diff --git a/doc/pub/Intro2Course/html/._Intro2Course-bs012.html b/doc/pub/Intro2Course/html/._Intro2Course-bs012.html new file mode 100644 index 000000000..5b48ea678 --- /dev/null +++ b/doc/pub/Intro2Course/html/._Intro2Course-bs012.html @@ -0,0 +1,175 @@ + + + + + + + + +Applied Data Analysis and Machine Learning: Introduction to the course, Logistics and Practicalities + + + + + + + + + + + + + + + + +
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+ + Applied Data Analysis and Machine Learning: Introduction to the course, Logistics and Practicalities +
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Other courses on Data science and Machine Learning at UiO

+ +

+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. + +

    +
  1. STK2100 Machine learning and statistical methods for prediction and classification.
    2. +
  2. IN3050 Introduction to Artificial Intelligence and Machine Learning. Introductory course in machine learning and AI with an algorithmic approach.
    3. +
  3. STK-INF3000/4000 Selected Topics in Data Science. The course provides insight into selected contemporary relevant topics within Data Science.
    4. +
  4. IN4080 Natural Language Processing. Probabilistic and machine learning techniques applied to natural language processing.
    5. +
  5. STK-IN4300 Statistical learning methods in Data Science. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.
    6. +
  6. INF4490 Biologically Inspired Computing. An introduction to self-adapting methods also called artificial intelligence or machine learning.
    7. +
  7. IN-STK5000 Adaptive Methods for Data-Based Decision Making. Methods for adaptive collection and processing of data based on machine learning techniques.
    8. +
  8. IN5400/INF5860 Machine Learning for Image Analysis. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.
    9. +
  9. TEK5040 Deep learning for autonomous systems. 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.
    10. +
  10. STK4051 Computational Statistics
    11. +
  11. STK4021 Applied Bayesian Analysis and Numerical Methods
    12. +
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+ + + + + + diff --git a/doc/pub/Introduction/html/._Introduction-bs000.html b/doc/pub/Introduction/html/._Introduction-bs000.html new file mode 100644 index 000000000..bfac8f3e2 --- /dev/null +++ b/doc/pub/Introduction/html/._Introduction-bs000.html @@ -0,0 +1,605 @@ + + + + + + + + +Introduction to Applied Data Analysis and Machine Learning + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Introduction to Applied Data Analysis and Machine Learning +
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+ + +
+ +

 

 

 

+ + + + + + +
+

Introduction to Applied Data Analysis and Machine Learning

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Sep 19, 2020

+
+

+

+ +

Introduction

+ +

+During the last two decades there has been a swift and amazing +development of Machine Learning techniques and algorithms that impact +many areas in not only Science and Technology but also the Humanities, +Social Sciences, Medicine, Law, indeed, almost all possible +disciplines. The applications are incredibly many, from self-driving +cars to solving high-dimensional differential equations or complicated +quantum mechanical many-body problems. Machine Learning is perceived +by many as one of the main disruptive techniques nowadays. + +

+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, +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. + +

Learning outcomes

+ +

+These sets of lectures aim at giving you an overview of central aspects of +statistical data analysis as well as some of the central algorithms +used in machine learning. We will introduce a variety of central +algorithms and methods essential for studies of data analysis and +machine learning. + +

+Hands-on projects and experimenting with data and algorithms plays a central role in +these lectures, and our hope is, through the various +projects and exercises, to expose you to fundamental +research problems in these fields, with the aim to reproduce state of +the art scientific results. You will learn to develop and +structure codes for studying these systems, 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, you will + +

    +
  1. Learn about basic data analysis, Bayesian statistics, Monte Carlo methods, data optimization and machine learning;
    2. +
  2. Be capable of extending the acquired knowledge to other systems and cases;
    3. +
  3. Have an understanding of central algorithms used in data analysis and machine learning;
    4. +
  4. Gain knowledge of central aspects of Monte Carlo methods, Markov chains, Gibbs samplers and their possible applications, from numerical integration to simulation of stock markets;
    5. +
  5. Understand methods for regression and classification;
    6. +
  6. Learn about neural network, genetic algorithms and Boltzmann machines;
    7. +
  7. 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++, in addition to a basic knowledge of linear algebra (typically taught during the first one or two years of undergraduate studies).
    8. +
+ +There are several topics we will cover here, spanning from +statistical data analysis and its basic concepts such as expectation +values, variance, covariance, correlation functions and errors, via +well-known probability distribution functions like the uniform +distribution, the binomial distribution, the Poisson distribution and +simple and multivariate normal distributions to central elements of +Bayesian statistics and modeling. We will also remind the reader about +central elements from linear algebra and standard methods based on +linear algebra used to optimize (minimize) functions (the family of gradient descent methods) +and the Singular-value decomposition and +least square methods for parameterizing data. + +

+We will also cover Monte Carlo methods, Markov chains, well-known +algorithms for sampling stochastic events like the Metropolis-Hastings +and Gibbs sampling methods. An important aspect of all our +calculations is a proper estimation of errors. Here we will also +discuss famous resampling techniques like the blocking, the bootstrapping +and the jackknife methods and the infamous bias-variance tradeoff. + +

+The second part of the material covers several algorithms used in +machine learning. + +

Machine Learning, a small (and probably biased) introduction

+ +

+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, an extremely rich field

+ +

+Machine learning is an extremely rich field, in spite of its young +age. The increases we have seen during the last 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 libraries +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. + +

A multidisciplinary approach

+ +

+Not all the +algorithms and methods can be given a rigorous mathematical +justification (for example decision trees and random forests), 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 the algorithms and methods we will discuss. + +

Types of Machine Learning

+ +

+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: + +

+ + +

+ + + +

Essential elements of ML

+ +

+The methods we cover have three main topics in common, irrespective of +whether we deal with supervised or unsupervised learning. + + +

+ + + +

An optimization/minimization problem

+ +

+At the heart of basically all Machine Learning algorithms we will encounter so-called minimization or optimization algorithms. A large family of such methods are so-called gradient methods. + +

A Frequentist approach to data analysis

+ +

+When you hear phrases like predictions and estimations and +correlations and causations, what do you think of? May be you think +of the difference between classifying new data points and generating +new data points. +Or perhaps you consider that correlations represent some kind of symmetric statements like +if \( A \) is correlated with \( B \), then \( B \) is correlated with +\( A \). Causation on the other hand is directional, that is if \( A \) causes \( B \), \( B \) does not +necessarily cause \( A \). + +

+These concepts are in some sense the difference between machine +learning and statistics. In machine learning and prediction based +tasks, we are often interested in developing algorithms that are +capable of learning patterns from given data in an automated fashion, +and then using these learned patterns to make predictions or +assessments of newly given data. In many cases, our primary concern +is the quality of the predictions or assessments, and we are less +concerned about the underlying patterns that were learned in order +to make these predictions. + +

+In machine learning we normally use a so-called frequentist approach, +where the aim is to make predictions and find correlations. We focus +less on for example extracting a probability distribution function (PDF). The PDF can be +used in turn to make estimations and find causations such as given \( A \) +what is the likelihood of finding \( B \). + +

What is a good model?

+ +

+In science and engineering we often end up in situations where we want to infer (or learn) a +quantitative model \( M \) for a given set of sample points \( \boldsymbol{X} \in [x_1, x_2,\dots x_N] \). + +

+As we will see repeatedely in these lectures, we could try to fit these data points to a model given by a +straight line, or if we wish to be more sophisticated to a more complex +function. + +

+The reason for inferring such a model is that it +serves many useful purposes. On the one hand, the model can reveal information +encoded in the data or underlying mechanisms from which the data were generated. For instance, we could discover important +corelations that relate interesting physics interpretations. + +

+In addition, it can simplify the representation of the given data set and help +us in making predictions about future data samples. + +

+A first important consideration to keep in mind is that inferring the correct model +for a given data set is an elusive, if not impossible, task. The fundamental difficulty +is that if we are not specific about what we mean by a correct model, there +could easily be many different models that fit the given data set equally well. + +

What is a good model? Can we define it?

+ +

+The central question is this: what leads us to say that a model is correct or +optimal for a given data set? To make the model inference problem well posed, i.e., +to guarantee that there is a unique optimal model for the given data, we need to +impose additional assumptions or restrictions on the class of models considered. To +this end, we should not be looking for just any model that can describe the data. +Instead, we should look for a model \( M \) that is the best among a restricted class +of models. In addition, to make the model inference problem computationally +tractable, we need to specify how restricted the class of models needs to be. A +common strategy is to start +with the simplest possible class of models that is just necessary to describe the data +or solve the problem at hand. More precisely, the model class should be rich enough +to contain at least one model that can fit the data to a desired accuracy and yet be +restricted enough that it is relatively simple to find the best model for the given data. + +

+Thus, the most popular strategy is to start from the +simplest class of models and increase the complexity of the models only when the +simpler models become inadequate. For instance, if we work with a regression problem to fit a set of sample points, one +may first try the simplest class of models, namely linear models, followed obviously by more complex models. + +

+How to evaluate which model fits best the data is something we will come back to over and over again in these set of lectures. + +

Choice of Programming Language

+ +

+Python plays nowadays a central role in the development of machine +learning techniques and tools for data analysis. In particular, seen +the wealth of machine learning and data analysis libraries written in +Python, easy to use libraries with immediate visualization(and not the +least impressive galleries of existing examples), the popularity of the +Jupyter notebook framework with the possibility to run R codes or +compiled programs written in C++, and much more made our choice of +programming language for this series of lectures easy. However, +since the focus here is not only on using existing Python libraries such +as Scikit-Learn, Tensorflow and Pytorch, but also on developing your own +algorithms and codes, we will as far as possible present many of these +algorithms either as a Python codes or C++ or Fortran (or other languages) codes. + +

Data handling, machine learning and ethical aspects

+ +

+In most of the cases we will study, we will either generate the data +to analyze ourselves (both for supervised learning and unsupervised +learning) or we will recur again and again to data present in say +Scikit-Learn or Tensorflow. Many of the examples we end up +dealing with are from a privacy and data protection point of view, +rather inoccuous and boring results of numerical +calculations. However, this does not hinder us from developing a sound +ethical attitude to the data we use, how we analyze the data and how +we handle the data. + +

+The most immediate and simplest possible ethical aspects deal with our +approach to the scientific process. Nowadays, with version control +software like Git and various online +repositories like Github, +Gitlab etc, we can easily make our codes +and data sets we have used, freely and easily accessible to a wider +community. This helps us almost automagically in making our science +reproducible. The large open-source development communities involved +in say Scikit-Learn, +Tensorflow, +PyTorch and Keras, are +all excellent examples of this. The codes can be tested and improved +upon continuosly, helping thereby our scientific community at large in +developing data analysis and machine learning tools. It is much +easier today to gain traction and acceptance for making your science +reproducible. From a societal stand, this is an important element +since many of the developers are employees of large public institutions like +universities and research labs. Our fellow taxpayers do deserve to get +something back for their bucks. + +

+However, this more mechanical aspect of the ethics of science (in +particular the reproducibility of scientific results) is something +which is obvious and everybody should do so as part of the dialectics of +science. The fact that many scientists are not willing to share their codes or +data is detrimental to the scientific discourse. + +

+Before we proceed, we should add a disclaimer. Even though +we may dream of computers developing some kind of higher learning +capabilities, at the end (even if the artificial intelligence +community keeps touting our ears full of fancy futuristic avenues), it is we, yes you reading these lines, +who end up constructing and instructing, via various algorithms, the +machine learning approaches. Self-driving cars for example, rely on sofisticated +programs which take into account all possible situations a car can +encounter. In addition, extensive usage of training data from GPS +information, maps etc, are typically fed into the software for +self-driving cars. Adding to this various sensors and cameras that +feed information to the programs, there are zillions of ethical issues +which arise from this. + +

+For self-driving cars, where basically many of the standard machine +learning algorithms discussed here enter into the codes, at a certain +stage we have to make choices. Yes, we , the lads and lasses who wrote +a program for a specific brand of a self-driving car. As an example, +all carmakers have as their utmost priority the security of the +driver and the accompanying passengers. A famous European carmaker, which is +one of the leaders in the market of self-driving cars, had if +statements of the following type: suppose there are two obstacles in +front of you and you cannot avoid to collide with one of them. One of +the obstacles is a monstertruck while the other one is a kindergarten +class trying to cross the road. The self-driving car algo would then +opt for the hitting the small folks instead of the monstertruck, since +the likelihood of surving a collision with our future citizens, is +much higher. + +

+This leads to serious ethical aspects. Why should we opt for such an +option? Who decides and who is entitled to make such choices? Keep in +mind that many of the algorithms you will encounter in this series of +lectures or hear about later, are indeed based on simple programming +instructions. And you are very likely to be one of the people who may +end up writing such a code. Thus, developing a sound ethical attitude +to what we do, an approach well beyond the simple mechanistic one of +making our science available and reproducible, is much needed. The +example of the self-driving cars is just one of infinitely many cases +where we have to make choices. When you analyze data on economic +inequalities, who guarantees that you are not weighting some data in a +particular way, perhaps because you dearly want a specific conclusion +which may support your political views? Or what about the recent +claims that a famous IT company like Apple has a sexist bias on the +their recently launched credit card? + +

+We do not have the answers here, nor will we venture into a deeper +discussions of these aspects, but we want you think over these topics +in a more overarching way. A statistical data analysis with its dry +numbers and graphs meant to guide the eye, does not necessarily +reflect the truth, whatever that is. As a scientist, and after a +university education, you are supposedly a better citizen, with an +improved critical view and understanding of the scientific method, and +perhaps some deeper understanding of the ethics of science at +large. Use these insights. Be a critical citizen. You owe it to our +society. + +

+ +

+ +

+ + +
+ + + + + + + +
+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/Introduction/ipynb/.ipynb_checkpoints/Introduction-checkpoint.ipynb b/doc/pub/Introduction/ipynb/.ipynb_checkpoints/Introduction-checkpoint.ipynb new file mode 100644 index 000000000..56c63bf60 --- /dev/null +++ b/doc/pub/Introduction/ipynb/.ipynb_checkpoints/Introduction-checkpoint.ipynb @@ -0,0 +1,370 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Introduction to Applied Data Analysis and Machine Learning\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Nov 19, 2019**\n", + "\n", + "Copyright 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Introduction\n", + "\n", + "During the last two decades there has been a swift and amazing\n", + "development of Machine Learning techniques and algorithms that impact\n", + "many areas in not only Science and Technology but also the Humanities,\n", + "Social Sciences, Medicine, Law, indeed, almost all possible\n", + "disciplines. The applications are incredibly many, from self-driving\n", + "cars to solving high-dimensional differential equations or complicated\n", + "quantum mechanical many-body problems. Machine Learning is perceived\n", + "by many as one of the main disruptive techniques nowadays. \n", + "\n", + "Statistics, Data science and Machine Learning form important\n", + "fields of research in modern science. They describe how to learn and\n", + "make predictions from data, as well as allowing us to extract\n", + "important correlations about physical process and the underlying laws\n", + "of motion in large data sets. The latter, big data sets, appear\n", + "frequently in essentially all disciplines, from the traditional\n", + "Science, Technology, Mathematics and Engineering fields to Life\n", + "Science, Law, education research, the Humanities and the Social\n", + "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", + "## Learning outcomes\n", + "\n", + "These sets of lectures aim at giving you an overview of central aspects of\n", + "statistical data analysis as well as some of the central algorithms\n", + "used in machine learning. We will introduce a variety of central\n", + "algorithms and methods essential for studies of data analysis and\n", + "machine learning. \n", + "\n", + "Hands-on projects and experimenting with data and algorithms plays a central role in\n", + "these lectures, and our hope is, through the various\n", + "projects and exercises, to expose you to fundamental\n", + "research problems in these fields, with the aim to reproduce state of\n", + "the art scientific results. You will learn to develop and\n", + "structure codes for studying these systems, get acquainted with\n", + "computing facilities and learn to handle large scientific projects. A\n", + "good scientific and ethical conduct is emphasized throughout the\n", + "course. More specifically, you will\n", + "\n", + "1. Learn about basic data analysis, Bayesian statistics, Monte Carlo methods, data optimization and machine learning;\n", + "\n", + "2. Be capable of extending the acquired knowledge to other systems and cases;\n", + "\n", + "3. Have an understanding of central algorithms used in data analysis and machine learning;\n", + "\n", + "4. Gain knowledge of central aspects of Monte Carlo methods, Markov chains, Gibbs samplers and their possible applications, from numerical integration to simulation of stock markets;\n", + "\n", + "5. Understand methods for regression and classification;\n", + "\n", + "6. Learn about neural network, genetic algorithms and Boltzmann machines;\n", + "\n", + "7. 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++, in addition to a basic knowledge of linear algebra (typically taught during the first one or two years of undergraduate studies).\n", + "\n", + "There are several topics we will cover here, spanning from \n", + "statistical data analysis and its basic concepts such as expectation\n", + "values, variance, covariance, correlation functions and errors, via\n", + "well-known probability distribution functions like the uniform\n", + "distribution, the binomial distribution, the Poisson distribution and\n", + "simple and multivariate normal distributions to central elements of\n", + "Bayesian statistics and modeling. We will also remind the reader about\n", + "central elements from linear algebra and standard methods based on\n", + "linear algebra used to optimize (minimize) functions (the family of gradient descent methods)\n", + "and the Singular-value decomposition and\n", + "least square methods for parameterizing data.\n", + "\n", + "We will also cover Monte Carlo methods, Markov chains, well-known\n", + "algorithms for sampling stochastic events like the Metropolis-Hastings\n", + "and Gibbs sampling methods. An important aspect of all our\n", + "calculations is a proper estimation of errors. Here we will also\n", + "discuss famous resampling techniques like the blocking, the bootstrapping\n", + "and the jackknife methods and the infamous bias-variance tradeoff. \n", + "\n", + "The second part of the material covers several algorithms used in\n", + "machine learning.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Types of Machine Learning\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\n", + "a function of some parameters. The model reflects our knowledge of\n", + "the system (or lack thereof). As an example, if we know that our data\n", + "show a behavior similar to what would be predicted by a polynomial,\n", + "fitting our data to a polynomial of some degree would then determin\n", + "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", + "\n", + "Here we will build our machine learning approach on elements of the\n", + "statistical foundation discussed above, with elements from data\n", + "analysis, stochastic processes etc. We will discuss the following\n", + "machine learning algorithms\n", + "\n", + "1. Linear regression and its variants\n", + "\n", + "2. Decision tree algorithms, from single trees to random forests\n", + "\n", + "3. Bayesian statistics and regression\n", + "\n", + "4. Support vector machines and finally various variants of\n", + "\n", + "5. Artifical neural networks and deep learning, including convolutional neural networks and Bayesian neural networks\n", + "\n", + "6. Networks for unsupervised learning using for example reduced Boltzmann machines.\n", + "\n", + "## Choice of programming language\n", + "\n", + "Python plays nowadays a central role in the development of machine\n", + "learning techniques and tools for data analysis. In particular, seen\n", + "the wealth of machine learning and data analysis libraries written in\n", + "Python, easy to use libraries with immediate visualization(and not the\n", + "least impressive galleries of existing examples), the popularity of the\n", + "Jupyter notebook framework with the possibility to run **R** codes or\n", + "compiled programs written in C++, and much more made our choice of\n", + "programming language for this series of lectures easy. However,\n", + "since the focus here is not only on using existing Python libraries such\n", + "as **Scikit-Learn** or **Tensorflow**, but also on developing your own\n", + "algorithms and codes, we will as far as possible present many of these\n", + "algorithms either as a Python codes or C++ or Fortran (or other languages) codes. \n", + "\n", + "The reason we also focus on compiled languages like C++ (or\n", + "Fortran), is that Python is still notoriously slow when we do not\n", + "utilize highly streamlined computational libraries like\n", + "[Lapack](http://www.netlib.org/lapack/) or other numerical libraries\n", + "written in compiled languages (many of these libraries are written in\n", + "Fortran). Although a project like [Numba](https://numba.pydata.org/)\n", + "holds great promise for speeding up the unrolling of lengthy loops, C++\n", + "and Fortran are presently still the performance winners. Numba gives\n", + "you potentially the power to speed up your applications with high\n", + "performance functions written directly in Python. In particular,\n", + "array-oriented and math-heavy Python code can achieve similar\n", + "performance to C, C++ and Fortran. However, even with these speed-ups,\n", + "for codes involving heavy Markov Chain Monte Carlo analyses and\n", + "optimizations of cost functions, C++/C or Fortran codes tend to\n", + "outperform Python codes. \n", + "\n", + "Presently thus, the community tends to let\n", + "code written in C++/C or Fortran do the heavy duty numerical\n", + "number crunching and leave the post-analysis of the data to the above\n", + "mentioned Python modules or software packages. However, with the developments taking place in for example the Python community, and seen\n", + "the changes during the last decade, the above situation may change swiftly in the not too distant future. \n", + "\n", + "Many of the examples we discuss in this series of lectures come with\n", + "existing data files or provide code examples which produce the data to\n", + "be analyzed. Most of the applications we will discuss deal with\n", + "small data sets (less than a terabyte of information) and can easily\n", + "be analyzed and tested on standard off the shelf laptops you find in general \n", + "stores.\n", + "\n", + "## Data handling, machine learning and ethical aspects\n", + "\n", + "In most of the cases we will study, we will either generate the data\n", + "to analyze ourselves (both for supervised learning and unsupervised\n", + "learning) or we will recur again and again to data present in say\n", + "**Scikit-Learn** or **Tensorflow**. Many of the examples we end up\n", + "dealing with are from a privacy and data protection point of view,\n", + "rather inoccuous and boring results of numerical\n", + "calculations. However, this does not hinder us from developing a sound\n", + "ethical attitude to the data we use, how we analyze the data and how\n", + "we handle the data.\n", + "\n", + "The most immediate and simplest possible ethical aspects deal with our\n", + "approach to the scientific process. Nowadays, with version control\n", + "software like [Git](https://git-scm.com/) and various online\n", + "repositories like [Github](https://github.com/),\n", + "[Gitlab](https://about.gitlab.com/) etc, we can easily make our codes\n", + "and data sets we have used, freely and easily accessible to a wider\n", + "community. This helps us almost automagically in making our science\n", + "reproducible. The large open-source development communities involved\n", + "in say [Scikit-Learn](http://scikit-learn.org/stable/),\n", + "[Tensorflow](https://www.tensorflow.org/),\n", + "[PyTorch](http://pytorch.org/) and [Keras](https://keras.io/), are\n", + "all excellent examples of this. The codes can be tested and improved\n", + "upon continuosly, helping thereby our scientific community at large in\n", + "developing data analysis and machine learning tools. It is much\n", + "easier today to gain traction and acceptance for making your science\n", + "reproducible. From a societal stand, this is an important element\n", + "since many of the developers are employees of large public institutions like\n", + "universities and research labs. Our fellow taxpayers do deserve to get\n", + "something back for their bucks.\n", + "\n", + "However, this more mechanical aspect of the ethics of science (in\n", + "particular the reproducibility of scientific results) is something\n", + "which is obvious and everybody should do so as part of the dialectics of\n", + "science. The fact that many scientists are not willing to share their codes or \n", + "data is detrimental to the scientific discourse.\n", + "\n", + "Before we proceed, we should add a disclaimer. Even though\n", + "we may dream of computers developing some kind of higher learning\n", + "capabilities, at the end (even if the artificial intelligence\n", + "community keeps touting our ears full of fancy futuristic avenues), it is we, yes you reading these lines,\n", + "who end up constructing and instructing, via various algorithms, the\n", + "machine learning approaches. Self-driving cars for example, rely on sofisticated\n", + "programs which take into account all possible situations a car can\n", + "encounter. In addition, extensive usage of training data from GPS\n", + "information, maps etc, are typically fed into the software for\n", + "self-driving cars. Adding to this various sensors and cameras that\n", + "feed information to the programs, there are zillions of ethical issues\n", + "which arise from this.\n", + "\n", + "For self-driving cars, where basically many of the standard machine\n", + "learning algorithms discussed here enter into the codes, at a certain\n", + "stage we have to make choices. Yes, we , the lads and lasses who wrote\n", + "a program for a specific brand of a self-driving car. As an example,\n", + "all carmakers have as their utmost priority the security of the\n", + "driver and the accompanying passengers. A famous European carmaker, which is\n", + "one of the leaders in the market of self-driving cars, had **if**\n", + "statements of the following type: suppose there are two obstacles in\n", + "front of you and you cannot avoid to collide with one of them. One of\n", + "the obstacles is a monstertruck while the other one is a kindergarten\n", + "class trying to cross the road. The self-driving car algo would then\n", + "opt for the hitting the small folks instead of the monstertruck, since\n", + "the likelihood of surving a collision with our future citizens, is\n", + "much higher.\n", + "\n", + "This leads to serious ethical aspects. Why should we opt for such an\n", + "option? Who decides and who is entitled to make such choices? Keep in\n", + "mind that many of the algorithms you will encounter in this series of\n", + "lectures or hear about later, are indeed based on simple programming\n", + "instructions. And you are very likely to be one of the people who may\n", + "end up writing such a code. Thus, developing a sound ethical attitude\n", + "to what we do, an approach well beyond the simple mechanistic one of\n", + "making our science available and reproducible, is much needed. The\n", + "example of the self-driving cars is just one of infinitely many cases\n", + "where we have to make choices. When you analyze data on economic\n", + "inequalities, who guarantees that you are not weighting some data in a\n", + "particular way, perhaps because you dearly want a specific conclusion\n", + "which may support your political views? Or what about the recent\n", + "claims that a famous IT company like Apple has a sexist bias on the\n", + "their recently [launched credit card](https://qz.com/1748321/the-role-of-goldman-sachs-algorithms-in-the-apple-credit-card-scandal/)?\n", + "\n", + "We do not have the answers here, nor will we venture into a deeper\n", + "discussions of these aspects, but we want you think over these topics\n", + "in a more overarching way. A statistical data analysis with its dry\n", + "numbers and graphs meant to guide the eye, does not necessarily\n", + "reflect the truth, whatever that is. As a scientist, and after a\n", + "university education, you are supposedly a better citizen, with an\n", + "improved critical view and understanding of the scientific method, and\n", + "perhaps some deeper understanding of the ethics of science at\n", + "large. Use these insights. Be a critical citizen. You owe it to our\n", + "society." + ] + } + ], + "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584552722.Mortens-MacBook-Pro.local b/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584552722.Mortens-MacBook-Pro.local new file mode 100644 index 000000000..3914b7df6 Binary files /dev/null and b/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584552722.Mortens-MacBook-Pro.local differ diff --git a/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584554765.Mortens-MacBook-Pro.local b/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584554765.Mortens-MacBook-Pro.local new file mode 100644 index 000000000..d90e12c05 Binary files /dev/null and b/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584554765.Mortens-MacBook-Pro.local differ diff --git a/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584555636.Mortens-MacBook-Pro.local b/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584555636.Mortens-MacBook-Pro.local new file mode 100644 index 000000000..f99ee9187 Binary files /dev/null and b/doc/pub/NeuralNet/ipynb/logs/events.out.tfevents.1584555636.Mortens-MacBook-Pro.local differ diff --git a/doc/pub/Statistics/html/Statistics.do.txt-bs.html b/doc/pub/Statistics/html/Statistics.do.txt-bs.html new file mode 100644 index 000000000..b8cde1a33 --- /dev/null +++ b/doc/pub/Statistics/html/Statistics.do.txt-bs.html @@ -0,0 +1,459 @@ + + + + + + + + +Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ + Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis +
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Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Sep 20, 2020

+
+

+ + +

Read »

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+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/Statistics/html/Statistics.do.txt-reveal.html b/doc/pub/Statistics/html/Statistics.do.txt-reveal.html new file mode 100644 index 000000000..6b9ebfaee --- /dev/null +++ b/doc/pub/Statistics/html/Statistics.do.txt-reveal.html @@ -0,0 +1,2713 @@ + + + + + + + +Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + + + + + + + + + + +
+ + + + +

Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

 
+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

 
+

Sep 20, 2020

+
+

+ +

+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+
+ + +
+

To do list

+ +
    +

  • add math about MVN and define MLE and other quantities
    • +

  • rewrite about covariance matrix
    • +

  • add KL theorem
    • +
+
+ + +
+

Domains and probabilities

+
+ +

+Consider the following simple example, namely the tossing of two dice, resulting in the following possible values +

 
+$$ +\begin{equation*} +\{2,3,4,5,6,7,8,9,10,11,12\}. +\end{equation*} +$$ +

 
+ +These values are called the domain. +To this domain we have the corresponding probabilities +

