include \n", - " using namespace std;\n", - " using namespace arma;\n", - " // output file\n", - " ofstream ofile;\n", - " \n", - " // Main function begins here \n", - " int main(int argc, char* argv[])\n", - " {\n", - " int MonteCarloCycles;\n", - " string filename;\n", - " if (argc > 1) {\n", - " filename=argv[1];\n", - " MonteCarloCycles = atoi(argv[2]);\n", - " string fileout = filename;\n", - " string argument = to_string(MonteCarloCycles);\n", - " fileout.append(argument);\n", - " ofile.open(fileout);\n", - " }\n", - " \n", - " // Compute the variance and the mean value of the uniform distribution\n", - " // Compute also the specific values x for each cycle in order to be able to\n", - " // compute the covariance and the correlation function \n", - " \n", - " vec X = zeros(MonteCarloCycles);\n", - " double MCint = 0.; double MCintsqr2=0.;\n", - " std::random_device rd;\n", - " std::mt19937_64 gen(rd());\n", - " // Set up the uniform distribution for x \\in [[0, 1]\n", - " std::uniform_real_distribution RandomNumberGenerator(0.0,1.0);\n", - " for (int i = 0; i < MonteCarloCycles; i++){\n", - " double x = RandomNumberGenerator(gen); \n", - " X(i) = x;\n", - " MCint += x;\n", - " MCintsqr2 += x*x;\n", - " }\n", - " double Mean = MCint/((double) MonteCarloCycles );\n", - " MCintsqr2 = MCintsqr2/((double) MonteCarloCycles );\n", - " double STDev = sqrt(MCintsqr2-Mean*Mean);\n", - " double Variance = MCintsqr2-Mean*Mean;\n", - " // Write mean value and variance\n", - " cout << \" Sample variance= \" << Variance << \" Mean value = \" << Mean << endl;\n", - " // Now we compute the autocorrelation function\n", - " vec autocorrelation = zeros(MonteCarloCycles);\n", - " for (int j = 0; j < MonteCarloCycles; j++){\n", - " double sum = 0.0;\n", - " for (int k = 0; k < (MonteCarloCycles-j); k++){\n", - " sum += (X(k)-Mean)*(X(k+j)-Mean); \n", - " }\n", - " autocorrelation(j) = sum/Variance/((double) MonteCarloCycles );\n", - " ofile << setiosflags(ios::showpoint | ios::uppercase);\n", - " ofile << setw(15) << setprecision(8) << j;\n", - " ofile << setw(15) << setprecision(8) << autocorrelation(j) << endl;\n", - " }\n", - " // Now compute the exact covariance using the autocorrelation function\n", - " double Covariance = 0.0;\n", - " for (int j = 0; j < MonteCarloCycles; j++){\n", - " Covariance += autocorrelation(j);\n", - " }\n", - " Covariance *= 2.0/((double) MonteCarloCycles);\n", - " // Compute now the total variance, including the covariance, and obtain the standard deviation\n", - " double TotalVariance = (Variance/((double) MonteCarloCycles ))+Covariance;\n", - " cout << \"Covariance =\" << Covariance << \"Totalvariance= \" << TotalVariance << \"Sample Variance/n= \" << (Variance/((double) MonteCarloCycles )) << endl;\n", - " cout << \" STD from sample variance= \" << sqrt(Variance/((double) MonteCarloCycles )) << \" STD with covariance = \" << sqrt(TotalVariance) << endl;\n", - " \n", - " ofile.close(); // close output file\n", - " return 0;\n", - " } // end of main program \n", - " \n", - " \n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## What is blocking?\n", - "**Blocking.**\n", - "\n", - " * Say that we have a set of samples from a Monte Carlo experiment\n", - "\n", - " * Assuming (wrongly) that our samples are uncorrelated our best estimate of the standard deviation of the mean $\\langle \\mathbf{M}\\rangle$ is given by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma=\\sqrt{\\frac{1}{n}\\left(\\langle \\mathbf{M}^2\\rangle-\\langle \\mathbf{M}\\rangle^2\\right)}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "* If the samples are correlated we can rewrite our results to show that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\sigma=\\sqrt{\\frac{1+2\\tau/\\Delta t}{n}\\left(\\langle \\mathbf{M}^2\\rangle-\\langle \\mathbf{M}\\rangle^2\\right)}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\tau$ is the correlation time (the time between a sample and the next uncorrelated sample) and $\\Delta t$ is time between each sample\n", - "\n", - " \n", - "\n", - "\n", - "## What is blocking?\n", - "**Blocking.**\n", - "\n", - " * If $\\Delta t\\gg\\tau$ our first estimate of $\\sigma$ still holds\n", - "\n", - " * Much more common that $\\Delta t<\\tau$\n", - "\n", - " * In the method of data blocking we divide the sequence of samples into blocks\n", - "\n", - " * We then take the mean $\\langle \\mathbf{M}_i\\rangle$ of block $i=1\\ldots n_{blocks}$ to calculate the total mean and variance\n", - "\n", - " * The size of each block must be so large that sample $j$ of block $i$ is not correlated with sample $j$ of block $i+1$\n", - "\n", - " * The correlation time $\\tau$ would be a good choice\n", - "\n", - "\n", - "\n", - "\n", - "## What is blocking?\n", - "**Blocking.**\n", - "\n", - " * Problem: We don't know $\\tau$ or it is too expensive to compute\n", - "\n", - " * Solution: Make a plot of std. dev. as a function of blocksize\n", - "\n", - " * The estimate of std. dev. of correlated data is too low $\\to$ the error will increase with increasing block size until the blocks are uncorrelated, where we reach a plateau\n", - "\n", - " * When the std. dev. stops increasing the blocks are uncorrelated\n", - "\n", - "\n", - "\n", - "\n", - "## Implementation\n", - " * Do a Monte Carlo simulation, storing all samples to file\n", - "\n", - " * Do the statistical analysis on this file, independently of your Monte Carlo program\n", - "\n", - " * Read the file into an array\n", - "\n", - " * Loop over various block sizes\n", - "\n", - " * For each block size $n_b$, loop over the array in steps of $n_b$ taking the mean of elements $i n_b,\\ldots,(i+1) n_b$\n", - "\n", - " * Take the mean and variance of the resulting array\n", - "\n", - " * Write the results for each block size to file for later\n", - " analysis\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "## Actual implementation with code, main function\n", - "When the file gets large, it can be useful to write your data in binary mode instead of ascii characters.\n", - "The [following python file](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) reads data from file with the output from every Monte Carlo cycle." - ] - }, - { - "cell_type": "code", - "execution_count": 6, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [ - { - "ename": "IndentationError", - "evalue": "unexpected indent (, line 2)", - "output_type": "error", - "traceback": [ - "\u001b[0;36m File \u001b[0;32m\"\"\u001b[0;36m, line \u001b[0;32m2\u001b[0m\n\u001b[0;31m @timeFunction\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mIndentationError\u001b[0m\u001b[0;31m:\u001b[0m unexpected indent\n" - ] - } - ], - "source": [ - "# Blocking\n", - " @timeFunction\n", - " def blocking(self, blockSizeMax = 500):\n", - " blockSizeMin = 1\n", - "\n", - " self.blockSizes = []\n", - " self.meanVec = []\n", - " self.varVec = []\n", - "\n", - " for i in range(blockSizeMin, blockSizeMax):\n", - " if(len(self.data) % i != 0):\n", - " pass#continue\n", - " blockSize = i\n", - " meanTempVec = []\n", - " varTempVec = []\n", - " startPoint = 0\n", - " endPoint = blockSize\n", - "\n", - " while endPoint <= len(self.data):\n", - " meanTempVec.append(np.average(self.data[startPoint:endPoint]))\n", - " startPoint = endPoint\n", - " endPoint += blockSize\n", - " mean, var = np.average(meanTempVec), np.var(meanTempVec)/len(meanTempVec)\n", - " self.meanVec.append(mean)\n", - " self.varVec.append(var)\n", - " self.blockSizes.append(blockSize)\n", - "\n", - " self.blockingAvg = np.average(self.meanVec[-200:])\n", - " self.blockingVar = (np.average(self.varVec[-200:]))\n", - " self.blockingStd = np.sqrt(self.blockingVar)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The Bootstrap method\n", - "\n", - "The Bootstrap resampling method is also very popular. It is very simple:\n", - "\n", - "1. Start with your sample of measurements and compute the sample variance and the mean values\n", - "\n", - "2. Then start again but pick in a random way the numbers in the sample and recalculate the mean and the sample variance.\n", - "\n", - "3. Repeat this $K$ times.\n", - "\n", - "It can be shown, see the article by [Efron](https://projecteuclid.org/download/pdf_1/euclid.aos/1176344552)\n", - "that it produces the correct standard deviation.\n", - "\n", - "This method is very useful for small ensembles of data points. \n", - "\n", - "\n", - "## Bootstrapping\n", - "Given a set of $N$ data, assume that we are interested in some \n", - "observable $\\theta$ which may be estimated from that set. This observable can also be for example the result of a fit based on all $N$ raw data. \n", - "Let us call the value of the observable obtained from the original \n", - "data set $\\hat{\\theta}$. One recreates from the sample repeatedly \n", - "other samples by choosing randomly $N$ data out of the original set. \n", - "This costs essentially nothing, since we just recycle the original data set for the building of new sets. \n", - "\n", - "\n", - "## Bootstrapping, recipe\n", - "Let us assume we have done this $K$ times and thus have $K$ sets of $N$ \n", - "data values each. \n", - "Of course some values will enter more than once in the new sets. For each of these sets one computes the observable $\\theta$ resulting in values $\\theta_k$ with $k = 1,...,K$. Then one determines" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\tilde{\\theta} = \\frac{1}{K} \\sum_{k=1}^K \\theta_k,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "sigma^2_{\\tilde{\\theta}} = \\frac{1}{K} \\sum_{k=1}^K \\left(\\theta_k-\\tilde{\\theta}\\right)^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "These are estimators for $\\angle\\theta\\rangle$ and its variance. They are not unbiased and therefore \n", - "$\\tilde{\\theta}\\neq\\hat{\\theta}$ for finite K. \n", - "\n", - "The difference is called bias and gives an idea on how far away the result may be from \n", - "the true $\\angle\\theta\\rangle$. As final result for the observable one quotes $\\angle\\theta\\rangle = \\tilde{\\theta} \\pm \\sigma_{\\tilde{\\theta}}$ .\n", - "\n", - "\n", - "\n", - "## Bootstrapping, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " # Bootstrap\n", - " @timeFunction\n", - " def bootstrap(self, nBoots = 1000):\n", - " bootVec = np.zeros(nBoots)\n", - " for k in range(0,nBoots):\n", - " bootVec[k] = np.average(np.random.choice(self.data, len(self.data)))\n", - " self.bootAvg = np.average(bootVec)\n", - " self.bootVar = np.var(bootVec)\n", - " self.bootStd = np.std(bootVec)\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Jackknife, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - " # Jackknife\n", - " @timeFunction\n", - " def jackknife(self):\n", - " jackknVec = np.zeros(len(self.data))\n", - " for k in range(0,len(self.data)):\n", - " jackknVec[k] = np.average(np.delete(self.data, k))\n", - " self.jackknAvg = self.avg - (len(self.data) - 1) * (np.average(jackknVec) - self.avg)\n", - " self.jackknVar = float(len(self.data) - 1) * np.var(jackknVec)\n", - " self.jackknStd = np.sqrt(self.jackknVar)\n" - ] - } - ], - "metadata": { - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.8.5" - } - }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/statistics.py b/doc/LectureNotes/_build/jupyter_execute/statistics.py deleted file mode 100644 index de12f2a6f..000000000 --- a/doc/LectureNotes/_build/jupyter_execute/statistics.py +++ /dev/null @@ -1,1824 +0,0 @@ -# Elements of Probability Theory and Statistical Data Analysis - - -## Domains and probabilities -Consider the following simple example, namely the tossing of two dice, resulting in the following possible values - -$$ -\{2,3,4,5,6,7,8,9,10,11,12\}. -$$ - -These values are called the *domain*. -To this domain we have the corresponding *probabilities* - -$$ -\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\}. -$$ - -## Tossing the dice -The numbers in the domain are the outcomes of the physical process of tossing say two dice. -We cannot tell beforehand whether the outcome is 3 or 5 or any other number in this domain. -This defines the randomness of the outcome, or unexpectedness or any other synonimous word which -encompasses the uncertitude of the final outcome. - -The only thing we can tell beforehand -is that say the outcome 2 has a certain probability. -If our favorite hobby is to spend an hour every evening throwing dice and -registering the sequence of outcomes, we will note that the numbers in the above domain - -$$ -\{2,3,4,5,6,7,8,9,10,11,12\}, -$$ - -appear in a random order. After 11 throws the results may look like - -$$ -\{10,8,6,3,6,9,11,8,12,4,5\}. -$$ - -## Stochastic variables - -**Random variables are characterized by a domain which contains all possible values that the random value may take. This domain has a corresponding probability distribution function(PDF)**. - - - - -## Stochastic variables and the main concepts, the discrete case -There are two main concepts associated with a stochastic variable. The -*domain* is the set $\mathbb D = \{x\}$ of all accessible values -the variable can assume, so that $X \in \mathbb D$. An example of a -discrete domain is the set of six different numbers that we may get by -throwing of a dice, $x\in\{1,\,2,\,3,\,4,\,5,\,6\}$. - -The *probability distribution function (PDF)* is a function -$p(x)$ on the domain which, in the discrete case, gives us the -probability or relative frequency with which these values of $X$ -occur - -$$ -p(x) = \mathrm{Prob}(X=x). -$$ - -## Stochastic variables and the main concepts, the continuous case -In the continuous case, the PDF does not directly depict the -actual probability. Instead we define the probability for the -stochastic variable to assume any value on an infinitesimal interval -around $x$ to be $p(x)dx$. The continuous function $p(x)$ then gives us -the *density* of the probability rather than the probability -itself. The probability for a stochastic variable to assume any value -on a non-infinitesimal interval $[a,\,b]$ is then just the integral - -$$ -\mathrm{Prob}(a\leq X\leq b) = \int_a^b p(x)dx. -$$ - -Qualitatively speaking, a stochastic variable represents the values of -numbers chosen as if by chance from some specified PDF so that the -selection of a large set of these numbers reproduces this PDF. - - - - -## The cumulative probability -Of interest to us is the *cumulative probability -distribution function* (**CDF**), $P(x)$, which is just the probability -for a stochastic variable $X$ to assume any value less than $x$ - -$$ -P(x)=\mathrm{Prob(}X\leq x\mathrm{)} = -\int_{-\infty}^x p(x^{\prime})dx^{\prime}. -$$ - -The relation between a CDF and its corresponding PDF is then - -$$ -p(x) = \frac{d}{dx}P(x). -$$ - -## Properties of PDFs - -There are two properties that all PDFs must satisfy. The first one is -positivity (assuming that the PDF is normalized) - -$$ -0 \leq p(x) \leq 1. -$$ - -Naturally, it would be nonsensical for any of the values of the domain -to occur with a probability greater than $1$ or less than $0$. Also, -the PDF must be normalized. That is, all the probabilities must add up -to unity. The probability of "anything" to happen is always unity. For -both discrete and continuous PDFs, this condition is - -$$ -\begin{align*} -\sum_{x_i\in\mathbb D} p(x_i) & = 1,\\ -\int_{x\in\mathbb D} p(x)\,dx & = 1. -\end{align*} -$$ - -## Important distributions, the uniform distribution -The first one -is the most basic PDF; namely the uniform distribution - - -
- -$$ -\begin{equation} -p(x) = \frac{1}{b-a}\theta(x-a)\theta(b-x). -\label{eq:unifromPDF} \tag{1} -\end{equation} -$$ - -For $a=0$ and $b=1$ we have - -$$ -\begin{array}{ll} -p(x)dx = dx & \in [0,1]. -\end{array} -$$ - -The latter distribution is used to generate random numbers. For other PDFs, one needs normally a mapping from this distribution to say for example the exponential distribution. - - - - -## Gaussian distribution -The second one is the Gaussian Distribution - -$$ -p(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{(x-\mu)^2}{2\sigma^2})}, -$$ - -with mean value $\mu$ and standard deviation $\sigma$. If $\mu=0$ and $\sigma=1$, it is normally called the **standard normal distribution** - -$$ -p(x) = \frac{1}{\sqrt{2\pi}} \exp{(-\frac{x^2}{2})}, -$$ - -The following simple Python code plots the above distribution for different values of $\mu$ and $\sigma$. - -%matplotlib inline - -import numpy as np -from math import acos, exp, sqrt -from matplotlib import pyplot as plt -from matplotlib import rc, rcParams -import matplotlib.units as units -import matplotlib.ticker as ticker -rc('text',usetex=True) -rc('font',**{'family':'serif','serif':['Gaussian distribution']}) -font = {'family' : 'serif', - 'color' : 'darkred', - 'weight' : 'normal', - 'size' : 16, - } -pi = acos(-1.0) -mu0 = 0.0 -sigma0 = 1.0 -mu1= 1.0 -sigma1 = 2.0 -mu2 = 2.0 -sigma2 = 4.0 - -x = np.linspace(-20.0, 20.0) -v0 = np.exp(-(x*x-2*x*mu0+mu0*mu0)/(2*sigma0*sigma0))/sqrt(2*pi*sigma0*sigma0) -v1 = np.exp(-(x*x-2*x*mu1+mu1*mu1)/(2*sigma1*sigma1))/sqrt(2*pi*sigma1*sigma1) -v2 = np.exp(-(x*x-2*x*mu2+mu2*mu2)/(2*sigma2*sigma2))/sqrt(2*pi*sigma2*sigma2) -plt.plot(x, v0, 'b-', x, v1, 'r-', x, v2, 'g-') -plt.title(r'{\bf Gaussian distributions}', fontsize=20) -plt.text(-19, 0.3, r'Parameters: $\mu = 0$, $\sigma = 1$', fontdict=font) -plt.text(-19, 0.18, r'Parameters: $\mu = 1$, $\sigma = 2$', fontdict=font) -plt.text(-19, 0.08, r'Parameters: $\mu = 2$, $\sigma = 4$', fontdict=font) -plt.xlabel(r'$x$',fontsize=20) -plt.ylabel(r'$p(x)$ [MeV]',fontsize=20) - -# Tweak spacing to prevent clipping of ylabel -plt.subplots_adjust(left=0.15) -plt.savefig('gaussian.pdf', format='pdf') -plt.show() - -## Exponential distribution -Another important distribution in science is the exponential distribution - -$$ -p(x) = \alpha\exp{-(\alpha x)}. -$$ - -## Expectation values -Let $h(x)$ be an arbitrary continuous function on the domain of the stochastic -variable $X$ whose PDF is $p(x)$. We define the *expectation value* -of $h$ with respect to $p$ as follows - - -
- -$$ -\begin{equation} -\langle h \rangle_X \equiv \int\! h(x)p(x)\,dx -\label{eq:expectation_value_of_h_wrt_p} \tag{2} -\end{equation} -$$ - -Whenever the PDF is known implicitly, like in this case, we will drop -the index $X$ for clarity. -A particularly useful class of special expectation values are the -*moments*. The $n$-th moment of the PDF $p$ is defined as -follows - -$$ -\langle x^n \rangle \equiv \int\! x^n p(x)\,dx -$$ - -## Stochastic variables and the main concepts, mean values -The zero-th moment $\langle 1\rangle$ is just the normalization condition of -$p$. The first moment, $\langle x\rangle$, is called the *mean* of $p$ -and often denoted by the letter $\mu$ - -$$ -\langle x\rangle = \mu \equiv \int x p(x)dx, -$$ - -for a continuous distribution and - -$$ -\langle x\rangle = \mu \equiv \sum_{i=1}^N x_i p(x_i), -$$ - -for a discrete distribution. -Qualitatively it represents the centroid or the average value of the -PDF and is therefore simply called the expectation value of $p(x)$. - - - - -## Stochastic variables and the main concepts, central moments, the variance - -A special version of the moments is the set of *central moments*, the n-th central moment defined as - -$$ -\langle (x-\langle x\rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx -$$ - -The zero-th and first central moments are both trivial, equal $1$ and -$0$, respectively. But the second central moment, known as the -*variance* of $p$, is of particular interest. For the stochastic -variable $X$, the variance is denoted as $\sigma^2_X$ or $\mathrm{Var}(X)$ - -$$ -\begin{align*} -\sigma^2_X &=\mathrm{Var}(X) = \langle (x-\langle x\rangle)^2\rangle = -\int (x-\langle x\rangle)^2 p(x)dx\\ -& = \int\left(x^2 - 2 x \langle x\rangle^{2} +\langle x\rangle^2\right)p(x)dx\\ -& = \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2\\ -& = \langle x^2 \rangle - \langle x\rangle^2 -\end{align*} -$$ - -The square root of the variance, $\sigma =\sqrt{\langle (x-\langle x\rangle)^2\rangle}$ is called the -**standard deviation** of $p$. It is the RMS (root-mean-square) -value of the deviation of the PDF from its mean value, interpreted -qualitatively as the "spread" of $p$ around its mean. - - - - - - -## Probability Distribution Functions - -The following table collects properties of probability distribution functions. -In our notation we reserve the label $p(x)$ for the probability of a certain event, -while $P(x)$ is the cumulative probability. - - - - - - - - - - - - - - - -
Discrete PDF Continuous PDF
Domain $\left\{x_1, x_2, x_3, \dots, x_N\right\}$ $[a,b]$
Probability $p(x_i)$ $p(x)dx$
Cumulative $P_i=\sum_{l=1}^ip(x_l)$ $P(x)=\int_a^xp(t)dt$
Positivity $0 \le p(x_i) \le 1$ $p(x) \ge 0$
Positivity $0 \le P_i \le 1$ $0 \le P(x) \le 1$
Monotonic $P_i \ge P_j$ if $x_i \ge x_j$ $P(x_i) \ge P(x_j)$ if $x_i \ge x_j$
Normalization $P_N=1$ $P(b)=1$
- - - - - -## Probability Distribution Functions -With a PDF we can compute expectation values of selected quantities such as - -$$ -\langle x^k\rangle=\sum_{i=1}^{N}x_i^kp(x_i), -$$ - -if we have a discrete PDF or - -$$ -\langle x^k\rangle=\int_a^b x^kp(x)dx, -$$ - -in the case of a continuous PDF. We have already defined the mean value $\mu$ -and the variance $\sigma^2$. - - - - -## The three famous Probability Distribution Functions - -There are at least three PDFs which one may encounter. These are the - -**Uniform distribution** - -$$ -p(x)=\frac{1}{b-a}\Theta(x-a)\Theta(b-x), -$$ - -yielding probabilities different from zero in the interval $[a,b]$. - -**The exponential distribution** - -$$ -p(x)=\alpha \exp{(-\alpha x)}, -$$ - -yielding probabilities different from zero in the interval $[0,\infty)$ and with mean value - -$$ -\mu = \int_0^{\infty}xp(x)dx=\int_0^{\infty}x\alpha \exp{(-\alpha x)}dx=\frac{1}{\alpha}, -$$ - -with variance - -$$ -\sigma^2=\int_0^{\infty}x^2p(x)dx-\mu^2 = \frac{1}{\alpha^2}. -$$ - -## Probability Distribution Functions, the normal distribution -Finally, we have the so-called univariate normal distribution, or just the **normal distribution** - -$$ -p(x)=\frac{1}{b\sqrt{2\pi}}\exp{\left(-\frac{(x-a)^2}{2b^2}\right)} -$$ - -with probabilities different from zero in the interval $(-\infty,\infty)$. -The integral $\int_{-\infty}^{\infty}\exp{\left(-(x^2\right)}dx$ appears in many calculations, its value -is $\sqrt{\pi}$, a result we will need when we compute the mean value and the variance. -The mean value is - -$$ -\mu = \int_0^{\infty}xp(x)dx=\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}x \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, -$$ - -which becomes with a suitable change of variables - -$$ -\mu =\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(a+b\sqrt{2}y)\exp{-y^2}dy=a. -$$ - -## Probability Distribution Functions, the normal distribution -Similarly, the variance becomes - -$$ -\sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}(x-\mu)^2 \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, -$$ - -and inserting the mean value and performing a variable change we obtain - -$$ -\sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(b\sqrt{2}y)^2\exp{\left(-y^2\right)}dy= -\frac{2b^2}{\sqrt{\pi}}\int_{-\infty}^{\infty}y^2\exp{\left(-y^2\right)}dy, -$$ - -and performing a final integration by parts we obtain the well-known result $\sigma^2=b^2$. -It is useful to introduce the standard normal distribution as well, defined by $\mu=a=0$, viz. a distribution -centered around zero and with a variance $\sigma^2=1$, leading to - - -
- -$$ -\begin{equation} - p(x)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right)}. -\label{_auto1} \tag{3} -\end{equation} -$$ - -## Probability Distribution Functions, the cumulative distribution - -The exponential and uniform distributions have simple cumulative functions, -whereas the normal distribution does not, being proportional to the so-called -error function $erf(x)$, given by - -$$ -P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x\exp{\left(-\frac{t^2}{2}\right)}dt, -$$ - -which is difficult to evaluate in a quick way. - - - - - -## Probability Distribution Functions, other important distribution - -Some other PDFs which one encounters often in the natural sciences are the binomial distribution - -$$ -p(x) = \left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} \hspace{0.5cm}x=0,1,\dots,n, -$$ - -where $y$ is the probability for a specific event, such as the tossing of a coin or moving left or right -in case of a random walker. Note that $x$ is a discrete stochastic variable. - -The sequence of binomial trials is characterized by the following definitions - - * Every experiment is thought to consist of $N$ independent trials. - - * In every independent trial one registers if a specific situation happens or not, such as the jump to the left or right of a random walker. - - * The probability for every outcome in a single trial has the same value, for example the outcome of tossing (either heads or tails) a coin is always $1/2$. - - - - -## Probability Distribution Functions, the binomial distribution - -In order to compute the mean and variance we need to recall Newton's binomial -formula - -$$ -(a+b)^m=\sum_{n=0}^m \left(\begin{array}{c} m \\ n\end{array}\right)a^nb^{m-n}, -$$ - -which can be used to show that - -$$ -\sum_{x=0}^n\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1, -$$ - -the PDF is normalized to one. -The mean value is - -$$ -\mu = \sum_{x=0}^n x\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = -\sum_{x=0}^n x\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x}, -$$ - -resulting in - -$$ -\mu = -\sum_{x=0}^n x\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)}, -$$ - -which we rewrite as - -$$ -\mu=ny\sum_{\nu=0}^n\left(\begin{array}{c} n-1 \\ \nu\end{array}\right)y^{\nu}(1-y)^{n-1-\nu} =ny(y+1-y)^{n-1}=ny. -$$ - -The variance is slightly trickier to get. It reads $\sigma^2=ny(1-y)$. - - -## Probability Distribution Functions, Poisson's distribution - -Another important distribution with discrete stochastic variables $x$ is -the Poisson model, which resembles the exponential distribution and reads - -$$ -p(x) = \frac{\lambda^x}{x!} e^{-\lambda} \hspace{0.5cm}x=0,1,\dots,;\lambda > 0. -$$ - -In this case both the mean value and the variance are easier to calculate, - -$$ -\mu = \sum_{x=0}^{\infty} x \frac{\lambda^x}{x!} e^{-\lambda} = \lambda e^{-\lambda}\sum_{x=1}^{\infty} -\frac{\lambda^{x-1}}{(x-1)!}=\lambda, -$$ - -and the variance is $\sigma^2=\lambda$. - - - - - - -## Probability Distribution Functions, Poisson's distribution -An example of applications of the Poisson distribution could be the counting -of the number of $\alpha$-particles emitted from a radioactive source in a given time interval. -In the limit of $n\rightarrow \infty$ and for small probabilities $y$, the binomial distribution -approaches the Poisson distribution. Setting $\lambda = ny$, with $y$ the probability for an event in -the binomial distribution we can show that - -$$ -\lim_{n\rightarrow \infty}\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} e^{-\lambda}=\sum_{x=1}^{\infty}\frac{\lambda^x}{x!} e^{-\lambda}. -$$ - -## Meet the covariance! -An important quantity in a statistical analysis is the so-called covariance. - -Consider the set $\{X_i\}$ of $n$ -stochastic variables (not necessarily uncorrelated) with the -multivariate PDF $P(x_1,\dots,x_n)$. The *covariance* of two -of the stochastic variables, $X_i$ and $X_j$, is defined as follows - - -
- -$$ -\begin{equation} -\mathrm{Cov}(X_i,\,X_j) = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle -\label{_auto2} \tag{4} -\end{equation} -$$ - - -
- -$$ -\begin{equation} -=\int\cdots\int (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)P(x_1,\dots,x_n)\,dx_1\dots dx_n, -\label{eq:def_covariance} \tag{5} -\end{equation} -$$ - -with - -$$ -\langle x_i\rangle = -\int\cdots\int x_i P(x_1,\dots,x_n)\,dx_1\dots dx_n. -$$ - -## Meet the covariance in matrix disguise -If we consider the above covariance as a matrix - -$$ -C_{ij} =\mathrm{Cov}(X_i,\,X_j), -$$ - -then the diagonal elements are just the familiar -variances, $C_{ii} = \mathrm{Cov}(X_i,\,X_i) = \mathrm{Var}(X_i)$. It turns out that -all the off-diagonal elements are zero if the stochastic variables are -uncorrelated. - - - - -## Covariance - -# Importing various packages -from math import exp, sqrt -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt - -def covariance(x, y, n): - sum = 0.0 - mean_x = np.mean(x) - mean_y = np.mean(y) - for i in range(0, n): - sum += (x[(i)]-mean_x)*(y[i]-mean_y) - return sum/n - -n = 10 - -x=np.random.normal(size=n) -y = 4+3*x+np.random.normal(size=n) -covxy = covariance(x,y,n) -print(covxy) -z = np.vstack((x, y)) -c = np.cov(z.T) - -print(c) - -## Meet the covariance, uncorrelated events - -Consider the stochastic variables $X_i$ and $X_j$, ($i\neq j$). We have - -$$ -\begin{align*} -Cov(X_i,\,X_j) &= \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle\\ -&=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle\\ -&=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j \rangle + -\langle \langle x_i\rangle\langle x_j\rangle\rangle \\ -&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle + -\langle x_i\rangle\langle x_j\rangle \\ -&=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle -\end{align*} -$$ - -If $X_i$ and $X_j$ are independent (assuming $i \neq j$), we have that - -$$ -\langle x_i x_j\rangle = \langle x_i\rangle\langle x_j\rangle, -$$ - -leading to - -$$ -Cov(X_i, X_j) = 0 \hspace{0.1cm} (i\neq j). -$$ - -## Numerical experiments and the covariance - -Now that we have constructed an idealized mathematical framework, let -us try to apply it to empirical observations. Examples of relevant -physical phenomena may be spontaneous decays of nuclei, or a purely -mathematical set of numbers produced by some deterministic -mechanism. It is the latter we will deal with, using so-called pseudo-random -number generators. In general our observations will contain only a limited set of -observables. We remind the reader that -a *stochastic process* is a process that produces sequentially a -chain of values - -$$ -\{x_1, x_2,\dots\,x_k,\dots\}. -$$ - -## Numerical experiments and the covariance -We will call these -values our *measurements* and the entire set as our measured -*sample*. The action of measuring all the elements of a sample -we will call a stochastic *experiment* (since, operationally, -they are often associated with results of empirical observation of -some physical or mathematical phenomena; precisely an experiment). We -assume that these values are distributed according to some -PDF $p_X^{\phantom X}(x)$, where $X$ is just the formal symbol for the -stochastic variable whose PDF is $p_X^{\phantom X}(x)$. Instead of -trying to determine the full distribution $p$ we are often only -interested in finding the few lowest moments, like the mean -$\mu_X^{\phantom X}$ and the variance $\sigma_X^{\phantom X}$. - - - - - -## Numerical experiments and the covariance, actual situations -In practical situations however, a sample is always of finite size. Let that -size be $n$. The expectation value of a sample $\alpha$, the **sample mean**, is then defined as follows - -$$ -\langle x_{\alpha} \rangle \equiv \frac{1}{n}\sum_{k=1}^n x_{\alpha,k}. -$$ - -The *sample variance* is: - -$$ -\mathrm{Var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_{\alpha,k} - \langle x_{\alpha} \rangle)^2, -$$ - -with its square root being the *standard deviation of the sample*. - - - - - -## Numerical experiments and the covariance, our observables -You can think of the above observables as a set of quantities which define -a given experiment. This experiment is then repeated several times, say $m$ times. -The total average is then - - -
- -$$ -\begin{equation} -\langle X_m \rangle= \frac{1}{m}\sum_{\alpha=1}^mx_{\alpha}=\frac{1}{mn}\sum_{\alpha, k} x_{\alpha,k}, -\label{eq:exptmean} \tag{6} -\end{equation} -$$ - -where the last sums end at $m$ and $n$. -The total variance is - -$$ -\sigma^2_m= \frac{1}{mn^2}\sum_{\alpha=1}^m(\langle x_{\alpha} \rangle-\langle X_m \rangle)^2, -$$ - -which we rewrite as - - -
- -$$ -\begin{equation} -\sigma^2_m=\frac{1}{m}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle). -\label{eq:exptvariance} \tag{7} -\end{equation} -$$ - -## Numerical experiments and the covariance, the sample variance - -We define also the sample variance $\sigma^2$ of all $mn$ individual experiments as - - -
- -$$ -\begin{equation} -\sigma^2=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n (x_{\alpha,k}-\langle X_m \rangle)^2. -\label{eq:sampleexptvariance} \tag{8} -\end{equation} -$$ - -These quantities, being known experimental values or the results from our calculations, -may differ, in some cases -significantly, from the similarly named -exact values for the mean value $\mu_X$, the variance $\mathrm{Var}(X)$ -and the covariance $\mathrm{Cov}(X,Y)$. - - - - -## Numerical experiments and the covariance, central limit theorem - -The central limit theorem states that the PDF $\tilde{p}(z)$ of -the average of $m$ random values corresponding to a PDF $p(x)$ -is a normal distribution whose mean is the -mean value of the PDF $p(x)$ and whose variance is the variance -of the PDF $p(x)$ divided by $m$, the number of values used to compute $z$. - -The central limit theorem leads then to the well-known expression for the -standard deviation, given by - -$$ -\sigma_m= -\frac{\sigma}{\sqrt{m}}. -$$ - -In many cases the above estimate for the standard deviation, in particular if correlations are strong, may be too simplistic. We need therefore a more precise defintion of the error and the variance in our results. - - - - -## Definition of Correlation Functions and Standard Deviation -Our estimate of the true average $\mu_{X}$ is the sample mean $\langle X_m \rangle$ - -$$ -\mu_{X}^{\phantom X} \approx X_m=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n x_{\alpha,k}. -$$ - -We can then use Eq. ([7](#eq:exptvariance)) - -$$ -\sigma^2_m=\frac{1}{mn^2}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), -$$ - -and rewrite it as - -$$ -\sigma^2_m=\frac{\sigma^2}{n}+\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k -
- -$$ -\begin{equation} -\kappa_d = \frac{f_d}{\sigma^2} -\label{eq:autocorrelformal} \tag{9} -\end{equation} -$$ - -which gives us a useful measure of the correlation pair correlation -starting always at $1$ for $d=0$. - - - - -## Definition of Correlation Functions and Standard Deviation, sample variance - -The sample variance of the $mn$ experiments can now be -written in terms of the autocorrelation function - - -
- -$$ -\begin{equation} -\sigma_m^2=\frac{\sigma^2}{n}+\frac{2}{n}\cdot\sigma^2\sum_{d=1}^{n-1} -\frac{f_d}{\sigma^2}=\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\sigma^2=\frac{\tau}{n}\cdot\sigma^2 -\label{eq:error_estimate_corr_time} \tag{10} -\end{equation} -$$ - -and we see that $\sigma_m$ can be expressed in terms of the -uncorrelated sample variance times a correction factor $\tau$ which -accounts for the correlation between measurements. We call this -correction factor the *autocorrelation time* - - -
- -$$ -\begin{equation} -\tau = 1+2\sum_{d=1}^{n-1}\kappa_d -\label{eq:autocorrelation_time} \tag{11} -\end{equation} -$$ - - - -For a correlation free experiment, $\tau$ -equals 1. - - - - - - -## Definition of Correlation Functions and Standard Deviation -From the point of view of -Eq. ([10](#eq:error_estimate_corr_time)) we can interpret a sequential -correlation as an effective reduction of the number of measurements by -a factor $\tau$. The effective number of measurements becomes - -$$ -n_\mathrm{eff} = \frac{n}{\tau} -$$ - -To neglect the autocorrelation time $\tau$ will always cause our -simple uncorrelated estimate of $\sigma_m^2\approx \sigma^2/n$ to -be less than the true sample error. The estimate of the error will be -too "good". On the other hand, the calculation of the full -autocorrelation time poses an efficiency problem if the set of -measurements is very large. The solution to this problem is given by -more practically oriented methods like the blocking technique. - - - - - - -## Code to compute the Covariance matrix and the Covariance - -# Importing various packages -from math import exp, sqrt -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt - -# Sample covariance, note the factor 1/(n-1) -def covariance(x, y, n): - sum = 0.0 - mean_x = np.mean(x) - mean_y = np.mean(y) - for i in range(0, n): - sum += (x[(i)]-mean_x)*(y[i]-mean_y) - return sum/(n-1.) - -n = 100 -x = np.random.normal(size=n) -print(np.mean(x)) -y = 4+3*x+np.random.normal(size=n) -print(np.mean(y)) -z = x**3+np.random.normal(size=n) -print(np.mean(z)) -covxx = covariance(x,x,n) -covyy = covariance(y,y,n) -covzz = covariance(z,z,n) -covxy = covariance(x,y,n) -covxz = covariance(x,z,n) -covyz = covariance(y,z,n) -print(covxx,covyy, covzz) -print(covxy,covxz, covyz) -w = np.vstack((x, y, z)) -#print(w) -c = np.cov(w) -print(c) -#eigen = np.zeros(n) -Eigvals, Eigvecs = np.linalg.eig(c) -print(Eigvals) - -## Random Numbers - -Uniform deviates are just random numbers that lie within a specified range -(typically 0 to 1), with any one number in the range just as likely as any other. They -are, in other words, what you probably think random numbers are. However, -we want to distinguish uniform deviates from other sorts of random numbers, for -example numbers drawn from a normal (Gaussian) distribution of specified mean -and standard deviation. These other sorts of deviates are almost always generated by -performing appropriate operations on one or more uniform deviates, as we will see -in subsequent sections. So, a reliable source of random uniform deviates, the subject -of this section, is an essential building block for any sort of stochastic modeling -or Monte Carlo computer work. - - - - - -## Random Numbers, better name: pseudo random numbers - -A disclaimer is however appropriate. It should be fairly obvious that -something as deterministic as a computer cannot generate purely random numbers. - -Numbers generated by any of the standard algorithms are in reality pseudo random -numbers, hopefully abiding to the following criteria: - - * they produce a uniform distribution in the interval [0,1]. - - * correlations between random numbers are negligible - - * the period before the same sequence of random numbers is repeated is as large as possible and finally - - * the algorithm should be fast. - - - - - -## Random number generator RNG - The most common random number generators are based on so-called -Linear congruential relations of the type - -$$ -N_i=(aN_{i-1}+c) \mathrm{MOD} (M), -$$ - -which yield a number in the interval [0,1] through - -$$ -x_i=N_i/M -$$ - -The number -$M$ is called the period and it should be as large as possible - and -$N_0$ is the starting value, or seed. The function $\mathrm{MOD}$ means the remainder, -that is if we were to evaluate $(13)\mathrm{MOD}(9)$, the outcome is the remainder -of the division $13/9$, namely $4$. - - - - -## Random number generator RNG and periodic outputs - -The problem with such generators is that their outputs are periodic; -they -will start to repeat themselves with a period that is at most $M$. If however -the parameters $a$ and $c$ are badly chosen, the period may be even shorter. - -Consider the following example - -$$ -N_i=(6N_{i-1}+7) \mathrm{MOD} (5), -$$ - -with a seed $N_0=2$. This generator produces the sequence -$4,1,3,0,2,4,1,3,0,2,...\dots$, i.e., a sequence with period $5$. -However, increasing $M$ may not guarantee a larger period as the following -example shows - -$$ -N_i=(27N_{i-1}+11) \mathrm{MOD} (54), -$$ - -which still, with $N_0=2$, results in $11,38,11,38,11,38,\dots$, a period of -just $2$. - - - - -## Random number generator RNG and its period -Typical periods for the random generators provided in the program library -are of the order of $\sim 10^9$ or larger. Other random number generators which have -become increasingly popular are so-called shift-register generators. -In these generators each successive number depends on many preceding -values (rather than the last values as in the linear congruential -generator). -For example, you could make a shift register generator whose $l$th -number is the sum of the $l-i$th and $l-j$th values with modulo $M$, - -$$ -N_l=(aN_{l-i}+cN_{l-j})\mathrm{MOD}(M). -$$ - -## Random number generator RNG, other examples -Such a generator again produces a sequence of pseudorandom numbers -but this time with a period much larger than $M$. -It is also possible to construct more elaborate algorithms by including -more than two past terms in the sum of each iteration. -One example is the generator of [Marsaglia and Zaman](http://dl.acm.org/citation.cfm?id=187154) -which consists of two congruential relations - - -