 
+$$ +\begin{equation*} +\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\}. +\end{equation*} +$$ +

 
+

+
+ + +
+

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 +

 
+$$ +\begin{equation*} +\{2,3,4,5,6,7,8,9,10,11,12\}, +\end{equation*} +$$ +

 
+ +appear in a random order. After 11 throws the results may look like + +

 
+$$ +\begin{equation*} +\{10,8,6,3,6,9,11,8,12,4,5\}. +\end{equation*} +$$ +

 
+

+
+ + +
+

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 +

 
+$$ +\begin{equation*} +p(x) = \mathrm{Prob}(X=x). +\end{equation*} +$$ +

 
+

+
+ + +
+

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 + +

 
+$$ +\begin{equation*} +\mathrm{Prob}(a\leq X\leq b) = \int_a^b p(x)dx. +\end{equation*} +$$ +

 
+ +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 \( x \) +

 
+$$ +\begin{equation*} +P(x)=\mathrm{Prob(}X\leq x\mathrm{)} = +\int_{-\infty}^x p(x^{\prime})dx^{\prime}. +\end{equation*} +$$ +

 
+ +The relation between a CDF and its corresponding PDF is then + +

 
+$$ +\begin{equation*} +p(x) = \frac{d}{dx}P(x). +\end{equation*} +$$ +

 
+

+
+ + +
+

Properties of PDFs

+
+ +

+There are two properties that all PDFs must satisfy. The first one is +positivity (assuming that the PDF is normalized) + +

 
+$$ +\begin{equation*} +0 \leq p(x) \leq 1. +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation} +p(x) = \frac{1}{b-a}\theta(x-a)\theta(b-x). +\tag{1} +\end{equation} +$$ +

 
+ +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 +

 
+$$ +\begin{equation*} +p(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{(x-\mu)^2}{2\sigma^2})}, +\end{equation*} +$$ +

 
+ +with mean value \( \mu \) and standard deviation \( \sigma \). If \( \mu=0 \) and \( \sigma=1 \), it is normally called the standard normal distribution +

 
+$$ +\begin{equation*} +p(x) = \frac{1}{\sqrt{2\pi}} \exp{(-\frac{x^2}{2})}, +\end{equation*} +$$ +

 
+ +

+The following simple Python code plots the above distribution for different values of \( \mu \) and \( \sigma \). +

+ + +

+ +
+
+ + +
+

Exponential distribution

+
+ +

+Another important distribution in science is the exponential distribution +

 
+$$ +\begin{equation*} +p(x) = \alpha\exp{-(\alpha x)}. +\end{equation*} +$$ +

 
+

+
+ + +
+

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 + +

 
+$$ +\begin{equation} +\langle h \rangle_X \equiv \int\! h(x)p(x)\,dx +\tag{2} +\end{equation} +$$ +

 
+ +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 +

 
+$$ +\begin{equation*} +\langle x^n \rangle \equiv \int\! x^n p(x)\,dx +\end{equation*} +$$ +

 
+

+
+ + +
+

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 \( \mu \) +

 
+$$ +\begin{equation*} +\langle x\rangle = \mu \equiv \int x p(x)dx, +\end{equation*} +$$ +

 
+ +for a continuous distribution and +

 
+$$ +\begin{equation*} +\langle x\rangle = \mu \equiv \sum_{i=1}^N x_i p(x_i), +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation*} +\langle (x-\langle x\rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx +\end{equation*} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation*} + \langle x^k\rangle=\sum_{i=1}^{N}x_i^kp(x_i), +\end{equation*} +$$ +

 
+ +if we have a discrete PDF or + +

 
+$$ +\begin{equation*} + \langle x^k\rangle=\int_a^b x^kp(x)dx, +\end{equation*} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation*} +p(x)=\frac{1}{b-a}\Theta(x-a)\Theta(b-x), +\end{equation*} +$$ +

 
+ +yielding probabilities different from zero in the interval \( [a,b] \). + +

+The exponential distribution +

 
+$$ +\begin{equation*} +p(x)=\alpha \exp{(-\alpha x)}, +\end{equation*} +$$ +

 
+ +yielding probabilities different from zero in the interval \( [0,\infty) \) and with mean value +

 
+$$ +\begin{equation*} +\mu = \int_0^{\infty}xp(x)dx=\int_0^{\infty}x\alpha \exp{(-\alpha x)}dx=\frac{1}{\alpha}, +\end{equation*} +$$ +

 
+

+ +with variance +

 
+$$ +\begin{equation*} +\sigma^2=\int_0^{\infty}x^2p(x)dx-\mu^2 = \frac{1}{\alpha^2}. +\end{equation*} +$$ +

 
+

+ + +
+

Probability Distribution Functions, the normal distribution

+
+ +

+Finally, we have the so-called univariate normal distribution, or just the normal distribution +

 
+$$ +\begin{equation*} +p(x)=\frac{1}{b\sqrt{2\pi}}\exp{\left(-\frac{(x-a)^2}{2b^2}\right)} +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation*} + \mu = \int_0^{\infty}xp(x)dx=\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}x \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, +\end{equation*} +$$ +

 
+ +which becomes with a suitable change of variables +

 
+$$ +\begin{equation*} + \mu =\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(a+b\sqrt{2}y)\exp{-y^2}dy=a. +\end{equation*} +$$ +

 
+

+
+ + +
+

Probability Distribution Functions, the normal distribution

+
+ +

+Similarly, the variance becomes +

 
+$$ +\begin{equation*} + \sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}(x-\mu)^2 \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, +\end{equation*} +$$ +

 
+ +and inserting the mean value and performing a variable change we obtain + +

 
+$$ +\begin{equation*} + \sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(b\sqrt{2}y)^2\exp{\left(-y^2\right)}dy= +\frac{2b^2}{\sqrt{\pi}}\int_{-\infty}^{\infty}y^2\exp{\left(-y^2\right)}dy, +\end{equation*} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation} + p(x)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right)}. +\tag{3} +\end{equation} +$$ +

 
+

+
+ + +
+

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 + +

 
+$$ +\begin{equation*} +P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x\exp{\left(-\frac{t^2}{2}\right)}dt, +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation*} + p(x) = \left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} \hspace{0.5cm}x=0,1,\dots,n, +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation*} + (a+b)^m=\sum_{n=0}^m \left(\begin{array}{c} m \\ n\end{array}\right)a^nb^{m-n}, +\end{equation*} +$$ +

 
+ +which can be used to show that + +

 
+$$ +\begin{equation*} +\sum_{x=0}^n\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1, +\end{equation*} +$$ +

 
+ +the PDF is normalized to one. +The mean value is +

 
+$$ +\begin{equation*} +\mu = \sum_{x=0}^n x\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = +\sum_{x=0}^n x\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x}, +\end{equation*} +$$ +

 
+ +resulting in +

 
+$$ +\begin{equation*} +\mu = +\sum_{x=0}^n x\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)}, +\end{equation*} +$$ +

 
+ +which we rewrite as + +

 
+$$ +\begin{equation*} +\mu=ny\sum_{\nu=0}^n\left(\begin{array}{c} n-1 \\ \nu\end{array}\right)y^{\nu}(1-y)^{n-1-\nu} =ny(y+1-y)^{n-1}=ny. +\end{equation*} +$$ +

 
+

+ +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 +

 
+$$ +\begin{equation*} + p(x) = \frac{\lambda^x}{x!} e^{-\lambda} \hspace{0.5cm}x=0,1,\dots,;\lambda > 0. +\end{equation*} +$$ +

 
+ +In this case both the mean value and the variance are easier to calculate, + +

 
+$$ +\begin{equation*} +\mu = \sum_{x=0}^{\infty} x \frac{\lambda^x}{x!} e^{-\lambda} = \lambda e^{-\lambda}\sum_{x=1}^{\infty} +\frac{\lambda^{x-1}}{(x-1)!}=\lambda, +\end{equation*} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation*} +\lim_{n\rightarrow \infty}\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} e^{-\lambda}=\sum_{x=1}^{\infty}\frac{\lambda^x}{x!} e^{-\lambda}. +\end{equation*} +$$ +

 
+

+
+ + +
+

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 + +

 
+$$ +\begin{align} +\mathrm{Cov}(X_i,\,X_j) & = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle +\tag{4}\\ +&=\int\cdots\int (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)P(x_1,\dots,x_n)\,dx_1\dots dx_n, +\tag{5} +\end{align} +$$ +

 
+ +with +

 
+$$ +\begin{equation*} +\langle x_i\rangle = +\int\cdots\int x_i P(x_1,\dots,x_n)\,dx_1\dots dx_n. +\end{equation*} +$$ +

 
+

+
+ + +
+

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 +

 
+$$ +\begin{equation*} +\{x_1, x_2,\dots\,x_k,\dots\}. +\end{equation*} +$$ +

 
+

+
+ + +
+

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 +

 
+$$ +\begin{equation*} +\langle x_{\alpha} \rangle \equiv \frac{1}{n}\sum_{k=1}^n x_{\alpha,k}. +\end{equation*} +$$ +

 
+ +The sample variance is: +

 
+$$ +\begin{equation*} +\mathrm{Var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_{\alpha,k} - \langle x_{\alpha} \rangle)^2, +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation} +\langle X_m \rangle= \frac{1}{m}\sum_{\alpha=1}^mx_{\alpha}=\frac{1}{mn}\sum_{\alpha, k} x_{\alpha,k}, +\tag{6} +\end{equation} +$$ +

 
+ +where the last sums end at \( m \) and \( n \). +The total variance is +

 
+$$ +\begin{equation*} +\sigma^2_m= \frac{1}{mn^2}\sum_{\alpha=1}^m(\langle x_{\alpha} \rangle-\langle X_m \rangle)^2, +\end{equation*} +$$ +

 
+ +which we rewrite as +

 
+$$ +\begin{equation} +\sigma^2_m=\frac{1}{m}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle). +\tag{7} +\end{equation} +$$ +

 
+

+
+ + +
+

Numerical experiments and the covariance, the sample variance

+
+ +

+We define also the sample variance \( \sigma^2 \) of all \( mn \) individual experiments as +

 
+$$ +\begin{equation} +\sigma^2=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n (x_{\alpha,k}-\langle X_m \rangle)^2. +\tag{8} +\end{equation} +$$ +

 
+ +

+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 +

 
+$$ +\begin{equation*} + \sigma_m= +\frac{\sigma}{\sqrt{m}}. +\end{equation*} +$$ +

 
+ +

+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 \( \langle X_m \rangle \) + +

 
+$$ +\begin{equation*} +\mu_{X}^{\phantom X} \approx X_m=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n x_{\alpha,k}. +\end{equation*} +$$ +

 
+ +

+We can then use Eq. (7) +

 
+$$ +\begin{equation*} +\sigma^2_m=\frac{1}{mn^2}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), +\end{equation*} +$$ +

 
+ +and rewrite it as +

 
+$$ +\begin{equation*} +\sigma^2_m=\frac{\sigma^2}{n}+\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k < l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), +\end{equation*} +$$ +

 
+ +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 \( k\ne l \). +

+
+ + +
+

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. +

+
+ + +
+

Definition of Correlation Functions and Standard Deviation

+
+ +

+Let us analyze the problem by splitting up the correlation term into +partial sums of the form + +

 
+$$ +\begin{equation*} +f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), +\end{equation*} +$$ +

 
+ +The correlation term of the total variance can now be rewritten in terms of +\( f_d \) + +

 
+$$ +\begin{equation*} +\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k < l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle)= +\frac{2}{n}\sum_{d=1}^{n-1} f_d +\end{equation*} +$$ +

 
+

+
+ + +
+

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 + +

 
+$$ +\begin{equation} +\kappa_d = \frac{f_d}{\sigma^2} +\tag{9} +\end{equation} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation} +\sigma_m^2=\frac{\sigma^2}{n}+\frac{2}{n}\cdot\sigma^2\sum_{d=1}^{n-1} +\frac{f_d}{\sigma^2}=\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\sigma^2=\frac{\tau}{n}\cdot\sigma^2 +\tag{10} +\end{equation} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation} +\tau = 1+2\sum_{d=1}^{n-1}\kappa_d +\tag{11} +\end{equation} +$$ +

 
+ + + +For a correlation free experiment, \( \tau \) +equals 1. +

+
+ + +
+

Definition of Correlation Functions and Standard Deviation

+
+ +

+From the point of view of +Eq. (10) 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 +

 
+$$ +\begin{equation*} +n_\mathrm{eff} = \frac{n}{\tau} +\end{equation*} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation*} + N_i=(aN_{i-1}+c) \mathrm{MOD} (M), +\end{equation*} +$$ +

 
+ +which yield a number in the interval [0,1] through + +

 
+$$ +\begin{equation*} + x_i=N_i/M +\end{equation*} +$$ +

 
+ +

+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 + +

 
+$$ +\begin{equation*} + N_i=(6N_{i-1}+7) \mathrm{MOD} (5), +\end{equation*} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation*} + N_i=(27N_{i-1}+11) \mathrm{MOD} (54), +\end{equation*} +$$ +

 
+ +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 \( M \), +

 
+$$ +\begin{equation*} + N_l=(aN_{l-i}+cN_{l-j})\mathrm{MOD}(M). +\end{equation*} +$$ +

 
+

+
+ + +
+

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 + +

 
+$$ +\begin{equation} + N_l=(N_{l-3}-N_{l-1})\mathrm{MOD}(2^{31}-69), +\tag{12} +\end{equation} +$$ +

 
+ +followed by +

 
+$$ +\begin{equation} + N_l=(69069N_{l-1}+1013904243)\mathrm{MOD}(2^{32}), +\tag{13} +\end{equation} +$$ +

 
+ +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 + +

 
+$$ +\begin{equation*} + N_l=(N_{l-i})\oplus (N_{l-j}) +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation*} + N_i=(aN_{i-1}) \mathrm{MOD} (M). +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation*} + M=aq+r, +\end{equation*} +$$ +

 
+ +where we have defined + +

 
+$$ +\begin{equation*} + q=[M/a], +\end{equation*} +$$ +

 
+ +and +

 
+$$ +\begin{equation*} + r = M\hspace{0.1cm}\mathrm{MOD} \hspace{0.1cm}a. +\end{equation*} +$$ +

 
+ +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 +

 
+$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\mathrm{MOD} (M), +\tag{14} +\end{equation} +$$ +

 
+ +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) as + +

 
+$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\mathrm{MOD} (M), +\tag{15} +\end{equation} +$$ +

 
+ +which results +in + +

 
+$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\right)\mathrm{MOD} (M), +\tag{16} +\end{equation} +$$ +

 
+ +yielding +

 
+$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}\mathrm{MOD} (q)) -[N_{i-1}/q]r)\right)\mathrm{MOD} (M). +\tag{17} +\end{equation} +$$ +

 
+

+
+ + +
+

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 + +

 
+$$ +\begin{equation*} + \mu=\langle x\rangle=\frac{1}{2} +\end{equation*} +$$ +

 
+ +while the standard deviation is + +

 
+$$ +\begin{equation*} + \sigma=\sqrt{\langle x^2\rangle-\mu^2}=\frac{1}{\sqrt{12}}=0.2886. +\end{equation*} +$$ +

 
+

+
+ + +
+

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. +

+ + +

+ +
+
+ + +
+

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 +

 
+$$ +\begin{equation*} + C_k=\frac{f_d} + {\sigma^2}, +\end{equation*} +$$ +

 
+ +with \( C_0=1 \). Recall that +\( \sigma^2=\langle x_i^2\rangle-\langle x_i\rangle^2 \) and that +

 
+$$ +\begin{equation*} +f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), +\end{equation*} +$$ +

 
+ +

+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 <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 
+
+ +
+
+ + +
+

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
    • +
+
+
+ + +
+

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);
+
+ +
+
+ + +
+

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). + + +

+
+ + +
+

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 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 
+
+
+ + +
+

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 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. +

  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. +

+ + +
+

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

+

+ + +

# 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

+

+ + +

# 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)
+
+
+ + + +
+
+ + + + + + + + + + + + diff --git a/doc/pub/Statistics/html/Statistics.do.txt-solarized.html b/doc/pub/Statistics/html/Statistics.do.txt-solarized.html new file mode 100644 index 000000000..d6af08719 --- /dev/null +++ b/doc/pub/Statistics/html/Statistics.do.txt-solarized.html @@ -0,0 +1,2633 @@ + + + + + + + + +Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Sep 20, 2020

+
+

+









+ +

To do list

+ + + +









+ +

Domains and probabilities

+
+ +

+Consider the following simple example, namely the tossing of two dice, resulting in the following possible values +$$ +\begin{equation*} +\{2,3,4,5,6,7,8,9,10,11,12\}. +\end{equation*} +$$ + +These values are called the domain. +To this domain we have the corresponding probabilities +$$ +\begin{equation*} +\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\}. +\end{equation*} +$$ +

+ + +

+









+ +

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 +$$ +\begin{equation*} +\{2,3,4,5,6,7,8,9,10,11,12\}, +\end{equation*} +$$ + +appear in a random order. After 11 throws the results may look like + +$$ +\begin{equation*} +\{10,8,6,3,6,9,11,8,12,4,5\}. +\end{equation*} +$$ +

+ + +

+









+ +

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 +$$ +\begin{equation*} +p(x) = \mathrm{Prob}(X=x). +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation*} +\mathrm{Prob}(a\leq X\leq b) = \int_a^b p(x)dx. +\end{equation*} +$$ + +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 \( x \) +$$ +\begin{equation*} +P(x)=\mathrm{Prob(}X\leq x\mathrm{)} = +\int_{-\infty}^x p(x^{\prime})dx^{\prime}. +\end{equation*} +$$ + +The relation between a CDF and its corresponding PDF is then + +$$ +\begin{equation*} +p(x) = \frac{d}{dx}P(x). +\end{equation*} +$$ +

+ + +

+









+ +

Properties of PDFs

+
+ +

+ +

+There are two properties that all PDFs must satisfy. The first one is +positivity (assuming that the PDF is normalized) + +$$ +\begin{equation*} +0 \leq p(x) \leq 1. +\end{equation*} +$$ + +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 +$$ +\begin{equation} +p(x) = \frac{1}{b-a}\theta(x-a)\theta(b-x). +\label{eq:unifromPDF} +\end{equation} +$$ + +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 +$$ +\begin{equation*} +p(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{(x-\mu)^2}{2\sigma^2})}, +\end{equation*} +$$ + +with mean value \( \mu \) and standard deviation \( \sigma \). If \( \mu=0 \) and \( \sigma=1 \), it is normally called the standard normal distribution +$$ +\begin{equation*} +p(x) = \frac{1}{\sqrt{2\pi}} \exp{(-\frac{x^2}{2})}, +\end{equation*} +$$ + +

+The following simple Python code plots the above distribution for different values of \( \mu \) and \( \sigma \). +

+ + +

+ +
+ + +

+









+ +

Exponential distribution

+
+ +

+Another important distribution in science is the exponential distribution +$$ +\begin{equation*} +p(x) = \alpha\exp{-(\alpha x)}. +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation} +\langle h \rangle_X \equiv \int\! h(x)p(x)\,dx +\label{eq:expectation_value_of_h_wrt_p} +\end{equation} +$$ + +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 +$$ +\begin{equation*} +\langle x^n \rangle \equiv \int\! x^n p(x)\,dx +\end{equation*} +$$ +

+ + +

+









+ +

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 \( \mu \) +$$ +\begin{equation*} +\langle x\rangle = \mu \equiv \int x p(x)dx, +\end{equation*} +$$ + +for a continuous distribution and +$$ +\begin{equation*} +\langle x\rangle = \mu \equiv \sum_{i=1}^N x_i p(x_i), +\end{equation*} +$$ + +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 +$$ +\begin{equation*} +\langle (x-\langle x\rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx +\end{equation*} +$$ + +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 + +$$ +\begin{equation*} + \langle x^k\rangle=\sum_{i=1}^{N}x_i^kp(x_i), +\end{equation*} +$$ + +if we have a discrete PDF or + +$$ +\begin{equation*} + \langle x^k\rangle=\int_a^b x^kp(x)dx, +\end{equation*} +$$ + +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 +$$ +\begin{equation*} +p(x)=\frac{1}{b-a}\Theta(x-a)\Theta(b-x), +\end{equation*} +$$ + +yielding probabilities different from zero in the interval \( [a,b] \). + +

+The exponential distribution +$$ +\begin{equation*} +p(x)=\alpha \exp{(-\alpha x)}, +\end{equation*} +$$ + +yielding probabilities different from zero in the interval \( [0,\infty) \) and with mean value +$$ +\begin{equation*} +\mu = \int_0^{\infty}xp(x)dx=\int_0^{\infty}x\alpha \exp{(-\alpha x)}dx=\frac{1}{\alpha}, +\end{equation*} +$$ +

+ +with variance +$$ +\begin{equation*} +\sigma^2=\int_0^{\infty}x^2p(x)dx-\mu^2 = \frac{1}{\alpha^2}. +\end{equation*} +$$ + +

+









+ +

Probability Distribution Functions, the normal distribution

+
+ +

+Finally, we have the so-called univariate normal distribution, or just the normal distribution +$$ +\begin{equation*} +p(x)=\frac{1}{b\sqrt{2\pi}}\exp{\left(-\frac{(x-a)^2}{2b^2}\right)} +\end{equation*} +$$ + +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 +$$ +\begin{equation*} + \mu = \int_0^{\infty}xp(x)dx=\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}x \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, +\end{equation*} +$$ + +which becomes with a suitable change of variables +$$ +\begin{equation*} + \mu =\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(a+b\sqrt{2}y)\exp{-y^2}dy=a. +\end{equation*} +$$ +

+ + +

+









+ +

Probability Distribution Functions, the normal distribution

+
+ +

+Similarly, the variance becomes +$$ +\begin{equation*} + \sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}(x-\mu)^2 \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, +\end{equation*} +$$ + +and inserting the mean value and performing a variable change we obtain + +$$ +\begin{equation*} + \sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(b\sqrt{2}y)^2\exp{\left(-y^2\right)}dy= +\frac{2b^2}{\sqrt{\pi}}\int_{-\infty}^{\infty}y^2\exp{\left(-y^2\right)}dy, +\end{equation*} +$$ + +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 + +$$ +\begin{equation} + p(x)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right)}. +\label{_auto1} +\end{equation} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation*} +P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x\exp{\left(-\frac{t^2}{2}\right)}dt, +\end{equation*} +$$ + +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 +$$ +\begin{equation*} + p(x) = \left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} \hspace{0.5cm}x=0,1,\dots,n, +\end{equation*} +$$ + +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 + +

+
+ + +

+









+ +

Probability Distribution Functions, the binomial distribution

+
+ +

+ +

+In order to compute the mean and variance we need to recall Newton's binomial +formula +$$ +\begin{equation*} + (a+b)^m=\sum_{n=0}^m \left(\begin{array}{c} m \\ n\end{array}\right)a^nb^{m-n}, +\end{equation*} +$$ + +which can be used to show that + +$$ +\begin{equation*} +\sum_{x=0}^n\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1, +\end{equation*} +$$ + +the PDF is normalized to one. +The mean value is +$$ +\begin{equation*} +\mu = \sum_{x=0}^n x\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = +\sum_{x=0}^n x\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x}, +\end{equation*} +$$ + +resulting in +$$ +\begin{equation*} +\mu = +\sum_{x=0}^n x\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)}, +\end{equation*} +$$ + +which we rewrite as + +$$ +\begin{equation*} +\mu=ny\sum_{\nu=0}^n\left(\begin{array}{c} n-1 \\ \nu\end{array}\right)y^{\nu}(1-y)^{n-1-\nu} =ny(y+1-y)^{n-1}=ny. +\end{equation*} +$$ +

+ +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 +$$ +\begin{equation*} + p(x) = \frac{\lambda^x}{x!} e^{-\lambda} \hspace{0.5cm}x=0,1,\dots,;\lambda > 0. +\end{equation*} +$$ + +In this case both the mean value and the variance are easier to calculate, + +$$ +\begin{equation*} +\mu = \sum_{x=0}^{\infty} x \frac{\lambda^x}{x!} e^{-\lambda} = \lambda e^{-\lambda}\sum_{x=1}^{\infty} +\frac{\lambda^{x-1}}{(x-1)!}=\lambda, +\end{equation*} +$$ + +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 + +$$ +\begin{equation*} +\lim_{n\rightarrow \infty}\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} e^{-\lambda}=\sum_{x=1}^{\infty}\frac{\lambda^x}{x!} e^{-\lambda}. +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{align} +\mathrm{Cov}(X_i,\,X_j) & = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle +\label{_auto2}\\ +&=\int\cdots\int (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)P(x_1,\dots,x_n)\,dx_1\dots dx_n, +\label{eq:def_covariance} +\end{align} +$$ + +with +$$ +\begin{equation*} +\langle x_i\rangle = +\int\cdots\int x_i P(x_1,\dots,x_n)\,dx_1\dots dx_n. +\end{equation*} +$$ +

+ + +

+









+ +

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 +$$ +\begin{equation*} +\{x_1, x_2,\dots\,x_k,\dots\}. +\end{equation*} +$$ +

+ + +

+









+ +

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 +$$ +\begin{equation*} +\langle x_{\alpha} \rangle \equiv \frac{1}{n}\sum_{k=1}^n x_{\alpha,k}. +\end{equation*} +$$ + +The sample variance is: +$$ +\begin{equation*} +\mathrm{Var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_{\alpha,k} - \langle x_{\alpha} \rangle)^2, +\end{equation*} +$$ + +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 +$$ +\begin{equation} +\langle X_m \rangle= \frac{1}{m}\sum_{\alpha=1}^mx_{\alpha}=\frac{1}{mn}\sum_{\alpha, k} x_{\alpha,k}, +\label{eq:exptmean} +\end{equation} +$$ + +where the last sums end at \( m \) and \( n \). +The total variance is +$$ +\begin{equation*} +\sigma^2_m= \frac{1}{mn^2}\sum_{\alpha=1}^m(\langle x_{\alpha} \rangle-\langle X_m \rangle)^2, +\end{equation*} +$$ + +which we rewrite as +$$ +\begin{equation} +\sigma^2_m=\frac{1}{m}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle). +\label{eq:exptvariance} +\end{equation} +$$ +

+ + +

+









+ +

Numerical experiments and the covariance, the sample variance

+
+ +

+ +

+We define also the sample variance \( \sigma^2 \) of all \( mn \) individual experiments as +$$ +\begin{equation} +\sigma^2=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n (x_{\alpha,k}-\langle X_m \rangle)^2. +\label{eq:sampleexptvariance} +\end{equation} +$$ + +

+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 +$$ +\begin{equation*} + \sigma_m= +\frac{\sigma}{\sqrt{m}}. +\end{equation*} +$$ + +

+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 \( \langle X_m \rangle \) + +$$ +\begin{equation*} +\mu_{X}^{\phantom X} \approx X_m=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n x_{\alpha,k}. +\end{equation*} +$$ + +

+We can then use Eq. \eqref{eq:exptvariance} +$$ +\begin{equation*} +\sigma^2_m=\frac{1}{mn^2}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), +\end{equation*} +$$ + +and rewrite it as +$$ +\begin{equation*} +\sigma^2_m=\frac{\sigma^2}{n}+\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k < l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), +\end{equation*} +$$ + +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 \( k\ne l \). +

+ + +

+









+ +

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. +

+ + +

+









+ +

Definition of Correlation Functions and Standard Deviation

+
+ +

+ +

+Let us analyze the problem by splitting up the correlation term into +partial sums of the form + +$$ +\begin{equation*} +f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), +\end{equation*} +$$ + +The correlation term of the total variance can now be rewritten in terms of +\( f_d \) + +$$ +\begin{equation*} +\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k < l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle)= +\frac{2}{n}\sum_{d=1}^{n-1} f_d +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation} +\kappa_d = \frac{f_d}{\sigma^2} +\label{eq:autocorrelformal} +\end{equation} +$$ + +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 + +$$ +\begin{equation} +\sigma_m^2=\frac{\sigma^2}{n}+\frac{2}{n}\cdot\sigma^2\sum_{d=1}^{n-1} +\frac{f_d}{\sigma^2}=\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\sigma^2=\frac{\tau}{n}\cdot\sigma^2 +\label{eq:error_estimate_corr_time} +\end{equation} +$$ + +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 + +$$ +\begin{equation} +\tau = 1+2\sum_{d=1}^{n-1}\kappa_d +\label{eq:autocorrelation_time} +\end{equation} +$$ + + + +For a correlation free experiment, \( \tau \) +equals 1. +

+ + +

+









+ +

Definition of Correlation Functions and Standard Deviation

+
+ +

+From the point of view of +Eq. \eqref{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 +$$ +\begin{equation*} +n_\mathrm{eff} = \frac{n}{\tau} +\end{equation*} +$$ + +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: + +

+
+ + +

+









+ +

Random number generator RNG

+
+ +

+ The most common random number generators are based on so-called +Linear congruential relations of the type + +$$ +\begin{equation*} + N_i=(aN_{i-1}+c) \mathrm{MOD} (M), +\end{equation*} +$$ + +which yield a number in the interval [0,1] through + +$$ +\begin{equation*} + x_i=N_i/M +\end{equation*} +$$ + +

+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 + +$$ +\begin{equation*} + N_i=(6N_{i-1}+7) \mathrm{MOD} (5), +\end{equation*} +$$ + +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 + +$$ +\begin{equation*} + N_i=(27N_{i-1}+11) \mathrm{MOD} (54), +\end{equation*} +$$ + +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 \( M \), +$$ +\begin{equation*} + N_l=(aN_{l-i}+cN_{l-j})\mathrm{MOD}(M). +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation} + N_l=(N_{l-3}-N_{l-1})\mathrm{MOD}(2^{31}-69), +\label{eq:mz1} +\end{equation} +$$ + +followed by +$$ +\begin{equation} + N_l=(69069N_{l-1}+1013904243)\mathrm{MOD}(2^{32}), +\label{eq:mz2} +\end{equation} +$$ + +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 + +$$ +\begin{equation*} + N_l=(N_{l-i})\oplus (N_{l-j}) +\end{equation*} +$$ + +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 +$$ +\begin{equation*} + N_i=(aN_{i-1}) \mathrm{MOD} (M). +\end{equation*} +$$ + +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 +$$ +\begin{equation*} + M=aq+r, +\end{equation*} +$$ + +where we have defined + +$$ +\begin{equation*} + q=[M/a], +\end{equation*} +$$ + +and +$$ +\begin{equation*} + r = M\hspace{0.1cm}\mathrm{MOD} \hspace{0.1cm}a. +\end{equation*} +$$ + +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 +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\mathrm{MOD} (M), +\label{eq:rntrick1} +\end{equation} +$$ + +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. \eqref{eq:rntrick1} as + +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\mathrm{MOD} (M), +\label{eq:rntrick2} +\end{equation} +$$ + +which results +in + +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\right)\mathrm{MOD} (M), +\label{eq:rntrick3} +\end{equation} +$$ + +yielding +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}\mathrm{MOD} (q)) -[N_{i-1}/q]r)\right)\mathrm{MOD} (M). +\label{eq:rntrick4} +\end{equation} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation*} + \mu=\langle x\rangle=\frac{1}{2} +\end{equation*} +$$ + +while the standard deviation is + +$$ +\begin{equation*} + \sigma=\sqrt{\langle x^2\rangle-\mu^2}=\frac{1}{\sqrt{12}}=0.2886. +\end{equation*} +$$ +

+ + +

+









+ +

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. +

+ + +

+ +
+ + +

+









+ +

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. \eqref{eq:autocorrelformal}. +We rewrite it here as +$$ +\begin{equation*} + C_k=\frac{f_d} + {\sigma^2}, +\end{equation*} +$$ + +with \( C_0=1 \). Recall that +\( \sigma^2=\langle x_i^2\rangle-\langle x_i\rangle^2 \) and that +$$ +\begin{equation*} +f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), +\end{equation*} +$$ + +

+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 <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 
+
+ +
+ + +

+









+ +

Which RNG should I use?