- -$$ -\begin{equation} - N_l=(N_{l-3}-N_{l-1})\mathrm{MOD}(2^{31}-69), -\label{eq:mz1} \tag{12} -\end{equation} -$$ - -followed by - - -
- -$$ -\begin{equation} - N_l=(69069N_{l-1}+1013904243)\mathrm{MOD}(2^{32}), -\label{eq:mz2} \tag{13} -\end{equation} -$$ - -which according to the authors has a period larger than $2^{94}$. - - - - -## Random number generator RNG, other examples -Instead of using modular addition, we could use the bitwise -exclusive-OR ($\oplus$) operation so that - -$$ -N_l=(N_{l-i})\oplus (N_{l-j}) -$$ - -where the bitwise action of $\oplus$ means that if $N_{l-i}=N_{l-j}$ the result is -$0$ whereas if $N_{l-i}\ne N_{l-j}$ the result is -$1$. As an example, consider the case where $N_{l-i}=6$ and $N_{l-j}=11$. The first -one has a bit representation (using 4 bits only) which reads $0110$ whereas the -second number is $1011$. Employing the $\oplus$ operator yields -$1101$, or $2^3+2^2+2^0=13$. - -In Fortran90, the bitwise $\oplus$ operation is coded through the intrinsic -function $\mathrm{IEOR}(m,n)$ where $m$ and $n$ are the input numbers, while in $C$ -it is given by $m\wedge n$. - - - - - -## Random number generator RNG, RAN0 - -We show here how the linear congruential algorithm can be implemented, namely - -$$ -N_i=(aN_{i-1}) \mathrm{MOD} (M). -$$ - -However, since $a$ and $N_{i-1}$ are integers and their multiplication -could become greater than the standard 32 bit integer, there is a trick via -Schrage's algorithm which approximates the multiplication -of large integers through the factorization - -$$ -M=aq+r, -$$ - -where we have defined - -$$ -q=[M/a], -$$ - -and - -$$ -r = M\hspace{0.1cm}\mathrm{MOD} \hspace{0.1cm}a. -$$ - -where the brackets denote integer division. In the code below the numbers -$q$ and $r$ are chosen so that $r < q$. - - - - - - -## Random number generator RNG, RAN0 - -To see how this works we note first that - - -
- -$$ -\begin{equation} -(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\mathrm{MOD} (M), -\label{eq:rntrick1} \tag{14} -\end{equation} -$$ - -since we can add or subtract any integer multiple of $M$ from $aN_{i-1}$. -The last term $[N_{i-1}/q]M\mathrm{MOD}(M)$ is zero since the integer division -$[N_{i-1}/q]$ just yields a constant which is multiplied with $M$. - - - - - -## Random number generator RNG, RAN0 -We can now rewrite Eq. ([14](#eq:rntrick1)) as - - -
- -$$ -\begin{equation} -(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\mathrm{MOD} (M), -\label{eq:rntrick2} \tag{15} -\end{equation} -$$ - -which results -in - - -
- -$$ -\begin{equation} -(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\right)\mathrm{MOD} (M), -\label{eq:rntrick3} \tag{16} -\end{equation} -$$ - -yielding - - -
- -$$ -\begin{equation} -(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}\mathrm{MOD} (q)) -[N_{i-1}/q]r)\right)\mathrm{MOD} (M). -\label{eq:rntrick4} \tag{17} -\end{equation} -$$ - -## Random number generator RNG, RAN0 -The term $[N_{i-1}/q]r$ is always smaller or equal $N_{i-1}(r/q)$ and with $r < q$ we obtain always a -number smaller than $N_{i-1}$, which is smaller than $M$. -And since the number $N_{i-1}\mathrm{MOD} (q)$ is between zero and $q-1$ then -$a(N_{i-1}\mathrm{MOD} (q))< aq$. Combined with our definition of $q=[M/a]$ ensures that -this term is also smaller than $M$ meaning that both terms fit into a -32-bit signed integer. None of these two terms can be negative, but their difference could. -The algorithm below adds $M$ if their difference is negative. -Note that the program uses the bitwise $\oplus$ operator to generate -the starting point for each generation of a random number. The period -of $ran0$ is $\sim 2.1\times 10^{9}$. A special feature of this -algorithm is that is should never be called with the initial seed -set to $0$. - - - - - -## Random number generator RNG, RAN0 code - - /* - ** The function - ** ran0() - ** is an "Minimal" random number generator of Park and Miller - ** Set or reset the input value - ** idum to any integer value (except the unlikely value MASK) - ** to initialize the sequence; idum must not be altered between - ** calls for sucessive deviates in a sequence. - ** The function returns a uniform deviate between 0.0 and 1.0. - */ - double ran0(long &idum) - { - const int a = 16807, m = 2147483647, q = 127773; - const int r = 2836, MASK = 123459876; - const double am = 1./m; - long k; - double ans; - idum ^= MASK; - k = (*idum)/q; - idum = a*(idum - k*q) - r*k; - // add m if negative difference - if(idum < 0) idum += m; - ans=am*(idum); - idum ^= MASK; - return ans; - } // End: function ran0() - - -## Properties of Selected Random Number Generators - -As mentioned previously, the underlying PDF for the generation of -random numbers is the uniform distribution, meaning that the -probability for finding a number $x$ in the interval [0,1] is $p(x)=1$. - -A random number generator should produce numbers which are uniformly distributed -in this interval. The table shows the distribution of $N=10000$ random -numbers generated by the functions in the program library. -We note in this table that the number of points in the various -intervals $0.0-0.1$, $0.1-0.2$ etc are fairly close to $1000$, with some minor -deviations. - -Two additional measures are the standard deviation $\sigma$ and the mean -$\mu=\langle x\rangle$. - - - - -## Properties of Selected Random Number Generators -For the uniform distribution, the mean value $\mu$ is then - -$$ -\mu=\langle x\rangle=\frac{1}{2} -$$ - -while the standard deviation is - -$$ -\sigma=\sqrt{\langle x^2\rangle-\mu^2}=\frac{1}{\sqrt{12}}=0.2886. -$$ - -## Properties of Selected Random Number Generators -The various random number generators produce results which agree rather well with -these limiting values. - - - - - - - - - - - - - - - - - - - -
$x$-bin ran0 ran1 ran2 ran3
0.0-0.1 1013 991 938 1047
0.1-0.2 1002 1009 1040 1030
0.2-0.3 989 999 1030 993
0.3-0.4 939 960 1023 937
0.4-0.5 1038 1001 1002 992
0.5-0.6 1037 1047 1009 1009
0.6-0.7 1005 989 1003 989
0.7-0.8 986 962 985 954
0.8-0.9 1000 1027 1009 1023
0.9-1.0 991 1015 961 1026
$\mu$ 0.4997 0.5018 0.4992 0.4990
$\sigma$ 0.2882 0.2892 0.2861 0.2915
- - - - -## Simple demonstration of RNGs using python -The following simple Python code plots the distribution of the produced random numbers using the linear congruential RNG employed by Python. The trend displayed in the previous table is seen rather clearly. - -#!/usr/bin/env python -import numpy as np -import matplotlib.mlab as mlab -import matplotlib.pyplot as plt -import random - -# initialize the rng with a seed -random.seed() -counts = 10000 -values = np.zeros(counts) -for i in range (1, counts, 1): - values[i] = random.random() - -# the histogram of the data -n, bins, patches = plt.hist(values, 10, facecolor='green') - -plt.xlabel('$x$') -plt.ylabel('Number of counts') -plt.title(r'Test of uniform distribution') -plt.axis([0, 1, 0, 1100]) -plt.grid(True) -plt.show() - -## Properties of Selected Random Number Generators -Since our random numbers, which are typically generated via a linear congruential algorithm, -are never fully independent, we can then define -an important test which measures the degree of correlation, namely the so-called -auto-correlation function defined previously, see again Eq. ([9](#eq:autocorrelformal)). -We rewrite it here as - -$$ -C_k=\frac{f_d} - {\sigma^2}, -$$ - -with $C_0=1$. Recall that -$\sigma^2=\langle x_i^2\rangle-\langle x_i\rangle^2$ and that - -$$ -f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), -$$ - -The non-vanishing of $C_k$ for $k\ne 0$ means that the random -numbers are not independent. The independence of the random numbers is crucial -in the evaluation of other expectation values. If they are not independent, our -assumption for approximating $\sigma_N$ is no longer valid. - - - - - -## Autocorrelation function -This program computes the autocorrelation function as discussed in the equation on the previous slide for random numbers generated with the normal distribution $N(0,1)$. - -# Importing various packages -from math import exp, sqrt -from random import random, seed -import numpy as np -import matplotlib.pyplot as plt - -def autocovariance(x, n, k, mean_x): - sum = 0.0 - for i in range(0, n-k): - sum += (x[(i+k)]-mean_x)*(x[i]-mean_x) - return sum/n - -n = 1000 -x=np.random.normal(size=n) -autocor = np.zeros(n) -figaxis = np.zeros(n) -mean_x=np.mean(x) -var_x = np.var(x) -print(mean_x, var_x) -for i in range (0, n): - figaxis[i] = i - autocor[i]=(autocovariance(x, n, i, mean_x))/var_x - -plt.plot(figaxis, autocor, "r-") -plt.axis([0,n,-0.1, 1.0]) -plt.xlabel(r'$i$') -plt.ylabel(r'$\gamma_i$') -plt.title(r'Autocorrelation function') -plt.show() - -As can be seen from the plot, the first point gives back the variance and a value of one. -For the remaining values we notice that there are still non-zero values for the auto-correlation function. - - - - - - - - - - - - - - - - - -## Correlation function and which random number generators should I use -The program here computes the correlation function for one of the standard functions included with the c++ compiler. - - // This function computes the autocorrelation function for - // the standard c++ random number generator - - #include - #include - #include - #include - using namespace std; - // output file as global variable - ofstream ofile; - - // Main function begins here - int main(int argc, char* argv[]) - { - int n; - char *outfilename; - - cin >> n; - double MCint = 0.; double MCintsqr2=0.; - double invers_period = 1./RAND_MAX; // initialise the random number generator - srand(time(NULL)); // This produces the so-called seed in MC jargon - // Compute the variance and the mean value of the uniform distribution - // Compute also the specific values x for each cycle in order to be able to - // the covariance and the correlation function - // Read in output file, abort if there are too few command-line arguments - if( argc <= 2 ){ - cout << "Bad Usage: " << argv[0] << - " read also output file and number of cycles on same line" << endl; - exit(1); - } - else{ - outfilename=argv[1]; - } - ofile.open(outfilename); - // Get the number of Monte-Carlo samples - n = atoi(argv[2]); - double *X; - X = new double[n]; - for (int i = 0; i < n; i++){ - double x = double(rand())*invers_period; - X[i] = x; - MCint += x; - MCintsqr2 += x*x; - } - double Mean = MCint/((double) n ); - MCintsqr2 = MCintsqr2/((double) n ); - double STDev = sqrt(MCintsqr2-Mean*Mean); - double Variance = MCintsqr2-Mean*Mean; - // Write mean value and standard deviation - cout << " Standard deviation= " << STDev << " Integral = " << Mean << endl; - - // Now we compute the autocorrelation function - double *autocor; autocor = new double[n]; - for (int j = 0; j < n; j++){ - double sum = 0.0; - for (int k = 0; k < (n-j); k++){ - sum += (X[k]-Mean)*(X[k+j]-Mean); - } - autocor[j] = sum/Variance/((double) n ); - ofile << setiosflags(ios::showpoint | ios::uppercase); - ofile << setw(15) << setprecision(8) << j; - ofile << setw(15) << setprecision(8) << autocor[j] << endl; - } - ofile.close(); // close output file - return 0; - } // end of main program - - -## Which RNG should I use? -* C++ has a class called **random**. The [random class](http://www.cplusplus.com/reference/random/) contains a large selection of RNGs and is highly recommended. Some of these RNGs have very large periods making it thereby very safe to use these RNGs in case one is performing large calculations. In particular, the [Mersenne twister random number engine](http://www.cplusplus.com/reference/random/mersenne_twister_engine/) has a period of $2^{19937}$. - -* Add RNGs in Python - - - - - -## How to use the Mersenne generator -The following part of a c++ code (from project 4) sets up the uniform distribution for $x\in [0,1]$. - - /* - - // You need this - #include - - // Initialize the seed and call the Mersienne algo - std::random_device rd; - std::mt19937_64 gen(rd()); - // Set up the uniform distribution for x \in [[0, 1] - std::uniform_real_distribution RandomNumberGenerator(0.0,1.0); - - // Now use the RNG - int ix = (int) (RandomNumberGenerator(gen)*NSpins); - - -## Why blocking? -**Statistical analysis.** - - * Monte Carlo simulations can be treated as *computer experiments* - - * The results can be analysed with the same statistical tools as we would use analysing experimental data. - - * As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors. - -A very good article which explains blocking is H. Flyvbjerg and H. G. Petersen, *Error estimates on averages of correlated data*, [Journal of Chemical Physics 91, 461-466 (1989)](http://scitation.aip.org/content/aip/journal/jcp/91/1/10.1063/1.457480). - - - - - - -## Why blocking? -**Statistical analysis.** - - * As in other experiments, Monte Carlo experiments have two classes of errors: - - * Statistical errors - - * Systematical errors - - - * Statistical errors can be estimated using standard tools from statistics - - * Systematical errors are method specific and must be treated differently from case to case. (In VMC a common source is the step length or time step in importance sampling) - - - - - -## Code to demonstrate the calculation of the autocorrelation function -The following code computes the autocorrelation function, the covariance and the standard deviation -for standard RNG. -The [following file](https://github.com/CompPhysics/ComputationalPhysics2/tree/gh-pages/doc/Programs/LecturePrograms/programs/Blocking/autocorrelation.cpp) gives the code. - - // This function computes the autocorrelation function for - // the Mersenne random number generator with a uniform distribution - #include - #include - #include - #include - #include - #include - #include - #include - using namespace std; - using namespace arma; - // output file - ofstream ofile; - - // Main function begins here - int main(int argc, char* argv[]) - { - int MonteCarloCycles; - string filename; - if (argc > 1) { - filename=argv[1]; - MonteCarloCycles = atoi(argv[2]); - string fileout = filename; - string argument = to_string(MonteCarloCycles); - fileout.append(argument); - ofile.open(fileout); - } - - // Compute the variance and the mean value of the uniform distribution - // Compute also the specific values x for each cycle in order to be able to - // compute the covariance and the correlation function - - vec X = zeros(MonteCarloCycles); - double MCint = 0.; double MCintsqr2=0.; - std::random_device rd; - std::mt19937_64 gen(rd()); - // Set up the uniform distribution for x \in [[0, 1] - std::uniform_real_distribution RandomNumberGenerator(0.0,1.0); - for (int i = 0; i < MonteCarloCycles; i++){ - double x = RandomNumberGenerator(gen); - X(i) = x; - MCint += x; - MCintsqr2 += x*x; - } - double Mean = MCint/((double) MonteCarloCycles ); - MCintsqr2 = MCintsqr2/((double) MonteCarloCycles ); - double STDev = sqrt(MCintsqr2-Mean*Mean); - double Variance = MCintsqr2-Mean*Mean; - // Write mean value and variance - cout << " Sample variance= " << Variance << " Mean value = " << Mean << endl; - // Now we compute the autocorrelation function - vec autocorrelation = zeros(MonteCarloCycles); - for (int j = 0; j < MonteCarloCycles; j++){ - double sum = 0.0; - for (int k = 0; k < (MonteCarloCycles-j); k++){ - sum += (X(k)-Mean)*(X(k+j)-Mean); - } - autocorrelation(j) = sum/Variance/((double) MonteCarloCycles ); - ofile << setiosflags(ios::showpoint | ios::uppercase); - ofile << setw(15) << setprecision(8) << j; - ofile << setw(15) << setprecision(8) << autocorrelation(j) << endl; - } - // Now compute the exact covariance using the autocorrelation function - double Covariance = 0.0; - for (int j = 0; j < MonteCarloCycles; j++){ - Covariance += autocorrelation(j); - } - Covariance *= 2.0/((double) MonteCarloCycles); - // Compute now the total variance, including the covariance, and obtain the standard deviation - double TotalVariance = (Variance/((double) MonteCarloCycles ))+Covariance; - cout << "Covariance =" << Covariance << "Totalvariance= " << TotalVariance << "Sample Variance/n= " << (Variance/((double) MonteCarloCycles )) << endl; - cout << " STD from sample variance= " << sqrt(Variance/((double) MonteCarloCycles )) << " STD with covariance = " << sqrt(TotalVariance) << endl; - - ofile.close(); // close output file - return 0; - } // end of main program - - - - -## What is blocking? -**Blocking.** - - * Say that we have a set of samples from a Monte Carlo experiment - - * Assuming (wrongly) that our samples are uncorrelated our best estimate of the standard deviation of the mean $\langle \mathbf{M}\rangle$ is given by - -$$ -\sigma=\sqrt{\frac{1}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} -$$ - -* If the samples are correlated we can rewrite our results to show that - -$$ -\sigma=\sqrt{\frac{1+2\tau/\Delta t}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} -$$ - -where $\tau$ is the correlation time (the time between a sample and the next uncorrelated sample) and $\Delta t$ is time between each sample - - - - -## What is blocking? -**Blocking.