+
+ +

+ +

+
+ + +

+









+ +

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);
+
+ +
+ + +

+









+ +

Why blocking?

+
+Statistical analysis. +

+ +

+ +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). + + +
+ + +

+









+ +

Why blocking?

+
+Statistical analysis. +

+ +

+
+ + +

+









+ +

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 
+
+

+









+ +

What is blocking?

+
+Blocking. +

+ +

+ +$$ +\sigma=\sqrt{\frac{1}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} +$$ + + + + +$$ +\sigma=\sqrt{\frac{1+2\tau/\Delta t}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} +$$ + + 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. +

+ +

+
+ + +

+









+ +

What is blocking?

+
+Blocking. +

+ +

+
+ + +

+









+ +

Implementation

+
+ +

+ +

+
+ + +

+









+ +

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)
+
+

+









+ +

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. + +

+ + +

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

+

+ + +

# 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

+

+ + +

# 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)
+
+

+ + + + +

+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/Statistics/html/Statistics.do.txt.html b/doc/pub/Statistics/html/Statistics.do.txt.html new file mode 100644 index 000000000..30b253550 --- /dev/null +++ b/doc/pub/Statistics/html/Statistics.do.txt.html @@ -0,0 +1,2638 @@ + + + + + + + + +Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Sep 20, 2020

+
+

+









+ +

To do list

+ + + +









+ +

Domains and probabilities

+
+ +

+Consider the following simple example, namely the tossing of two dice, resulting in the following possible values +$$ +\begin{equation*} +\{2,3,4,5,6,7,8,9,10,11,12\}. +\end{equation*} +$$ + +These values are called the domain. +To this domain we have the corresponding probabilities +$$ +\begin{equation*} +\{1/36,2/36/,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36\}. +\end{equation*} +$$ +

+ + +

+









+ +

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 +$$ +\begin{equation*} +\{2,3,4,5,6,7,8,9,10,11,12\}, +\end{equation*} +$$ + +appear in a random order. After 11 throws the results may look like + +$$ +\begin{equation*} +\{10,8,6,3,6,9,11,8,12,4,5\}. +\end{equation*} +$$ +

+ + +

+









+ +

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 +$$ +\begin{equation*} +p(x) = \mathrm{Prob}(X=x). +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation*} +\mathrm{Prob}(a\leq X\leq b) = \int_a^b p(x)dx. +\end{equation*} +$$ + +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 \( x \) +$$ +\begin{equation*} +P(x)=\mathrm{Prob(}X\leq x\mathrm{)} = +\int_{-\infty}^x p(x^{\prime})dx^{\prime}. +\end{equation*} +$$ + +The relation between a CDF and its corresponding PDF is then + +$$ +\begin{equation*} +p(x) = \frac{d}{dx}P(x). +\end{equation*} +$$ +

+ + +

+









+ +

Properties of PDFs

+
+ +

+ +

+There are two properties that all PDFs must satisfy. The first one is +positivity (assuming that the PDF is normalized) + +$$ +\begin{equation*} +0 \leq p(x) \leq 1. +\end{equation*} +$$ + +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 +$$ +\begin{equation} +p(x) = \frac{1}{b-a}\theta(x-a)\theta(b-x). +\label{eq:unifromPDF} +\end{equation} +$$ + +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 +$$ +\begin{equation*} +p(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{(x-\mu)^2}{2\sigma^2})}, +\end{equation*} +$$ + +with mean value \( \mu \) and standard deviation \( \sigma \). If \( \mu=0 \) and \( \sigma=1 \), it is normally called the standard normal distribution +$$ +\begin{equation*} +p(x) = \frac{1}{\sqrt{2\pi}} \exp{(-\frac{x^2}{2})}, +\end{equation*} +$$ + +

+The following simple Python code plots the above distribution for different values of \( \mu \) and \( \sigma \). +

+ + +

+ +
+ + +

+









+ +

Exponential distribution

+
+ +

+Another important distribution in science is the exponential distribution +$$ +\begin{equation*} +p(x) = \alpha\exp{-(\alpha x)}. +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation} +\langle h \rangle_X \equiv \int\! h(x)p(x)\,dx +\label{eq:expectation_value_of_h_wrt_p} +\end{equation} +$$ + +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 +$$ +\begin{equation*} +\langle x^n \rangle \equiv \int\! x^n p(x)\,dx +\end{equation*} +$$ +

+ + +

+









+ +

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 \( \mu \) +$$ +\begin{equation*} +\langle x\rangle = \mu \equiv \int x p(x)dx, +\end{equation*} +$$ + +for a continuous distribution and +$$ +\begin{equation*} +\langle x\rangle = \mu \equiv \sum_{i=1}^N x_i p(x_i), +\end{equation*} +$$ + +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 +$$ +\begin{equation*} +\langle (x-\langle x\rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx +\end{equation*} +$$ + +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 + +$$ +\begin{equation*} + \langle x^k\rangle=\sum_{i=1}^{N}x_i^kp(x_i), +\end{equation*} +$$ + +if we have a discrete PDF or + +$$ +\begin{equation*} + \langle x^k\rangle=\int_a^b x^kp(x)dx, +\end{equation*} +$$ + +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 +$$ +\begin{equation*} +p(x)=\frac{1}{b-a}\Theta(x-a)\Theta(b-x), +\end{equation*} +$$ + +yielding probabilities different from zero in the interval \( [a,b] \). + +

+The exponential distribution +$$ +\begin{equation*} +p(x)=\alpha \exp{(-\alpha x)}, +\end{equation*} +$$ + +yielding probabilities different from zero in the interval \( [0,\infty) \) and with mean value +$$ +\begin{equation*} +\mu = \int_0^{\infty}xp(x)dx=\int_0^{\infty}x\alpha \exp{(-\alpha x)}dx=\frac{1}{\alpha}, +\end{equation*} +$$ +

+ +with variance +$$ +\begin{equation*} +\sigma^2=\int_0^{\infty}x^2p(x)dx-\mu^2 = \frac{1}{\alpha^2}. +\end{equation*} +$$ + +

+









+ +

Probability Distribution Functions, the normal distribution

+
+ +

+Finally, we have the so-called univariate normal distribution, or just the normal distribution +$$ +\begin{equation*} +p(x)=\frac{1}{b\sqrt{2\pi}}\exp{\left(-\frac{(x-a)^2}{2b^2}\right)} +\end{equation*} +$$ + +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 +$$ +\begin{equation*} + \mu = \int_0^{\infty}xp(x)dx=\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}x \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, +\end{equation*} +$$ + +which becomes with a suitable change of variables +$$ +\begin{equation*} + \mu =\frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(a+b\sqrt{2}y)\exp{-y^2}dy=a. +\end{equation*} +$$ +

+ + +

+









+ +

Probability Distribution Functions, the normal distribution

+
+ +

+Similarly, the variance becomes +$$ +\begin{equation*} + \sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}(x-\mu)^2 \exp{\left(-\frac{(x-a)^2}{2b^2}\right)}dx, +\end{equation*} +$$ + +and inserting the mean value and performing a variable change we obtain + +$$ +\begin{equation*} + \sigma^2 = \frac{1}{b\sqrt{2\pi}}\int_{-\infty}^{\infty}b\sqrt{2}(b\sqrt{2}y)^2\exp{\left(-y^2\right)}dy= +\frac{2b^2}{\sqrt{\pi}}\int_{-\infty}^{\infty}y^2\exp{\left(-y^2\right)}dy, +\end{equation*} +$$ + +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 + +$$ +\begin{equation} + p(x)=\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\right)}. +\label{_auto1} +\end{equation} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation*} +P(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x\exp{\left(-\frac{t^2}{2}\right)}dt, +\end{equation*} +$$ + +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 +$$ +\begin{equation*} + p(x) = \left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} \hspace{0.5cm}x=0,1,\dots,n, +\end{equation*} +$$ + +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 + +

+
+ + +

+









+ +

Probability Distribution Functions, the binomial distribution

+
+ +

+ +

+In order to compute the mean and variance we need to recall Newton's binomial +formula +$$ +\begin{equation*} + (a+b)^m=\sum_{n=0}^m \left(\begin{array}{c} m \\ n\end{array}\right)a^nb^{m-n}, +\end{equation*} +$$ + +which can be used to show that + +$$ +\begin{equation*} +\sum_{x=0}^n\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = (y+1-y)^n = 1, +\end{equation*} +$$ + +the PDF is normalized to one. +The mean value is +$$ +\begin{equation*} +\mu = \sum_{x=0}^n x\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} = +\sum_{x=0}^n x\frac{n!}{x!(n-x)!}y^x(1-y)^{n-x}, +\end{equation*} +$$ + +resulting in +$$ +\begin{equation*} +\mu = +\sum_{x=0}^n x\frac{(n-1)!}{(x-1)!(n-1-(x-1))!}y^{x-1}(1-y)^{n-1-(x-1)}, +\end{equation*} +$$ + +which we rewrite as + +$$ +\begin{equation*} +\mu=ny\sum_{\nu=0}^n\left(\begin{array}{c} n-1 \\ \nu\end{array}\right)y^{\nu}(1-y)^{n-1-\nu} =ny(y+1-y)^{n-1}=ny. +\end{equation*} +$$ +

+ +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 +$$ +\begin{equation*} + p(x) = \frac{\lambda^x}{x!} e^{-\lambda} \hspace{0.5cm}x=0,1,\dots,;\lambda > 0. +\end{equation*} +$$ + +In this case both the mean value and the variance are easier to calculate, + +$$ +\begin{equation*} +\mu = \sum_{x=0}^{\infty} x \frac{\lambda^x}{x!} e^{-\lambda} = \lambda e^{-\lambda}\sum_{x=1}^{\infty} +\frac{\lambda^{x-1}}{(x-1)!}=\lambda, +\end{equation*} +$$ + +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 + +$$ +\begin{equation*} +\lim_{n\rightarrow \infty}\left(\begin{array}{c} n \\ x\end{array}\right)y^x(1-y)^{n-x} e^{-\lambda}=\sum_{x=1}^{\infty}\frac{\lambda^x}{x!} e^{-\lambda}. +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{align} +\mathrm{Cov}(X_i,\,X_j) & = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle +\label{_auto2}\\ +&=\int\cdots\int (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)P(x_1,\dots,x_n)\,dx_1\dots dx_n, +\label{eq:def_covariance} +\end{align} +$$ + +with +$$ +\begin{equation*} +\langle x_i\rangle = +\int\cdots\int x_i P(x_1,\dots,x_n)\,dx_1\dots dx_n. +\end{equation*} +$$ +

+ + +

+









+ +

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 +$$ +\begin{equation*} +\{x_1, x_2,\dots\,x_k,\dots\}. +\end{equation*} +$$ +

+ + +

+









+ +

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 +$$ +\begin{equation*} +\langle x_{\alpha} \rangle \equiv \frac{1}{n}\sum_{k=1}^n x_{\alpha,k}. +\end{equation*} +$$ + +The sample variance is: +$$ +\begin{equation*} +\mathrm{Var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_{\alpha,k} - \langle x_{\alpha} \rangle)^2, +\end{equation*} +$$ + +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 +$$ +\begin{equation} +\langle X_m \rangle= \frac{1}{m}\sum_{\alpha=1}^mx_{\alpha}=\frac{1}{mn}\sum_{\alpha, k} x_{\alpha,k}, +\label{eq:exptmean} +\end{equation} +$$ + +where the last sums end at \( m \) and \( n \). +The total variance is +$$ +\begin{equation*} +\sigma^2_m= \frac{1}{mn^2}\sum_{\alpha=1}^m(\langle x_{\alpha} \rangle-\langle X_m \rangle)^2, +\end{equation*} +$$ + +which we rewrite as +$$ +\begin{equation} +\sigma^2_m=\frac{1}{m}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle). +\label{eq:exptvariance} +\end{equation} +$$ +

+ + +

+









+ +

Numerical experiments and the covariance, the sample variance

+
+ +

+ +

+We define also the sample variance \( \sigma^2 \) of all \( mn \) individual experiments as +$$ +\begin{equation} +\sigma^2=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n (x_{\alpha,k}-\langle X_m \rangle)^2. +\label{eq:sampleexptvariance} +\end{equation} +$$ + +

+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 +$$ +\begin{equation*} + \sigma_m= +\frac{\sigma}{\sqrt{m}}. +\end{equation*} +$$ + +

+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 \( \langle X_m \rangle \) + +$$ +\begin{equation*} +\mu_{X}^{\phantom X} \approx X_m=\frac{1}{mn}\sum_{\alpha=1}^m\sum_{k=1}^n x_{\alpha,k}. +\end{equation*} +$$ + +

+We can then use Eq. \eqref{eq:exptvariance} +$$ +\begin{equation*} +\sigma^2_m=\frac{1}{mn^2}\sum_{\alpha=1}^m\sum_{kl=1}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), +\end{equation*} +$$ + +and rewrite it as +$$ +\begin{equation*} +\sigma^2_m=\frac{\sigma^2}{n}+\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k < l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle), +\end{equation*} +$$ + +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 \( k\ne l \). +

+ + +

+









+ +

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. +

+ + +

+









+ +

Definition of Correlation Functions and Standard Deviation

+
+ +

+ +

+Let us analyze the problem by splitting up the correlation term into +partial sums of the form + +$$ +\begin{equation*} +f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), +\end{equation*} +$$ + +The correlation term of the total variance can now be rewritten in terms of +\( f_d \) + +$$ +\begin{equation*} +\frac{2}{mn^2}\sum_{\alpha=1}^m\sum_{k < l}^n (x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,l}-\langle X_m \rangle)= +\frac{2}{n}\sum_{d=1}^{n-1} f_d +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation} +\kappa_d = \frac{f_d}{\sigma^2} +\label{eq:autocorrelformal} +\end{equation} +$$ + +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 + +$$ +\begin{equation} +\sigma_m^2=\frac{\sigma^2}{n}+\frac{2}{n}\cdot\sigma^2\sum_{d=1}^{n-1} +\frac{f_d}{\sigma^2}=\left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\sigma^2=\frac{\tau}{n}\cdot\sigma^2 +\label{eq:error_estimate_corr_time} +\end{equation} +$$ + +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 + +$$ +\begin{equation} +\tau = 1+2\sum_{d=1}^{n-1}\kappa_d +\label{eq:autocorrelation_time} +\end{equation} +$$ + + + +For a correlation free experiment, \( \tau \) +equals 1. +

+ + +

+









+ +

Definition of Correlation Functions and Standard Deviation

+
+ +

+From the point of view of +Eq. \eqref{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 +$$ +\begin{equation*} +n_\mathrm{eff} = \frac{n}{\tau} +\end{equation*} +$$ + +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: + +

+
+ + +

+









+ +

Random number generator RNG

+
+ +

+ The most common random number generators are based on so-called +Linear congruential relations of the type + +$$ +\begin{equation*} + N_i=(aN_{i-1}+c) \mathrm{MOD} (M), +\end{equation*} +$$ + +which yield a number in the interval [0,1] through + +$$ +\begin{equation*} + x_i=N_i/M +\end{equation*} +$$ + +

+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 + +$$ +\begin{equation*} + N_i=(6N_{i-1}+7) \mathrm{MOD} (5), +\end{equation*} +$$ + +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 + +$$ +\begin{equation*} + N_i=(27N_{i-1}+11) \mathrm{MOD} (54), +\end{equation*} +$$ + +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 \( M \), +$$ +\begin{equation*} + N_l=(aN_{l-i}+cN_{l-j})\mathrm{MOD}(M). +\end{equation*} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation} + N_l=(N_{l-3}-N_{l-1})\mathrm{MOD}(2^{31}-69), +\label{eq:mz1} +\end{equation} +$$ + +followed by +$$ +\begin{equation} + N_l=(69069N_{l-1}+1013904243)\mathrm{MOD}(2^{32}), +\label{eq:mz2} +\end{equation} +$$ + +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 + +$$ +\begin{equation*} + N_l=(N_{l-i})\oplus (N_{l-j}) +\end{equation*} +$$ + +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 +$$ +\begin{equation*} + N_i=(aN_{i-1}) \mathrm{MOD} (M). +\end{equation*} +$$ + +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 +$$ +\begin{equation*} + M=aq+r, +\end{equation*} +$$ + +where we have defined + +$$ +\begin{equation*} + q=[M/a], +\end{equation*} +$$ + +and +$$ +\begin{equation*} + r = M\hspace{0.1cm}\mathrm{MOD} \hspace{0.1cm}a. +\end{equation*} +$$ + +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 +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q]M)\mathrm{MOD} (M), +\label{eq:rntrick1} +\end{equation} +$$ + +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. \eqref{eq:rntrick1} as + +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= (aN_{i-1}-[N_{i-1}/q](aq+r))\mathrm{MOD} (M), +\label{eq:rntrick2} +\end{equation} +$$ + +which results +in + +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}-[N_{i-1}/q]q)-[N_{i-1}/q]r)\right)\mathrm{MOD} (M), +\label{eq:rntrick3} +\end{equation} +$$ + +yielding +$$ +\begin{equation} +(aN_{i-1}) \mathrm{MOD} (M)= \left(a(N_{i-1}\mathrm{MOD} (q)) -[N_{i-1}/q]r)\right)\mathrm{MOD} (M). +\label{eq:rntrick4} +\end{equation} +$$ +

+ + +

+









+ +

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 + +$$ +\begin{equation*} + \mu=\langle x\rangle=\frac{1}{2} +\end{equation*} +$$ + +while the standard deviation is + +$$ +\begin{equation*} + \sigma=\sqrt{\langle x^2\rangle-\mu^2}=\frac{1}{\sqrt{12}}=0.2886. +\end{equation*} +$$ +

+ + +

+









+ +

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. +

+ + +

+ +
+ + +

+









+ +

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. \eqref{eq:autocorrelformal}. +We rewrite it here as +$$ +\begin{equation*} + C_k=\frac{f_d} + {\sigma^2}, +\end{equation*} +$$ + +with \( C_0=1 \). Recall that +\( \sigma^2=\langle x_i^2\rangle-\langle x_i\rangle^2 \) and that +$$ +\begin{equation*} +f_d = \frac{1}{nm}\sum_{\alpha=1}^m\sum_{k=1}^{n-d}(x_{\alpha,k}-\langle X_m \rangle)(x_{\alpha,k+d}-\langle X_m \rangle), +\end{equation*} +$$ + +

+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 <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 
+
+ +
+ + +

+









+ +

Which RNG should I use?

+
+ +

+ +

+
+ + +

+









+ +

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);
+
+ +
+ + +

+









+ +

Why blocking?

+
+Statistical analysis. +

+ +

+ +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). + + +
+ + +

+









+ +

Why blocking?

+
+Statistical analysis. +

+ +

+
+ + +

+









+ +

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 
+
+

+









+ +

What is blocking?

+
+Blocking. +

+ +

+ +$$ +\sigma=\sqrt{\frac{1}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} +$$ + + + + +$$ +\sigma=\sqrt{\frac{1+2\tau/\Delta t}{n}\left(\langle \mathbf{M}^2\rangle-\langle \mathbf{M}\rangle^2\right)} +$$ + + 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. +

+ +

+
+ + +

+









+ +

What is blocking?