** - - * If $\Delta t\gg\tau$ our first estimate of $\sigma$ still holds - - * Much more common that $\Delta t<\tau$ - - * In the method of data blocking we divide the sequence of samples into blocks - - * We then take the mean $\langle \mathbf{M}_i\rangle$ of block $i=1\ldots n_{blocks}$ to calculate the total mean and variance - - * The size of each block must be so large that sample $j$ of block $i$ is not correlated with sample $j$ of block $i+1$ - - * The correlation time $\tau$ would be a good choice - - - - -## What is blocking? -**Blocking.** - - * Problem: We don't know $\tau$ or it is too expensive to compute - - * Solution: Make a plot of std. dev. as a function of blocksize - - * The estimate of std. dev. of correlated data is too low $\to$ the error will increase with increasing block size until the blocks are uncorrelated, where we reach a plateau - - * When the std. dev. stops increasing the blocks are uncorrelated - - - - -## Implementation - * Do a Monte Carlo simulation, storing all samples to file - - * Do the statistical analysis on this file, independently of your Monte Carlo program - - * Read the file into an array - - * Loop over various block sizes - - * For each block size $n_b$, loop over the array in steps of $n_b$ taking the mean of elements $i n_b,\ldots,(i+1) n_b$ - - * Take the mean and variance of the resulting array - - * Write the results for each block size to file for later - analysis - - - - - - - - -## Actual implementation with code, main function -When the file gets large, it can be useful to write your data in binary mode instead of ascii characters. -The [following python file](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) reads data from file with the output from every Monte Carlo cycle. - -# Blocking - @timeFunction - def blocking(self, blockSizeMax = 500): - blockSizeMin = 1 - - self.blockSizes = [] - self.meanVec = [] - self.varVec = [] - - for i in range(blockSizeMin, blockSizeMax): - if(len(self.data) % i != 0): - pass#continue - blockSize = i - meanTempVec = [] - varTempVec = [] - startPoint = 0 - endPoint = blockSize - - while endPoint <= len(self.data): - meanTempVec.append(np.average(self.data[startPoint:endPoint])) - startPoint = endPoint - endPoint += blockSize - mean, var = np.average(meanTempVec), np.var(meanTempVec)/len(meanTempVec) - self.meanVec.append(mean) - self.varVec.append(var) - self.blockSizes.append(blockSize) - - self.blockingAvg = np.average(self.meanVec[-200:]) - self.blockingVar = (np.average(self.varVec[-200:])) - self.blockingStd = np.sqrt(self.blockingVar) - -## The Bootstrap method - -The Bootstrap resampling method is also very popular. It is very simple: - -1. Start with your sample of measurements and compute the sample variance and the mean values - -2. Then start again but pick in a random way the numbers in the sample and recalculate the mean and the sample variance. - -3. Repeat this $K$ times. - -It can be shown, see the article by [Efron](https://projecteuclid.org/download/pdf_1/euclid.aos/1176344552) -that it produces the correct standard deviation. - -This method is very useful for small ensembles of data points. - - -## Bootstrapping -Given a set of $N$ data, assume that we are interested in some -observable $\theta$ which may be estimated from that set. This observable can also be for example the result of a fit based on all $N$ raw data. -Let us call the value of the observable obtained from the original -data set $\hat{\theta}$. One recreates from the sample repeatedly -other samples by choosing randomly $N$ data out of the original set. -This costs essentially nothing, since we just recycle the original data set for the building of new sets. - - -## Bootstrapping, recipe -Let us assume we have done this $K$ times and thus have $K$ sets of $N$ -data values each. -Of course some values will enter more than once in the new sets. For each of these sets one computes the observable $\theta$ resulting in values $\theta_k$ with $k = 1,...,K$. Then one determines - -$$ -\tilde{\theta} = \frac{1}{K} \sum_{k=1}^K \theta_k, -$$ - -and - -$$ -sigma^2_{\tilde{\theta}} = \frac{1}{K} \sum_{k=1}^K \left(\theta_k-\tilde{\theta}\right)^2. -$$ - -These are estimators for $\angle\theta\rangle$ and its variance. They are not unbiased and therefore -$\tilde{\theta}\neq\hat{\theta}$ for finite K. - -The difference is called bias and gives an idea on how far away the result may be from -the true $\angle\theta\rangle$. As final result for the observable one quotes $\angle\theta\rangle = \tilde{\theta} \pm \sigma_{\tilde{\theta}}$ . - - - -## Bootstrapping, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) - - # Bootstrap - @timeFunction - def bootstrap(self, nBoots = 1000): - bootVec = np.zeros(nBoots) - for k in range(0,nBoots): - bootVec[k] = np.average(np.random.choice(self.data, len(self.data))) - self.bootAvg = np.average(bootVec) - self.bootVar = np.var(bootVec) - self.bootStd = np.std(bootVec) - - -## Jackknife, [code](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Programs/Sampling/analysis.py) - - # Jackknife - @timeFunction - def jackknife(self): - jackknVec = np.zeros(len(self.data)) - for k in range(0,len(self.data)): - jackknVec[k] = np.average(np.delete(self.data, k)) - self.jackknAvg = self.avg - (len(self.data) - 1) * (np.average(jackknVec) - self.avg) - self.jackknVar = float(len(self.data) - 1) * np.var(jackknVec) - 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a/doc/LectureNotes/_toc.yml +++ b/doc/LectureNotes/_toc.yml @@ -34,5 +34,4 @@ parts: chapters: - file: chapter9.ipynb - file: chapter10.ipynb - - file: chapter11.ipynb - - file: chapter12.ipynb + diff --git a/doc/LectureNotes/chapter11.ipynb b/doc/LectureNotes/chapter11.ipynb deleted file mode 100644 index c03e4a368..000000000 --- a/doc/LectureNotes/chapter11.ipynb +++ /dev/null @@ -1,3023 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Solving Differential Equations with Deep Learning\n", - "\n", - "The Universal Approximation Theorem states that a neural network can\n", - "approximate any function at a single hidden layer along with one input\n", - "and output layer to any given precision. \n", - "\n", - "\n", - "An ordinary differential equation (ODE) is an equation involving functions having one variable.\n", - "\n", - "In general, an ordinary differential equation looks like" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "

\n", - "\n", - "$$\n", - "\\begin{equation} \\label{ode} \\tag{1}\n", - "f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right) = 0\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(x)$ is the function to find, and $g^{(n)}(x)$ is the $n$-th derivative of $g(x)$.\n", - "\n", - "The $f\\left(x, g(x), g'(x), g''(x), \\, \\dots \\, , g^{(n)}(x)\\right)$ is just a way to write that there is an expression involving $x$ and $g(x), \\ g'(x), \\ g''(x), \\, \\dots \\, , \\text{ and } g^{(n)}(x)$ on the left side of the equality sign in ([1](#ode)).\n", - "The highest order of derivative, that is the value of $n$, determines to the order of the equation.\n", - "The equation is referred to as a $n$-th order ODE.\n", - "Along with ([1](#ode)), some additional conditions of the function $g(x)$ are typically given\n", - "for the solution to be unique.\n", - "\n", - "\n", - "\n", - "Let the trial solution $g_t(x)$ be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - "\tg_t(x) = h_1(x) + h_2(x,N(x,P))\n", - "\\label{_auto1} \\tag{2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $h_1(x)$ is a function that makes $g_t(x)$ satisfy a given set\n", - "of conditions, $N(x,P)$ a neural network with weights and biases\n", - "described by $P$ and $h_2(x, N(x,P))$ some expression involving the\n", - "neural network. The role of the function $h_2(x, N(x,P))$, is to\n", - "ensure that the output from $N(x,P)$ is zero when $g_t(x)$ is\n", - "evaluated at the values of $x$ where the given conditions must be\n", - "satisfied. The function $h_1(x)$ should alone make $g_t(x)$ satisfy\n", - "the conditions.\n", - "\n", - "But what about the network $N(x,P)$?\n", - "\n", - "\n", - "As described previously, an optimization method could be used to minimize the parameters of a neural network, that being its weights and biases, through backward propagation.\n", - "\n", - "\n", - "\n", - "For the minimization to be defined, we need to have a cost function at hand to minimize.\n", - "\n", - "It is given that $f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)$ should be equal to zero in ([1](#ode)).\n", - "We can choose to consider the mean squared error as the cost function for an input $x$.\n", - "Since we are looking at one input, the cost function is just $f$ squared.\n", - "The cost function $c\\left(x, P \\right)$ can therefore be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(x, P\\right) = \\big(f\\left(x, \\, g(x), \\, g'(x), \\, g''(x), \\, \\dots \\, , \\, g^{(n)}(x)\\right)\\big)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If $N$ inputs are given as a vector $\\boldsymbol{x}$ with elements $x_i$ for $i = 1,\\dots,N$,\n", - "the cost function becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{cost} \\tag{3}\n", - "\tC\\left(\\boldsymbol{x}, P\\right) = \\frac{1}{N} \\sum_{i=1}^N \\big(f\\left(x_i, \\, g(x_i), \\, g'(x_i), \\, g''(x_i), \\, \\dots \\, , \\, g^{(n)}(x_i)\\right)\\big)^2\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The neural net should then find the parameters $P$ that minimizes the cost function in\n", - "([3](#cost)) for a set of $N$ training samples $x_i$.\n", - "\n", - "\n", - "\n", - "To perform the minimization using gradient descent, the gradient of $C\\left(\\boldsymbol{x}, P\\right)$ is needed.\n", - "It might happen so that finding an analytical expression of the gradient of $C(\\boldsymbol{x}, P)$ from ([3](#cost)) gets too messy, depending on which cost function one desires to use.\n", - "\n", - "Luckily, there exists libraries that makes the job for us through automatic differentiation.\n", - "Automatic differentiation is a method of finding the derivatives numerically with very high precision.\n", - "\n", - "\n", - "### Example: Exponential decay\n", - "\n", - "An exponential decay of a quantity $g(x)$ is described by the equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solve_expdec} \\tag{4}\n", - " g'(x) = -\\gamma g(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $g(0) = g_0$ for some chosen initial value $g_0$.\n", - "\n", - "The analytical solution of ([4](#solve_expdec)) is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " g(x) = g_0 \\exp\\left(-\\gamma x\\right)\n", - "\\label{_auto2} \\tag{5}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Having an analytical solution at hand, it is possible to use it to compare how well a neural network finds a solution of ([4](#solve_expdec)).\n", - "\n", - "\n", - "\n", - "The program will use a neural network to solve" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solveode} \\tag{6}\n", - "g'(x) = -\\gamma g(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(0) = g_0$ with $\\gamma$ and $g_0$ being some chosen values.\n", - "\n", - "In this example, $\\gamma = 2$ and $g_0 = 10$.\n", - "\n", - "\n", - "To begin with, a trial solution $g_t(t)$ must be chosen. A general trial solution for ordinary differential equations could be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x, P) = h_1(x) + h_2(x, N(x, P))\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $h_1(x)$ ensuring that $g_t(x)$ satisfies some conditions and $h_2(x,N(x, P))$ an expression involving $x$ and the output from the neural network $N(x,P)$ with $P $ being the collection of the weights and biases for each layer. For now, it is assumed that the network consists of one input layer, one hidden layer, and one output layer.\n", - "\n", - "\n", - "\n", - "In this network, there are no weights and bias at the input layer, so $P = \\{ P_{\\text{hidden}}, P_{\\text{output}} \\}$.\n", - "If there are $N_{\\text{hidden} }$ neurons in the hidden layer, then $P_{\\text{hidden}}$ is a $N_{\\text{hidden} } \\times (1 + N_{\\text{input}})$ matrix, given that there are $N_{\\text{input}}$ neurons in the input layer.\n", - "\n", - "The first column in $P_{\\text{hidden} }$ represents the bias for each neuron in the hidden layer and the second column represents the weights for each neuron in the hidden layer from the input layer.\n", - "If there are $N_{\\text{output} }$ neurons in the output layer, then $P_{\\text{output}} $ is a $N_{\\text{output} } \\times (1 + N_{\\text{hidden} })$ matrix.\n", - "\n", - "Its first column represents the bias of each neuron and the remaining columns represents the weights to each neuron.\n", - "\n", - "It is given that $g(0) = g_0$. The trial solution must fulfill this condition to be a proper solution of ([6](#solveode)). A possible way to ensure that $g_t(0, P) = g_0$, is to let $F(N(x,P)) = x \\cdot N(x,P)$ and $A(x) = g_0$. This gives the following trial solution:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{trial} \\tag{7}\n", - "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Reformulating the problem\n", - "\n", - "We wish that our neural network manages to minimize a given cost function.\n", - "\n", - "A reformulation of out equation, ([6](#solveode)), must therefore be done,\n", - "such that it describes the problem a neural network can solve for.\n", - "\n", - "The neural network must find the set of weights and biases $P$ such that the trial solution in ([7](#trial)) satisfies ([6](#solveode)).\n", - "\n", - "The trial solution" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x, P) = g_0 + x \\cdot N(x, P)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "has been chosen such that it already solves the condition $g(0) = g_0$. What remains, is to find $P$ such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{nnmin} \\tag{8}\n", - "g_t'(x, P) = - \\gamma g_t(x, P)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "is fulfilled as *best as possible*.\n", - "\n", - "\n", - "The left hand side and right hand side of ([8](#nnmin)) must be computed separately, and then the neural network must choose weights and biases, contained in $P$, such that the sides are equal as best as possible.\n", - "This means that the absolute or squared difference between the sides must be as close to zero, ideally equal to zero.\n", - "In this case, the difference squared shows to be an appropriate measurement of how erroneous the trial solution is with respect to $P$ of the neural network.\n", - "\n", - "This gives the following cost function our neural network must solve for:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P}\\Big\\{ \\big(g_t'(x, P) - ( -\\gamma g_t(x, P) \\big)^2 \\Big\\}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "(the notation $\\min_{P}\\{ f(x, P) \\}$ means that we desire to find $P$ that yields the minimum of $f(x, P)$)\n", - "\n", - "or, in terms of weights and biases for the hidden and output layer in our network:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }}\\Big\\{ \\big(g_t'(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) - ( -\\gamma g_t(x, \\{ P_{\\text{hidden} }, P_{\\text{output} }\\}) \\big)^2 \\Big\\}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for an input value $x$.\n", - "\n", - "\n", - "\n", - "If the neural network evaluates $g_t(x, P)$ at more values for $x$, say $N$ values $x_i$ for $i = 1, \\dots, N$, then the *total* error to minimize becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{min} \\tag{9}\n", - "\\min_{P}\\Big\\{\\frac{1}{N} \\sum_{i=1}^N \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2 \\Big\\}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Letting $\\boldsymbol{x}$ be a vector with elements $x_i$ and $C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2$ denote the cost function, the minimization problem that our network must solve, becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\min_{P} C(\\boldsymbol{x}, P)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In terms of $P_{\\text{hidden} }$ and $P_{\\text{output} }$, this could also be expressed as\n", - "\n", - "$$\n", - "\\min_{P_{\\text{hidden} }, \\ P_{\\text{output} }} C(\\boldsymbol{x}, \\{P_{\\text{hidden} }, P_{\\text{output} }\\})\n", - "$$\n", - "\n", - "\n", - "For simplicity, it is assumed that the input is an array $\\boldsymbol{x} = (x_1, \\dots, x_N)$ with $N$ elements. It is at these points the neural network should find $P$ such that it fulfills ([9](#min)).\n", - "\n", - "First, the neural network must feed forward the inputs.\n", - "This means that $\\boldsymbol{x}s$ must be passed through an input layer, a hidden layer and a output layer. The input layer in this case, does not need to process the data any further.\n", - "The input layer will consist of $N_{\\text{input} }$ neurons, passing its element to each neuron in the hidden layer. The number of neurons in the hidden layer will be $N_{\\text{hidden} }$.\n", - "\n", - "\n", - "For the $i$-th in the hidden layer with weight $w_i^{\\text{hidden} }$ and bias $b_i^{\\text{hidden} }$, the weighting from the $j$-th neuron at the input layer is:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "z_{i,j}^{\\text{hidden}} &= b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_j \\\\\n", - "&=\n", - "\\begin{pmatrix}\n", - "b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 \\\\\n", - "x_j\n", - "\\end{pmatrix}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The result after weighting the inputs at the $i$-th hidden neuron can be written as a vector:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "\\boldsymbol{z}_{i}^{\\text{hidden}} &= \\Big( b_i^{\\text{hidden}} + w_i^{\\text{hidden}}x_1 , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_2, \\ \\dots \\, , \\ b_i^{\\text{hidden}} + w_i^{\\text{hidden}} x_N\\Big) \\\\\n", - "&=\n", - "\\begin{pmatrix}\n", - " b_i^{\\text{hidden}} & w_i^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 & 1 & \\dots & 1 \\\\\n", - "x_1 & x_2 & \\dots & x_N\n", - "\\end{pmatrix} \\\\\n", - "&= \\boldsymbol{p}_{i, \\text{hidden}}^T X\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The vector $\\boldsymbol{p}_{i, \\text{hidden}}^T$ constitutes each row in $P_{\\text{hidden} }$, which contains the weights for the neural network to minimize according to ([9](#min)).\n", - "\n", - "After having found $\\boldsymbol{z}_{i}^{\\text{hidden}} $ for every $i$-th neuron within the hidden layer, the vector will be sent to an activation function $a_i(\\boldsymbol{z})$.\n", - "\n", - "In this example, the sigmoid function has been chosen to be the activation function for each hidden neuron:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "f(z) = \\frac{1}{1 + \\exp{(-z)}}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "It is possible to use other activations functions for the hidden layer also.\n", - "\n", - "The output $\\boldsymbol{x}_i^{\\text{hidden}}$ from each $i$-th hidden neuron is:\n", - "\n", - "$$\n", - "\\boldsymbol{x}_i^{\\text{hidden} } = f\\big( \\boldsymbol{z}_{i}^{\\text{hidden}} \\big)\n", - "$$\n", - "\n", - "The outputs $\\boldsymbol{x}_i^{\\text{hidden} } $ are then sent to the output layer.\n", - "\n", - "The output layer consists of one neuron in this case, and combines the\n", - "output from each of the neurons in the hidden layers. The output layer\n", - "combines the results from the hidden layer using some weights $w_i^{\\text{output}}$\n", - "and biases $b_i^{\\text{output}}$. In this case,\n", - "it is assumes that the number of neurons in the output layer is one.