+
+Blocking. +

+ +

+
+ + +

+









+ +

Implementation

+
+ +

+ +

+
+ + +

+









+ +

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)
+
+

+









+ +

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. + +

+ + +

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

+

+ + +

# 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

+

+ + +

# 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)
+
+

+ + + + +

+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/Statistics/ipynb/Statistics.ipynb b/doc/pub/Statistics/ipynb/Statistics.ipynb new file mode 100644 index 000000000..9d3125d58 --- /dev/null +++ b/doc/pub/Statistics/ipynb/Statistics.ipynb @@ -0,0 +1,2855 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Data Analysis and Machine Learning: Elements of Probability Theory and Statistical Data Analysis\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Sep 20, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "## To do list\n", + "\n", + "* add math about MVN and define MLE and other quantities\n", + "\n", + "* rewrite about covariance matrix\n", + "\n", + "* add KL theorem\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", + "## 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", + "## 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", + "## 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 + }, + "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", + "## 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", + "## 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", + "## 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", + "## 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", + "## 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", + "## 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", + "## 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", + "## 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", + "## Covariance" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "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", + "## 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", + "## 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", + "## 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", + "## 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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which gives us a useful measure of the correlation pair correlation\n", + "starting always at $1$ for $d=0$.\n", + "\n", + "\n", + "\n", + "## Definition of Correlation Functions and Standard Deviation, sample variance\n", + "\n", + "The sample variance of the $mn$ experiments can now be\n", + "written in terms of the autocorrelation function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "\n", + "For a correlation free experiment, $\\tau$\n", + "equals 1.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Definition of Correlation Functions and Standard Deviation\n", + "From the point of view of\n", + "Eq. ([10](#eq:error_estimate_corr_time)) we can interpret a sequential\n", + "correlation as an effective reduction of the number of measurements by\n", + "a factor $\\tau$. The effective number of measurements becomes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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 + }, + "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", + "# 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", + "# 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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The number \n", + "$M$ is called the period and it should be as large as possible \n", + " and \n", + "$N_0$ is the starting value, or seed. The function $\\mathrm{MOD}$ means the remainder,\n", + "that is if we were to evaluate $(13)\\mathrm{MOD}(9)$, the outcome is the remainder\n", + "of the division $13/9$, namely $4$.\n", + "\n", + "\n", + "\n", + "# Random number generator RNG and periodic outputs\n", + "\n", + "The problem with such generators is that their outputs are periodic;\n", + "they \n", + "will start to repeat themselves with a period that is at most $M$. If however\n", + "the parameters $a$ and $c$ are badly chosen, the period may be even shorter.\n", + "\n", + "Consider the following example" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which still, with $N_0=2$, results in $11,38,11,38,11,38,\\dots$, a period of\n", + "just $2$.\n", + "\n", + "\n", + "\n", + "# Random number generator RNG and its period\n", + "Typical periods for the random generators provided in the program library \n", + "are of the order of $\\sim 10^9$ or larger. Other random number generators which have\n", + "become increasingly popular are so-called shift-register generators.\n", + "In these generators each successive number depends on many preceding\n", + "values (rather than the last values as in the linear congruential\n", + "generator).\n", + "For example, you could make a shift register generator whose $l$th \n", + "number is the sum of the $l-i$th and $l-j$th values with modulo $M$," + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which according to the authors has a period larger than $2^{94}$.\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, other examples\n", + "Instead of using modular addition, we could use the bitwise\n", + "exclusive-OR ($\\oplus$) operation so that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "N_l=(N_{l-i})\\oplus (N_{l-j})\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the bitwise action of $\\oplus$ means that if $N_{l-i}=N_{l-j}$ the result is\n", + "$0$ whereas if $N_{l-i}\\ne N_{l-j}$ the result is\n", + "$1$. As an example, consider the case where $N_{l-i}=6$ and $N_{l-j}=11$. The first\n", + "one has a bit representation (using 4 bits only) which reads $0110$ whereas the \n", + "second number is $1011$. Employing the $\\oplus$ operator yields \n", + "$1101$, or $2^3+2^2+2^0=13$.\n", + "\n", + "In Fortran90, the bitwise $\\oplus$ operation is coded through the intrinsic\n", + "function $\\mathrm{IEOR}(m,n)$ where $m$ and $n$ are the input numbers, while in $C$\n", + "it is given by $m\\wedge n$.\n", + "\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, RAN0\n", + "\n", + "We show here how the linear congruential algorithm can be implemented, namely" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the brackets denote integer division. In the code below the numbers \n", + "$q$ and $r$ are chosen so that $r < q$.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, RAN0\n", + "\n", + "To see how this works we note first that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "since we can add or subtract any integer multiple of $M$ from $aN_{i-1}$.\n", + "The last term $[N_{i-1}/q]M\\mathrm{MOD}(M)$ is zero since the integer division \n", + "$[N_{i-1}/q]$ just yields a constant which is multiplied with $M$.\n", + "\n", + "\n", + "\n", + "\n", + "# Random number generator RNG, RAN0\n", + "We can now rewrite Eq. ([14](#eq:rntrick1)) as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\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", + "# 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", + "## 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", + "## 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 + }, + "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", + "## 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 + }, + "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", + "## 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", + "## 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", + "## 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", + "## 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", + "## 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", + "## 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", + "## 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", + "## 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 + }, + "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": [ + "\n", + "## 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/pub/Statistics/ipynb/ipynb-Statistics-src.tar.gz b/doc/pub/Statistics/ipynb/ipynb-Statistics-src.tar.gz new file mode 100644 index 000000000..6f6a9b5fe Binary files /dev/null and b/doc/pub/Statistics/ipynb/ipynb-Statistics-src.tar.gz differ diff --git a/doc/pub/week38/ipynb/.ipynb_checkpoints/week38-checkpoint.ipynb b/doc/pub/week38/ipynb/.ipynb_checkpoints/week38-checkpoint.ipynb new file mode 100644 index 000000000..50914c1bb --- /dev/null +++ b/doc/pub/week38/ipynb/.ipynb_checkpoints/week38-checkpoint.ipynb @@ -0,0 +1,2608 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Data Analysis and Machine Learning: Logistic Regression\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Sep 18, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "## Plans for week 38\n", + "\n", + "* Thursday: Summary of regression methods and discussion of project 1. We revisit also cross-validation and bootstrap as resampling techniques with examples. Recommended reading: [Hastie et al](https://www.springer.com/gp/book/9780387848570) chapters 3 and 7.1-7.6 and 7.10-7.12.\n", + "\n", + "* Friday: Logistic Regression. Recommended reading: [Hastie et al](https://www.springer.com/gp/book/9780387848570) chapters 4.1-4.4 and [Murphy](https://mitpress.mit.edu/books/machine-learning-1) chapter 8.1-8.2\n", + "\n", + "## Thursday September 17\n", + "\n", + "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember17.mp4?vrtx=view-as-webpage) and [link to handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember17.pdf).\n", + "\n", + "## Ridge and LASSO Regression, reminder\n", + "\n", + "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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Various steps in cross-validation\n", + "\n", + "When the repetitive splitting of the data set is done randomly,\n", + "samples may accidently end up in a fast majority of the splits in\n", + "either training or test set. Such samples may have an unbalanced\n", + "influence on either model building or prediction evaluation. To avoid\n", + "this $k$-fold cross-validation structures the data splitting. The\n", + "samples are divided into $k$ more or less equally sized exhaustive and\n", + "mutually exclusive subsets. In turn (at each split) one of these\n", + "subsets plays the role of the test set while the union of the\n", + "remaining subsets constitutes the training set. Such a splitting\n", + "warrants a balanced representation of each sample in both training and\n", + "test set over the splits. Still the division into the $k$ subsets\n", + "involves a degree of randomness. This may be fully excluded when\n", + "choosing $k=n$. This particular case is referred to as leave-one-out\n", + "cross-validation (LOOCV). \n", + "\n", + "\n", + "## How to set up the cross-validation for Ridge and/or Lasso\n", + "\n", + "* Define a range of interest for the penalty parameter.\n", + "\n", + "* Divide the data set into training and test set comprising samples $\\{1, \\ldots, n\\} \\setminus i$ and $\\{ i \\}$, respectively.\n", + "\n", + "* Fit the linear regression model by means of ridge estimation for each $\\lambda$ in the grid using the training set, and the corresponding estimate of the error variance $\\boldsymbol{\\sigma}_{-i}^2(\\lambda)$, as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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": [ + "## Cross-validation in brief\n", + "\n", + "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", + "## Code Example for Cross-validation and $k$-fold Cross-validation\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": 1, + "metadata": {}, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "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": [ + "## Bias-Variance tradeoff with Bootstrap" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "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": [ + "## Another Example from Scikit-Learn's Repository" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "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": "markdown", + "metadata": {}, + "source": [ + "## Cross-validation with Ridge" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "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()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The Ising model\n", + "\n", + "The one-dimensional Ising model with nearest neighbor interaction, no\n", + "external field and a constant coupling constant $J$ is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = -J \\sum_{k}^L s_k s_{k + 1},\n", + "\\label{_auto1} \\tag{1}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins\n", + "in the system is determined by $L$. For the one-dimensional system\n", + "there is no phase transition.\n", + "\n", + "We will look at a system of $L = 40$ spins with a coupling constant of\n", + "$J = 1$. To get enough training data we will generate 10000 states\n", + "with their respective energies." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from mpl_toolkits.axes_grid1 import make_axes_locatable\n", + "import seaborn as sns\n", + "import scipy.linalg as scl\n", + "from sklearn.model_selection import train_test_split\n", + "import tqdm\n", + "sns.set(color_codes=True)\n", + "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n", + "\n", + "L = 40\n", + "n = int(1e4)\n", + "\n", + "spins = np.random.choice([-1, 1], size=(n, L))\n", + "J = 1.0\n", + "\n", + "energies = np.zeros(n)\n", + "\n", + "for i in range(n):\n", + " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we use ordinary least squares\n", + "regression to predict the energy for the nearest neighbor\n", + "one-dimensional Ising model on a ring, i.e., the endpoints wrap\n", + "around. We will use linear regression to fit a value for\n", + "the coupling constant to achieve this.\n", + "\n", + "## Reformulating the problem to suit regression\n", + "\n", + "A more general form for the one-dimensional Ising model is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n", + "\\label{_auto2} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we allow for interactions beyond the nearest neighbors and a state dependent\n", + "coupling constant. This latter expression can be formulated as\n", + "a matrix-product" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\boldsymbol{H} = \\boldsymbol{X} J,\n", + "\\label{_auto3} \\tag{3}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $X_{jk} = s_j s_k$ and $J$ is a matrix which consists of the\n", + "elements $-J_{jk}$. This form of writing the energy fits perfectly\n", + "with the form utilized in linear regression, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon},\n", + "\\label{_auto4} \\tag{4}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We split the data in training and test data as discussed in the previous example" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "X = np.zeros((n, L ** 2))\n", + "for i in range(n):\n", + " X[i] = np.outer(spins[i], spins[i]).ravel()\n", + "y = energies\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Linear regression\n", + "\n", + "In the ordinary least squares method we choose the cost function" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " C(\\boldsymbol{X}, \\boldsymbol{\\beta})= \\frac{1}{n}\\left\\{(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})\\right\\}.\n", + "\\label{_auto5} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We then find the extremal point of $C$ by taking the derivative with respect to $\\boldsymbol{\\beta}$ as discussed above.\n", + "This yields the expression for $\\boldsymbol{\\beta}$ to be" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta} = \\frac{\\boldsymbol{X}^T \\boldsymbol{y}}{\\boldsymbol{X}^T \\boldsymbol{X}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which immediately imposes some requirements on $\\boldsymbol{X}$ as there must exist\n", + "an inverse of $\\boldsymbol{X}^T \\boldsymbol{X}$. If the expression we are modeling contains an\n", + "intercept, i.e., a constant term, we must make sure that the\n", + "first column of $\\boldsymbol{X}$ consists of $1$. We do this here" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "X_train_own = np.concatenate(\n", + " (np.ones(len(X_train))[:, np.newaxis], X_train),\n", + " axis=1\n", + ")\n", + "X_test_own = np.concatenate(\n", + " (np.ones(len(X_test))[:, np.newaxis], X_test),\n", + " axis=1\n", + ")" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [], + "source": [ + "def ols_inv(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n", + " return scl.inv(x.T @ x) @ (x.T @ y)\n", + "beta = ols_inv(X_train_own, y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Singular Value decomposition\n", + "\n", + "Doing the inversion directly turns out to be a bad idea since the matrix\n", + "$\\boldsymbol{X}^T\\boldsymbol{X}$ is singular. An alternative approach is to use the **singular\n", + "value decomposition**. Using the definition of the Moore-Penrose\n", + "pseudoinverse we can write the equation for $\\boldsymbol{\\beta}$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\beta} = \\boldsymbol{X}^{+}\\boldsymbol{y},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the pseudoinverse of $\\boldsymbol{X}$ is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{X}^{+} = \\frac{\\boldsymbol{X}^T}{\\boldsymbol{X}^T\\boldsymbol{X}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Using singular value decomposition we can decompose the matrix $\\boldsymbol{X} = \\boldsymbol{U}\\boldsymbol{\\Sigma} \\boldsymbol{V}^T$,\n", + "where $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are orthogonal(unitary) matrices and $\\boldsymbol{\\Sigma}$ contains the singular values (more details below).\n", + "where $X^{+} = V\\Sigma^{+} U^T$. This reduces the equation for\n", + "$\\omega$ to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\boldsymbol{\\beta} = \\boldsymbol{V}\\boldsymbol{\\Sigma}^{+} \\boldsymbol{U}^T \\boldsymbol{y}.\n", + "\\label{_auto6} \\tag{6}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that solving this equation by actually doing the pseudoinverse\n", + "(which is what we will do) is not a good idea as this operation scales\n", + "as $\\mathcal{O}(n^3)$, where $n$ is the number of elements in a\n", + "general matrix. Instead, doing $QR$-factorization and solving the\n", + "linear system as an equation would reduce this down to\n", + "$\\mathcal{O}(n^2)$ operations." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "def ols_svd(x: np.ndarray, y: np.ndarray) -> np.ndarray:\n", + " u, s, v = scl.svd(x)\n", + " return v.T @ scl.pinv(scl.diagsvd(s, u.shape[0], v.shape[0])) @ u.T @ y" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [], + "source": [ + "beta = ols_svd(X_train_own,y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "When extracting the $J$-matrix we need to make sure that we remove the intercept, as is done here" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [], + "source": [ + "J = beta[1:].reshape(L, L)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "A way of looking at the coefficients in $J$ is to plot the matrices as images." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J, **cmap_args)\n", + "plt.title(\"OLS\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is interesting to note that OLS\n", + "considers both $J_{j, j + 1} = -0.5$ and $J_{j, j - 1} = -0.5$ as\n", + "valid matrix elements for $J$.\n", + "In our discussion below on hyperparameters and Ridge and Lasso regression we will see that\n", + "this problem can be removed, partly and only with Lasso regression. \n", + "\n", + "In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## The one-dimensional Ising model\n", + "\n", + "Let us bring back the Ising model again, but now with an additional\n", + "focus on Ridge and Lasso regression as well. We repeat some of the\n", + "basic parts of the Ising model and the setup of the training and test\n", + "data. The one-dimensional Ising model with nearest neighbor\n", + "interaction, no external field and a constant coupling constant $J$ is\n", + "given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = -J \\sum_{k}^L s_k s_{k + 1},\n", + "\\label{_auto7} \\tag{7}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $s_i \\in \\{-1, 1\\}$ and $s_{N + 1} = s_1$. The number of spins in the system is determined by $L$. For the one-dimensional system there is no phase transition.\n", + "\n", + "We will look at a system of $L = 40$ spins with a coupling constant of $J = 1$. To get enough training data we will generate 10000 states with their respective energies." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from mpl_toolkits.axes_grid1 import make_axes_locatable\n", + "import seaborn as sns\n", + "import scipy.linalg as scl\n", + "from sklearn.model_selection import train_test_split\n", + "import sklearn.linear_model as skl\n", + "import tqdm\n", + "sns.set(color_codes=True)\n", + "cmap_args=dict(vmin=-1., vmax=1., cmap='seismic')\n", + "\n", + "L = 40\n", + "n = int(1e4)\n", + "\n", + "spins = np.random.choice([-1, 1], size=(n, L))\n", + "J = 1.0\n", + "\n", + "energies = np.zeros(n)\n", + "\n", + "for i in range(n):\n", + " energies[i] = - J * np.dot(spins[i], np.roll(spins[i], 1))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "A more general form for the one-dimensional Ising model is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = - \\sum_j^L \\sum_k^L s_j s_k J_{jk}.\n", + "\\label{_auto8} \\tag{8}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we allow for interactions beyond the nearest neighbors and a more\n", + "adaptive coupling matrix. This latter expression can be formulated as\n", + "a matrix-product on the form" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " H = X J,\n", + "\\label{_auto9} \\tag{9}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $X_{jk} = s_j s_k$ and $J$ is the matrix consisting of the\n", + "elements $-J_{jk}$. This form of writing the energy fits perfectly\n", + "with the form utilized in linear regression, viz." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " \\boldsymbol{y} = \\boldsymbol{X}\\boldsymbol{\\beta} + \\boldsymbol{\\epsilon}.\n", + "\\label{_auto10} \\tag{10}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We organize the data as we did above" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [], + "source": [ + "X = np.zeros((n, L ** 2))\n", + "for i in range(n):\n", + " X[i] = np.outer(spins[i], spins[i]).ravel()\n", + "y = energies\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.96)\n", + "\n", + "X_train_own = np.concatenate(\n", + " (np.ones(len(X_train))[:, np.newaxis], X_train),\n", + " axis=1\n", + ")\n", + "\n", + "X_test_own = np.concatenate(\n", + " (np.ones(len(X_test))[:, np.newaxis], X_test),\n", + " axis=1\n", + ")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We will do all fitting with **Scikit-Learn**," + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [], + "source": [ + "clf = skl.LinearRegression().fit(X_train, y_train)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "When extracting the $J$-matrix we make sure to remove the intercept" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "J_sk = clf.coef_.reshape(L, L)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "And then we plot the results" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_sk, **cmap_args)\n", + "plt.title(\"LinearRegression from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The results perfectly with our previous discussion where we used our own code.\n", + "\n", + "## Ridge regression\n", + "\n", + "Having explored the ordinary least squares we move on to ridge\n", + "regression. In ridge regression we include a **regularizer**. This\n", + "involves a new cost function which leads to a new estimate for the\n", + "weights $\\boldsymbol{\\beta}$. This results in a penalized regression problem. The\n", + "cost function is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "2\n", + "2\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": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [], + "source": [ + "_lambda = 0.1\n", + "clf_ridge = skl.Ridge(alpha=_lambda).fit(X_train, y_train)\n", + "J_ridge_sk = clf_ridge.coef_.reshape(L, L)\n", + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_ridge_sk, **cmap_args)\n", + "plt.title(\"Ridge from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## LASSO regression\n", + "\n", + "In the **Least Absolute Shrinkage and Selection Operator** (LASSO)-method we get a third cost function." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + " C(\\boldsymbol{X}, \\boldsymbol{\\beta}; \\lambda) = (\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y})^T(\\boldsymbol{X}\\boldsymbol{\\beta} - \\boldsymbol{y}) + \\lambda \\sqrt{\\boldsymbol{\\beta}^T\\boldsymbol{\\beta}}.\n", + "\\label{_auto12} \\tag{12}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Finding the extremal point of this cost function is not so straight-forward as in least squares and ridge. We will therefore rely solely on the function ``Lasso`` from **Scikit-Learn**." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [], + "source": [ + "clf_lasso = skl.Lasso(alpha=_lambda).fit(X_train, y_train)\n", + "J_lasso_sk = clf_lasso.coef_.reshape(L, L)\n", + "fig = plt.figure(figsize=(20, 14))\n", + "im = plt.imshow(J_lasso_sk, **cmap_args)\n", + "plt.title(\"Lasso from Scikit-learn\", fontsize=18)\n", + "plt.xticks(fontsize=18)\n", + "plt.yticks(fontsize=18)\n", + "cb = fig.colorbar(im)\n", + "cb.ax.set_yticklabels(cb.ax.get_yticklabels(), fontsize=18)\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "It is quite striking how LASSO breaks the symmetry of the coupling\n", + "constant as opposed to ridge and OLS. We get a sparse solution with\n", + "$J_{j, j + 1} = -1$.\n", + "\n", + "\n", + "\n", + "## Performance as function of the regularization parameter\n", + "\n", + "We see how the different models perform for a different set of values for $\\lambda$." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "lambdas = np.logspace(-4, 5, 10)\n", + "\n", + "train_errors = {\n", + " \"ols_sk\": np.zeros(lambdas.size),\n", + " \"ridge_sk\": np.zeros(lambdas.size),\n", + " \"lasso_sk\": np.zeros(lambdas.size)\n", + "}\n", + "\n", + "test_errors = {\n", + " \"ols_sk\": np.zeros(lambdas.size),\n", + " \"ridge_sk\": np.zeros(lambdas.size),\n", + " \"lasso_sk\": np.zeros(lambdas.size)\n", + "}\n", + "\n", + "plot_counter = 1\n", + "\n", + "fig = plt.figure(figsize=(32, 54))\n", + "\n", + "for i, _lambda in enumerate(tqdm.tqdm(lambdas)):\n", + " for key, method in zip(\n", + " [\"ols_sk\", \"ridge_sk\", \"lasso_sk\"],\n", + " [skl.LinearRegression(), skl.Ridge(alpha=_lambda), skl.Lasso(alpha=_lambda)]\n", + " ):\n", + " method = method.fit(X_train, y_train)\n", + "\n", + " train_errors[key][i] = method.score(X_train, y_train)\n", + " test_errors[key][i] = method.score(X_test, y_test)\n", + "\n", + " omega = method.coef_.reshape(L, L)\n", + "\n", + " plt.subplot(10, 5, plot_counter)\n", + " plt.imshow(omega, **cmap_args)\n", + " plt.title(r\"%s, $\\lambda = %.4f$\" % (key, _lambda))\n", + " plot_counter += 1\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We see that LASSO reaches a good solution for low\n", + "values of $\\lambda$, but will \"wither\" when we increase $\\lambda$ too\n", + "much. Ridge is more stable over a larger range of values for\n", + "$\\lambda$, but eventually also fades away.\n", + "\n", + "## Finding the optimal value of $\\lambda$\n", + "\n", + "To determine which value of $\\lambda$ is best we plot the accuracy of\n", + "the models when predicting the training and the testing set. We expect\n", + "the accuracy of the training set to be quite good, but if the accuracy\n", + "of the testing set is much lower this tells us that we might be\n", + "subject to an overfit model. The ideal scenario is an accuracy on the\n", + "testing set that is close to the accuracy of the training set." + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(20, 14))\n", + "\n", + "colors = {\n", + " \"ols_sk\": \"r\",\n", + " \"ridge_sk\": \"y\",\n", + " \"lasso_sk\": \"c\"\n", + "}\n", + "\n", + "for key in train_errors:\n", + " plt.semilogx(\n", + " lambdas,\n", + " train_errors[key],\n", + " colors[key],\n", + " label=\"Train {0}\".format(key),\n", + " linewidth=4.0\n", + " )\n", + "\n", + "for key in test_errors:\n", + " plt.semilogx(\n", + " lambdas,\n", + " test_errors[key],\n", + " colors[key] + \"--\",\n", + " label=\"Test {0}\".format(key),\n", + " linewidth=4.0\n", + " )\n", + "plt.legend(loc=\"best\", fontsize=18)\n", + "plt.xlabel(r\"$\\lambda$\", fontsize=18)\n", + "plt.ylabel(r\"$R^2$\", fontsize=18)\n", + "plt.tick_params(labelsize=18)\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "From the above figure we can see that LASSO with $\\lambda = 10^{-2}$\n", + "achieves a very good accuracy on the test set. This by far surpasses the\n", + "other models for all values of $\\lambda$.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Friday September 18: Intro to Logistic Regression\n", + "\n", + "[Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureSeptember18.mp4?vrtx=view-as-webpage) and [link to handwritten notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/NotesSeptember18.pdf).\n", + "\n", + "\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\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", + "## Optimization and Deep learning\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Linear classifier\n", + "\n", + "Before moving to the logistic model, let us try to use our linear\n", + "regression model to classify these two outcomes. We could for example\n", + "fit a linear model to the default case if $y_i > 0.5$ and the no\n", + "default case $y_i \\leq 0.5$.\n", + "\n", + "We would then have our \n", + "weighted linear combination, namely" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\hat{y} = \\hat{X}^T\\hat{\\beta} + \\hat{\\epsilon},\n", + "\\label{_auto13} \\tag{13}\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", + "## Some selected properties\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", + "## Simple example\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": {}, + "outputs": [ + { + "data": { + "text/html": [ + "
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\n", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "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", + "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": [ + "## Plotting the mean value for each group\n", + "\n", + "What we could attempt however is to plot the mean value for each group." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This expression implies however that $f(y_i\\vert x_i)$ could take any\n", + "value from minus infinity to plus infinity. If we however let\n", + "$f(y\\vert y)$ be represented by the mean value, the above example\n", + "shows us that we can constrain the function to take values between\n", + "zero and one, that is we have $0 \\le f(y_i\\vert x_i) \\le 1$. Looking\n", + "at our last curve we see also that it has an S-shaped form. This leads\n", + "us to a very popular model for the function $f$, namely the so-called\n", + "Sigmoid function or logistic model. We will consider this function as\n", + "representing the probability for finding a value of $y_i$ with a given\n", + "$x_i$.\n", + "\n", + "## The logistic function\n", + "\n", + "Another widely studied model, is the so-called \n", + "perceptron model, which is an example of a \"hard classification\" model. We\n", + "will encounter this model when we discuss neural networks as\n", + "well. Each datapoint is deterministically assigned to a category (i.e\n", + "$y_i=0$ or $y_i=1$). In many cases, and the coronary heart disease data forms one of many such examples, it is favorable to have a \"soft\"\n", + "classifier that outputs the probability of a given category rather\n", + "than a single value. For example, given $x_i$, the classifier\n", + "outputs the probability of being in a category $k$. Logistic regression\n", + "is the most common example of a so-called soft classifier. In logistic\n", + "regression, the probability that a data point $x_i$\n", + "belongs to a category $y_i=\\{0,1\\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event," + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", 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\n", 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5cyduv/32dvuzcnJyAAAqlQqDBw9Gfn5+h0He2GjxqTa9Php1dS0+jQ037I137I13YvemyWTD5l1nEatW4qeDkwLq/5PYvQlk7I13vvbmemHv99T6sGHDUF1dDbvdDgDIz8/HhAkTYDQaYTKZ2o3dtGkTpkyZ4rmdl5eHPXv2eG6Xl5cjNZVrJBPRtf1tXxnsDjceHJuJCCUvT0oEdMEWuUqlwuLFi/Hmm29Cq9UiKysLubm5WLZsGeLi4jBz5kwAQFFRETIyMtpNw8fHx2PVqlU4deoULl26hEmTJmHkyJH+lkREIehSowV7jlYjSavC2ByebkZ0hUQQBEHsIjrL1ykaTud4x954x954J2ZvNnxVjN1Hq/HsA9kYPSRJlBquh9833rE33gXE1DoRUXdrMtnw3YkaJMapMGpQotjlEAUUBjkRBbxvDlfC6XLjZ6PTIJVy8ReiH2KQE1FAs7Q68e+CSsSolfjp0F5il0MUcBjkRBTQdh+tgtXmwl0j+0Dh4/oRROGEQU5EAcvhdGPH4QpEKmW4fXiK2OUQBSQGOREFrLzCGjSZ7JgwLAVRkQqxyyEKSAxyIgpIgiDg64MXIJNKcNcoLhRF5A2DnIgCUskFIy4aLBg1KBHa6AixyyEKWAxyIgpIu45WAQAmcN840XUxyIko4DSZ7ThSUocUvRoD+sSKXQ5RQGOQE1HA+fZ4NVxuAROGpbS7WiIRXY1BTkQBxe0WsPtoNZQKKXKzuQAMUUcY5EQUUE6WNqC+qRVjhiQhKtLvCzQShTwGOREFlF0FPMiNqDMY5EQUMBqaW3HsXD0ye0cjo1eM2OUQBQUGOREFjO9OXIQgAOOHcWucyFcMciIKCIIgIK+wFkq5lNccJ+oEBjkRBYSymhbUNFgwbEACVBE8yI3IVwxyIgoIeYU1AMBTzog6iUFORKJzud04eKoWGpUC2ZnxYpdDFFQY5EQkusLSRjRbHBg9OAlyGX8tEXUGf2KISHT7L0+rj7kpSeRKiIIPg5yIRGW1OZF/ug6JWhX69ua540SdxSAnIlEVnKmD3elGbnYvXiCF6AYwyIlIVHmFtQCAMdmcVie6EQxyIhJNi8WOU2UNyOwdgyRtlNjlEAUlBjkRiabgTD0EAVzJjcgPDHIiEs2RkjoAwIgsvciVEAUvBjkRicLS6sSpsgakJWqQGKcSuxyioNUlCxrv27cPO3bsgE6ng0QiwaxZs9o9vmnTJnz66aeIiIgAAEybNg0PPfQQAGDr1q0oKiqCVCpFWloapk+f3hUlEVGAO3auHi63gFu4NU7kF7+D3Gq1YtGiRdi+fTuUSiVmz56NvLw85Obmthv3zjvvoE+fPu3uq6mpwQcffIAtW7ZAIpFg2rRpGDNmDDIyMvwti4gC3PfT6tw/TuQPv6fWjx49iuTkZCiVSgDAiBEjsGvXrqvGbdy4EevXr8eqVatgNBoBAHv37kV2drbn3NHhw4djz549/pZERAHOZnfh5HkDeuuikJKgFrscoqDm9xa5wWCAWv39D6JGo4HBYGg3ZtSoUZgwYQLi4+Oxe/duvPDCC9iwYQMaGhraPVetVl/1XCIKPSfOG2B3ujmtTtQF/A5ynU4Hs9nsuW0ymaDT6dqNSU1N9Xw9ZswYPPfcc3C5XIiPj0d5ebnnMbPZjLS0tA7fU6uNglwu86k+vT7ap3HhiL3xjr3xrit6c/LrEgDAxNEZIdXrUPosXY298c7f3vgd5MOGDUN1dTXsdjuUSiXy8/Px6KOPwmg0Qi6XQ6PRYMWKFXjhhRcgl8tRVlaGPn36QCaTYdy4cfjTn/4EQRAgkUhQUFCAxx9/vMP3bGy0+FSbXh+NuroWfz9iSGJvvGNvvOuK3jicbhwsrEFCbCRilNKQ6TW/b7xjb7zztTfXC3u/g1ylUmHx4sV48803odVqkZWVhdzcXCxbtgxxcXGYOXMmEhISsHjxYvTp0wenT5/GsmXLAAC9evXC008/jSVLlkAmk+Hhhx/mgW5EIe5UWQNa7S6MH5bMtdWJuoBEEARB7CI6y9e/7PhXoHfsjXfsjXdd0ZuP/lGMPceq8fLjIzCgT1wXVSY+ft94x9541xVb5FwQhoh6jCAIOHHeAI1KgX7JsWKXQxQSGORE1GMqLpnQ2GLD0L7xkEo5rU7UFRjkRNRjjp1rO700p1+CyJUQhQ4GORH1mOPn6iGVSHBT33ixSyEKGQxyIuoRLRY7zlc1o39KDNSRCrHLIQoZDHIi6hEnzhsgAMjpz2l1oq7EICeiHnHcs39c18FIIuoMBjkRdTuX242T5xugi4ngRVKIuhiDnIi63dnKJlhsTuT0S+BqbkRdjEFORN2O0+pE3YdBTkTd7vg5AxRyKQala8UuhSjkMMiJqFs1NLeiqt6MwelaRCh8u/wwEfmOQU5E3aqwtAEAkJ3JRWCIugODnIi6VWHZ5SDPYJATdQcGORF1G7cg4FRZI7TREeitixK7HKKQxCAnom5zobYFJqsD2RnxPO2MqJswyImo21zZPz4kk0erE3UXBjkRdZtTZY0AgCHcP07UbRjkRNQtbA4XzlQakZakQUyUUuxyiEIWg5yIusXpCiOcLoGnnRF1MwY5EXULz/njnFYn6lYMciLqFoVlDVDIpRjQJ1bsUohCGoOciLqc0WRDVZ0ZWalxUMi5LCtRd2KQE1GX85x2xml1om7HICeiLnfltDMe6EbU/RjkRNSlBEFA8YVGREcpkKJXi10OUchjkBNRl7rUaEVjiw1ZaVpIuSwrUbdjkBNRlyq60DatPjgtTuRKiMIDg5yIulRxeVuQD0rn+upEPYFBTkRdpm3/uBGxGiV6xfOypUQ9Qd4VL7Jv3z7s2LEDOp0OEokEs2bNavf42rVrUV9fj4SEBBQWFmLOnDno168fAOCOO+5ASkoKACAxMRErVqzoipKISATVBguazXaMGZLEy5YS9RC/g9xqtWLRokXYvn07lEolZs+ejby8POTm5nrGWCwWvPzyy5BIJPj73/+Ot99+G++//z4AYMqUKZg9e7a/ZRBRAOC0OlHP83tq/ejRo0hOToZS2XZ1oxEjRmDXrl3txrz44ouev87dbjeior6fcjt06BDWrVuHlStXIj8/399yiEhEDHKinuf3FrnBYIBa/f25ohqNBgaD4Zpj7XY7Nm/ejEWLFnnumzdvHnJycmC1WjFlyhSsWbMG6enp131PrTYKch+XfdTro30aF47YG+/YG++89cbtFnC60gi9VoUh/fVhObXO7xvv2Bvv/O2N30Gu0+lgNps9t00mE3Q63VXj7HY7Fi9ejF/96ldIS0vz3J+TkwMAUKlUGDx4MPLz8zsM8sZGi0+16fXRqKtr8WlsuGFvvGNvvLteby7UtqDF4kBOPx3q6009XJn4+H3jHXvjna+9uV7Y+z21PmzYMFRXV8NutwMA8vPzMWHCBBiNRphMbT/Mra2tWLRoEf7zP/8TN910E77++msAQF5eHvbs2eN5rfLycqSmpvpbEhGJwDOtnsZpdaKe5PcWuUqlwuLFi/Hmm29Cq9UiKysLubm5WLZsGeLi4jBz5kzMmzcPZ86cQWVlJYC2g98mTZqE+Ph4rFq1CqdOncKlS5cwadIkjBw50u8PRUQ9r/iCEQAwmPvHiXqURBAEQewiOsvXKRpO53jH3njH3njnrTcutxtzfrcX0VFK/ObZ3Gs8M/Tx+8Y79sa7gJhaJyK6UGuC1ebitDqRCBjkROS30xVt0+pZXF+dqMcxyInIbyWX949npTLIiXoag5yI/OIWBJypNCIhNhLxMZFil0MUdhjkROSXqjozzK1OTqsTiYRBTkR+Kbl8/fGBnFYnEgWDnIj88v2BbjxinUgMDHIiumGCIOB0hRHa6AjoY7l/nEgMDHIiumE1DRY0WxzISo0Ly4ukEAUCBjkR3bArp50N5IFuRKJhkBPRDfPsH+eBbkSiYZAT0Q0RBAElFUbERCnQKz5K7HKIwhaDnIhuSJ3RisYWGwZy/ziRqBjkRHRDSnjaGVFAYJAT0Q05zfXViQICg5yIbsjpSiPUkXIk69Vil0IU1hjkRNRpjS021BlbMaBPHKTcP04kKgY5EXXaldPOBqTGilwJETHIiajTTldeXgimD/ePE4mNQU5EnXamwgilXIr0XtFil0IU9hjkRNQpJqsDVXVm9E2OgVzGXyFEYuNPIRF1ytnKJgjg9ceJAgWDnIg6xbN/nEFOFBAY5ETUKWcqjJBJJeiXzCPWiQIBg5yIfNZqd6KspgVpSdGIUMrELoeIwCAnok44faERLreAgTx/nChgMMiJyGeF5wwAeP44USBhkBORzwpL24J8AA90IwoYDHIi8onT5UZxeSOSE9TQqBRil0NEl8m74kX27duHHTt2QKfTQSKRYNasWe0et9lsWLp0KZKSklBWVoaZM2ciMzMTALB161YUFRVBKpUiLS0N06dP74qSiKiLXag1wWZ3YWAf7h8nCiR+B7nVasWiRYuwfft2KJVKzJ49G3l5ecjNzfWM2bBhA3r37o1nnnkGJSUlWLhwIT755BPU1NTggw8+wJYtWyCRSDBt2jSMGTMGGRkZ/pZFRF3s+wulcFqdKJD4PbV+9OhRJCcnQ6lUAgBGjBiBXbt2tRuza9cuDB8+HACQlZWF4uJimEwm7N27F9nZ2ZBcvgzi8OHDsWfPHn9LIqJucIYXSiEKSH5vkRsMBqjVas9tjUYDg8Hg05iGhoZ296vV6queey1abRTkct/OYdXreVEHb9gb79ib9txuAWermqHXqjCov17scgIWv2+8Y2+887c3fge5TqeD2Wz23DaZTNDpdD6NiY+PR3l5ued+s9mMtLS0Dt+zsdHiU216fTTq6lp8Ghtu2Bvv2JurVdWb0WKx45ZBfdgbL/h94x17452vvble2Ps9tT5s2DBUV1fDbrcDAPLz8zFhwgQYjUaYTCYAwIQJE1BQUAAAKCkpwaBBg6DRaDBu3DgUFhZCEAQAQEFBAW677TZ/SyKiLnbm8v7xIX11HYwkop7m9xa5SqXC4sWL8eabb0Kr1SIrKwu5ublYtmwZ4uLiMHPmTDz55JNYunQpVq9ejQsXLuCtt94CAPTq1QtPP/00lixZAplMhocffpgHuhEFoCsXSsnOjBe5EiL6MYlwZXM4iPg6RcPpHO/YG+/Ym6v9z+rvYHO48cn/uwf19SaxywlI/L7xjr3xLiCm1okotBmaWmFotmFAn1jPGSZEFDgY5ER0XVem1QfwtDOigMQgJ6LrunKg20AuBEMUkBjkRHRdpyuboFRIkZakEbsUIroGBjkReWWyOlBdb0a/5FjIZfx1QRSI+JNJRF55lmXltDpRwGKQE5FXZyqaAAADeMUzooDFICcir05XGiGTStAvmUFOFKgY5ER0Ta12J8prWpDeKxoRSt8uUkREPY9BTkTXdK66GS63wP3jRAGOQU5E13T6Ag90IwoGDHIiuqbTFUZIwAPdiAIdg5yIruJwunGuuhl9EjVQRyrELoeIroNBTkRXKb3YDKfLzWl1oiDAICeiq1xZCCaLQU4U8BjkRHSVkssXShnAICcKeAxyImrH5XbjbGUTesVHIVatFLscIuoAg5yI2qm4ZEKr3cX940RBgkFORO1cOX+c+8eJggODnIjaubJ/nFvkRMGBQU5EHm5BwJnKJuhiIqGLjRS7HCLyAYOciDwu1pthsjq4NU4URBjkRORxZVo9K41BThQsGORE5FF8gUFOFGwY5EQEABAEASUXGqGNjkBinErscojIRwxyIgIAVNeb0WJxYFBaHCQSidjlEJGPGOREBOCH0+pakSshos5gkBMRAKDkQiMAYBD3jxMFFQY5EcEtCCi+YIQ2OgJ67h8nCioMciJC9eXzxwelabl/nCjIyP15stFoxIoVK5CamoqysjLMnTsXCQkJ7cYcP34cGzZswJAhQ1BaWoqcnBw88sgjAIDXX38dpaWlnrGvvvoqsrKy/CmJiG5AyeX945xWJwo+fgX5O++8g9zcXEyePBn/+te/sHTpUrz99tvtxtTV1eHnP/85cnJy4HA48JOf/AR33nkn4uPjodfr8cYbb/j1AYjIf8WX949npfNAN6Jg41eQ7969G8899xwAYMSIEb7Wz8AAABeNSURBVFiwYMFVYyZOnNjutkwmg0KhAACYzWa89957kMlkiIqKwvTp0yGX+1USEXWSWxBQcsGI+JgI6Lm+OlHQ6TA1Z8yYgfr6+qvunzNnDgwGA9RqNQBAo9GgqakJTqfTaxhv3LgRv/jFLxAdHQ0AuP/++5GVlQW5XI5ly5ZhzZo1eP755zssWquNglwu63AcAOj10T6NC0fsjXfh1Juyi80wWR24IzsViYkxHY4Pp950FnvjHXvjnb+96TDI169f7/UxnU4Hs9mMmJgYmEwmxMbGeg3xbdu2wWKx4Je//KXnvuzsbM/XY8aMwbp163wK8sZGS4djgLbm1NW1+DQ23LA33oVbb/KOVgIA0hPVHX7ucOtNZ7A33rE33vnam+uFvV9HrY8fPx4FBQUAgPz8fIwfPx4A4Ha7UV1d7Rn3+eefw2Aw4Je//CVKSko8B7gtXbrUM6a8vBzp6en+lENEN+D7A924f5woGPm1Q3ru3LlYvnw5ysrKUFFRgfnz5wMASkpK8NJLL2Hbtm345z//id/85jcYMmQIdu7cCaPRiFdffRWZmZlobGzE8uXLERkZidLSUrz88std8qGIyDdt5483QhcTgQTuHycKShJBEASxi+gsX6doOJ3jHXvjXTj1prymBb/+6BDGDu2Np+8d3OH4cOpNZ7E33rE33ok+tU5Ewe1UWQMAYEgGp9WJghWDnCiMXQnywRnxIldCRDeKQU4UphxOF05XNqGPXo1YtVLscojoBjHIicLU2apmOJxuDOHWOFFQY5AThSnuHycKDQxyojB1qqwRMqkEA1N5oRSiYMYgJwpD5lYHymqa0Tc5BpFKXt+AKJgxyInCUHG5EYIA7h8nCgEMcqIwdKqc+8eJQgWDnCgMnSprRIRShszeHV/tjIgCG4OcKMw0NLeitsGCQalxkMv4K4Ao2PGnmCjMFHpOO+P+caJQwCAnCjMnz18O8kwGOVEoYJAThRGX243C0gboYiKQrIsSuxwi6gIMcqIwcr66GRabE0P76iCRSMQuh4i6AIOcKIycOG8AAAztqxO5EiLqKgxyojBy4lwDZFIJBqXz/HGiUMEgJwoTTSYbymtbMDA1DqoILstKFCoY5ERh4mRp29HqnFYnCi0McqIw8f3+cZ52RhRKGOREYeDKaWfxMRFITlCLXQ4RdSEGOVEYKK1ugbmVp50RhSIGOVEYOM7TzohCFoOcKAycOG+ATCrBYJ52RhRyGOREIc5osqG8pgUD+sTytDOiEMQgJwpxR8/UAwCGD9CLXAkRdQcGOVGIK/AEeYLIlRBRd2CQE4Uwq82JovIGpCVqkBCnErscIuoGDHKiEHbivAFOl4DhAzmtThSq/DryxWg0YsWKFUhNTUVZWRnmzp2LhISrp+/uuOMOpKSkAAASExOxYsUKAEBlZSVWr16N9PR0VFVVYf78+VCruVgFUVfhtDpR6PNri/ydd95Bbm4uZs6ciTvvvBNLly695rgpU6bg448/xscff+wJcQBYtGgRpk+fjmeffRYDBgzAunXr/CmHiH7A6XLj+Ll66GIikZqoEbscIuomfgX57t27MXz4cADAiBEjsHv37muOO3ToENatW4eVK1ciPz8fAOBwOHDgwAEMHTq0w+cTUecVX2iE1ebC8IEJXM2NKIR1OLU+Y8YM1NfXX3X/nDlzYDAYPFPhGo0GTU1NcDqdkMvbv+y8efOQk5MDq9WKKVOmYM2aNVCpVIiMjPT8gtFoNDAYDD4VrdVGQS6X+TRWr4/2aVw4Ym+8C4XeFO05DwC4Y1R6l36eUOhNd2FvvGNvvPO3Nx0G+fr1670+ptPpYDabERMTA5PJhNjY2KtCHABycnIAACqVCoMHD0Z+fj7uu+8+tLa2QhAESCQSmEwm6HS+LR/Z2GjxaZxeH426uhafxoYb9sa7UOiNWxCQd7wa6kg59NGKLvs8odCb7sLeeMfeeOdrb64X9n5NrY8fPx4FBQUAgPz8fIwfPx4A4Ha7UV1dDQDIy8vDnj17PM8pLy9HamoqFAoFRo8ejRMnTlz1fCLyT3lNC4wmO4b1T4BMypNTiEKZX0etz507F8uXL0dZWRkqKiowf/58AEBJSQleeuklbNu2DfHx8Vi1ahVOnTqFS5cuYdKkSRg5ciQA4Ne//jX+8Ic/4Ntvv8XFixexYMEC/z8RESH/dB0A8LQzojAgEQRBELuIzvJ1iobTOd6xN94Fe28EQcCCNXloNjuwcs5YRCh8O57EF8Hem+7E3njH3ngn+tQ6EQWe8xebUWdsxfCBCV0a4kQUmBjkRCHmwKlaAMCtg5NEroSIegKDnCiEuN0CDhVfgjpSjpsy48Uuh4h6AIOcKISUVBjRZLLjlqxEyGX88SYKB/xJJwohV6bVRw9OFLkSIuopDHKiEOF0uXGk5BJiNUpkpWnFLoeIegiDnChEFJY2wNzqxKhBiZBKubY6UbhgkBOFiANFV6bVebQ6UThhkBOFAJvDhYIz9UiIjUTf5BixyyGiHsQgJwoBR0ouwWZ3YUx2Ei9ZShRmGOREIWDPsYsAgLE5ySJXQkQ9jUFOFORqGiw4XWHE4HQtEuNUYpdDRD2MQU4U5PYea7tk8Libe4tcCRGJgUFOFMScLje+O1kDdaQct/CSpURhiUFOFMSOnzOg2WxHbnYvKOS80hlROGKQEwWxPZ5pdR7kRhSuGOREQaqhuRUnzhuQ2TsGqYkascshIpEwyImC1HcnLkIQeJAbUbhjkBMFIafLjX8XVCFCKeOSrERhjkFOFIQOFV+C0WTHuJzeUEXIxS6HiETEICcKMoIgYMfBCkgkwJ0jU8Uuh4hExiAnCjKnK4wor23BiIF6ruRGRAxyomCz41AFAODuUdwaJyIGOVFQqW204OiZemT2jkH/lFixyyGiAMAgJwoi/zxUCQHApFtTeblSIgLAICcKGuZWB/aeqIYuJgK3ZHFddSJqwyAnChJfH6yA3eHGnSNTIZPyR5eI2vC3AVEQaLHY8c3hCsSolZgwPEXscogogDDIiYLAVwcuwGZ34d7cdEQoeJUzIvqeX0tCGY1GrFixAqmpqSgrK8PcuXORkJDQbsyBAwfwxhtvID4+HgBgMBhwzz33YPbs2Xj99ddRWlrqGfvqq68iKyvLn5KIQk6TyYadRyqhjY7AhGG8yhkRtedXkL/zzjvIzc3F5MmT8a9//QtLly7F22+/3W5MYmIi3n77bQwZMgQA8Morr2Dq1KkAAL1ejzfeeMOfEohC3vb95bA73fi/P8ngNceJ6Cp+Bfnu3bvx3HPPAQBGjBiBBQsWXDUmMzPT83V9fT3sdjtSUtr28ZnNZrz33nuQyWSIiorC9OnTIZdz3WiiKxqaW7GroBoJsZEYl8OrnBHR1TpMzRkzZqC+vv6q++fMmQODwQC1Wg0A0Gg0aGpqgtPp9BrGn3zyCaZPn+65ff/99yMrKwtyuRzLli3DmjVr8Pzzz3dYtFYbBbmPWyZ6fbRP48IRe+NdoPTm893n4XS58eikQejdKzAWgAmU3gQi9sY79sY7f3vTYZCvX7/e62M6nQ5msxkxMTEwmUyIjY31GuJ2ux0nT57EnDlzPPdlZ2d7vh4zZgzWrVvnU5A3Nlo6HAO0NaeursWnseGGvfEuUHpTccmEr/aXISk+CkMz4gKipkDpTSBib7xjb7zztTfXC3u/jlofP348CgoKAAD5+fkYP348AMDtdqO6urrd2G3btuHee+9td9/SpUs9X5eXlyM9Pd2fcohChiAI+NOOEggC8NidA3jeOBF55dcO6blz52L58uUoKytDRUUF5s+fDwAoKSnBSy+9hG3btnnGfvXVV1i9enW75zc2NmL58uWIjIxEaWkpXn75ZX/KIQoZ+0/V4kxlE0YM1OOmvjqxyyGiACYRBEEQu4jO8nWKhtM53rE33ondG6vNiVfW7ofF5sRb/zUaCQF0qVKxexPI2Bvv2BvvRJ9aJ6Kut+27MjSZ7bh3THpAhTgRBSYGOVEAqaoz4ZvDFdDHReKeMWlil0NEQYBBThQgnC431m07BZdbwKN3DuTiL0TkEwY5UYDYsrcUFy6ZMC6nN27un9DxE4iIwCAnCginK4z4x/5y6OMiMX3iALHLIaIgwiAnEpnV5sQfvzwFSIBn7suGKoLLFBOR7xjkRCISBAEbvzmN+qZW3Jubjv59AmMZViIKHgxyIhHtPFKJfSdrkNErGg/8NLPjJxAR/QiDnEgkhWUN+HTnWcSolZg1dSjkMv44ElHn8TcHkQhqGyx4b/NJSKXArKlDER8TKXZJRBSkGOREPczS6sTvvzgOi82JJycNQv8U7hcnohvHICfqQa12J1Z+fgwXDRbcPSoVY3N6i10SEQU5BjlRD7HZXVj5+XGcrWrCrYMT8cjt/cUuiYhCAIOcqAfYHS78/ovjOF1hxC1Zejxz/xBIpRKxyyKiEMCVJ4i6WavdiT9sPomi8kYMH5CAZx/IhkzKv6GJqGswyIm6UWOLDb/7/BguXDIhp58Ov3jwJp5mRkRdikFO1E0u1Lbgd389jsYWG8YPS8Zjdw1kiBNRl2OQE3WDw8WXsH57EewOFx65vT8m3ZoKiYT7xImo6zHIibqQze7Cn3eexp5jF6GUS/HcQzdh5KBEscsiohDGICfqIuU1LXj/b4WobbAgLVGDmQ9kIzlBLXZZRBTiGOREfrLanNj6bSl2HqmEyy3g7lGpmDa+HxRy7g8nou7HICe6QYIgYH9hLf7y77NoMtuhj4vEE5OycFOmTuzSiCiMMMiJOkkQBBw7a8DfvitFWU0LFHIpHhqXiXtGp0Ehl4ldHhGFGQY5kY/cbgFHz9Zj23dlKK9tAQCMHJSIRyb0Q0KcSuTqiChcMciJOtBktuPb49XYVVANQ3MrJABuHZyI+36SgT56jdjlEVGYY5ATXYPN7sKxc/U4VHQJR8/Ww+UWoFRIcdvNybhrVCpSeDQ6EQUIBjnRZU0mG06WNqC4shiHTtXA7nADAFIS1JgwPAW52b0QFckfGSIKLPytRGGr2WzHmcomnK0yoqisERcumTyPJWlVGDU4CbcOSkSKXs1V2YgoYDHIKeQJgoAmsx2Vl0wor21BxSUTympacKnR6hkjl0mQnaFFdqYOt92SCpUMDG8iCgoMcgoJTpcbTSY76pusqG9qRZ3RijpjK2oazKhpsMBqc7UbHxUhx9C+OvTvE4sBKbHITI5BhKLt1DG9Php1dS1ifAwiok7zK8jdbjf+8pe/4He/+x02bNiAgQMHXnPc1q1bUVRUBKlUirS0NEyfPh0AUFlZidWrVyM9PR1VVVWYP38+1GoeRBTu3G4BVrsTVpsTVpsLllYHzK1OmK1t/7ZY7Gi22NFsdqDJbIOxxYYWiwPCNV5LLpMgSRuFXhlRSElQIy0pGmmJGuhiI7nFTUQhwa8gLy4uxs033wyVyvs5tDU1Nfjggw+wZcsWSCQSTJs2DWPGjEFGRgYWLVqEF154ATk5Ofj444+xbt06vPjiix2+b4vF7lN9SpPN57E/dK1AuJHBVz0kCF4fE673Oj98nnDl+Z4vIFx5vcsPCj+8/8p9QtvXVx4zOdxoaDBDEAD35TFuQfCMc7sFuK98LQhwu3/4ddtjLrcbbrcA1+X/3G4BLpcAp9sNp0uAy9X2r9Pl9vzncLphd17+1+GC/fK/NocLrXYXbPa2+3yllEsRFx2B3jo1tNERiI+JREJcJPSxKiTERSIhNhIyKZdKJaLQ5VeQDxkypMMxe/fuRXZ2tmfrZ/jw4dizZw9SUlJw4MABDB06FAAwYsQIvPrqqz4F+Qu//9afsimAyGVSRCikUCpkUEcqoIuJRIRCBlWE/PJ/MkRFyqGJVECtUkAdqUB0lALRaiViohSIUMi4ZU1EYa3DIJ8xYwbq6+uvun/OnDmYOHFih2/Q0NDQbrpcrVbDYDCgsbERkZHfT29qNBoYDAafit624kGfxhHdKL0+WuwSAhZ74x174x17452/vekwyNevX+/XG8THx6O8vNxz22w2Iy0tDVqtFq2trRAEARKJBCaTCTodLzZBRETUGd2y89DtdqO6uhoAMG7cOBQWFnr21RYUFOC2226DQqHA6NGjceLECQBAfn4+xo8f3x3lEBERhSyJIFzvMKvra2pqwsaNG/Hhhx/iwQcfxH333Ydhw4ahqKgIL730ErZt2wag7aj1kydPQiaTISMjo91R63/4wx+QmpqKixcvYsGCBTxqnYiIqBP8CnIiIiISF8/LISIiCmIMciIioiAW8ku02u12fPjhh1CpVDh79iy0Wi1+9atfiV1WQFm9ejU2bNiAAwcOiF1KwFiyZAlUKhWioqJQXFyMV155BXq9XuyyRLNv3z7s2LEDOp0OEokEs2bNErukgHHhwgWsXLkSQ4YMQU1NDeLi4tifH2htbcXDDz+MsWPHYv78+WKXE1DOnz+P7du3IyIiAocOHcLs2bORk5PT6dcJ+SBft24dbr31VowaNQpA22p09L0DBw6gublZ7DICjkql8vzBt3btWrz//vt47bXXRK5KHFarFYsWLcL27duhVCoxe/Zs5OXlITc3V+zSAoLRaMTkyZNx5513AgAmT56MCRMm4KabbhK5ssBw5Y8cas/lcuE3v/kN3n//fUilUjz00EOQy28skkM+yL/88kskJyejsLAQRqMRTzzxhNglBYz6+nps374dM2fOxObNm8UuJ6D8cNZGEARERUWJWI24jh49iuTkZCiVSgBtqzDu2rWLQX7Zj7eg3G73dZetDidbtmzBiBEjUFJSAovFInY5AeXEiRMQBAEff/wxWltbERcXh0ceeeSGXiskgvx6q89VVVVBIpHgqaeewr59+/Diiy/i448/FqFKcVyvNzt37sT8+fPR0hKeV/ryZdXC5uZmfPvtt3j33Xd7uryAYTAY2p0W2plVGMPNN998g7Fjx6Jfv35ilyK6s2fP4vz585g7dy5KSkrELifgVFdX4+jRo3jnnXcQHR2NefPmQaFQYOrUqZ1+rZAI8uutPqfRaDx/Md9yyy04fPgwXC4XZDJZT5UnKm+9OXHiBORyOT777DM0NTXBZrNh7dq1uPvuu5GRkdGzRYqko1ULW1pa8Otf/xpLlixBXFxcD1UVeHQ6Hcxms+c2V2G8tv379+PAgQN45ZVXxC4lIHzzzTdQKpVYu3Ytjhw5AofDgY8++ghPPfWU2KUFBLVajb59+yI6um151ltuuQUHDx4M3yC/ntzcXFRUVKBv376oqqpCWlpa2IT49QwdOtRzwZrKykr89a9/xcyZM0WuKnA0NDRgyZIleOmll5CUlISvv/4akyZNErssUQwbNgzV1dWw2+1QKpXIz8/Ho48+KnZZAWXXrl04fPgwFi5ciEuXLqG6uhrDhw8XuyxRPffcc56vbTYbLBYLQ/wHbr75ZhiNRs+GZXV19Q1vRIX8gjC1tbX4/e9/j7S0NJw7dw6PP/74DR0VGKrKy8vx6aef4s9//jNmzpyJp556Kqz3B18xZcoUOJ1Oz5a4Wq3G+++/L3JV4vnuu+/w9ddfQ6vVQqFQ8KjsHzh58iSeeOIJz8FtFosFjz322A1tWYWir7/+Ghs3boTD4cBjjz2G++67T+ySAsY333yD/fv3Q6vV4uLFi3jttdcQGRnZ6dcJ+SAnIiIKZVwQhoiIKIgxyImIiIIYg5yIiCiIMciJiIiCGIOciIgoiDHIiYiIghiDnIiIKIiF/MpuROS/TZs2YefOnUhKSkJTUxP+/ve/Y+vWrRg4cKDYpRGFPS4IQ0QdKigoQFxcHDIzMzF79mykpaXhf/7nf8Qui4jAICeiTvj888/xySef4LPPPvNc1pSIxMWpdSLySWlpKVasWIGNGzcyxIkCCA92I6IOORwO/Pd//zdmz56Nfv364dy5czh8+LDYZRERuEVORD746KOPcO7cOZw7dw5vvPEGamtrMXHiRIwcOVLs0ojCHveRExERBTFOrRMREQUxBjkREVEQY5ATEREFMQY5ERFREGOQExERBTEGORERURBjkBMREQUxBjkREVEQ+/+RA+hyZdAJPAAAAABJRU5ErkJggg==\n", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "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": [ + "## Two parameters\n", + "\n", + "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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Maximum likelihood\n", + "\n", + "In order to define the total likelihood for all possible outcomes from a \n", + "dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n", + "$y_i\\in\\{0,1\\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. \n", + "We aim thus at maximizing \n", + "the probability of seeing the observed data. We can then approximate the \n", + "likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## The cost function rewritten\n", + "\n", + "Reordering the logarithms, we can rewrite the **cost/loss** function as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n", + "in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n", + "\n", + "## Minimizing the cross entropy\n", + "\n", + "The cross entropy is a convex function of the weights $\\hat{\\beta}$ and,\n", + "therefore, any local minimizer is a global minimizer. \n", + "\n", + "\n", + "Minimizing this\n", + "cost function with respect to the two parameters $\\beta_0$ and $\\beta_1$ we obtain" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## A more compact expression\n", + "\n", + "Let us now define a vector $\\hat{y}$ with $n$ elements $y_i$, an\n", + "$n\\times p$ matrix $\\hat{X}$ which contains the $x_i$ values and a\n", + "vector $\\hat{p}$ of fitted probabilities $p(y_i\\vert x_i,\\hat{\\beta})$. We can rewrite in a more compact form the first\n", + "derivative of cost function as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Extending to more predictors\n", + "\n", + "Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Including more classes\n", + "\n", + "Till now we have mainly focused on two classes, the so-called binary\n", + "system. Suppose we wish to extend to $K$ classes. Let us for the sake\n", + "of simplicity assume we have only two predictors. We have then following model" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and the model is specified in term of $K-1$ so-called log-odds or\n", + "**logit** transformations.\n", + "\n", + "\n", + "## More classes\n", + "\n", + "In our discussion of neural networks we will encounter the above again\n", + "in terms of a slightly modified function, the so-called **Softmax** function.\n", + "\n", + "The softmax function is used in various multiclass classification\n", + "methods, such as multinomial logistic regression (also known as\n", + "softmax regression), multiclass linear discriminant analysis, naive\n", + "Bayes classifiers, and artificial neural networks. Specifically, in\n", + "multinomial logistic regression and linear discriminant analysis, the\n", + "input to the function is the result of $K$ distinct linear functions,\n", + "and the predicted probability for the $k$-th class given a sample\n", + "vector $\\hat{x}$ and a weighting vector $\\hat{\\beta}$ is (with two\n", + "predictors):" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "This will be discussed next week. Before we develop our own codes for logistic regression, we end this lecture by studying the functionality that **Scikit-learn** offers. \n", + "\n", + "\n", + "\n", + "\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": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(426, 30)\n", + "(143, 30)\n", + "Test set accuracy with Logistic Regression: 0.95\n", + "Test set accuracy Logistic Regression with scaled data: 0.96\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/hjensen/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", + "\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": [ + "## Using the correlation matrix\n", + "\n", + "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": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "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": [ + "## Discussing the correlation data\n", + "\n", + "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": 27, + "metadata": {}, + "outputs": [], + "source": [ + "cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and then" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "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)).\n", + "\n", + "\n", + "\n", + "## Other measures in classification studies: Cancer Data again" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(426, 30)\n", + "(143, 30)\n", + "Test set accuracy with Logistic Regression: 0.95\n", + "Test set accuracy Logistic Regression with scaled data: 0.96\n", + "[1. 1. 1. 1. 1. 1.\n", + " 1. 1. 0.92857143 0.92857143]\n", + "Test set accuracy with Logistic Regression and scaled data: 0.96\n" + ] + }, + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/Users/hjensen/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" + ] + }, + { + "ename": "ModuleNotFoundError", + "evalue": "No module named 'scikitplot'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mModuleNotFoundError\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 34\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 35\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 36\u001b[0;31m \u001b[0;32mimport\u001b[0m \u001b[0mscikitplot\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mskplt\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 37\u001b[0m \u001b[0my_pred\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlogreg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpredict\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mX_test_scaled\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 38\u001b[0m \u001b[0mskplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmetrics\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot_confusion_matrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my_test\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my_pred\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnormalize\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'scikitplot'" + ] + } + ], + "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": "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/doc/pub/week39/ipynb/.ipynb_checkpoints/week39-checkpoint.ipynb b/doc/pub/week39/ipynb/.ipynb_checkpoints/week39-checkpoint.ipynb new file mode 100644 index 000000000..d689b88df --- /dev/null +++ b/doc/pub/week39/ipynb/.ipynb_checkpoints/week39-checkpoint.ipynb @@ -0,0 +1,2891 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Week 39: Optimization and Gradient Methods\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Sep 25, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "## Plan for week 39\n", + "\n", + "* Thursday: Repetition of Logistic regression equations and discussion of Gradient methods\n", + "\n", + "* Friday: Stochastic Gradient descent with examples and automatic differeantion\n", + "\n", + "Reading suggestions for both days: [Aurelien Geron's chapter 4](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf) and [Murphy sections 8.3 and 8.5](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/MachineLearningMurphy.pdf) \n", + "\n", + "## Thursday September 24\n", + "\n", + "[Overview Video, why do we care about gradient methods?](https://www.uio.no/studier/emner/matnat/fys/FYS-STK3155/h20/forelesningsvideoer/OverarchingAimsWeek39.mp4?vrtx=view-as-webpage)\n", + "\n", + "## 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", + "## 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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\boldsymbol{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n", + "\n", + "## The equations to solve\n", + "\n", + "Our compact equations used a definition of a vector $\\boldsymbol{y}$ with $n$\n", + "elements $y_i$, an $n\\times p$ matrix $\\boldsymbol{X}$ which contains the\n", + "$x_i$ values and a vector $\\boldsymbol{p}$ of fitted probabilities\n", + "$p(y_i\\vert x_i,\\boldsymbol{\\beta})$. We rewrote in a more compact form\n", + "the first derivative of the cost function as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This defines what is called the Hessian matrix.\n", + "\n", + "## Solving using Newton-Raphson's method\n", + "\n", + "If we can set up these equations, Newton-Raphson's iterative method is normally the method of choice. It requires however that we can compute in an efficient way the matrices that define the first and second derivatives. \n", + "\n", + "Our iterative scheme is then given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The right-hand side is computed with the old values of $\\beta$. \n", + "\n", + "If we can compute these matrices, in particular the Hessian, the above is often the easiest method to implement. \n", + "\n", + "\n", + "## Brief reminder on Newton-Raphson's method\n", + "\n", + "Let us quickly remind ourselves how we derive the above method.\n", + "\n", + "Perhaps the most celebrated of all one-dimensional root-finding\n", + "routines is Newton's method, also called the Newton-Raphson\n", + "method. This method requires the evaluation of both the\n", + "function $f$ and its derivative $f'$ at arbitrary points. \n", + "If you can only calculate the derivative\n", + "numerically and/or your function is not of the smooth type, we\n", + "normally discourage the use of this method.\n", + "\n", + "## The equations\n", + "\n", + "The Newton-Raphson formula consists geometrically of extending the\n", + "tangent line at a current point until it crosses zero, then setting\n", + "the next guess to the abscissa of that zero-crossing. The mathematics\n", + "behind this method is rather simple. Employing a Taylor expansion for\n", + "$x$ sufficiently close to the solution $s$, we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "f(s)=0=f(x)+(s-x)f'(x)+\\frac{(s-x)^2}{2}f''(x) +\\dots.\n", + " \\label{eq:taylornr} \\tag{1}\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Simple geometric interpretation\n", + "\n", + "The above is Newton-Raphson's method. It has a simple geometric\n", + "interpretation, namely $x_{n+1}$ is the point where the tangent from\n", + "$(x_n,f(x_n))$ crosses the $x$-axis. Close to the solution,\n", + "Newton-Raphson converges fast to the desired result. However, if we\n", + "are far from a root, where the higher-order terms in the series are\n", + "important, the Newton-Raphson formula can give grossly inaccurate\n", + "results. For instance, the initial guess for the root might be so far\n", + "from the true root as to let the search interval include a local\n", + "maximum or minimum of the function. If an iteration places a trial\n", + "guess near such a local extremum, so that the first derivative nearly\n", + "vanishes, then Newton-Raphson may fail totally\n", + "\n", + "\n", + "## Extending to more than one variable\n", + "\n", + "Newton's method can be generalized to systems of several non-linear equations\n", + "and variables. Consider the case with two equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We need thus to compute the inverse of the Jacobian matrix and it\n", + "is to understand that difficulties may\n", + "arise in case ${\\bf \\boldsymbol{J}}$ is nearly singular.\n", + "\n", + "It is rather straightforward to extend the above scheme to systems of\n", + "more than two non-linear equations. In our case, the Jacobian matrix is given by the Hessian that represents the second derivative of cost function. \n", + "\n", + "\n", + "\n", + "## Steepest descent\n", + "\n", + "The basic idea of gradient descent is\n", + "that a function $F(\\mathbf{x})$, \n", + "$\\mathbf{x} \\equiv (x_1,\\cdots,x_n)$, decreases fastest if one goes from $\\bf {x}$ in the\n", + "direction of the negative gradient $-\\nabla F(\\mathbf{x})$.\n", + "\n", + "It can be shown that if" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "## 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) $$ for all $x_1, x_2 \\in X$ and for all $t \\in [0,1]$. If $\\leq$ is replaced with a strict inequaltiy in the definition, we demand $x_1 \\neq x_2$ and $t\\in(0,1)$ then $f$ is said to be strictly convex. For a single variable function, convexity means that if you draw a straight line connecting $f(x_1)$ and $f(x_2)$, the value of the function on the interval $[x_1,x_2]$ is always below the line as illustrated below.\n", + "\n", + "## Conditions on convex functions\n", + "\n", + "In the following we state first and second-order conditions which\n", + "ensures convexity of a function $f$. We write $D_f$ to denote the\n", + "domain of $f$, i.e the subset of $R^n$ where $f$ is defined. For more\n", + "details and proofs we refer to: [S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press](http://stanford.edu/boyd/cvxbook/, 2004).\n", + "\n", + "**First order condition.**\n", + "\n", + "Suppose $f$ is differentiable (i.e $\\nabla f(x)$ is well defined for\n", + "all $x$ in the domain of $f$). Then $f$ is convex if and only if $D_f$\n", + "is a convex set and $$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", + "## 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", + "## 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", + "## 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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\boldsymbol{r}$ is the so-called residual or error in the iterative process.\n", + "\n", + "When we have found the exact solution, $\\boldsymbol{r}=0$.\n", + "\n", + "## Gradient method\n", + "\n", + "The residual is zero when we reach the minimum of the quadratic equation" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "P(\\boldsymbol{x})=\\frac{1}{2}\\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x} - \\boldsymbol{x}^T\\boldsymbol{b},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with the constraint that the matrix $\\boldsymbol{A}$ is positive definite and\n", + "symmetric. This defines also the Hessian and we want it to be positive definite. \n", + "\n", + "\n", + "## Steepest descent method\n", + "\n", + "We denote the initial guess for $\\boldsymbol{x}$ as $\\boldsymbol{x}_0$. \n", + "We can assume without loss of generality that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "instead.\n", + "\n", + "\n", + "## Steepest descent method\n", + "One can show that the solution $\\boldsymbol{x}$ is also the unique minimizer of the quadratic form" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and \n", + "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", + "\n", + "\n", + "\n", + "## Final expressions\n", + "We can compute the residual iteratively as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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": 1, + "metadata": {}, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "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": 2, + "metadata": {}, + "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": 3, + "metadata": {}, + "outputs": [], + "source": [ + "x = guesses[-1]\n", + "s = -df(x)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Run it!" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "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": 5, + "metadata": {}, + "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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Two vectors are conjugate if they are orthogonal with respect to \n", + "this inner product. Being conjugate is a symmetric relation: if $\\boldsymbol{s}$ is conjugate to $\\boldsymbol{t}$, then $\\boldsymbol{t}$ is conjugate to $\\boldsymbol{s}$.\n", + "\n", + "\n", + "\n", + "## Conjugate gradient method\n", + "An example is given by the eigenvectors of the matrix" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{v}_i^T\\boldsymbol{A}\\boldsymbol{v}_j= \\lambda\\boldsymbol{v}_i^T\\boldsymbol{v}_j,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which is zero unless $i=j$.\n", + "\n", + "\n", + "\n", + "\n", + "## Conjugate gradient method\n", + "Assume now that we have a symmetric positive-definite matrix $\\boldsymbol{A}$ of size\n", + "$n\\times n$. At each iteration $i+1$ we obtain the conjugate direction of a vector" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "instead.\n", + "\n", + "\n", + "\n", + "\n", + "## Conjugate gradient method\n", + "One can show that the solution $\\boldsymbol{x}$ is also the unique minimizer of the quadratic form" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and \n", + "$\\boldsymbol{x}_0=0$ it is equal $-\\boldsymbol{b}$.\n", + "The other vectors in the basis will be conjugate to the gradient, \n", + "hence the name conjugate gradient method.\n", + "\n", + "\n", + "\n", + "\n", + "## Conjugate gradient method\n", + "Let $\\boldsymbol{r}_k$ be the residual at the $k$-th step:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Revisiting our first homework\n", + "\n", + "We will use linear regression as a case study for the gradient descent\n", + "methods. Linear regression is a great test case for the gradient\n", + "descent methods discussed in the lectures since it has several\n", + "desirable properties such as:\n", + "\n", + "1. An analytical solution (recall homework set 1).\n", + "\n", + "2. The gradient can be computed analytically.\n", + "\n", + "3. The cost function is convex which guarantees that gradient descent converges for small enough learning rates\n", + "\n", + "We revisitan example similar to what we had in the first homework set. We had a function of the type" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [], + "source": [ + "x = 2*np.random.rand(m,1)\n", + "y = 4+3*x+np.random.randn(m,1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $x_i \\in [0,1] $ is chosen randomly using a uniform distribution. Additionally we have a stochastic noise chosen according to a normal distribution $\\cal {N}(0,1)$. \n", + "The linear regression model is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Gradient descent example\n", + "\n", + "Let $\\mathbf{y} = (y_1,\\cdots,y_n)^T$, $\\mathbf{\\boldsymbol{y}} = (\\boldsymbol{y}_1,\\cdots,\\boldsymbol{y}_n)^T$ and $\\beta = (\\beta_0, \\beta_1)^T$\n", + "\n", + "It is convenient to write $\\mathbf{\\boldsymbol{y}} = X\\beta$ where $X \\in \\mathbb{R}^{100 \\times 2} $ is the design matrix given by (we keep the intercept here)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "and we want to find $\\beta$ such that $C(\\beta)$ is minimized.\n", + "\n", + "## The derivative of the cost/loss function\n", + "\n", + "Computing $\\partial C(\\beta) / \\partial \\beta_0$ and $\\partial C(\\beta) / \\partial \\beta_1$ we can show that the gradient can be written as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $X$ is the design matrix defined above.\n", + "\n", + "## The Hessian matrix\n", + "The Hessian matrix of $C(\\beta)$ is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This result implies that $C(\\beta)$ is a convex function since the matrix $X^T X$ always is positive semi-definite.\n", + "\n", + "\n", + "\n", + "\n", + "## Simple program\n", + "\n", + "We can now write a program that minimizes $C(\\beta)$ using the gradient descent method with a constant learning rate $\\gamma$ according to" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "## Gradient Descent Example\n", + "\n", + "Here our simple example" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[3.67083588]\n", + " [3.14477055]]\n", + "[[3.67083588]\n", + " [3.14477055]]\n" + ] + }, + { + "data": { + "image/png": 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" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "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", + "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 = 0.1\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": 2, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[4.12702041]\n", + " [3.00710913]]\n", + "[4.12131119] [2.98875841]\n" + ] + } + ], + "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", + "x = 2*np.random.rand(100,1)\n", + "y = 4+3*x+np.random.randn(100,1)\n", + "\n", + "xb = np.c_[np.ones((100,1)), x]\n", + "beta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(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": [ + "\n", + "## 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) = ||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) = 2\\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": 9, + "metadata": {}, + "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", + "m = 100\n", + "x = 2*np.random.rand(m,1)\n", + "y = 4+3*x+np.random.randn(m,1)\n", + "\n", + "xb = np.c_[np.ones((m,1)), x]\n", + "XT_X = xb.T @ xb\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) @ xb.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/m*xb.T @ (xb @ (beta)-y)+2*lmbda*beta\n", + " beta -= eta*gradients\n", + "\n", + "print(beta)\n", + "ypredict = xb @ beta\n", + "ypredict2 = xb @ 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 energy 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 energy 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", + "## Friday September 25\n", + "\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Stochasticity/randomness is introduced by only taking the\n", + "gradient on a subset of the data called minibatches. If there are $n$\n", + "data points and the size of each minibatch is $M$, there will be $n/M$\n", + "minibatches. We denote these minibatches by $B_k$ where\n", + "$k=1,\\cdots,n/M$.\n", + "\n", + "## SGD example\n", + "As an example, suppose we have $10$ data points $(\\mathbf{x}_1,\\cdots, \\mathbf{x}_{10})$ \n", + "and we choose to have $M=5$ minibathces,\n", + "then each minibatch contains two data points. In particular we have\n", + "$B_1 = (\\mathbf{x}_1,\\mathbf{x}_2), \\cdots, B_5 =\n", + "(\\mathbf{x}_9,\\mathbf{x}_{10})$. Note that if you choose $M=1$ you\n", + "have only a single batch with all data points and on the other extreme,\n", + "you may choose $M=n$ resulting in a minibatch for each datapoint, i.e\n", + "$B_k = \\mathbf{x}_k$.\n", + "\n", + "The idea is now to approximate the gradient by replacing the sum over\n", + "all data points with a sum over the data points in one the minibatches\n", + "picked at random in each gradient descent step" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "## Simple example code" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "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", + "## 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", + "## 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": 11, + "metadata": {}, + "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": 12, + "metadata": {}, + "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", + "xb = np.c_[np.ones((m,1)), x]\n", + "theta_linreg = np.linalg.inv(xb.T.dot(xb)).dot(xb.T).dot(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*xb.T @ ((xb @ theta)-y)\n", + " theta -= eta*gradients\n", + "print(\"theta frm own gd\")\n", + "print(theta)\n", + "\n", + "xnew = np.array([[0],[2]])\n", + "xbnew = np.c_[np.ones((2,1)), xnew]\n", + "ypredict = xbnew.dot(theta)\n", + "ypredict2 = xbnew.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 = xb[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.\n", + "\n", + "\n", + "## Momentum based GD\n", + "\n", + "The stochastic gradient descent (SGD) is almost always used with a\n", + "*momentum* or inertia term that serves as a memory of the direction we\n", + "are moving in parameter space. This is typically implemented as\n", + "follows" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t) \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \n", + "\\boldsymbol{\\theta}_{t+1}= \\boldsymbol{\\theta}_t -\\mathbf{v}_{t},\n", + "\\label{_auto1} \\tag{2}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have introduced a momentum parameter $\\gamma$, with\n", + "$0\\le\\gamma\\le 1$, and for brevity we dropped the explicit notation to\n", + "indicate the gradient is to be taken over a different mini-batch at\n", + "each step. We call this algorithm gradient descent with momentum\n", + "(GDM). From these equations, it is clear that $\\mathbf{v}_t$ is a\n", + "running average of recently encountered gradients and\n", + "$(1-\\gamma)^{-1}$ sets the characteristic time scale for the memory\n", + "used in the averaging procedure. Consistent with this, when\n", + "$\\gamma=0$, this just reduces down to ordinary SGD as discussed\n", + "earlier. An equivalent way of writing the updates is" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\Delta \\boldsymbol{\\theta}_{t+1} = \\gamma \\Delta \\boldsymbol{\\theta}_t -\\ \\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have defined $\\Delta \\boldsymbol{\\theta}_{t}= \\boldsymbol{\\theta}_t-\\boldsymbol{\\theta}_{t-1}$.\n", + "\n", + "## More on momentum based approaches\n", + "\n", + "Let us try to get more intuition from these equations. It is helpful\n", + "to consider a simple physical analogy with a particle of mass $m$\n", + "moving in a viscous medium with drag coefficient $\\mu$ and potential\n", + "$E(\\mathbf{w})$. If we denote the particle's position by $\\mathbf{w}$,\n", + "then its motion is described by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "m {d^2 \\mathbf{w} \\over dt^2} + \\mu {d \\mathbf{w} \\over dt }= -\\nabla_w E(\\mathbf{w}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can discretize this equation in the usual way to get" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "m { \\mathbf{w}_{t+\\Delta t}-2 \\mathbf{w}_{t} +\\mathbf{w}_{t-\\Delta t} \\over (\\Delta t)^2}+\\mu {\\mathbf{w}_{t+\\Delta t}- \\mathbf{w}_{t} \\over \\Delta t} = -\\nabla_w E(\\mathbf{w}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Rearranging this equation, we can rewrite this as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\Delta \\mathbf{w}_{t +\\Delta t}= - { (\\Delta t)^2 \\over m +\\mu \\Delta t} \\nabla_w E(\\mathbf{w})+ {m \\over m +\\mu \\Delta t} \\Delta \\mathbf{w}_t.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Momentum parameter\n", + "\n", + "Notice that this equation is identical to previous one if we identify\n", + "the position of the particle, $\\mathbf{w}$, with the parameters\n", + "$\\boldsymbol{\\theta}$. This allows us to identify the momentum\n", + "parameter and learning rate with the mass of the particle and the\n", + "viscous drag as:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\gamma= {m \\over m +\\mu \\Delta t }, \\qquad \\eta = {(\\Delta t)^2 \\over m +\\mu \\Delta t}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Thus, as the name suggests, the momentum parameter is proportional to\n", + "the mass of the particle and effectively provides inertia.\n", + "Furthermore, in the large viscosity/small learning rate limit, our\n", + "memory time scales as $(1-\\gamma)^{-1} \\approx m/(\\mu \\Delta t)$.\n", + "\n", + "Why is momentum useful? SGD momentum helps the gradient descent\n", + "algorithm gain speed in directions with persistent but small gradients\n", + "even in the presence of stochasticity, while suppressing oscillations\n", + "in high-curvature directions. This becomes especially important in\n", + "situations where the landscape is shallow and flat in some directions\n", + "and narrow and steep in others. It has been argued that first-order\n", + "methods (with appropriate initial conditions) can perform comparable\n", + "to more expensive second order methods, especially in the context of\n", + "complex deep learning models.\n", + "\n", + "These beneficial properties of momentum can sometimes become even more\n", + "pronounced by using a slight modification of the classical momentum\n", + "algorithm called Nesterov Accelerated Gradient (NAG).\n", + "\n", + "In the NAG algorithm, rather than calculating the gradient at the\n", + "current parameters, $\\nabla_\\theta E(\\boldsymbol{\\theta}_t)$, one\n", + "calculates the gradient at the expected value of the parameters given\n", + "our current momentum, $\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma\n", + "\\mathbf{v}_{t-1})$. This yields the NAG update rule" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{v}_{t}=\\gamma \\mathbf{v}_{t-1}+\\eta_{t}\\nabla_\\theta E(\\boldsymbol{\\theta}_t +\\gamma \\mathbf{v}_{t-1}) \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \n", + "\\boldsymbol{\\theta}_{t+1}= \\boldsymbol{\\theta}_t -\\mathbf{v}_{t}.\n", + "\\label{_auto2} \\tag{3}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "One of the major advantages of NAG is that it allows for the use of a larger learning rate than GDM for the same choice of $\\gamma$.\n", + "\n", + "\n", + "## Second moment of the gradient\n", + "\n", + "\n", + "In stochastic gradient descent, with and without momentum, we still\n", + "have to specify a schedule for tuning the learning rates $\\eta_t$\n", + "as a function of time. As discussed in the context of Newton's\n", + "method, this presents a number of dilemmas. The learning rate is\n", + "limited by the steepest direction which can change depending on the\n", + "current position in the landscape. To circumvent this problem, ideally\n", + "our algorithm would keep track of curvature and take large steps in\n", + "shallow, flat directions and small steps in steep, narrow directions.\n", + "Second-order methods accomplish this by calculating or approximating\n", + "the Hessian and normalizing the learning rate by the\n", + "curvature. However, this is very computationally expensive for\n", + "extremely large models. Ideally, we would like to be able to\n", + "adaptively change the step size to match the landscape without paying\n", + "the steep computational price of calculating or approximating\n", + "Hessians.\n", + "\n", + "Recently, a number of methods have been introduced that accomplish\n", + "this by tracking not only the gradient, but also the second moment of\n", + "the gradient. These methods include AdaGrad, AdaDelta, RMS-Prop, and\n", + "ADAM.\n", + "\n", + "## RMS prop\n", + "\n", + "In RMS prop, in addition to keeping a running average of the first\n", + "moment of the gradient, we also keep track of the second moment\n", + "denoted by $\\mathbf{s}_t=\\mathbb{E}[\\mathbf{g}_t^2]$. The update rule\n", + "for RMS prop is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\mathbf{g}_t = \\nabla_\\theta E(\\boldsymbol{\\theta}) \n", + "\\label{_auto3} \\tag{4}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{s}_t =\\beta \\mathbf{s}_{t-1} +(1-\\beta)\\mathbf{g}_t^2 \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\mathbf{g}_t \\over \\sqrt{\\mathbf{s}_t +\\epsilon}}, \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\beta$ controls the averaging time of the second moment and is\n", + "typically taken to be about $\\beta=0.9$, $\\eta_t$ is a learning rate\n", + "typically chosen to be $10^{-3}$, and $\\epsilon\\sim 10^{-8} $ is a\n", + "small regularization constant to prevent divergences. Multiplication\n", + "and division by vectors is understood as an element-wise operation. It\n", + "is clear from this formula that the learning rate is reduced in\n", + "directions where the norm of the gradient is consistently large. This\n", + "greatly speeds up the convergence by allowing us to use a larger\n", + "learning rate for flat directions.\n", + "\n", + "\n", + "## ADAM optimizer\n", + "\n", + "A related algorithm is the ADAM optimizer. In ADAM, we keep a running\n", + "average of both the first and second moment of the gradient and use\n", + "this information to adaptively change the learning rate for different\n", + "parameters. In addition to keeping a running average of the first and\n", + "second moments of the gradient\n", + "(i.e. $\\mathbf{m}_t=\\mathbb{E}[\\mathbf{g}_t]$ and\n", + "$\\mathbf{s}_t=\\mathbb{E}[\\mathbf{g}^2_t]$, respectively), ADAM\n", + "performs an additional bias correction to account for the fact that we\n", + "are estimating the first two moments of the gradient using a running\n", + "average (denoted by the hats in the update rule below). The update\n", + "rule for ADAM is given by (where multiplication and division are once\n", + "again understood to be element-wise operations below)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation}\n", + "\\mathbf{g}_t = \\nabla_\\theta E(\\boldsymbol{\\theta}) \n", + "\\label{_auto4} \\tag{5}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{m}_t = \\beta_1 \\mathbf{m}_{t-1} + (1-\\beta_1) \\mathbf{g}_t \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbf{s}_t =\\beta_2 \\mathbf{s}_{t-1} +(1-\\beta_2)\\mathbf{g}_t^2 \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\mathbf{m}}_t={\\mathbf{m}_t \\over 1-\\beta_1^t} \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\mathbf{s}}_t ={\\mathbf{s}_t \\over1-\\beta_2^t} \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{\\theta}_{t+1}=\\boldsymbol{\\theta}_t - \\eta_t { \\boldsymbol{\\mathbf{m}}_t \\over \\sqrt{\\boldsymbol{\\mathbf{s}}_t} +\\epsilon}, \\nonumber\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "
\n", + "\n", + "$$\n", + "\\begin{equation} \n", + "\\label{_auto5} \\tag{6}\n", + "\\end{equation}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $\\beta_1$ and $\\beta_2$ set the memory lifetime of the first and\n", + "second moment and are typically taken to be $0.9$ and $0.99$\n", + "respectively, and $\\eta$ and $\\epsilon$ are identical to RMSprop.\n", + "\n", + "Like in RMSprop, the effective step size of a parameter depends on the\n", + "magnitude of its gradient squared. To understand this better, let us\n", + "rewrite this expression in terms of the variance\n", + "$\\boldsymbol{\\sigma}_t^2 = \\boldsymbol{\\mathbf{s}}_t -\n", + "(\\boldsymbol{\\mathbf{m}}_t)^2$. Consider a single parameter $\\theta_t$. The\n", + "update rule for this parameter is given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\Delta \\theta_{t+1}= -\\eta_t { \\boldsymbol{m}_t \\over \\sqrt{\\sigma_t^2 + m_t^2 }+\\epsilon}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Practical tips\n", + "\n", + "* **Randomize the data when making mini-batches**. It is always important to randomly shuffle the data when forming mini-batches. Otherwise, the gradient descent method can fit spurious correlations resulting from the order in which data is presented.\n", + "\n", + "* **Transform your inputs**. Learning becomes difficult when our landscape has a mixture of steep and flat directions. One simple trick for minimizing these situations is to standardize the data by subtracting the mean and normalizing the variance of input variables. Whenever possible, also decorrelate the inputs. To understand why this is helpful, consider the case of linear regression. It is easy to show that for the squared error cost function, the Hessian of the energy matrix is just the correlation matrix between the inputs. Thus, by standardizing the inputs, we are ensuring that the landscape looks homogeneous in all directions in parameter space. Since most deep networks can be viewed as linear transformations followed by a non-linearity at each layer, we expect this intuition to hold beyond the linear case.\n", + "\n", + "* **Monitor the out-of-sample performance.** Always monitor the performance of your model on a validation set (a small portion of the training data that is held out of the training process to serve as a proxy for the test set. If the validation error starts increasing, then the model is beginning to overfit. Terminate the learning process. This *early stopping* significantly improves performance in many settings.\n", + "\n", + "* **Adaptive optimization methods don't always have good generalization.** Recent studies have shown that adaptive methods such as ADAM, RMSPorp, and AdaGrad tend to have poor generalization compared to SGD or SGD with momentum, particularly in the high-dimensional limit (i.e. the number of parameters exceeds the number of data points). Although it is not clear at this stage why these methods perform so well in training deep neural networks, simpler procedures like properly-tuned SGD may work as well or better in these applications.\n", + "\n", + "Geron's text, see chapter 11, has several interesting discussions.\n", + "\n", + "\n", + "\n", + "## Automatic differentiation\n", + "\n", + "[Automatic differentiation (AD)](https://en.wikipedia.org/wiki/Automatic_differentiation), \n", + "also called algorithmic\n", + "differentiation or computational differentiation,is a set of\n", + "techniques to numerically evaluate the derivative of a function\n", + "specified by a computer program. AD exploits the fact that every\n", + "computer program, no matter how complicated, executes a sequence of\n", + "elementary arithmetic operations (addition, subtraction,\n", + "multiplication, division, etc.) and elementary functions (exp, log,\n", + "sin, cos, etc.). By applying the chain rule repeatedly to these\n", + "operations, derivatives of arbitrary order can be computed\n", + "automatically, accurately to working precision, and using at most a\n", + "small constant factor more arithmetic operations than the original\n", + "program.\n", + "\n", + "Automatic differentiation is neither:\n", + "\n", + "* Symbolic differentiation, nor\n", + "\n", + "* Numerical differentiation (the method of finite differences).\n", + "\n", + "Symbolic differentiation can lead to inefficient code and faces the\n", + "difficulty of converting a computer program into a single expression,\n", + "while numerical differentiation can introduce round-off errors in the\n", + "discretization process and cancellation\n", + "\n", + "\n", + "\n", + "Python has tools for so-called **automatic differentiation**.\n", + "Consider the following example" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "f(x) = \\sin\\left(2\\pi x + x^2\\right)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which has the following derivative" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "f'(x) = \\cos\\left(2\\pi x + x^2\\right)\\left(2\\pi + 2x\\right)\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Using **autograd** we have" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "\n", + "# To do elementwise differentiation:\n", + "from autograd import elementwise_grad as egrad \n", + "\n", + "# To plot:\n", + "import matplotlib.pyplot as plt \n", + "\n", + "\n", + "def f(x):\n", + " return np.sin(2*np.pi*x + x**2)\n", + "\n", + "def f_grad_analytic(x):\n", + " return np.cos(2*np.pi*x + x**2)*(2*np.pi + 2*x)\n", + "\n", + "# Do the comparison:\n", + "x = np.linspace(0,1,1000)\n", + "\n", + "f_grad = egrad(f)\n", + "\n", + "computed = f_grad(x)\n", + "analytic = f_grad_analytic(x)\n", + "\n", + "plt.title('Derivative computed from Autograd compared with the analytical derivative')\n", + "plt.plot(x,computed,label='autograd')\n", + "plt.plot(x,analytic,label='analytic')\n", + "\n", + "plt.xlabel('x')\n", + "plt.ylabel('y')\n", + "plt.legend()\n", + "\n", + "plt.show()\n", + "\n", + "print(\"The max absolute difference is: %g\"%(np.max(np.abs(computed - analytic))))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Using autograd\n", + "\n", + "Here we\n", + "experiment with what kind of functions Autograd is capable\n", + "of finding the gradient of. The following Python functions are just\n", + "meant to illustrate what Autograd can do, but please feel free to\n", + "experiment with other, possibly more complicated, functions as well." + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "\n", + "def f1(x):\n", + " return x**3 + 1\n", + "\n", + "f1_grad = grad(f1)\n", + "\n", + "# Remember to send in float as argument to the computed gradient from Autograd!\n", + "a = 1.0\n", + "\n", + "# See the evaluated gradient at a using autograd:\n", + "print(\"The gradient of f1 evaluated at a = %g using autograd is: %g\"%(a,f1_grad(a)))\n", + "\n", + "# Compare with the analytical derivative, that is f1'(x) = 3*x**2 \n", + "grad_analytical = 3*a**2\n", + "print(\"The gradient of f1 evaluated at a = %g by finding the analytic expression is: %g\"%(a,grad_analytical))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Autograd with more complicated functions\n", + "\n", + "To differentiate with respect to two (or more) arguments of a Python\n", + "function, Autograd need to know at which variable the function if\n", + "being differentiated with respect to." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f2(x1,x2):\n", + " return 3*x1**3 + x2*(x1 - 5) + 1\n", + "\n", + "# By sending the argument 0, Autograd will compute the derivative w.r.t the first variable, in this case x1\n", + "f2_grad_x1 = grad(f2,0)\n", + "\n", + "# ... and differentiate w.r.t x2 by sending 1 as an additional arugment to grad\n", + "f2_grad_x2 = grad(f2,1)\n", + "\n", + "x1 = 1.0\n", + "x2 = 3.0 \n", + "\n", + "print(\"Evaluating at x1 = %g, x2 = %g\"%(x1,x2))\n", + "print(\"-\"*30)\n", + "\n", + "# Compare with the analytical derivatives:\n", + "\n", + "# Derivative of f2 w.r.t x1 is: 9*x1**2 + x2:\n", + "f2_grad_x1_analytical = 9*x1**2 + x2\n", + "\n", + "# Derivative of f2 w.r.t x2 is: x1 - 5:\n", + "f2_grad_x2_analytical = x1 - 5\n", + "\n", + "# See the evaluated derivations:\n", + "print(\"The derivative of f2 w.r.t x1: %g\"%( f2_grad_x1(x1,x2) ))\n", + "print(\"The analytical derivative of f2 w.r.t x1: %g\"%( f2_grad_x1(x1,x2) ))\n", + "\n", + "print()\n", + "\n", + "print(\"The derivative of f2 w.r.t x2: %g\"%( f2_grad_x2(x1,x2) ))\n", + "print(\"The analytical derivative of f2 w.r.t x2: %g\"%( f2_grad_x2(x1,x2) ))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that the grad function will not produce the true gradient of the function. The true gradient of a function with two or more variables will produce a vector, where each element is the function differentiated w.r.t a variable.\n", + "\n", + "\n", + "## More complicated functions using the elements of their arguments directly" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f3(x): # Assumes x is an array of length 5 or higher\n", + " return 2*x[0] + 3*x[1] + 5*x[2] + 7*x[3] + 11*x[4]**2\n", + "\n", + "f3_grad = grad(f3)\n", + "\n", + "x = np.linspace(0,4,5)\n", + "\n", + "# Print the computed gradient:\n", + "print(\"The computed gradient of f3 is: \", f3_grad(x))\n", + "\n", + "# The analytical gradient is: (2, 3, 5, 7, 22*x[4])\n", + "f3_grad_analytical = np.array([2, 3, 5, 7, 22*x[4]])\n", + "\n", + "# Print the analytical gradient:\n", + "print(\"The analytical gradient of f3 is: \", f3_grad_analytical)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that in this case, when sending an array as input argument, the\n", + "output from Autograd is another array. This is the true gradient of\n", + "the function, as opposed to the function in the previous example. By\n", + "using arrays to represent the variables, the output from Autograd\n", + "might be easier to work with, as the output is closer to what one\n", + "could expect form a gradient-evaluting function.\n", + "\n", + "\n", + "## Functions using mathematical functions from Numpy" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f4(x):\n", + " return np.sqrt(1+x**2) + np.exp(x) + np.sin(2*np.pi*x)\n", + "\n", + "f4_grad = grad(f4)\n", + "\n", + "x = 2.7\n", + "\n", + "# Print the computed derivative:\n", + "print(\"The computed derivative of f4 at x = %g is: %g\"%(x,f4_grad(x)))\n", + "\n", + "# The analytical derivative is: x/sqrt(1 + x**2) + exp(x) + cos(2*pi*x)*2*pi\n", + "f4_grad_analytical = x/np.sqrt(1 + x**2) + np.exp(x) + np.cos(2*np.pi*x)*2*np.pi\n", + "\n", + "# Print the analytical gradient:\n", + "print(\"The analytical gradient of f4 at x = %g is: %g\"%(x,f4_grad_analytical))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## More autograd" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f5(x):\n", + " if x >= 0:\n", + " return x**2\n", + " else:\n", + " return -3*x + 1\n", + "\n", + "f5_grad = grad(f5)\n", + "\n", + "x = 2.7\n", + "\n", + "# Print the computed derivative:\n", + "print(\"The computed derivative of f5 at x = %g is: %g\"%(x,f5_grad(x)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## And with loops" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "1\n", + "8\n", + " \n", + "<\n", + "<\n", + "<\n", + "!\n", + "!\n", + "C\n", + "O\n", + "D\n", + "E\n", + "_\n", + "B\n", + "L\n", + "O\n", + "C\n", + "K\n", + " \n", + " \n", + "p\n", + "y\n", + "c\n", + "o\n", + "d" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "# Both of the functions are implementation of the sum: sum(x**i) for i = 0, ..., 9\n", + "# The analytical derivative is: sum(i*x**(i-1)) \n", + "f6_grad_analytical = 0\n", + "for i in range(10):\n", + " f6_grad_analytical += i*x**(i-1)\n", + "\n", + "print(\"The analytical derivative of f6 at x = %g is: %g\"%(x,f6_grad_analytical))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using recursion" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "\n", + "def f7(n): # Assume that n is an integer\n", + " if n == 1 or n == 0:\n", + " return 1\n", + " else:\n", + " return n*f7(n-1)\n", + "\n", + "f7_grad = grad(f7)\n", + "\n", + "n = 2.0\n", + "\n", + "print(\"The computed derivative of f7 at n = %d is: %g\"%(n,f7_grad(n)))\n", + "\n", + "# The function f7 is an implementation of the factorial of n.\n", + "# By using the product rule, one can find that the derivative is:\n", + "\n", + "f7_grad_analytical = 0\n", + "for i in range(int(n)-1):\n", + " tmp = 1\n", + " for k in range(int(n)-1):\n", + " if k != i:\n", + " tmp *= (n - k)\n", + " f7_grad_analytical += tmp\n", + "\n", + "print(\"The analytical derivative of f7 at n = %d is: %g\"%(n,f7_grad_analytical))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that if n is equal to zero or one, Autograd will give an error message. This message appears when the output is independent on input.\n", + "\n", + "## Unsupported functions\n", + "Autograd supports many features. However, there are some functions that is not supported (yet) by Autograd.\n", + "\n", + "Assigning a value to the variable being differentiated with respect to" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f8(x): # Assume x is an array\n", + " x[2] = 3\n", + " return x*2\n", + "\n", + "f8_grad = grad(f8)\n", + "\n", + "x = 8.4\n", + "\n", + "print(\"The derivative of f8 is:\",f8_grad(x))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here, Autograd tells us that an 'ArrayBox' does not support item assignment. The item assignment is done when the program tries to assign x[2] to the value 3. However, Autograd has implemented the computation of the derivative such that this assignment is not possible.\n", + "\n", + "## The syntax a.dot(b) when finding the dot product" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f9(a): # Assume a is an array with 2 elements\n", + " b = np.array([1.0,2.0])\n", + " return a.dot(b)\n", + "\n", + "f9_grad = grad(f9)\n", + "\n", + "x = np.array([1.0,0.0])\n", + "\n", + "print(\"The derivative of f9 is:\",f9_grad(x))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Here we are told that the 'dot' function does not belong to Autograd's\n", + "version of a Numpy array. To overcome this, an alternative syntax\n", + "which also computed the dot product can be used:" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [], + "source": [ + "import autograd.numpy as np\n", + "from autograd import grad\n", + "def f9_alternative(x): # Assume a is an array with 2 elements\n", + " b = np.array([1.0,2.0])\n", + " return np.dot(x,b) # The same as x_1*b_1 + x_2*b_2\n", + "\n", + "f9_alternative_grad = grad(f9_alternative)\n", + "\n", + "x = np.array([3.0,0.0])\n", + "\n", + "print(\"The gradient of f9 is:\",f9_alternative_grad(x))\n", + "\n", + "# The analytical gradient of the dot product of vectors x and b with two elements (x_1,x_2) and (b_1, b_2) respectively\n", + "# w.r.t x is (b_1, b_2)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Recommended to avoid\n", + "The documentation recommends to avoid inplace operations such as" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "metadata": {}, + "outputs": [], + "source": [ + "a += b\n", + "a -= b\n", + "a*= b\n", + "a /=b" + ] + } + ], + "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/pub/week41/ipynb/.ipynb_checkpoints/week41-checkpoint.ipynb b/doc/pub/week41/ipynb/.ipynb_checkpoints/week41-checkpoint.ipynb new file mode 100644 index 000000000..8303560d2 --- /dev/null +++ b/doc/pub/week41/ipynb/.ipynb_checkpoints/week41-checkpoint.ipynb @@ -0,0 +1,2531 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Week 41 Tensor flow and Deep Learning, Convolutional Neural Networks\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Oct 8, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "## Plan for week 41\n", + "\n", + "* Thursday: Building our own Feed-forward Neural Network. [Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureOctober8.mp4?vrtx=view-as-webpage)\n", + "\n", + "* Friday: Playing around with our own Feed-forward Neural Network and introduction to TensorFlow. Start convolutional Neural Networks (CNN).\n", + "\n", + "Reading suggestions for both days: [Aurelien Geron's chapters 10-11](https://github.com/CompPhysics/MachineLearning/blob/master/doc/T\\\n", + "extbooks/TensorflowML.pdf) and Hastie et al chapter 11.\n", + "\n", + "\n", + "## Setting up the Back propagation algorithm\n", + "\n", + "\n", + "\n", + "The four equations derived last week 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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The parameter $\\eta$ is the learning parameter discussed in connection with the gradient descent methods.\n", + "Here it is convenient to use stochastic gradient descent (see the examples below) with mini-batches with an outer loop that steps through multiple epochs of training.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Setting up a Multi-layer perceptron model for classification\n", + "\n", + "We are now gong to develop an example based on the MNIST data\n", + "base. This is a classification problem and we need to use our\n", + "cross-entropy function we discussed in connection with logistic\n", + "regression. The cross-entropy defines our cost function for the\n", + "classificaton problems with neural networks.\n", + "\n", + "In binary classification with two classes $(0, 1)$ we define the\n", + "logistic/sigmoid function as the probability that a particular input\n", + "is in class $0$ or $1$. This is possible because the logistic\n", + "function takes any input from the real numbers and inputs a number\n", + "between 0 and 1, and can therefore be interpreted as a probability. It\n", + "also has other nice properties, such as a derivative that is simple to\n", + "calculate.\n", + "\n", + "For an input $\\boldsymbol{a}$ from the hidden layer, the probability that the input $\\boldsymbol{x}$\n", + "is in class 0 or 1 is just. We let $\\theta$ represent the unknown weights and biases to be adjusted by our equations). The variable $x$\n", + "represents our activation values $z$. We have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where $y \\in \\{0, 1\\}$ and $\\hat{\\theta}$ represents the weights and biases\n", + "of our network.\n", + "\n", + "\n", + "## Defining the cost function\n", + "\n", + "Our cost function is given as (see the Logistic regression lectures)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "See the logistic regression lectures for a full definition of the cost function.\n", + "\n", + "The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!\n", + "\n", + "## Example: binary classification problem\n", + "\n", + "As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters $\\beta$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In case we use another activation function than the logistic one, we need to evaluate other derivatives. \n", + "\n", + "\n", + "## The Softmax function\n", + "In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation $z_i^l$, that is we need" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\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": {}, + "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": 2, + "metadata": {}, + "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}} ,$$\n", + "\n", + "which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.\n", + "\n", + "\n", + "## Layers\n", + "\n", + "* Input \n", + "\n", + "Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons. \n", + "\n", + "* Hidden layer\n", + "\n", + "We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer. \n", + "Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer. \n", + "\n", + "* Output\n", + "\n", + "If we were building a binary classifier, it would be sufficient with a single neuron in the output layer,\n", + "which could output 0 or 1 according to the Heaviside function. This would be an example of a *hard* classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a *soft* classifier, which outputs the probability of being in class 0 or 1. \n", + "\n", + "For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class. \n", + "\n", + "Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons $j = 0,1,...,9$. The activation of each output neuron $j$ will be according to the *softmax* function: \n", + "\n", + "$$ 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.$$ \n", + "\n", + "Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500\n", + "weights to the output layer.\n", + "\n", + "\n", + "## Weights and biases\n", + "\n", + "Typically weights are initialized with small values distributed around zero, drawn from a uniform\n", + "or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless. \n", + "\n", + "Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range\n", + "of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron $j$, $b_j$: \n", + "\n", + "$$ z_j = \\sum_{i=1}^n w_ {ij} a_i + b_j.$$ \n", + "\n", + "The bias weights $\\hat{b}$ are often initialized to zero, but a small value like $0.01$ ensures all neurons have some output which can be backpropagated in the first training cycle." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "# building our neural network\n", + "\n", + "n_inputs, n_features = X_train.shape\n", + "n_hidden_neurons = 50\n", + "n_categories = 10\n", + "\n", + "# we make the weights normally distributed using numpy.random.randn\n", + "\n", + "# weights and bias in the hidden layer\n", + "hidden_weights = np.random.randn(n_features, n_hidden_neurons)\n", + "hidden_bias = np.zeros(n_hidden_neurons) + 0.01\n", + "\n", + "# weights and bias in the output layer\n", + "output_weights = np.random.randn(n_hidden_neurons, n_categories)\n", + "output_bias = np.zeros(n_categories) + 0.01" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Feed-forward pass\n", + "\n", + "Denote $F$ the number of features, $H$ the number of hidden neurons and $C$ the number of categories. \n", + "For each input image we calculate a weighted sum of input features (pixel values) to each neuron $j$ in the hidden layer $l$: \n", + "\n", + "$$ 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})}} .$$ \n", + "\n", + "\n", + "## Matrix multiplications\n", + "\n", + "Since our data has the dimensions $X = (n_{inputs}, n_{features})$ and our weights to the hidden\n", + "layer have the dimensions \n", + "$W_{hidden} = (n_{features}, n_{hidden})$,\n", + "we can easily feed the network all our training data in one go by taking the matrix product \n", + "\n", + "$$ 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": {}, + "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) ,$$ \n", + "\n", + "\n", + "i.e. a binary bit string of length $C$, where $C = 10$ is the number of classes in the MNIST dataset. \n", + "\n", + "Let $y_{ic}$ denote the $c$-th component of the $i$-th one-hot vector. \n", + "We define the cost function $\\mathcal{C}$ as a sum over the cross-entropy loss for each point $\\hat{x}_i$ in the dataset.\n", + "\n", + "In the one-hot representation only one of the terms in the loss function is non-zero, namely the\n", + "probability of the correct category $c'$ \n", + "(i.e. the category $c'$ such that $y_{ic'} = 1$). This means that the cross entropy loss only punishes you for how wrong\n", + "you got the correct label. The probability of category $c$ is given by the softmax function. The vector $\\hat{\\theta}$ represents the parameters of our network, i.e. all the weights and biases. \n", + "\n", + "\n", + "## Optimizing the cost function\n", + "\n", + "The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is *gradient descent* and its generalizations. The idea behind gradient descent\n", + "is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a *local* minimum of the cost function. \n", + "Each parameter $\\theta$ is iteratively adjusted according to the rule \n", + "\n", + "$$ \\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": {}, + "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", + "## Full object-oriented implementation\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": {}, + "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": {}, + "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": 8, + "metadata": {}, + "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": 9, + "metadata": {}, + "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": 10, + "metadata": {}, + "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": 11, + "metadata": {}, + "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", + "## Tensorflow\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": {}, + "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)" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [], + "source": [ + "conda install tensorflow" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Collect and pre-process data" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "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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+ "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# 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": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "ename": "ModuleNotFoundError", + "evalue": "No module named 'keras'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mModuleNotFoundError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32mfrom\u001b[0m \u001b[0mkeras\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mutils\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mto_categorical\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 2\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0msklearn\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmodel_selection\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mtrain_test_split\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;31m# one-hot representation of labels\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[0mlabels\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mto_categorical\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlabels\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'keras'" + ] + } + ], + "source": [ + "from keras.utils import to_categorical\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": "markdown", + "metadata": {}, + "source": [ + "## Using TensorFlow backend\n", + "\n", + "1. Define model and architecture\n", + "\n", + "2. Choose cost function and optimizer" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "ename": "ModuleNotFoundError", + "evalue": "No module named 'tensorflow'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mModuleNotFoundError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32mimport\u001b[0m \u001b[0mtensorflow\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mtf\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;32mclass\u001b[0m \u001b[0mNeuralNetworkTensorflow\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 def __init__(\n\u001b[1;32m 5\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'tensorflow'" + ] + } + ], + "source": [ + "import tensorflow as tf\n", + "\n", + "class NeuralNetworkTensorflow:\n", + " def __init__(\n", + " self,\n", + " X_train,\n", + " Y_train,\n", + " X_test,\n", + " Y_test,\n", + " n_neurons_layer1=100,\n", + " n_neurons_layer2=50,\n", + " n_categories=2,\n", + " epochs=10,\n", + " batch_size=100,\n", + " eta=0.1,\n", + " lmbd=0.0):\n", + " \n", + " # keep track of number of steps\n", + " self.global_step = tf.Variable(0, dtype=tf.int32, trainable=False, name='global_step')\n", + " \n", + " self.X_train = X_train\n", + " self.Y_train = Y_train\n", + " self.X_test = X_test\n", + " self.Y_test = Y_test\n", + " \n", + " self.n_inputs = X_train.shape[0]\n", + " self.n_features = X_train.shape[1]\n", + " self.n_neurons_layer1 = n_neurons_layer1\n", + " self.n_neurons_layer2 = n_neurons_layer2\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", + " # build network piece by piece\n", + " # name scopes (with) are used to enforce creation of new variables\n", + " # https://www.tensorflow.org/guide/variables\n", + " self.create_placeholders()\n", + " self.create_DNN()\n", + " self.create_loss()\n", + " self.create_optimiser()\n", + " self.create_accuracy()\n", + " \n", + " def create_placeholders(self):\n", + " # placeholders are fine here, but \"Datasets\" are the preferred method\n", + " # of streaming data into a model\n", + " with tf.name_scope('data'):\n", + " self.X = tf.placeholder(tf.float32, shape=(None, self.n_features), name='X_data')\n", + " self.Y = tf.placeholder(tf.float32, shape=(None, self.n_categories), name='Y_data')\n", + " \n", + " def create_DNN(self):\n", + " with tf.name_scope('DNN'):\n", + " # the weights are stored to calculate regularization loss later\n", + " \n", + " # Fully connected layer 1\n", + " self.W_fc1 = self.weight_variable([self.n_features, self.n_neurons_layer1], name='fc1', dtype=tf.float32)\n", + " b_fc1 = self.bias_variable([self.n_neurons_layer1], name='fc1', dtype=tf.float32)\n", + " a_fc1 = tf.nn.sigmoid(tf.matmul(self.X, self.W_fc1) + b_fc1)\n", + " \n", + " # Fully connected layer 2\n", + " self.W_fc2 = self.weight_variable([self.n_neurons_layer1, self.n_neurons_layer2], name='fc2', dtype=tf.float32)\n", + " b_fc2 = self.bias_variable([self.n_neurons_layer2], name='fc2', dtype=tf.float32)\n", + " a_fc2 = tf.nn.sigmoid(tf.matmul(a_fc1, self.W_fc2) + b_fc2)\n", + " \n", + " # Output layer\n", + " self.W_out = self.weight_variable([self.n_neurons_layer2, self.n_categories], name='out', dtype=tf.float32)\n", + " b_out = self.bias_variable([self.n_categories], name='out', dtype=tf.float32)\n", + " self.z_out = tf.matmul(a_fc2, self.W_out) + b_out\n", + " \n", + " def create_loss(self):\n", + " with tf.name_scope('loss'):\n", + " softmax_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=self.Y, logits=self.z_out))\n", + " \n", + " regularizer_loss_fc1 = tf.nn.l2_loss(self.W_fc1)\n", + " regularizer_loss_fc2 = tf.nn.l2_loss(self.W_fc2)\n", + " regularizer_loss_out = tf.nn.l2_loss(self.W_out)\n", + " regularizer_loss = self.lmbd*(regularizer_loss_fc1 + regularizer_loss_fc2 + regularizer_loss_out)\n", + " \n", + " self.loss = softmax_loss + regularizer_loss\n", + "\n", + " def create_accuracy(self):\n", + " with tf.name_scope('accuracy'):\n", + " probabilities = tf.nn.softmax(self.z_out)\n", + " predictions = tf.argmax(probabilities, axis=1)\n", + " labels = tf.argmax(self.Y, axis=1)\n", + " \n", + " correct_predictions = tf.equal(predictions, labels)\n", + " correct_predictions = tf.cast(correct_predictions, tf.float32)\n", + " self.accuracy = tf.reduce_mean(correct_predictions)\n", + " \n", + " def create_optimiser(self):\n", + " with tf.name_scope('optimizer'):\n", + " self.optimizer = tf.train.GradientDescentOptimizer(learning_rate=self.eta).minimize(self.loss, global_step=self.global_step)\n", + " \n", + " def weight_variable(self, shape, name='', dtype=tf.float32):\n", + " initial = tf.truncated_normal(shape, stddev=0.1)\n", + " return tf.Variable(initial, name=name, dtype=dtype)\n", + " \n", + " def bias_variable(self, shape, name='', dtype=tf.float32):\n", + " initial = tf.constant(0.1, shape=shape)\n", + " return tf.Variable(initial, name=name, dtype=dtype)\n", + " \n", + " def fit(self):\n", + " data_indices = np.arange(self.n_inputs)\n", + "\n", + " with tf.Session() as sess:\n", + " sess.run(tf.global_variables_initializer())\n", + " for i in range(self.epochs):\n", + " for j in range(self.iterations):\n", + " chosen_datapoints = np.random.choice(data_indices, size=self.batch_size, replace=False)\n", + " batch_X, batch_Y = self.X_train[chosen_datapoints], self.Y_train[chosen_datapoints]\n", + " \n", + " sess.run([DNN.loss, DNN.optimizer],\n", + " feed_dict={DNN.X: batch_X,\n", + " DNN.Y: batch_Y})\n", + " accuracy = sess.run(DNN.accuracy,\n", + " feed_dict={DNN.X: batch_X,\n", + " DNN.Y: batch_Y})\n", + " step = sess.run(DNN.global_step)\n", + " \n", + " self.train_loss, self.train_accuracy = sess.run([DNN.loss, DNN.accuracy],\n", + " feed_dict={DNN.X: self.X_train,\n", + " DNN.Y: self.Y_train})\n", + " \n", + " self.test_loss, self.test_accuracy = sess.run([DNN.loss, DNN.accuracy],\n", + " feed_dict={DNN.X: self.X_test,\n", + " DNN.Y: self.Y_test})" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Optimizing and using gradient descent" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "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)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "ename": "NameError", + "evalue": "name 'NeuralNetworkTensorflow' 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 3\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0meta\u001b[0m \u001b[0;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0meta_vals\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 4\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlmbd\u001b[0m \u001b[0;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlmbd_vals\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----> 5\u001b[0;31m DNN = NeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,\n\u001b[0m\u001b[1;32m 6\u001b[0m \u001b[0mn_neurons_layer1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn_neurons_layer2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn_categories\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 7\u001b[0m epochs=epochs, batch_size=batch_size, eta=eta, lmbd=lmbd)\n", + "\u001b[0;31mNameError\u001b[0m: name 'NeuralNetworkTensorflow' is not defined" + ] + } + ], + "source": [ + "DNN_tf = 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 = NeuralNetworkTensorflow(X_train, Y_train, X_test, Y_test,\n", + " n_neurons_layer1, n_neurons_layer2, n_categories,\n", + " epochs=epochs, batch_size=batch_size, eta=eta, lmbd=lmbd)\n", + " DNN.fit()\n", + " \n", + " DNN_tf[i][j] = DNN\n", + " \n", + " print(\"Learning rate = \", eta)\n", + " print(\"Lambda = \", lmbd)\n", + " print(\"Test accuracy: %.3f\" % DNN.test_accuracy)\n", + " print()" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "ename": "AttributeError", + "evalue": "'int' object has no attribute 'train_accuracy'", + "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 13\u001b[0m \u001b[0mDNN\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mDNN_tf\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 14\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 15\u001b[0;31m \u001b[0mtrain_accuracy\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mDNN\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtrain_accuracy\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 16\u001b[0m \u001b[0mtest_accuracy\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mDNN\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtest_accuracy\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 17\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mAttributeError\u001b[0m: 'int' object has no attribute 'train_accuracy'" + ] + } + ], + "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_tf[i][j]\n", + "\n", + " train_accuracy[i][j] = DNN.train_accuracy\n", + " test_accuracy[i][j] = DNN.test_accuracy\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": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "ename": "NameError", + "evalue": "name 'tf' 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[0;31m# optional\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;31m# we can use log files to visualize our graph in Tensorboard\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mwriter\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mtf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msummary\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mFileWriter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'logs/'\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 4\u001b[0m \u001b[0mwriter\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0madd_graph\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget_default_graph\u001b[0m\u001b[0;34m(\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;31mNameError\u001b[0m: name 'tf' is not defined" + ] + } + ], + "source": [ + "# optional\n", + "# we can use log files to visualize our graph in Tensorboard\n", + "writer = tf.summary.FileWriter('logs/')\n", + "writer.add_graph(tf.get_default_graph())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Using Keras\n", + "\n", + "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 Tensorflow installed Keras is available through the *tf.keras* module. \n", + "If you have Anaconda installed you may run the following command" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [], + "source": [ + "conda install keras" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Alternatively, if you have Tensorflow or one of the other supported backends install you may use the pip package manager:" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [], + "source": [ + "pip install keras" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "or look up the [instructions here](https://keras.io/)." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "ename": "ModuleNotFoundError", + "evalue": "No module named 'tensorflow'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mModuleNotFoundError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32mimport\u001b[0m \u001b[0mtensorflow\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mtf\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 2\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mtensorflow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkeras\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlayers\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mInput\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mtensorflow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkeras\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmodels\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mSequential\u001b[0m \u001b[0;31m#This allows appending layers to existing models\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mtensorflow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkeras\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlayers\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mDense\u001b[0m \u001b[0;31m#This allows defining the characteristics of a particular layer\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mtensorflow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkeras\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0moptimizers\u001b[0m \u001b[0;31m#This allows using whichever optimiser we want (sgd,adam,RMSprop)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'tensorflow'" + ] + } + ], + "source": [ + "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", + "\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": 24, + "metadata": {}, + "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": 25, + "metadata": {}, + "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": 9, + "metadata": {}, + "outputs": [ + { + "ename": "ModuleNotFoundError", + "evalue": "No module named 'tensorflow'", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mModuleNotFoundError\u001b[0m Traceback (most recent call last)", + "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32mimport\u001b[0m \u001b[0mtensorflow\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mtf\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 2\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mtensorflow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkeras\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlayers\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mInput\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mtensorflow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkeras\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmodels\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mSequential\u001b[0m \u001b[0;31m#This allows appending layers to existing models\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mtensorflow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkeras\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlayers\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mDense\u001b[0m \u001b[0;31m#This allows defining the characteristics of a particular layer\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 5\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0mtensorflow\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mkeras\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0moptimizers\u001b[0m \u001b[0;31m#This allows using whichever optimiser we want (sgd,adam,RMSprop)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", + "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'tensorflow'" + ] + } + ], + "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": [ + "\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", + "## Is the Logistic activation function (Sigmoid) our choice?\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", + "## The derivative of the Logistic funtion\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": [ + "## Which activation function should we use?\n", + "\n", + "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", + "## 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", + "## 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.\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Convolutional Neural Networks (recognizing images)\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", + "Here we provide only a superficial overview, for the more interested, we recommend highly the course\n", + "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n", + "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/).\n", + "\n", + "Another good read is the article here . \n", + "\n", + "## Regular NNs don’t scale well to full images\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", + "