\n", - "\n", - "\n", - "\n", - "The procedure of weighting the output neuron $j$ in the hidden layer to the $i$-th neuron in the output layer is similar as for the hidden layer described previously." - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "z_{1,j}^{\\text{output}} & =\n", - "\\begin{pmatrix}\n", - "b_1^{\\text{output}} & \\boldsymbol{w}_1^{\\text{output}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 \\\\\n", - "\\boldsymbol{x}_j^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Expressing $z_{1,j}^{\\text{output}}$ as a vector gives the following way of weighting the inputs from the hidden layer:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{z}_{1}^{\\text{output}} =\n", - "\\begin{pmatrix}\n", - "b_1^{\\text{output}} & \\boldsymbol{w}_1^{\\text{output}}\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "1 & 1 & \\dots & 1 \\\\\n", - "\\boldsymbol{x}_1^{\\text{hidden}} & \\boldsymbol{x}_2^{\\text{hidden}} & \\dots & \\boldsymbol{x}_N^{\\text{hidden}}\n", - "\\end{pmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this case we seek a continuous range of values since we are approximating a function. This means that after computing $\\boldsymbol{z}_{1}^{\\text{output}}$ the neural network has finished its feed forward step, and $\\boldsymbol{z}_{1}^{\\text{output}}$ is the final output of the network.\n", - "\n", - "\n", - "The next step is to decide how the parameters should be changed such that they minimize the cost function.\n", - "\n", - "The chosen cost function for this problem is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C(\\boldsymbol{x}, P) = \\frac{1}{N} \\sum_i \\big(g_t'(x_i, P) - ( -\\gamma g_t(x_i, P) \\big)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In order to minimize the cost function, an optimization method must be chosen.\n", - "\n", - "Here, gradient descent with a constant step size has been chosen.\n", - "\n", - "### Gradient descent\n", - "\n", - "The idea of the gradient descent algorithm is to update parameters in\n", - "a direction where the cost function decreases goes to a minimum.\n", - "\n", - "In general, the update of some parameters $\\boldsymbol{\\omega}$ given a cost\n", - "function defined by some weights $\\boldsymbol{\\omega}$, $C(\\boldsymbol{x},\n", - "\\boldsymbol{\\omega})$, goes as follows:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\omega}_{\\text{new} } = \\boldsymbol{\\omega} - \\lambda \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for a number of iterations or until $ \\big|\\big| \\boldsymbol{\\omega}_{\\text{new} } - \\boldsymbol{\\omega} \\big|\\big|$ becomes smaller than some given tolerance.\n", - "\n", - "The value of $\\lambda$ decides how large steps the algorithm must take\n", - "in the direction of $ \\nabla_{\\boldsymbol{\\omega}} C(\\boldsymbol{x}, \\boldsymbol{\\omega})$.\n", - "The notation $\\nabla_{\\boldsymbol{\\omega}}$ express the gradient with respect\n", - "to the elements in $\\boldsymbol{\\omega}$.\n", - "\n", - "In our case, we have to minimize the cost function $C(\\boldsymbol{x}, P)$ with\n", - "respect to the two sets of weights and biases, that is for the hidden\n", - "layer $P_{\\text{hidden} }$ and for the output layer $P_{\\text{output}\n", - "}$ .\n", - "\n", - "This means that $P_{\\text{hidden} }$ and $P_{\\text{output} }$ is updated by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "P_{\\text{hidden},\\text{new}} &= P_{\\text{hidden}} - \\lambda \\nabla_{P_{\\text{hidden}}} C(\\boldsymbol{x}, P) \\\\\n", - "P_{\\text{output},\\text{new}} &= P_{\\text{output}} - \\lambda \\nabla_{P_{\\text{output}}} C(\\boldsymbol{x}, P)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### The code for solving the ODE" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# Assuming one input, hidden, and output layer\n", - "def neural_network(params, x):\n", - "\n", - " # Find the weights (including and biases) for the hidden and output layer.\n", - " # Assume that params is a list of parameters for each layer.\n", - " # The biases are the first element for each array in params,\n", - " # and the weights are the remaning elements in each array in params.\n", - "\n", - " w_hidden = params[0]\n", - " w_output = params[1]\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " ## Hidden layer:\n", - "\n", - " # Add a row of ones to include bias\n", - " x_input = np.concatenate((np.ones((1,num_values)), x_input ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_input)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " ## Output layer:\n", - "\n", - " # Include bias:\n", - " x_hidden = np.concatenate((np.ones((1,num_values)), x_hidden ), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_hidden)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial(x,params, g0 = 10):\n", - " return g0 + x*neural_network(params,x)\n", - "\n", - "# The right side of the ODE:\n", - "def g(x, g_trial, gamma = 2):\n", - " return -gamma*g_trial\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the neural network\n", - " d_net_out = elementwise_grad(neural_network,1)(P,x)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = g(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# Solve the exponential decay ODE using neural network with one input, hidden, and output layer\n", - "def solve_ode_neural_network(x, num_neurons_hidden, num_iter, lmb):\n", - " ## Set up initial weights and biases\n", - "\n", - " # For the hidden layer\n", - " p0 = npr.randn(num_neurons_hidden, 2 )\n", - "\n", - " # For the output layer\n", - " p1 = npr.randn(1, num_neurons_hidden + 1 ) # +1 since bias is included\n", - "\n", - " P = [p0, p1]\n", - "\n", - " print('Initial cost: %g'%cost_function(P, x))\n", - "\n", - " ## Start finding the optimal weights using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of two arrays;\n", - " # one for the gradient w.r.t P_hidden and\n", - " # one for the gradient w.r.t P_output\n", - " cost_grad = cost_function_grad(P, x)\n", - "\n", - " P[0] = P[0] - lmb * cost_grad[0]\n", - " P[1] = P[1] - lmb * cost_grad[1]\n", - "\n", - " print('Final cost: %g'%cost_function(P, x))\n", - "\n", - " return P\n", - "\n", - "def g_analytic(x, gamma = 2, g0 = 10):\n", - " return g0*np.exp(-gamma*x)\n", - "\n", - "# Solve the given problem\n", - "if __name__ == '__main__':\n", - " # Set seed such that the weight are initialized\n", - " # with same weights and biases for every run.\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " N = 10\n", - " x = np.linspace(0, 1, N)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = 10\n", - " num_iter = 10000\n", - " lmb = 0.001\n", - "\n", - " # Use the network\n", - " P = solve_ode_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " # Print the deviation from the trial solution and true solution\n", - " res = g_trial(x,P)\n", - " res_analytical = g_analytic(x)\n", - "\n", - " print('Max absolute difference: %g'%np.max(np.abs(res - res_analytical)))\n", - "\n", - " # Plot the results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, res_analytical)\n", - " plt.plot(x, res[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The network with one input layer, specified number of hidden layers, and one output layer\n", - "\n", - "It is also possible to extend the construction of our network into a more general one, allowing the network to contain more than one hidden layers.\n", - "\n", - "The number of neurons within each hidden layer are given as a list of integers in the program below." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# The neural network with one input layer and one output layer,\n", - "# but with number of hidden layers specified by the user.\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - "\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consists of\n", - " # parameters to all the hidden\n", - " # layers AND the output layer.\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial_deep(x,params, g0 = 10):\n", - " return g0 + x*deep_neural_network(params, x)\n", - "\n", - "# The right side of the ODE:\n", - "def g(x, g_trial, gamma = 2):\n", - " return -gamma*g_trial\n", - "\n", - "# The same cost function as before, but calls deep_neural_network instead.\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the neural network\n", - " d_net_out = elementwise_grad(deep_neural_network,1)(P,x)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial_deep,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = g(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# Solve the exponential decay ODE using neural network with one input and one output layer,\n", - "# but with specified number of hidden layers from the user.\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # The number of elements in the list num_hidden_neurons thus represents\n", - " # the number of hidden layers.\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weights and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weights using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "def g_analytic(x, gamma = 2, g0 = 10):\n", - " return g0*np.exp(-gamma*x)\n", - "\n", - "# Solve the given problem\n", - "if __name__ == '__main__':\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " N = 10\n", - " x = np.linspace(0, 1, N)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = np.array([10,10])\n", - " num_iter = 10000\n", - " lmb = 0.001\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " res = g_trial_deep(x,P)\n", - " res_analytical = g_analytic(x)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of a deep neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, res_analytical)\n", - " plt.plot(x, res[0,:])\n", - " plt.legend(['analytical','dnn'])\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Example: Population growth\n", - "\n", - "A logistic model of population growth assumes that a population converges toward an equilibrium.\n", - "The population growth can be modeled by" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{log} \\tag{10}\n", - "\tg'(t) = \\alpha g(t)(A - g(t))\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(t)$ is the population density at time $t$, $\\alpha > 0$ the growth rate and $A > 0$ is the maximum population number in the environment.\n", - "Also, at $t = 0$ the population has the size $g(0) = g_0$, where $g_0$ is some chosen constant.\n", - "\n", - "In this example, similar network as for the exponential decay using Autograd has been used to solve the equation. However, as the implementation might suffer from e.g numerical instability\n", - "and high execution time (this might be more apparent in the examples solving PDEs),\n", - "using a library like TensorFlow is recommended.\n", - "Here, we stay with a more simple approach and implement for comparison, the simple forward Euler method.\n", - "\n", - "\n", - "\n", - "Here, we will model a population $g(t)$ in an environment having carrying capacity $A$.\n", - "The population follows the model" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{solveode_population} \\tag{11}\n", - "g'(t) = \\alpha g(t)(A - g(t))\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $g(0) = g_0$.\n", - "\n", - "In this example, we let $\\alpha = 2$, $A = 1$, and $g_0 = 1.2$.\n", - "\n", - "\n", - "We will get a slightly different trial solution, as the boundary conditions are different\n", - "compared to the case for exponential decay.\n", - "\n", - "A possible trial solution satisfying the condition $g(0) = g_0$ could be\n", - "\n", - "$$\n", - "h_1(t) = g_0 + t \\cdot N(t,P)\n", - "$$\n", - "\n", - "with $N(t,P)$ being the output from the neural network with weights and biases for each layer collected in the set $P$.\n", - "\n", - "The analytical solution is\n", - "\n", - "$$\n", - "g(t) = \\frac{Ag_0}{g_0 + (A - g_0)\\exp(-\\alpha A t)}\n", - "$$\n", - "\n", - "\n", - "\n", - "The network will be the similar as for the exponential decay example, but with some small modifications for our problem." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "# Function to get the parameters.\n", - "# Done such that one can easily change the paramaters after one's liking.\n", - "def get_parameters():\n", - " alpha = 2\n", - " A = 1\n", - " g0 = 1.2\n", - " return alpha, A, g0\n", - "\n", - "def deep_neural_network(P, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(P) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = P[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = P[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d_g_t = elementwise_grad(g_trial_deep,0)(x,P)\n", - "\n", - " # The right side of the ODE\n", - " func = f(x, g_t)\n", - "\n", - " err_sqr = (d_g_t - func)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum / np.size(err_sqr)\n", - "\n", - "# The right side of the ODE:\n", - "def f(x, g_trial):\n", - " alpha,A, g0 = get_parameters()\n", - " return alpha*g_trial*(A - g_trial)\n", - "\n", - "# The trial solution using the deep neural network:\n", - "def g_trial_deep(x, params):\n", - " alpha,A, g0 = get_parameters()\n", - " return g0 + x*deep_neural_network(params,x)\n", - "\n", - "# The analytical solution:\n", - "def g_analytic(t):\n", - " alpha,A, g0 = get_parameters()\n", - " return A*g0/(g0 + (A - g0)*np.exp(-alpha*A*t))\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nt = 10\n", - " T = 1\n", - " t = np.linspace(0,T, Nt)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [100, 50, 25]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(t,P)\n", - " g_analytical = g_analytic(t)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%diff_ag)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(t, g_analytical)\n", - " plt.plot(t, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Using forward Euler to solve the ODE\n", - "\n", - "A straightforward way of solving an ODE numerically, is to use Euler's method.\n", - "\n", - "Euler's method uses Taylor series to approximate the value at a function $f$ at a step $\\Delta x$ from $x$:\n", - "\n", - "$$\n", - "f(x + \\Delta x) \\approx f(x) + \\Delta x f'(x)\n", - "$$\n", - "\n", - "In our case, using Euler's method to approximate the value of $g$ at a step $\\Delta t$ from $t$ yields" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - " g(t + \\Delta t) &\\approx g(t) + \\Delta t g'(t) \\\\\n", - " &= g(t) + \\Delta t \\big(\\alpha g(t)(A - g(t))\\big)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "along with the condition that $g(0) = g_0$.\n", - "\n", - "Let $t_i = i \\cdot \\Delta t$ where $\\Delta t = \\frac{T}{N_t-1}$ where $T$ is the final time our solver must solve for and $N_t$ the number of values for $t \\in [0, T]$ for $i = 0, \\dots, N_t-1$.\n", - "\n", - "For $i \\geq 1$, we have that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "t_i &= i\\Delta t \\\\\n", - "&= (i - 1)\\Delta t + \\Delta t \\\\\n", - "&= t_{i-1} + \\Delta t\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Now, if $g_i = g(t_i)$ then" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\begin{aligned}\n", - " g_i &= g(t_i) \\\\\n", - " &= g(t_{i-1} + \\Delta t) \\\\\n", - " &\\approx g(t_{i-1}) + \\Delta t \\big(\\alpha g(t_{i-1})(A - g(t_{i-1}))\\big) \\\\\n", - " &= g_{i-1} + \\Delta t \\big(\\alpha g_{i-1}(A - g_{i-1})\\big)\n", - " \\end{aligned}\n", - "\\end{equation} \\label{odenum} \\tag{12}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for $i \\geq 1$ and $g_0 = g(t_0) = g(0) = g_0$.\n", - "\n", - "Equation ([12](#odenum)) could be implemented in the following way,\n", - "extending the program that uses the network using Autograd:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Assume that all function definitions from the example program using Autograd\n", - "# are located here.\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nt = 10\n", - " T = 1\n", - " t = np.linspace(0,T, Nt)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [100,50,25]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(t,P)\n", - " g_analytical = g_analytic(t)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " diff_ag = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%diff_ag)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(t, g_analytical)\n", - " plt.plot(t, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " ## Find an approximation to the funtion using forward Euler\n", - "\n", - " alpha, A, g0 = get_parameters()\n", - " dt = T/(Nt - 1)\n", - "\n", - " # Perform forward Euler to solve the ODE\n", - " g_euler = np.zeros(Nt)\n", - " g_euler[0] = g0\n", - "\n", - " for i in range(1,Nt):\n", - " g_euler[i] = g_euler[i-1] + dt*(alpha*g_euler[i-1]*(A - g_euler[i-1]))\n", - "\n", - " # Print the errors done by each method\n", - " diff1 = np.max(np.abs(g_euler - g_analytical))\n", - " diff2 = np.max(np.abs(g_dnn_ag[0,:] - g_analytical))\n", - "\n", - " print('Max absolute difference between Euler method and analytical: %g'%diff1)\n", - " print('Max absolute difference between deep neural network and analytical: %g'%diff2)\n", - "\n", - " # Plot results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.plot(t,g_euler)\n", - " plt.plot(t,g_analytical)\n", - " plt.plot(t,g_dnn_ag[0,:])\n", - "\n", - " plt.legend(['euler','analytical','dnn'])\n", - " plt.xlabel('Time t')\n", - " plt.ylabel('g(t)')\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Solving the one dimensional Poisson equation\n", - "\n", - "The Poisson equation for $g(x)$ in one dimension is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{poisson} \\tag{13}\n", - " -g''(x) = f(x)\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f(x)$ is a given function for $x \\in (0,1)$.\n", - "\n", - "The conditions that $g(x)$ is chosen to fulfill, are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " g(0) &= 0 \\\\\n", - " g(1) &= 0\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "This equation can be solved numerically using programs where e.g Autograd and TensorFlow are used.\n", - "The results from the networks can then be compared to the analytical solution.\n", - "In addition, it could be interesting to see how a typical method for numerically solving second order ODEs compares to the neural networks.\n", - "\n", - "\n", - "Here, the function $g(x)$ to solve for follows the equation" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "-g''(x) = f(x),\\qquad x \\in (0,1)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f(x)$ is a given function, along with the chosen conditions" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{aligned}\n", - "g(0) = g(1) = 0\n", - "\\end{aligned}\\label{cond} \\tag{14}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this example, we consider the case when $f(x) = (3x + x^2)\\exp(x)$.