A regular 3-layer Neural Network.

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## 3D volumes of neurons\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", + "

A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Layers used to build CNNs\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", + "## Transforming images\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", + "## 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", + "For more material on convolutional networks, we strongly recommend\n", + "the course\n", + "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n", + "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/) which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). [Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs](http://neuralnetworksanddeeplearning.com/chap6.html)." + ] + } + ], + "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/pub/week42/ipynb/.ipynb_checkpoints/week42-checkpoint.ipynb b/doc/pub/week42/ipynb/.ipynb_checkpoints/week42-checkpoint.ipynb new file mode 100644 index 000000000..d95ea9940 --- /dev/null +++ b/doc/pub/week42/ipynb/.ipynb_checkpoints/week42-checkpoint.ipynb @@ -0,0 +1,1251 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Week 42 Convolutional (CNN) and Recurrent (RNN) Neural Networks and Autoencoders\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Oct 15, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "## Plan for week 42\n", + "\n", + "* Thursday: Convolutional Neural Networks and examples. [Video of Lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureOctober15.mp4?vrtx=view-as-webpage)\n", + "\n", + "* Friday: Recurrent Neural Networks and Autoencoders\n", + "\n", + "Reading suggestions for both days: [Aurelien Geron's chapters 13 and 14](https://github.com/CompPhysics/MachineLearning/blob/master/doc/Textbooks/TensorflowML.pdf). Autoencoders are discussed in chapter 15 of Geron's text.\n", + "\n", + "**Excellent lectures on CNNs and RNNs.**\n", + "\n", + "* [Video on Convolutional Neural Networks from MIT](https://www.youtube.com/watch?v=iaSUYvmCekI&ab_channel=AlexanderAmini)\n", + "\n", + "* [Video on Recurrent Neural Networks from MIT](https://www.youtube.com/watch?v=SEnXr6v2ifU&ab_channel=AlexanderAmini)\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Convolutional Neural Networks (recognizing images)\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", + "Here we provide only a superficial overview, for the more interested, we recommend highly the course\n", + "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n", + "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/).\n", + "\n", + "Another good read is the article here . \n", + "\n", + "\n", + "\n", + "\n", + "## Neural Networks vs CNNs\n", + "\n", + "Neural networks are defined as **affine transformations**, that is \n", + "a vector is received as input and is multiplied with a matrix of so-called weights (our unknown paramters) to produce an\n", + "output (to which a bias vector is usually added before passing the result\n", + "through a nonlinear activation function). This is applicable to any type of input, be it an\n", + "image, a sound clip or an unordered collection of features: whatever their\n", + "dimensionality, their representation can always be flattened into a vector\n", + "before the transformation.\n", + "\n", + "\n", + "## Why CNNS for images, sound files, medical images from CT scans etc?\n", + "\n", + "However, when we consider images, sound clips and many other similar kinds of data, these data have an intrinsic\n", + "structure. More formally, they share these important properties:\n", + "* They are stored as multi-dimensional arrays (think of the pixels of a figure) .\n", + "\n", + "* They feature one or more axes for which ordering matters (e.g., width and height axes for an image, time axis for a sound clip).\n", + "\n", + "* One axis, called the channel axis, is used to access different views of the data (e.g., the red, green and blue channels of a color image, or the left and right channels of a stereo audio track).\n", + "\n", + "These properties are not exploited when an affine transformation is applied; in\n", + "fact, all the axes are treated in the same way and the topological information\n", + "is not taken into account. Still, taking advantage of the implicit structure of\n", + "the data may prove very handy in solving some tasks, like computer vision and\n", + "speech recognition, and in these cases it would be best to preserve it. This is\n", + "where discrete convolutions come into play.\n", + "\n", + "A discrete convolution is a linear transformation that preserves this notion of\n", + "ordering. It is sparse (only a few input units contribute to a given output\n", + "unit) and reuses parameters (the same weights are applied to multiple locations\n", + "in the input).\n", + "\n", + "\n", + "\n", + "\n", + "## Regular NNs don’t scale well to full images\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", + "

A regular 3-layer Neural Network.