\n", - "\n", - "For this case, a possible trial solution satisfying the conditions could be" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g_t(x) = x \\cdot (1-x) \\cdot N(P,x)\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The analytical solution for this problem is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "g(x) = x(1 - x)\\exp(x)\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "## Set up the cost function specified for this Poisson equation:\n", - "\n", - "# The right side of the ODE\n", - "def f(x):\n", - " return (3*x + x**2)*np.exp(x)\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)\n", - "\n", - " right_side = f(x)\n", - "\n", - " err_sqr = (-d2_g_t - right_side)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum/np.size(err_sqr)\n", - "\n", - "# The trial solution:\n", - "def g_trial_deep(x,P):\n", - " return x*(1-x)*deep_neural_network(P,x)\n", - "\n", - "# The analytic solution;\n", - "def g_analytic(x):\n", - " return x*(1-x)*np.exp(x)\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10\n", - " x = np.linspace(0,1, Nx)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [200,100]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(x,P)\n", - " g_analytical = g_analytic(x)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - " max_diff = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " print(\"The max absolute difference between the solutions is: %g\"%max_diff)\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, g_analytical)\n", - " plt.plot(x, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Comparing with a numerical scheme\n", - "\n", - "The Poisson equation is possible to solve using Taylor series to approximate the second derivative.\n", - "\n", - "Using Taylor series, the second derivative can be expressed as\n", - "\n", - "$$\n", - "g''(x) = \\frac{g(x + \\Delta x) - 2g(x) + g(x-\\Delta x)}{\\Delta x^2} + E_{\\Delta x}(x)\n", - "$$\n", - "\n", - "where $\\Delta x$ is a small step size and $E_{\\Delta x}(x)$ being the error term.\n", - "\n", - "Looking away from the error terms gives an approximation to the second derivative:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{approx} \\tag{15}\n", - "g''(x) \\approx \\frac{g(x + \\Delta x) - 2g(x) + g(x-\\Delta x)}{\\Delta x^2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If $x_i = i \\Delta x = x_{i-1} + \\Delta x$ and $g_i = g(x_i)$ for $i = 1,\\dots N_x - 2$ with $N_x$ being the number of values for $x$, ([15](#approx)) becomes" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "g''(x_i) &\\approx \\frac{g(x_i + \\Delta x) - 2g(x_i) + g(x_i -\\Delta x)}{\\Delta x^2} \\\\\n", - "&= \\frac{g_{i+1} - 2g_i + g_{i-1}}{\\Delta x^2}\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Since we know from our problem that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "-g''(x) &= f(x) \\\\\n", - "&= (3x + x^2)\\exp(x)\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "along with the conditions $g(0) = g(1) = 0$,\n", - "the following scheme can be used to find an approximate solution for $g(x)$ numerically:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\begin{aligned}\n", - " -\\Big( \\frac{g_{i+1} - 2g_i + g_{i-1}}{\\Delta x^2} \\Big) &= f(x_i) \\\\\n", - " -g_{i+1} + 2g_i - g_{i-1} &= \\Delta x^2 f(x_i)\n", - " \\end{aligned}\n", - "\\end{equation} \\label{odesys} \\tag{16}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "for $i = 1, \\dots, N_x - 2$ where $g_0 = g_{N_x - 1} = 0$ and $f(x_i) = (3x_i + x_i^2)\\exp(x_i)$, which is given for our specific problem.\n", - "\n", - "The equation can be rewritten into a matrix equation:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{aligned}\n", - "\\begin{pmatrix}\n", - "2 & -1 & 0 & \\dots & 0 \\\\\n", - "-1 & 2 & -1 & \\dots & 0 \\\\\n", - "\\vdots & & \\ddots & & \\vdots \\\\\n", - "0 & \\dots & -1 & 2 & -1 \\\\\n", - "0 & \\dots & 0 & -1 & 2\\\\\n", - "\\end{pmatrix}\n", - "\\begin{pmatrix}\n", - "g_1 \\\\\n", - "g_2 \\\\\n", - "\\vdots \\\\\n", - "g_{N_x - 3} \\\\\n", - "g_{N_x - 2}\n", - "\\end{pmatrix}\n", - "&=\n", - "\\Delta x^2\n", - "\\begin{pmatrix}\n", - "f(x_1) \\\\\n", - "f(x_2) \\\\\n", - "\\vdots \\\\\n", - "f(x_{N_x - 3}) \\\\\n", - "f(x_{N_x - 2})\n", - "\\end{pmatrix} \\\\\n", - "\\boldsymbol{A}\\boldsymbol{g} &= \\boldsymbol{f},\n", - "\\end{aligned}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "which makes it possible to solve for the vector $\\boldsymbol{g}$.\n", - "\n", - "\n", - "We can then compare the result from this numerical scheme with the output from our network using Autograd:" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import grad, elementwise_grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import pyplot as plt\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assumes input x being an one-dimensional array\n", - " num_values = np.size(x)\n", - " x = x.reshape(-1, num_values)\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - "\n", - " # Due to multiple hidden layers, define a variable referencing to the\n", - " # output of the previous layer:\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_values)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output\n", - "\n", - "def solve_ode_deep_neural_network(x, num_neurons, num_iter, lmb):\n", - " # num_hidden_neurons is now a list of number of neurons within each hidden layer\n", - "\n", - " # Find the number of hidden layers:\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 )\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: %g'%cost_function_deep(P, x))\n", - "\n", - " ## Start finding the optimal weigths using gradient descent\n", - "\n", - " # Find the Python function that represents the gradient of the cost function\n", - " # w.r.t the 0-th input argument -- that is the weights and biases in the hidden and output layer\n", - " cost_function_deep_grad = grad(cost_function_deep,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " # Evaluate the gradient at the current weights and biases in P.\n", - " # The cost_grad consist now of N_hidden + 1 arrays; the gradient w.r.t the weights and biases\n", - " # in the hidden layers and output layers evaluated at x.\n", - " cost_deep_grad = cost_function_deep_grad(P, x)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_deep_grad[l]\n", - "\n", - " print('Final cost: %g'%cost_function_deep(P, x))\n", - "\n", - " return P\n", - "\n", - "## Set up the cost function specified for this Poisson equation:\n", - "\n", - "# The right side of the ODE\n", - "def f(x):\n", - " return (3*x + x**2)*np.exp(x)\n", - "\n", - "def cost_function_deep(P, x):\n", - "\n", - " # Evaluate the trial function with the current parameters P\n", - " g_t = g_trial_deep(x,P)\n", - "\n", - " # Find the derivative w.r.t x of the trial function\n", - " d2_g_t = elementwise_grad(elementwise_grad(g_trial_deep,0))(x,P)\n", - "\n", - " right_side = f(x)\n", - "\n", - " err_sqr = (-d2_g_t - right_side)**2\n", - " cost_sum = np.sum(err_sqr)\n", - "\n", - " return cost_sum/np.size(err_sqr)\n", - "\n", - "# The trial solution:\n", - "def g_trial_deep(x,P):\n", - " return x*(1-x)*deep_neural_network(P,x)\n", - "\n", - "# The analytic solution;\n", - "def g_analytic(x):\n", - " return x*(1-x)*np.exp(x)\n", - "\n", - "if __name__ == '__main__':\n", - " npr.seed(4155)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10\n", - " x = np.linspace(0,1, Nx)\n", - "\n", - " ## Set up the initial parameters\n", - " num_hidden_neurons = [200,100]\n", - " num_iter = 1000\n", - " lmb = 1e-3\n", - "\n", - " P = solve_ode_deep_neural_network(x, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " g_dnn_ag = g_trial_deep(x,P)\n", - " g_analytical = g_analytic(x)\n", - "\n", - " # Find the maximum absolute difference between the solutons:\n", - "\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.title('Performance of neural network solving an ODE compared to the analytical solution')\n", - " plt.plot(x, g_analytical)\n", - " plt.plot(x, g_dnn_ag[0,:])\n", - " plt.legend(['analytical','nn'])\n", - " plt.xlabel('x')\n", - " plt.ylabel('g(x)')\n", - "\n", - " ## Perform the computation using the numerical scheme\n", - "\n", - " dx = 1/(Nx - 1)\n", - "\n", - " # Set up the matrix A\n", - " A = np.zeros((Nx-2,Nx-2))\n", - "\n", - " A[0,0] = 2\n", - " A[0,1] = -1\n", - "\n", - " for i in range(1,Nx-3):\n", - " A[i,i-1] = -1\n", - " A[i,i] = 2\n", - " A[i,i+1] = -1\n", - "\n", - " A[Nx - 3, Nx - 4] = -1\n", - " A[Nx - 3, Nx - 3] = 2\n", - "\n", - " # Set up the vector f\n", - " f_vec = dx**2 * f(x[1:-1])\n", - "\n", - " # Solve the equation\n", - " g_res = np.linalg.solve(A,f_vec)\n", - "\n", - " g_vec = np.zeros(Nx)\n", - " g_vec[1:-1] = g_res\n", - "\n", - " # Print the differences between each method\n", - " max_diff1 = np.max(np.abs(g_dnn_ag - g_analytical))\n", - " max_diff2 = np.max(np.abs(g_vec - g_analytical))\n", - " print(\"The max absolute difference between the analytical solution and DNN Autograd: %g\"%max_diff1)\n", - " print(\"The max absolute difference between the analytical solution and numerical scheme: %g\"%max_diff2)\n", - "\n", - " # Plot the results\n", - " plt.figure(figsize=(10,10))\n", - "\n", - " plt.plot(x,g_vec)\n", - " plt.plot(x,g_analytical)\n", - " plt.plot(x,g_dnn_ag[0,:])\n", - "\n", - " plt.legend(['numerical scheme','analytical','dnn'])\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Partial Differential Equations\n", - "\n", - "A partial differential equation (PDE) has a solution here the function\n", - "is defined by multiple variables. The equation may involve all kinds\n", - "of combinations of which variables the function is differentiated with\n", - "respect to.\n", - "\n", - "In general, a partial differential equation for a function $g(x_1,\\dots,x_N)$ with $N$ variables may be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{PDE} \\tag{17}\n", - " f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) = 0\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $f$ is an expression involving all kinds of possible mixed derivatives of $g(x_1,\\dots,x_N)$ up to an order $n$. In order for the solution to be unique, some additional conditions must also be given.\n", - "\n", - "### Type of problem\n", - "\n", - "The problem our network must solve for, is similar to the ODE case.\n", - "We must have a trial solution $g_t$ at hand.\n", - "\n", - "For instance, the trial solution could be expressed as" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - " g_t(x_1,\\dots,x_N) = h_1(x_1,\\dots,x_N) + h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $h_1(x_1,\\dots,x_N)$ is a function that ensures $g_t(x_1,\\dots,x_N)$ satisfies some given conditions.\n", - "The neural network $N(x_1,\\dots,x_N,P)$ has weights and biases described by $P$ and $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$ is an expression using the output from the neural network in some way.\n", - "\n", - "The role of the function $h_2(x_1,\\dots,x_N,N(x_1,\\dots,x_N,P))$, is to ensure that the output of $N(x_1,\\dots,x_N,P)$ is zero when $g_t(x_1,\\dots,x_N)$ is evaluated at the values of $x_1,\\dots,x_N$ where the given conditions must be satisfied. The function $h_1(x_1,\\dots,x_N)$ should alone make $g_t(x_1,\\dots,x_N)$ satisfy the conditions.\n", - "\n", - "\n", - "\n", - "### Network requirements\n", - "\n", - "The network tries then the minimize the cost function following the\n", - "same ideas as described for the ODE case, but now with more than one\n", - "variables to consider. The concept still remains the same; find a set\n", - "of parameters $P$ such that the expression $f$ in ([17](#PDE)) is as\n", - "close to zero as possible.\n", - "\n", - "As for the ODE case, the cost function is the mean squared error that\n", - "the network must try to minimize. The cost function for the network to\n", - "minimize is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(x_1, \\dots, x_N, P\\right) = \\left( f\\left(x_1, \\, \\dots \\, , x_N, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1}, \\dots , \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_N}, \\frac{\\partial g(x_1,\\dots,x_N) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(x_1,\\dots,x_N) }{\\partial x_N^n} \\right) \\right)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we let $\\boldsymbol{x} = \\big( x_1, \\dots, x_N \\big)$ be an array containing the values for $x_1, \\dots, x_N$ respectively, the cost function can be reformulated into the following:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(\\boldsymbol{x}, P\\right) = f\\left( \\left( \\boldsymbol{x}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}) }{\\partial x_N^n} \\right) \\right)^2\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "If we also have $M$ different sets of values for $x_1, \\dots, x_N$, that is $\\boldsymbol{x}_i = \\big(x_1^{(i)}, \\dots, x_N^{(i)}\\big)$ for $i = 1,\\dots,M$ being the rows in matrix $X$, the cost function can be generalized into" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "C\\left(X, P \\right) = \\sum_{i=1}^M f\\left( \\left( \\boldsymbol{x}_i, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1}, \\dots , \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_N}, \\frac{\\partial g(\\boldsymbol{x}_i) }{\\partial x_1\\partial x_2}, \\, \\dots \\, , \\frac{\\partial^n g(\\boldsymbol{x}_i) }{\\partial x_N^n} \\right) \\right)^2.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Example: The diffusion equation\n", - "\n", - "In one spatial dimension, the equation reads" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where a possible choice of conditions are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "g(0,t) &= 0 ,\\qquad t \\geq 0 \\\\\n", - "g(1,t) &= 0, \\qquad t \\geq 0 \\\\\n", - "g(x,0) &= u(x),\\qquad x\\in [0,1]\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $u(x)$ being some given function.\n", - "\n", - "\n", - "\n", - "For this case, we want to find $g(x,t)$ such that" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation}\n", - " \\frac{\\partial g(x,t)}{\\partial t} = \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "\\end{equation} \\label{diffonedim} \\tag{18}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "and" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "g(0,t) &= 0 ,\\qquad t \\geq 0 \\\\\n", - "g(1,t) &= 0, \\qquad t \\geq 0 \\\\\n", - "g(x,0) &= u(x),\\qquad x\\in [0,1]\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $u(x) = \\sin(\\pi x)$.\n", - "\n", - "First, let us set up the deep neural network.\n", - "The deep neural network will follow the same structure as discussed in the examples solving the ODEs.\n", - "First, we will look into how Autograd could be used in a network tailored to solve for bivariate functions.\n", - "\n", - "\n", - "\n", - "\n", - "The only change to do here, is to extend our network such that\n", - "functions of multiple parameters are correctly handled. In this case\n", - "we have two variables in our function to solve for, that is time $t$\n", - "and position $x$. The variables will be represented by a\n", - "one-dimensional array in the program. The program will evaluate the\n", - "network at each possible pair $(x,t)$, given an array for the desired\n", - "$x$-values and $t$-values to approximate the solution at." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The cost function must then iterate through the given arrays\n", - "containing values for $x$ and $t$, defines a point $(x,t)$ the deep\n", - "neural network and the trial solution is evaluated at, and then finds\n", - "the Jacobian of the trial solution.\n", - "\n", - "A possible trial solution for this PDE is\n", - "\n", - "$$\n", - "g_t(x,t) = h_1(x,t) + x(1-x)tN(x,t,P)\n", - "$$\n", - "\n", - "with $A(x,t)$ being a function ensuring that $g_t(x,t)$ satisfies our given conditions, and $N(x,t,P)$ being the output from the deep neural network using weights and biases for each layer from $P$.\n", - "\n", - "To fulfill the conditions, $A(x,t)$ could be:\n", - "\n", - "$$\n", - "h_1(x,t) = (1-t)\\Big(u(x) - \\big((1-x)u(0) + x u(1)\\big)\\Big) = (1-t)u(x) = (1-t)\\sin(\\pi x)\n", - "$$\n", - "since $(0) = u(1) = 0$ and $u(x) = \\sin(\\pi x)$.\n", - "\n", - "\n", - "\n", - "The Jacobian is used because the program must find the derivative of\n", - "the trial solution with respect to $x$ and $t$.\n", - "\n", - "This gives the necessity of computing the Jacobian matrix, as we want\n", - "to evaluate the gradient with respect to $x$ and $t$ (note that the\n", - "Jacobian of a scalar-valued multivariate function is simply its\n", - "gradient).\n", - "\n", - "In Autograd, the differentiation is by default done with respect to\n", - "the first input argument of your Python function. Since the points is\n", - "an array representing $x$ and $t$, the Jacobian is calculated using\n", - "the values of $x$ and $t$.\n", - "\n", - "To find the second derivative with respect to $x$ and $t$, the\n", - "Jacobian can be found for the second time. The result is a Hessian\n", - "matrix, which is the matrix containing all the possible second order\n", - "mixed derivatives of $g(x,t)$." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Set up the trial function:\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)\n", - "\n", - "# The right side of the ODE:\n", - "def f(point):\n", - " return 0.\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_jacobian_func = jacobian(g_trial)\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t = g_trial(point,P)\n", - " g_t_jacobian = g_t_jacobian_func(point,P)\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_dt = g_t_jacobian[1]\n", - " g_t_d2x = g_t_hessian[0][0]\n", - "\n", - " func = f(point)\n", - "\n", - " err_sqr = ( (g_t_dt - g_t_d2x) - func)**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Setting up the network using Autograd; The full program\n", - "\n", - "Having set up the network, along with the trial solution and cost function, we can now see how the deep neural network performs by comparing the results to the analytical solution.\n", - "\n", - "The analytical solution of our problem is\n", - "\n", - "$$\n", - "g(x,t) = \\exp(-\\pi^2 t)\\sin(\\pi x)\n", - "$$\n", - "\n", - "A possible way to implement a neural network solving the PDE, is given below.\n", - "Be aware, though, that it is fairly slow for the parameters used.\n", - "A better result is possible, but requires more iterations, and thus longer time to complete.\n", - "\n", - "\n", - "Indeed, the program below is not optimal in its implementation, but rather serves as an example on how to implement and use a neural network to solve a PDE.\n", - "Using TensorFlow results in a much better execution time. Try it!" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import jacobian,hessian,grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import cm\n", - "from matplotlib import pyplot as plt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "## Set up the network\n", - "\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]\n", - "\n", - "## Define the trial solution and cost function\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return (1-t)*u(x) + x*(1-x)*t*deep_neural_network(P,point)\n", - "\n", - "# The right side of the ODE:\n", - "def f(point):\n", - " return 0.\n", - "\n", - "# The cost function:\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_jacobian_func = jacobian(g_trial)\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t = g_trial(point,P)\n", - " g_t_jacobian = g_t_jacobian_func(point,P)\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_dt = g_t_jacobian[1]\n", - " g_t_d2x = g_t_hessian[0][0]\n", - "\n", - " func = f(point)\n", - "\n", - " err_sqr = ( (g_t_dt - g_t_d2x) - func)**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum /( np.size(x)*np.size(t) )\n", - "\n", - "## For comparison, define the analytical solution\n", - "def g_analytic(point):\n", - " x,t = point\n", - " return np.exp(-np.pi**2*t)*np.sin(np.pi*x)\n", - "\n", - "## Set up a function for training the network to solve for the equation\n", - "def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):\n", - " ## Set up initial weigths and biases\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: ',cost_function(P, x, t))\n", - "\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " cost_grad = cost_function_grad(P, x , t)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_grad[l]\n", - "\n", - " print('Final cost: ',cost_function(P, x, t))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " ### Use the neural network:\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10; Nt = 10\n", - " x = np.linspace(0, 1, Nx)\n", - " t = np.linspace(0,1,Nt)\n", - "\n", - " ## Set up the parameters for the network\n", - " num_hidden_neurons = [100, 25]\n", - " num_iter = 250\n", - " lmb = 0.01\n", - "\n", - " P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " ## Store the results\n", - " g_dnn_ag = np.zeros((Nx, Nt))\n", - " G_analytical = np.zeros((Nx, Nt))\n", - " for i,x_ in enumerate(x):\n", - " for j, t_ in enumerate(t):\n", - " point = np.array([x_, t_])\n", - " g_dnn_ag[i,j] = g_trial(point,P)\n", - "\n", - " G_analytical[i,j] = g_analytic(point)\n", - "\n", - " # Find the map difference between the analytical and the computed solution\n", - " diff_ag = np.abs(g_dnn_ag - G_analytical)\n", - " print('Max absolute difference between the analytical solution and the network: %g'%np.max(diff_ag))\n", - "\n", - " ## Plot the solutions in two dimensions, that being in position and time\n", - "\n", - " T,X = np.meshgrid(t,x)\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", - " s = ax.plot_surface(T,X,g_dnn_ag,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Analytical solution')\n", - " s = ax.plot_surface(T,X,G_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Difference')\n", - " s = ax.plot_surface(T,X,diff_ag,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " ## Take some slices of the 3D plots just to see the solutions at particular times\n", - " indx1 = 0\n", - " indx2 = int(Nt/2)\n", - " indx3 = Nt-1\n", - "\n", - " t1 = t[indx1]\n", - " t2 = t[indx2]\n", - " t3 = t[indx3]\n", - "\n", - " # Slice the results from the DNN\n", - " res1 = g_dnn_ag[:,indx1]\n", - " res2 = g_dnn_ag[:,indx2]\n", - " res3 = g_dnn_ag[:,indx3]\n", - "\n", - " # Slice the analytical results\n", - " res_analytical1 = G_analytical[:,indx1]\n", - " res_analytical2 = G_analytical[:,indx2]\n", - " res_analytical3 = G_analytical[:,indx3]\n", - "\n", - " # Plot the slices\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t1)\n", - " plt.plot(x, res1)\n", - " plt.plot(x,res_analytical1)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t2)\n", - " plt.plot(x, res2)\n", - " plt.plot(x,res_analytical2)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t3)\n", - " plt.plot(x, res3)\n", - " plt.plot(x,res_analytical3)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Solving the wave equation with Neural Networks\n", - "\n", - "The wave equation is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2\\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "with $c$ being the specified wave speed.\n", - "\n", - "Here, the chosen conditions are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{align*}\n", - "\tg(0,t) &= 0 \\\\\n", - "\tg(1,t) &= 0 \\\\\n", - "\tg(x,0) &= u(x) \\\\\n", - "\t\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0} &= v(x)\n", - "\\end{align*}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $\\frac{\\partial g(x,t)}{\\partial t} \\Big |_{t = 0}$ means the derivative of $g(x,t)$ with respect to $t$ is evaluated at $t = 0$, and $u(x)$ and $v(x)$ being given functions.\n", - "\n", - "\n", - "The wave equation to solve for, is" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{equation} \\label{wave} \\tag{19}\n", - "\\frac{\\partial^2 g(x,t)}{\\partial t^2} = c^2 \\frac{\\partial^2 g(x,t)}{\\partial x^2}\n", - "\\end{equation}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "where $c$ is the given wave speed.\n", - "The chosen conditions for this equation are" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "
\n", - "\n", - "$$\n", - "\\begin{aligned}\n", - "g(0,t) &= 0, &t \\geq 0 \\\\\n", - "g(1,t) &= 0, &t \\geq 0 \\\\\n", - "g(x,0) &= u(x), &x\\in[0,1] \\\\\n", - "\\frac{\\partial g(x,t)}{\\partial t}\\Big |_{t = 0} &= v(x), &x \\in [0,1]\n", - "\\end{aligned} \\label{condwave} \\tag{20}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "In this example, let $c = 1$ and $u(x) = \\sin(\\pi x)$ and $v(x) = -\\pi\\sin(\\pi x)$.\n", - "\n", - "\n", - "\n", - "Setting up the network is done in similar matter as for the example of solving the diffusion equation.\n", - "The only things we have to change, is the trial solution such that it satisfies the conditions from ([20](#condwave)) and the cost function.\n", - "\n", - "The trial solution becomes slightly different since we have other conditions than in the example of solving the diffusion equation. Here, a possible trial solution $g_t(x,t)$ is\n", - "\n", - "$$\n", - "g_t(x,t) = h_1(x,t) + x(1-x)t^2N(x,t,P)\n", - "$$\n", - "\n", - "where\n", - "\n", - "$$\n", - "h_1(x,t) = (1-t^2)u(x) + tv(x)\n", - "$$\n", - "\n", - "Note that this trial solution satisfies the conditions only if $u(0) = v(0) = u(1) = v(1) = 0$, which is the case in this example.\n", - "\n", - "\n", - "The analytical solution for our specific problem, is\n", - "\n", - "$$\n", - "g(x,t) = \\sin(\\pi x)\\cos(\\pi t) - \\sin(\\pi x)\\sin(\\pi t)\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import autograd.numpy as np\n", - "from autograd import hessian,grad\n", - "import autograd.numpy.random as npr\n", - "from matplotlib import cm\n", - "from matplotlib import pyplot as plt\n", - "from mpl_toolkits.mplot3d import axes3d\n", - "\n", - "## Set up the trial function:\n", - "def u(x):\n", - " return np.sin(np.pi*x)\n", - "\n", - "def v(x):\n", - " return -np.pi*np.sin(np.pi*x)\n", - "\n", - "def h1(point):\n", - " x,t = point\n", - " return (1 - t**2)*u(x) + t*v(x)\n", - "\n", - "def g_trial(point,P):\n", - " x,t = point\n", - " return h1(point) + x*(1-x)*t**2*deep_neural_network(P,point)\n", - "\n", - "## Define the cost function\n", - "def cost_function(P, x, t):\n", - " cost_sum = 0\n", - "\n", - " g_t_hessian_func = hessian(g_trial)\n", - "\n", - " for x_ in x:\n", - " for t_ in t:\n", - " point = np.array([x_,t_])\n", - "\n", - " g_t_hessian = g_t_hessian_func(point,P)\n", - "\n", - " g_t_d2x = g_t_hessian[0][0]\n", - " g_t_d2t = g_t_hessian[1][1]\n", - "\n", - " err_sqr = ( (g_t_d2t - g_t_d2x) )**2\n", - " cost_sum += err_sqr\n", - "\n", - " return cost_sum / (np.size(t) * np.size(x))\n", - "\n", - "## The neural network\n", - "def sigmoid(z):\n", - " return 1/(1 + np.exp(-z))\n", - "\n", - "def deep_neural_network(deep_params, x):\n", - " # x is now a point and a 1D numpy array; make it a column vector\n", - " num_coordinates = np.size(x,0)\n", - " x = x.reshape(num_coordinates,-1)\n", - "\n", - " num_points = np.size(x,1)\n", - "\n", - " # N_hidden is the number of hidden layers\n", - " N_hidden = np.size(deep_params) - 1 # -1 since params consist of parameters to all the hidden layers AND the output layer\n", - "\n", - " # Assume that the input layer does nothing to the input x\n", - " x_input = x\n", - " x_prev = x_input\n", - "\n", - " ## Hidden layers:\n", - "\n", - " for l in range(N_hidden):\n", - " # From the list of parameters P; find the correct weigths and bias for this layer\n", - " w_hidden = deep_params[l]\n", - "\n", - " # Add a row of ones to include bias\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev ), axis = 0)\n", - "\n", - " z_hidden = np.matmul(w_hidden, x_prev)\n", - " x_hidden = sigmoid(z_hidden)\n", - "\n", - " # Update x_prev such that next layer can use the output from this layer\n", - " x_prev = x_hidden\n", - "\n", - " ## Output layer:\n", - "\n", - " # Get the weights and bias for this layer\n", - " w_output = deep_params[-1]\n", - "\n", - " # Include bias:\n", - " x_prev = np.concatenate((np.ones((1,num_points)), x_prev), axis = 0)\n", - "\n", - " z_output = np.matmul(w_output, x_prev)\n", - " x_output = z_output\n", - "\n", - " return x_output[0][0]\n", - "\n", - "## The analytical solution\n", - "def g_analytic(point):\n", - " x,t = point\n", - " return np.sin(np.pi*x)*np.cos(np.pi*t) - np.sin(np.pi*x)*np.sin(np.pi*t)\n", - "\n", - "def solve_pde_deep_neural_network(x,t, num_neurons, num_iter, lmb):\n", - " ## Set up initial weigths and biases\n", - " N_hidden = np.size(num_neurons)\n", - "\n", - " ## Set up initial weigths and biases\n", - "\n", - " # Initialize the list of parameters:\n", - " P = [None]*(N_hidden + 1) # + 1 to include the output layer\n", - "\n", - " P[0] = npr.randn(num_neurons[0], 2 + 1 ) # 2 since we have two points, +1 to include bias\n", - " for l in range(1,N_hidden):\n", - " P[l] = npr.randn(num_neurons[l], num_neurons[l-1] + 1) # +1 to include bias\n", - "\n", - " # For the output layer\n", - " P[-1] = npr.randn(1, num_neurons[-1] + 1 ) # +1 since bias is included\n", - "\n", - " print('Initial cost: ',cost_function(P, x, t))\n", - "\n", - " cost_function_grad = grad(cost_function,0)\n", - "\n", - " # Let the update be done num_iter times\n", - " for i in range(num_iter):\n", - " cost_grad = cost_function_grad(P, x , t)\n", - "\n", - " for l in range(N_hidden+1):\n", - " P[l] = P[l] - lmb * cost_grad[l]\n", - "\n", - "\n", - " print('Final cost: ',cost_function(P, x, t))\n", - "\n", - " return P\n", - "\n", - "if __name__ == '__main__':\n", - " ### Use the neural network:\n", - " npr.seed(15)\n", - "\n", - " ## Decide the vales of arguments to the function to solve\n", - " Nx = 10; Nt = 10\n", - " x = np.linspace(0, 1, Nx)\n", - " t = np.linspace(0,1,Nt)\n", - "\n", - " ## Set up the parameters for the network\n", - " num_hidden_neurons = [50,20]\n", - " num_iter = 1000\n", - " lmb = 0.01\n", - "\n", - " P = solve_pde_deep_neural_network(x,t, num_hidden_neurons, num_iter, lmb)\n", - "\n", - " ## Store the results\n", - " res = np.zeros((Nx, Nt))\n", - " res_analytical = np.zeros((Nx, Nt))\n", - " for i,x_ in enumerate(x):\n", - " for j, t_ in enumerate(t):\n", - " point = np.array([x_, t_])\n", - " res[i,j] = g_trial(point,P)\n", - "\n", - " res_analytical[i,j] = g_analytic(point)\n", - "\n", - " diff = np.abs(res - res_analytical)\n", - " print(\"Max difference between analytical and solution from nn: %g\"%np.max(diff))\n", - "\n", - " ## Plot the solutions in two dimensions, that being in position and time\n", - "\n", - " T,X = np.meshgrid(t,x)\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Solution from the deep neural network w/ %d layer'%len(num_hidden_neurons))\n", - " s = ax.plot_surface(T,X,res,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Analytical solution')\n", - " s = ax.plot_surface(T,X,res_analytical,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - "\n", - " fig = plt.figure(figsize=(10,10))\n", - " ax = fig.gca(projection='3d')\n", - " ax.set_title('Difference')\n", - " s = ax.plot_surface(T,X,diff,linewidth=0,antialiased=False,cmap=cm.viridis)\n", - " ax.set_xlabel('Time $t$')\n", - " ax.set_ylabel('Position $x$');\n", - "\n", - " ## Take some slices of the 3D plots just to see the solutions at particular times\n", - " indx1 = 0\n", - " indx2 = int(Nt/2)\n", - " indx3 = Nt-1\n", - "\n", - " t1 = t[indx1]\n", - " t2 = t[indx2]\n", - " t3 = t[indx3]\n", - "\n", - " # Slice the results from the DNN\n", - " res1 = res[:,indx1]\n", - " res2 = res[:,indx2]\n", - " res3 = res[:,indx3]\n", - "\n", - " # Slice the analytical results\n", - " res_analytical1 = res_analytical[:,indx1]\n", - " res_analytical2 = res_analytical[:,indx2]\n", - " res_analytical3 = res_analytical[:,indx3]\n", - "\n", - " # Plot the slices\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t1)\n", - " plt.plot(x, res1)\n", - " plt.plot(x,res_analytical1)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t2)\n", - " plt.plot(x, res2)\n", - " plt.plot(x,res_analytical2)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.figure(figsize=(10,10))\n", - " plt.title(\"Computed solutions at time = %g\"%t3)\n", - " plt.plot(x, res3)\n", - " plt.plot(x,res_analytical3)\n", - " plt.legend(['dnn','analytical'])\n", - "\n", - " plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Resources on differential equations and deep learning\n", - "\n", - "1. [Artificial neural networks for solving ordinary and partial differential equations by I.E. Lagaris et al](https://pdfs.semanticscholar.org/d061/df393e0e8fbfd0ea24976458b7d42419040d.pdf)\n", - "\n", - "2. [Neural networks for solving differential equations by A. Honchar](https://becominghuman.ai/neural-networks-for-solving-differential-equations-fa230ac5e04c)\n", - "\n", - "3. [Solving differential equations using neural networks by M.M Chiaramonte and M. Kiener](http://cs229.stanford.edu/proj2013/ChiaramonteKiener-SolvingDifferentialEquationsUsingNeuralNetworks.pdf)\n", - "\n", - "4. [Introduction to Partial Differential Equations by A. Tveito, R. Winther](https://www.springer.com/us/book/9783540225515)" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -} diff --git a/doc/LectureNotes/chapter12.ipynb b/doc/LectureNotes/chapter12.ipynb deleted file mode 100644 index dfbc0e322..000000000 --- a/doc/LectureNotes/chapter12.ipynb +++ /dev/null @@ -1,718 +0,0 @@ -{ - "cells": [ - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "# Convolutional Neural Networks\n", - "\n", - "\n", - "Convolutional neural networks (CNNs) were developed during the last\n", - "decade of the previous century, with a focus on character recognition\n", - "tasks. Nowadays, CNNs are a central element in the spectacular success\n", - "of deep learning methods. The success in for example image\n", - "classifications have made them a central tool for most machine\n", - "learning practitioners.\n", - "\n", - "CNNs are very similar to ordinary Neural Networks.\n", - "They are made up of neurons that have learnable weights and\n", - "biases. Each neuron receives some inputs, performs a dot product and\n", - "optionally follows it with a non-linearity. The whole network still\n", - "expresses a single differentiable score function: from the raw image\n", - "pixels on one end to class scores at the other. And they still have a\n", - "loss function (for example Softmax) on the last (fully-connected) layer\n", - "and all the tips/tricks we developed for learning regular Neural\n", - "Networks still apply (back propagation, gradient descent etc etc).\n", - "\n", - "What is the difference? **CNN architectures make the explicit assumption that\n", - "the inputs are images, which allows us to encode certain properties\n", - "into the architecture. These then make the forward function more\n", - "efficient to implement and vastly reduce the amount of parameters in\n", - "the network.**\n", - "\n", - "\n", - "As an example, consider\n", - "an image of size $32\\times 32\\times 3$ (32 wide, 32 high, 3 color channels), so a\n", - "single fully-connected neuron in a first hidden layer of a regular\n", - "Neural Network would have $32\\times 32\\times 3 = 3072$ weights. This amount still\n", - "seems manageable, but clearly this fully-connected structure does not\n", - "scale to larger images. For example, an image of more respectable\n", - "size, say $200\\times 200\\times 3$, would lead to neurons that have \n", - "$200\\times 200\\times 3 = 120,000$ weights. \n", - "\n", - "We could have\n", - "several such neurons, and the parameters would add up quickly! Clearly,\n", - "this full connectivity is wasteful and the huge number of parameters\n", - "would quickly lead to possible overfitting.\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "Convolutional Neural Networks take advantage of the fact that the\n", - "input consists of images and they constrain the architecture in a more\n", - "sensible way. \n", - "\n", - "In particular, unlike a regular Neural Network, the\n", - "layers of a CNN have neurons arranged in 3 dimensions: width,\n", - "height, depth. (Note that the word depth here refers to the third\n", - "dimension of an activation volume, not to the depth of a full Neural\n", - "Network, which can refer to the total number of layers in a network.)\n", - "\n", - "To understand it better, the above example of an image \n", - "with an input volume of\n", - "activations has dimensions $32\\times 32\\times 3$ (width, height,\n", - "depth respectively). \n", - "\n", - "The neurons in a layer will\n", - "only be connected to a small region of the layer before it, instead of\n", - "all of the neurons in a fully-connected manner. Moreover, the final\n", - "output layer could for this specific image have dimensions $1\\times 1 \\times 10$, \n", - "because by the\n", - "end of the CNN architecture we will reduce the full image into a\n", - "single vector of class scores, arranged along the depth\n", - "dimension. \n", - "\n", - "\n", - "\n", - "\n", - "\n", - "\n", - "A simple CNN is a sequence of layers, and every layer of a CNN\n", - "transforms one volume of activations to another through a\n", - "differentiable function. We use three main types of layers to build\n", - "CNN architectures: Convolutional Layer, Pooling Layer, and\n", - "Fully-Connected Layer (exactly as seen in regular Neural Networks). We\n", - "will stack these layers to form a full CNN architecture.\n", - "\n", - "A simple CNN for image classification could have the architecture:\n", - "\n", - "* **INPUT** ($32\\times 32 \\times 3$) will hold the raw pixel values of the image, in this case an image of width 32, height 32, and with three color channels R,G,B.\n", - "\n", - "* **CONV** (convolutional )layer will compute the output of neurons that are connected to local regions in the input, each computing a dot product between their weights and a small region they are connected to in the input volume. This may result in volume such as $[32\\times 32\\times 12]$ if we decided to use 12 filters.\n", - "\n", - "* **RELU** layer will apply an elementwise activation function, such as the $max(0,x)$ thresholding at zero. This leaves the size of the volume unchanged ($[32\\times 32\\times 12]$).\n", - "\n", - "* **POOL** (pooling) layer will perform a downsampling operation along the spatial dimensions (width, height), resulting in volume such as $[16\\times 16\\times 12]$.\n", - "\n", - "* **FC** (i.e. fully-connected) layer will compute the class scores, resulting in volume of size $[1\\times 1\\times 10]$, where each of the 10 numbers correspond to a class score, such as among the 10 categories of the MNIST images we considered above . As with ordinary Neural Networks and as the name implies, each neuron in this layer will be connected to all the numbers in the previous volume.\n", - "\n", - "CNNs transform the original image layer by layer from the original\n", - "pixel values to the final class scores. \n", - "\n", - "Observe that some layers contain\n", - "parameters and other don’t. In particular, the CNN layers perform\n", - "transformations that are a function of not only the activations in the\n", - "input volume, but also of the parameters (the weights and biases of\n", - "the neurons). On the other hand, the RELU/POOL layers will implement a\n", - "fixed function. The parameters in the CONV/FC layers will be trained\n", - "with gradient descent so that the class scores that the CNN computes\n", - "are consistent with the labels in the training set for each image.\n", - "\n", - "\n", - "\n", - "### CNNs in brief\n", - "\n", - "In summary:\n", - "\n", - "* A CNN architecture is in the simplest case a list of Layers that transform the image volume into an output volume (e.g. holding the class scores)\n", - "\n", - "* There are a few distinct types of Layers (e.g. CONV/FC/RELU/POOL are by far the most popular)\n", - "\n", - "* Each Layer accepts an input 3D volume and transforms it to an output 3D volume through a differentiable function\n", - "\n", - "* Each Layer may or may not have parameters (e.g. CONV/FC do, RELU/POOL don’t)\n", - "\n", - "* Each Layer may or may not have additional hyperparameters (e.g. CONV/FC/POOL do, RELU doesn’t)\n", - "\n", - "## CNNs in more detail, building convolutional neural networks in Tensorflow and Keras\n", - "\n", - "\n", - "As discussed above, CNNs are neural networks built from the assumption that the inputs\n", - "to the network are 2D images. This is important because the number of features or pixels in images\n", - "grows very fast with the image size, and an enormous number of weights and biases are needed in order to build an accurate network. \n", - "\n", - "As before, we still have our input, a hidden layer and an output. What's novel about convolutional networks\n", - "are the **convolutional** and **pooling** layers stacked in pairs between the input and the hidden layer.\n", - "In addition, the data is no longer represented as a 2D feature matrix, instead each input is a number of 2D\n", - "matrices, typically 1 for each color dimension (Red, Green, Blue). \n", - "\n", - "\n", - "\n", - "It means that to represent the entire\n", - "dataset of images, we require a 4D matrix or **tensor**. This tensor has the dimensions:" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "$$\n", - "(n_{inputs},\\, n_{pixels, width},\\, n_{pixels, height},\\, depth) .\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The MNIST dataset consists of grayscale images with a pixel size of\n", - "$28\\times 28$, meaning we require $28 \\times 28 = 724$ weights to each\n", - "neuron in the first hidden layer.\n", - "\n", - "If we were to analyze images of size $128\\times 128$ we would require\n", - "$128 \\times 128 = 16384$ weights to each neuron. Even worse if we were\n", - "dealing with color images, as most images are, we have an image matrix\n", - "of size $128\\times 128$ for each color dimension (Red, Green, Blue),\n", - "meaning 3 times the number of weights $= 49152$ are required for every\n", - "single neuron in the first hidden layer.\n", - "\n", - "\n", - "\n", - "Images typically have strong local correlations, meaning that a small\n", - "part of the image varies little from its neighboring regions. If for\n", - "example we have an image of a blue car, we can roughly assume that a\n", - "small blue part of the image is surrounded by other blue regions.\n", - "\n", - "Therefore, instead of connecting every single pixel to a neuron in the\n", - "first hidden layer, as we have previously done with deep neural\n", - "networks, we can instead connect each neuron to a small part of the\n", - "image (in all 3 RGB depth dimensions). The size of each small area is\n", - "fixed, and known as a [receptive](https://en.wikipedia.org/wiki/Receptive_field).\n", - "\n", - "\n", - "\n", - "The layers of a convolutional neural network arrange neurons in 3D: width, height and depth. \n", - "The input image is typically a square matrix of depth 3. \n", - "\n", - "A **convolution** is performed on the image which outputs\n", - "a 3D volume of neurons. The weights to the input are arranged in a number of 2D matrices, known as **filters**.\n", - "\n", - "\n", - "Each filter slides along the input image, taking the dot product\n", - "between each small part of the image and the filter, in all depth\n", - "dimensions. This is then passed through a non-linear function,\n", - "typically the **Rectified Linear (ReLu)** function, which serves as the\n", - "activation of the neurons in the first convolutional layer. This is\n", - "further passed through a **pooling layer**, which reduces the size of the\n", - "convolutional layer, e.g. by taking the maximum or average across some\n", - "small regions, and this serves as input to the next convolutional\n", - "layer.\n", - "\n", - "\n", - "\n", - "By systematically reducing the size of the input volume, through\n", - "convolution and pooling, the network should create representations of\n", - "small parts of the input, and then from them assemble representations\n", - "of larger areas. The final pooling layer is flattened to serve as\n", - "input to a hidden layer, such that each neuron in the final pooling\n", - "layer is connected to every single neuron in the hidden layer. This\n", - "then serves as input to the output layer, e.g. a softmax output for\n", - "classification.\n", - "\n", - "\n", - "\n", - "### Prerequisites: Collect and pre-process data" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "%matplotlib inline\n", - "\n", - "# import necessary packages\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "from sklearn import datasets\n", - "\n", - "\n", - "# ensure the same random numbers appear every time\n", - "np.random.seed(0)\n", - "\n", - "# display images in notebook\n", - "%matplotlib inline\n", - "plt.rcParams['figure.figsize'] = (12,12)\n", - "\n", - "\n", - "# download MNIST dataset\n", - "digits = datasets.load_digits()\n", - "\n", - "# define inputs and labels\n", - "inputs = digits.images\n", - "labels = digits.target\n", - "\n", - "# RGB images have a depth of 3\n", - "# our images are grayscale so they should have a depth of 1\n", - "inputs = inputs[:,:,:,np.newaxis]\n", - "\n", - "print(\"inputs = (n_inputs, pixel_width, pixel_height, depth) = \" + str(inputs.shape))\n", - "print(\"labels = (n_inputs) = \" + str(labels.shape))\n", - "\n", - "\n", - "# choose some random images to display\n", - "n_inputs = len(inputs)\n", - "indices = np.arange(n_inputs)\n", - "random_indices = np.random.choice(indices, size=5)\n", - "\n", - "for i, image in enumerate(digits.images[random_indices]):\n", - " plt.subplot(1, 5, i+1)\n", - " plt.axis('off')\n", - " plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')\n", - " plt.title(\"Label: %d\" % digits.target[random_indices[i]])\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Importing Keras and Tensorflow" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "from tensorflow.keras import datasets, layers, models\n", - "from tensorflow.keras.layers import Input\n", - "from tensorflow.keras.models import Sequential #This allows appending layers to existing models\n", - "from tensorflow.keras.layers import Dense #This allows defining the characteristics of a particular layer\n", - "from tensorflow.keras import optimizers #This allows using whichever optimiser we want (sgd,adam,RMSprop)\n", - "from tensorflow.keras import regularizers #This allows using whichever regularizer we want (l1,l2,l1_l2)\n", - "from tensorflow.keras.utils import to_categorical #This allows using categorical cross entropy as the cost function\n", - "#from tensorflow.keras import Conv2D\n", - "#from tensorflow.keras import MaxPooling2D\n", - "#from tensorflow.keras import Flatten\n", - "\n", - "from sklearn.model_selection import train_test_split\n", - "\n", - "# representation of labels\n", - "labels = to_categorical(labels)\n", - "\n", - "# split into train and test data\n", - "# one-liner from scikit-learn library\n", - "train_size = 0.8\n", - "test_size = 1 - train_size\n", - "X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,\n", - " test_size=test_size)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "def create_convolutional_neural_network_keras(input_shape, receptive_field,\n", - " n_filters, n_neurons_connected, n_categories,\n", - " eta, lmbd):\n", - " model = Sequential()\n", - " model.add(layers.Conv2D(n_filters, (receptive_field, receptive_field), input_shape=input_shape, padding='same',\n", - " activation='relu', kernel_regularizer=regularizers.l2(lmbd)))\n", - " model.add(layers.MaxPooling2D(pool_size=(2, 2)))\n", - " model.add(layers.Flatten())\n", - " model.add(layers.Dense(n_neurons_connected, activation='relu', kernel_regularizer=regularizers.l2(lmbd)))\n", - " model.add(layers.Dense(n_categories, activation='softmax', kernel_regularizer=regularizers.l2(lmbd)))\n", - " \n", - " sgd = optimizers.SGD(lr=eta)\n", - " model.compile(loss='categorical_crossentropy', optimizer=sgd, metrics=['accuracy'])\n", - " \n", - " return model\n", - "\n", - "epochs = 100\n", - "batch_size = 100\n", - "input_shape = X_train.shape[1:4]\n", - "receptive_field = 3\n", - "n_filters = 10\n", - "n_neurons_connected = 50\n", - "n_categories = 10\n", - "\n", - "eta_vals = np.logspace(-5, 1, 7)\n", - "lmbd_vals = np.logspace(-5, 1, 7)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "CNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", - " \n", - "for i, eta in enumerate(eta_vals):\n", - " for j, lmbd in enumerate(lmbd_vals):\n", - " CNN = create_convolutional_neural_network_keras(input_shape, receptive_field,\n", - " n_filters, n_neurons_connected, n_categories,\n", - " eta, lmbd)\n", - " CNN.fit(X_train, Y_train, epochs=epochs, batch_size=batch_size, verbose=0)\n", - " scores = CNN.evaluate(X_test, Y_test)\n", - " \n", - " CNN_keras[i][j] = CNN\n", - " \n", - " print(\"Learning rate = \", eta)\n", - " print(\"Lambda = \", lmbd)\n", - " print(\"Test accuracy: %.3f\" % scores[1])\n", - " print()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "### Final visualization" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# visual representation of grid search\n", - "# uses seaborn heatmap, could probably do this in matplotlib\n", - "import seaborn as sns\n", - "\n", - "sns.set()\n", - "\n", - "train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))\n", - "\n", - "for i in range(len(eta_vals)):\n", - " for j in range(len(lmbd_vals)):\n", - " CNN = CNN_keras[i][j]\n", - "\n", - " train_accuracy[i][j] = CNN.evaluate(X_train, Y_train)[1]\n", - " test_accuracy[i][j] = CNN.evaluate(X_test, Y_test)[1]\n", - "\n", - " \n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(train_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Training Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()\n", - "\n", - "fig, ax = plt.subplots(figsize = (10, 10))\n", - "sns.heatmap(test_accuracy, annot=True, ax=ax, cmap=\"viridis\")\n", - "ax.set_title(\"Test Accuracy\")\n", - "ax.set_ylabel(\"$\\eta$\")\n", - "ax.set_xlabel(\"$\\lambda$\")\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## The CIFAR01 data set\n", - "\n", - "The CIFAR10 dataset contains 60,000 color images in 10 classes, with\n", - "6,000 images in each class. The dataset is divided into 50,000\n", - "training images and 10,000 testing images. The classes are mutually\n", - "exclusive and there is no overlap between them." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "import tensorflow as tf\n", - "\n", - "from tensorflow.keras import datasets, layers, models\n", - "import matplotlib.pyplot as plt\n", - "\n", - "# We import the data set\n", - "(train_images, train_labels), (test_images, test_labels) = datasets.cifar10.load_data()\n", - "\n", - "# Normalize pixel values to be between 0 and 1 by dividing by 255. \n", - "train_images, test_images = train_images / 255.0, test_images / 255.0" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "To verify that the dataset looks correct, let's plot the first 25 images from the training set and display the class name below each image." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',\n", - " 'dog', 'frog', 'horse', 'ship', 'truck']\n", - "​\n", - "plt.figure(figsize=(10,10))\n", - "for i in range(25):\n", - " plt.subplot(5,5,i+1)\n", - " plt.xticks([])\n", - " plt.yticks([])\n", - " plt.grid(False)\n", - " plt.imshow(train_images[i], cmap=plt.cm.binary)\n", - " # The CIFAR labels happen to be arrays, \n", - " # which is why you need the extra index\n", - " plt.xlabel(class_names[train_labels[i][0]])\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "The 6 lines of code below define the convolutional base using a common pattern: a stack of Conv2D and MaxPooling2D layers.\n", - "\n", - "As input, a CNN takes tensors of shape (image_height, image_width, color_channels), ignoring the batch size. If you are new to these dimensions, color_channels refers to (R,G,B). In this example, you will configure our CNN to process inputs of shape (32, 32, 3), which is the format of CIFAR images. You can do this by passing the argument input_shape to our first layer." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "model = models.Sequential()\n", - "model.add(layers.Conv2D(32, (3, 3), activation='relu', input_shape=(32, 32, 3)))\n", - "model.add(layers.MaxPooling2D((2, 2)))\n", - "model.add(layers.Conv2D(64, (3, 3), activation='relu'))\n", - "model.add(layers.MaxPooling2D((2, 2)))\n", - "model.add(layers.Conv2D(64, (3, 3), activation='relu'))\n", - "\n", - "# Let's display the architecture of our model so far.\n", - "\n", - "model.summary()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "You can see that the output of every Conv2D and MaxPooling2D layer is a 3D tensor of shape (height, width, channels). The width and height dimensions tend to shrink as you go deeper in the network. The number of output channels for each Conv2D layer is controlled by the first argument (e.g., 32 or 64). Typically, as the width and height shrink, you can afford (computationally) to add more output channels in each Conv2D layer.\n", - "\n", - "\n", - "\n", - "To complete our model, you will feed the last output tensor from the\n", - "convolutional base (of shape (4, 4, 64)) into one or more Dense layers\n", - "to perform classification. Dense layers take vectors as input (which\n", - "are 1D), while the current output is a 3D tensor. First, you will\n", - "flatten (or unroll) the 3D output to 1D, then add one or more Dense\n", - "layers on top. CIFAR has 10 output classes, so you use a final Dense\n", - "layer with 10 outputs and a softmax activation." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "model.add(layers.Flatten())\n", - "model.add(layers.Dense(64, activation='relu'))\n", - "model.add(layers.Dense(10))\n", - "Here's the complete architecture of our model.\n", - "\n", - "model.summary()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "As you can see, our (4, 4, 64) outputs were flattened into vectors of shape (1024) before going through two Dense layers.\n", - "\n", - "Compile and train the model." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "model.compile(optimizer='adam',\n", - " loss=tf.keras.losses.SparseCategoricalCrossentropy(from_logits=True),\n", - " metrics=['accuracy'])\n", - "​\n", - "history = model.fit(train_images, train_labels, epochs=10, \n", - " validation_data=(test_images, test_labels))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "Finally, we evaluate the model." - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "plt.plot(history.history['accuracy'], label='accuracy')\n", - "plt.plot(history.history['val_accuracy'], label = 'val_accuracy')\n", - "plt.xlabel('Epoch')\n", - "plt.ylabel('Accuracy')\n", - "plt.ylim([0.5, 1])\n", - "plt.legend(loc='lower right')\n", - "\n", - "test_loss, test_acc = model.evaluate(test_images, test_labels, verbose=2)\n", - "\n", - "print(test_acc)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Recurrent neural networks: Overarching view\n", - "\n", - "Till now our focus has been, including convolutional neural networks\n", - "as well, on feedforward neural networks. The output or the activations\n", - "flow only in one direction, from the input layer to the output layer.\n", - "\n", - "A recurrent neural network (RNN) looks very much like a feedforward\n", - "neural network, except that it also has connections pointing\n", - "backward. \n", - "\n", - "RNNs are used to analyze time series data such as stock prices, and\n", - "tell you when to buy or sell. In autonomous driving systems, they can\n", - "anticipate car trajectories and help avoid accidents. More generally,\n", - "they can work on sequences of arbitrary lengths, rather than on\n", - "fixed-sized inputs like all the nets we have discussed so far. For\n", - "example, they can take sentences, documents, or audio samples as\n", - "input, making them extremely useful for natural language processing\n", - "systems such as automatic translation and speech-to-text.\n", - "\n", - "\n", - "\n", - "\n", - "### A simple example" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false, - "editable": true - }, - "outputs": [], - "source": [ - "# Start importing packages\n", - "import pandas as pd\n", - "import numpy as np\n", - "import matplotlib.pyplot as plt\n", - "import tensorflow as tf\n", - "from tensorflow.keras import datasets, layers, models\n", - "from tensorflow.keras.layers import Input\n", - "from tensorflow.keras.models import Model, Sequential \n", - "from tensorflow.keras.layers import Dense, SimpleRNN, LSTM, GRU\n", - "from tensorflow.keras import optimizers \n", - "from tensorflow.keras import regularizers \n", - "from tensorflow.keras.utils import to_categorical \n", - "\n", - "\n", - "\n", - "# convert into dataset matrix\n", - "def convertToMatrix(data, step):\n", - " X, Y =[], []\n", - " for i in range(len(data)-step):\n", - " d=i+step \n", - " X.append(data[i:d,])\n", - " Y.append(data[d,])\n", - " return np.array(X), np.array(Y)\n", - "\n", - "step = 4\n", - "N = 1000 \n", - "Tp = 800 \n", - "\n", - "t=np.arange(0,N)\n", - "x=np.sin(0.02*t)+2*np.random.rand(N)\n", - "df = pd.DataFrame(x)\n", - "df.head()\n", - "\n", - "plt.plot(df)\n", - "plt.show()\n", - "\n", - "values=df.values\n", - "train,test = values[0:Tp,:], values[Tp:N,:]\n", - "\n", - "# add step elements into train and test\n", - "test = np.append(test,np.repeat(test[-1,],step))\n", - "train = np.append(train,np.repeat(train[-1,],step))\n", - " \n", - "trainX,trainY =convertToMatrix(train,step)\n", - "testX,testY =convertToMatrix(test,step)\n", - "trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1]))\n", - "testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1]))\n", - "\n", - "model = Sequential()\n", - "model.add(SimpleRNN(units=32, input_shape=(1,step), activation=\"relu\"))\n", - "model.add(Dense(8, activation=\"relu\")) \n", - "model.add(Dense(1))\n", - "model.compile(loss='mean_squared_error', optimizer='rmsprop')\n", - "model.summary()\n", - "\n", - "model.fit(trainX,trainY, epochs=100, batch_size=16, verbose=2)\n", - "trainPredict = model.predict(trainX)\n", - "testPredict= model.predict(testX)\n", - "predicted=np.concatenate((trainPredict,testPredict),axis=0)\n", - "\n", - "trainScore = model.evaluate(trainX, trainY, verbose=0)\n", - "print(trainScore)\n", - "\n", - "index = df.index.values\n", - "plt.plot(index,df)\n", - "plt.plot(index,predicted)\n", - "plt.axvline(df.index[Tp], c=\"r\")\n", - "plt.show()" - ] - } - ], - "metadata": {}, - "nbformat": 4, - "nbformat_minor": 4 -}