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## 3D volumes of neurons\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", + "

A CNN arranges its neurons in three dimensions (width, height, depth), as visualized in one of the layers. Every layer of a CNN transforms the 3D input volume to a 3D output volume of neuron activations. In this example, the red input layer holds the image, so its width and height would be the dimensions of the image, and the depth would be 3 (Red, Green, Blue channels).

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Layers used to build CNNs\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", + "## Transforming images\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", + "## 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", + "For more material on convolutional networks, we strongly recommend\n", + "the course\n", + "[IN5400 – Machine Learning for Image Analysis](https://www.uio.no/studier/emner/matnat/ifi/IN5400/index-eng.html)\n", + "and the slides of [CS231](http://cs231n.github.io/convolutional-networks/) which is taught at Stanford University (consistently ranked as one of the top computer science programs in the world). [Michael Nielsen's book is a must read, in particular chapter 6 which deals with CNNs](http://neuralnetworksanddeeplearning.com/chap6.html).\n", + "\n", + "\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", + "## Setting it up\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 again\n", + "\n", + "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", + "## Strong correlations\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", + "## Layers of a CNN\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", + "## Systematic reduction\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", + "## Prerequisites: Collect and pre-process data" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "inputs = (n_inputs, pixel_width, pixel_height, depth) = (1797, 8, 8, 1)\n", + "labels = (n_inputs) = (1797,)\n" + ] + }, + { + "data": { + "image/png": "iVBORw0KGgoAAAANSUhEUgAAAqwAAACRCAYAAAAGuepqAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjIsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+WH4yJAAALD0lEQVR4nO3dX2jl6VkH8OdpZ6HWP3O6KKJ2d6ItFbYXmxtRsTJnQBAESWBZEaw7GWnBK2eW6o0gyUi9EC82I17o1Wa6ghUtJKCLIDqJ1lW0sBnozYKUtGux2MqeuKsiWl8vTgbDMH+yz++cnDcznw8EMnvy/J73l/Oc93zPLyfZbK0FAAD06j2LXgAAADyIwAoAQNcEVgAAuiawAgDQNYEVAICuCawAAHTtkQ+smbmbmZ847VrOFnPCSZkVTsqscBLm5GTOTGDNzIPM/IlFr+N+curTmfnVzDw8GqKPLnpdj5szMCe/m5nvHPv4r8x8e9HrehyZFU7qDMzKz2bmG0fPPf+SmTcz8zsWva7HjTmZrzMTWM+A5yPiFyLixyPiyYj424h4ZaErojuttV9srX3bnY+I+IOI+KNFr4v+mBXehb+JiB9rrZ2PiB+IiHMR8enFLokOnek5OfOBNTM/kJl/kplfz8y3jj7/4F1f9qHM/PujVxU7mfnksfofyczXMnOSmbczc1xcyvdHxOdba19qrX0zIn4/Ip4pHosZ62hOjq/pWyPiuYi4OfRYzI5Z4aR6mZXW2puttW8c+0/fjIgPV47F7JmT2TjzgTWm5/ByRFyIiKcj4j8j4nfu+poXYnr183sj4n8i4rcjIjLz+yLiT2P6CuPJiPjliPhcZn7X3U0y8+mjYXn6Puv4bER8ODM/kplPRMTliPizgefG7PQyJ8c9FxFfj4i/qpwQc2NWOKluZiUzP5aZhxHxdkznZXPYqTFD5mQGznxgba39a2vtc621/2itvR0RvxERF+/6sldaa19srf17RPxaRPxMZr43Ij4eEa+21l5trf1va+3PI+ILEfFT9+jzldbaqLX2lfss5Z8j4q8j4o2YDuPzEfHiTE6SwTqak+MuR8RnWmtt0MkxU2aFk+ppVlprnz/6Ue8HI+K3IuJgJifJYOZkNs58YM3M92fm72XmlzPz32J6BWJ0dEff8eaxz78cEU9ExHfG9NXO80evSCaZOYmIj0XE9xSWsh4RPxQRT0XE+yLiekT8ZWa+v3AsZqyjObmznqdiumF9pnoM5sOscFK9zUpERGvtqzH96d5nhxyH2TEns3Fu0QuYgU9FxA9GxA+31r6WmcsR8XpE5LGveerY509HxH9HxDdiOiCvtNY+OYN1PBsRf9ha+6ejf29l5mZM38f6hRkcn2F6mZM7XoiI11prX5rhMZkNs8JJ9TYrd5yLiA/N4bjUmJMZOGtXWJ/IzPcd+zgXEd8e0x/BT47epLx+j7qPZ+YzR1c7fz0i/vjYL0b9dGb+ZGa+9+iY43u8Gfok/iGmr4K+OzPfk5k/H9NXSP9YOlOG6HlO7nghIrYG1DMbZoWT6nZWMvPnjt6/mJl5IaY/cv6L8pkyhDmZk7MWWF+N6Z1+52Mjpm8Y/paYvhL5u7j3Lzq9EtMN/2sx/XH9L0VMf2MuIlYi4ldj+gsNb0bEr8Q9vi9Hd/I7ef83M/9mRNyOiP2ImMT0/avPtdYm7/40GajnOYnM/NGYvn/InyhaPLPCSfU8K89ExGsR8U5M/3TRGxExjytyPJw5mZP0Hn4AAHp21q6wAgDwmBFYAQDomsAKAEDXBFYAALr2sL/Deuq/kbW1tVWq29jYKPccjUalus3N+v/RbDwel2sHyId/Sdmpz8ru7m6prjpjERHb29ulusPDw3LPW7duleoGzti8ZuXU52RnZ6dUd/Xq1Rmv5OGqMx0RsbS0NLN1vAvd7SkHBwflhtU9fcieUt0bzp8/X+65v79fqhs4Y13tKZNJ/Q/4LGJOquut3tcR/e0prrACANA1gRUAgK4JrAAAdE1gBQCgawIrAABdE1gBAOiawAoAQNcEVgAAuiawAgDQNYEVAICuCawAAHRNYAUAoGsCKwAAXTs3j4Pu7u6Wa69cuVKqW1lZKfccjUalutXV1XLPyWRSrmXq2rVrpboh3/u1tbVS3Y0bN8o9q/P5KDk4OCjXDnmcnrbt7e1ybfXx8KhZxPfh5s2b5dpbt26V6obsKZ5/hn0Pqo/TIXtRtefW1la558bGRrl2HlxhBQCgawIrAABdE1gBAOiawAoAQNcEVgAAuiawAgDQNYEVAICuCawAAHRNYAUAoGsCKwAAXRNYAQDomsAKAEDXBFYAALomsAIA0LVsrT3o9gfeeD/Xrl2rrSYiDg4OSnXb29vlnuPxuFQ3Go3KPYesd4Cc47FLszJEdVaG3G97e3ulusuXL5d7TiaTcu0A85qVU5+Tzc3NUt3y8nK556VLl0p1Fy9eLPfc3d0t1w7wSO0pi1B9vtzf3y/3fMRm5bGYk2pOGbKPVffOge47J66wAgDQNYEVAICuCawAAHRNYAUAoGsCKwAAXRNYAQDomsAKAEDXBFYAALomsAIA0DWBFQCArgmsAAB0TWAFAKBrAisAAF07N4+DLi0tlWsPDg5KdRsbG+Wee3t7pbrXX3+93JPhJpNJqa46YxER6+vrpbrRaFTuWV3vkMfho2Rtba1UN2RPqaruRRH19S7iPPl/y8vLpbqtra1yz+reOWQfe5RU9+TV1dXZLuQENjc3T73nvLjCCgBA1wRWAAC6JrACANA1gRUAgK4JrAAAdE1gBQCgawIrAABdE1gBAOiawAoAQNcEVgAAuiawAgDQNYEVAICuCawAAHRNYAUAoGvZWnvQ7Q+8cR6Wl5dLdbdv3y73vHz5cqlua2ur3HNBco7HLs3Kzs5OueHq6mq59ixZX18v1W1sbAxpO69ZKc3J/v5+ueF4PC7VHR4elntWVfeiiPr9vbS0VO4ZHe4pj4sh91t179zc3Cz3jM72lCGq+1F1L4qo70cvv/xyuefa2lq5doD7zokrrAAAdE1gBQCgawIrAABdE1gBAOiawAoAQNcEVgAAuiawAgDQNYEVAICuCawAAHRNYAUAoGsCKwAAXRNYAQDomsAKAEDXsrX2oNsfeOM8LC8vn3bLGI1Gpboha93c3CzXDpBzPHZpVnZ3d8sNt7e3S3X7+/vlngcHB6feszqfA81rVk59Ti5dulSurVpZWSnVVWd6gbrbUx4X4/H41HsOeRxGZ3vKZDKZ9ToeasheXr2/q89ZQ2sHuO+cuMIKAEDXBFYAALomsAIA0DWBFQCArgmsAAB0TWAFAKBrAisAAF0TWAEA6JrACgBA1wRWAAC6JrACANA1gRUAgK4JrAAAdE1gBQCga+cWvYC7jUajUt14PC733NjYKNVV17qonj0acr8dHh6W6ra2tso9V1dXS3WP2v122obMydWrV0t1N27cKPe8cuVKuZbF2dnZKdVduHCh3HN/f/9U6yLqzz+Pkr29vXLt+vp6qe769evlnmtra6W6IXvRZDIp1c3r+c4VVgAAuiawAgDQNYEVAICuCawAAHRNYAUAoGsCKwAAXRNYAQDomsAKAEDXBFYAALomsAIA0DWBFQCArgmsAAB0TWAFAKBr5xa9gLu9+OKLpbrV1dVyz+vXr5fqVlZWyj1Ho1G5lqm33nqrVHd4eFjuuba2Vq7lbHn22WfLtUP2BhbnpZdeKtXt7e2Ve54/f75UN2Qvso9FXLx4sVw7Ho9LddX5ioiYTCaluqtXr5Z79pZTXGEFAKBrAisAAF0TWAEA6JrACgBA1wRWAAC6JrACANA1gRUAgK4JrAAAdE1gBQCgawIrAABdE1gBAOiawAoAQNcEVgAAuiawAgDQtWytLXoNAABwX66wAgDQNYEVAICuCawAAHRNYAUAoGsCKwAAXRNYAQDo2v8BlXpliBTuHMEAAAAASUVORK5CYII=\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "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", + "# 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": 2, + "metadata": {}, + "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", + "#rt Cofrom tensorflow.keras imponv2D\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": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Running with Keras" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "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": "markdown", + "metadata": {}, + "source": [ + "## Final part" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "12/12 [==============================] - 0s 2ms/step - loss: 2.8826 - accuracy: 0.2444\n", + "Learning rate = 1e-05\n", + "Lambda = 1e-05\n", + "Test accuracy: 0.244\n", + "\n", + "12/12 [==============================] - 0s 2ms/step - loss: 3.3143 - accuracy: 0.0528\n", + "Learning rate = 1e-05\n", + "Lambda = 0.0001\n", + "Test accuracy: 0.053\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 2.6159 - accuracy: 0.1611\n", + "Learning rate = 1e-05\n", + "Lambda = 0.001\n", + "Test accuracy: 0.161\n", + "\n", + "12/12 [==============================] - 0s 2ms/step - loss: 4.6617 - accuracy: 0.1278\n", + "Learning rate = 1e-05\n", + "Lambda = 0.01\n", + "Test accuracy: 0.128\n", + "\n", + "12/12 [==============================] - 0s 2ms/step - loss: 12.1948 - accuracy: 0.1139\n", + "Learning rate = 1e-05\n", + "Lambda = 0.1\n", + "Test accuracy: 0.114\n", + "\n", + "12/12 [==============================] - 0s 2ms/step - loss: 91.3207 - accuracy: 0.0917\n", + "Learning rate = 1e-05\n", + "Lambda = 1.0\n", + "Test accuracy: 0.092\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 517.6693 - accuracy: 0.1250\n", + "Learning rate = 1e-05\n", + "Lambda = 10.0\n", + "Test accuracy: 0.125\n", + "\n", + "12/12 [==============================] - 0s 2ms/step - loss: 1.3215 - accuracy: 0.6111\n", + "Learning rate = 0.0001\n", + "Lambda = 1e-05\n", + "Test accuracy: 0.611\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 1.2700 - accuracy: 0.5889\n", + "Learning rate = 0.0001\n", + "Lambda = 0.0001\n", + "Test accuracy: 0.589\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 1.4245 - accuracy: 0.5806\n", + "Learning rate = 0.0001\n", + "Lambda = 0.001\n", + "Test accuracy: 0.581\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 2.6471 - accuracy: 0.4556\n", + "Learning rate = 0.0001\n", + "Lambda = 0.01\n", + "Test accuracy: 0.456\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 10.4180 - accuracy: 0.5139\n", + "Learning rate = 0.0001\n", + "Lambda = 0.1\n", + "Test accuracy: 0.514\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 54.1625 - accuracy: 0.2583\n", + "Learning rate = 0.0001\n", + "Lambda = 1.0\n", + "Test accuracy: 0.258\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 4.5475 - accuracy: 0.0889\n", + "Learning rate = 0.0001\n", + "Lambda = 10.0\n", + "Test accuracy: 0.089\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 0.2355 - accuracy: 0.9306\n", + "Learning rate = 0.001\n", + "Lambda = 1e-05\n", + "Test accuracy: 0.931\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 0.2488 - accuracy: 0.9333\n", + "Learning rate = 0.001\n", + "Lambda = 0.0001\n", + "Test accuracy: 0.933\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 0.3576 - accuracy: 0.9194\n", + "Learning rate = 0.001\n", + "Lambda = 0.001\n", + "Test accuracy: 0.919\n", + "\n", + "12/12 [==============================] - 0s 1ms/step - loss: 1.2576 - accuracy: 0.8778\n", + "Learning rate = 0.001\n", + "Lambda = 0.01\n", + "Test accuracy: 0.878\n", + "\n", + "12/12 [==============================] - 0s 2ms/step - loss: 5.8163 - accuracy: 0.9167\n", + "Learning rate = 0.001\n", + "Lambda = 0.1\n", + "Test accuracy: 0.917\n", + "\n" + ] + } + ], + "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": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "1437/1437 [==============================] - 0s 43us/sample - loss: 3.3022 - accuracy: 0.1872\n", + "360/360 [==============================] - 0s 121us/sample - loss: 3.4180 - accuracy: 0.1778\n", + "1437/1437 [==============================] - 0s 86us/sample - loss: 3.3955 - accuracy: 0.1093\n", + "360/360 [==============================] - 0s 142us/sample - loss: 3.4203 - accuracy: 0.0917\n", + "1437/1437 [==============================] - 0s 80us/sample - loss: 2.7250 - accuracy: 0.1587\n", + "360/360 [==============================] - 0s 216us/sample - loss: 2.7661 - accuracy: 0.1556\n", + "1437/1437 [==============================] - 0s 66us/sample - loss: 3.5698 - accuracy: 0.1343\n", + "360/360 [==============================] - 0s 46us/sample - loss: 3.5947 - accuracy: 0.1167\n", + "1437/1437 [==============================] - 0s 63us/sample - loss: 12.5837 - accuracy: 0.0946\n", + "360/360 [==============================] - 0s 60us/sample - loss: 12.5511 - accuracy: 0.1111\n", + "1437/1437 [==============================] - 0s 59us/sample - loss: 91.5210 - accuracy: 0.2408\n", + "360/360 [==============================] - 0s 53us/sample - loss: 91.5551 - accuracy: 0.2222\n", + "1437/1437 [==============================] - 0s 64us/sample - loss: 518.1178 - accuracy: 0.1969\n", + "360/360 [==============================] - 0s 48us/sample - loss: 518.1064 - accuracy: 0.1889\n", + "1437/1437 [==============================] - 0s 66us/sample - loss: 1.4465 - accuracy: 0.5623\n", + "360/360 [==============================] - 0s 37us/sample - loss: 1.4667 - accuracy: 0.5444\n", + "1437/1437 [==============================] - 0s 63us/sample - loss: 1.0335 - accuracy: 0.7015\n", + "360/360 [==============================] - 0s 75us/sample - loss: 1.0560 - accuracy: 0.6806\n", + "1437/1437 [==============================] - 0s 62us/sample - loss: 1.9454 - accuracy: 0.3730\n", + "360/360 [==============================] - 0s 48us/sample - loss: 2.0023 - accuracy: 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\n", 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "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": 7, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Downloading data from https://www.cs.toronto.edu/~kriz/cifar-10-python.tar.gz\n", + "170500096/170498071 [==============================] - 35s 0us/step\n" + ] + } + ], + "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": [ + "## Verifying the data set\n", + "\n", + "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": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "class_names = ['airplane', 'automobile', 'bird', 'cat', 'deer',\n", + " 'dog', 'frog', 'horse', 'ship', 'truck']\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": [ + "## Set up the model\n", + "\n", + "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": 10, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Model: \"sequential_49\"\n", + "_________________________________________________________________\n", + "Layer (type) Output Shape Param # \n", + "=================================================================\n", + "conv2d_49 (Conv2D) (None, 30, 30, 32) 896 \n", + "_________________________________________________________________\n", + "max_pooling2d_49 (MaxPooling (None, 15, 15, 32) 0 \n", + "_________________________________________________________________\n", + "conv2d_50 (Conv2D) (None, 13, 13, 64) 18496 \n", + "_________________________________________________________________\n", + "max_pooling2d_50 (MaxPooling (None, 6, 6, 64) 0 \n", + "_________________________________________________________________\n", + "conv2d_51 (Conv2D) (None, 4, 4, 64) 36928 \n", + "=================================================================\n", + "Total params: 56,320\n", + "Trainable params: 56,320\n", + "Non-trainable params: 0\n", + "_________________________________________________________________\n" + ] + } + ], + "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", + "\n", + "## Add Dense layers on top\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": 12, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Model: \"sequential_49\"\n", + "_________________________________________________________________\n", + "Layer (type) Output Shape Param # \n", + "=================================================================\n", + "conv2d_49 (Conv2D) (None, 30, 30, 32) 896 \n", + "_________________________________________________________________\n", + "max_pooling2d_49 (MaxPooling (None, 15, 15, 32) 0 \n", + "_________________________________________________________________\n", + "conv2d_50 (Conv2D) (None, 13, 13, 64) 18496 \n", + "_________________________________________________________________\n", + "max_pooling2d_50 (MaxPooling (None, 6, 6, 64) 0 \n", + "_________________________________________________________________\n", + "conv2d_51 (Conv2D) (None, 4, 4, 64) 36928 \n", + "_________________________________________________________________\n", + "flatten_49 (Flatten) (None, 1024) 0 \n", + "_________________________________________________________________\n", + "dense_98 (Dense) (None, 64) 65600 \n", + "_________________________________________________________________\n", + "dense_99 (Dense) (None, 10) 650 \n", + "=================================================================\n", + "Total params: 122,570\n", + "Trainable params: 122,570\n", + "Non-trainable params: 0\n", + "_________________________________________________________________\n" + ] + } + ], + "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": 13, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Train on 50000 samples, validate on 10000 samples\n", + "Epoch 1/10\n", + "50000/50000 [==============================] - 40s 793us/sample - loss: 1.5115 - accuracy: 0.4515 - val_loss: 1.2411 - val_accuracy: 0.5545\n", + "Epoch 2/10\n", + "50000/50000 [==============================] - 41s 826us/sample - loss: 1.1297 - accuracy: 0.6006 - val_loss: 1.0419 - val_accuracy: 0.6307\n", + "Epoch 3/10\n", + "50000/50000 [==============================] - 43s 870us/sample - loss: 0.9842 - accuracy: 0.6534 - val_loss: 1.0402 - val_accuracy: 0.6314\n", + "Epoch 4/10\n", + "50000/50000 [==============================] - 43s 869us/sample - loss: 0.8824 - accuracy: 0.6894 - val_loss: 0.9944 - val_accuracy: 0.6599\n", + "Epoch 5/10\n", + "50000/50000 [==============================] - 40s 803us/sample - loss: 0.8098 - accuracy: 0.7171 - val_loss: 0.9176 - val_accuracy: 0.6829\n", + "Epoch 6/10\n", + "50000/50000 [==============================] - 46s 925us/sample - loss: 0.7469 - accuracy: 0.7370 - val_loss: 0.8683 - val_accuracy: 0.7072\n", + "Epoch 7/10\n", + "50000/50000 [==============================] - 43s 857us/sample - loss: 0.6939 - accuracy: 0.7546 - val_loss: 0.8628 - val_accuracy: 0.7055\n", + "Epoch 8/10\n", + "50000/50000 [==============================] - 38s 770us/sample - loss: 0.6492 - accuracy: 0.7719 - val_loss: 0.8725 - val_accuracy: 0.7120\n", + "Epoch 9/10\n", + "50000/50000 [==============================] - 37s 743us/sample - loss: 0.6064 - accuracy: 0.7881 - val_loss: 0.8604 - val_accuracy: 0.7144\n", + "Epoch 10/10\n", + "50000/50000 [==============================] - 36s 715us/sample - loss: 0.5675 - accuracy: 0.8003 - val_loss: 0.8882 - val_accuracy: 0.7137\n" + ] + } + ], + "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, evaluate the model" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "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", + "## A simple example" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "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()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Set up of an RNN\n", + "\n", + "The figure here displays a simple example of an RNN, with inputs $x_t$\n", + "at a given time $t$ and outputs $y_t$. Introducing time as a variable\n", + "offers an intutitive way of understanding these networks. In addition\n", + "to the inputs $x_t$, the layer at a time $t$ receives also as input\n", + "the output from the previous layer $t-1$, that is $y_{t1}$.\n", + "\n", + "This means also that we need to have weights that link both the inputs\n", + "$x_t$ to the outputs $y_t$ as well as weights that link the output\n", + "from the previous time $y_{t-1}$ and $y_t$. The figure here shows an\n", + "example of a simple RNN.\n", + "\n", + "More material will be added here.\n", + "\n", + "\n", + "## Solving differential equations and eigenvalue problems with RNNs\n", + "\n", + "\n", + "\n", + "In our discussions of ordinary differential equations and partial\n", + "differential equations using neural networks. Here we will discuss how\n", + "we can solve say ordinary differential equations and eigenvalue\n", + "problems using RNNs. Eigenvalue problems can be solved using RNNs by\n", + "rewriting such a problems as a non-linear differential equation.\n", + "\n", + "Instead of starting with a well-known ordinary differential equation,\n", + "we start directly with an eigenvaule problem.\n", + "\n", + "\n", + "\n", + "## Long-Short Time Memory\n", + "\n", + "Discussions about dynamic unrolling through time. discuss memory cells, input and output\n", + "\n", + "\n", + "\n", + "\n", + "## Autoencoders: Overarching view\n", + "\n", + "Autoencoders are artificial neural networks capable of learning\n", + "efficient representations of the input data (these representations are called codings) without\n", + "any supervision (i.e., the training set is unlabeled). These codings\n", + "typically have a much lower dimensionality than the input data, making\n", + "autoencoders useful for dimensionality reduction. \n", + "\n", + "More importantly, autoencoders act as powerful feature detectors, and\n", + "they can be used for unsupervised pretraining of deep neural networks.\n", + "\n", + "Lastly, they are capable of randomly generating new data that looks\n", + "very similar to the training data; this is called a generative\n", + "model. For example, you could train an autoencoder on pictures of\n", + "faces, and it would then be able to generate new faces. Surprisingly,\n", + "autoencoders work by simply learning to copy their inputs to their\n", + "outputs. This may sound like a trivial task, but we will see that\n", + "constraining the network in various ways can make it rather\n", + "difficult. For example, you can limit the size of the internal\n", + "representation, or you can add noise to the inputs and train the\n", + "network to recover the original inputs. These constraints prevent the\n", + "autoencoder from trivially copying the inputs directly to the outputs,\n", + "which forces it to learn efficient ways of representing the data. In\n", + "short, the codings are byproducts of the autoencoder’s attempt to\n", + "learn the identity function under some constraints.\n", + "\n", + "## Simple examples of Autoencoders" + ] + } + ], + "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/pub/week42/ipynb/VaryDimensionPredicted12.csv b/doc/pub/week42/ipynb/VaryDimensionPredicted12.csv new file mode 100644 index 000000000..e4a19e3dd --- /dev/null +++ b/doc/pub/week42/ipynb/VaryDimensionPredicted12.csv @@ -0,0 +1,20 @@ +-3.077640548999999864e-02 +-8.336233265999999642e-02 +-1.446729566999999939e-01 +-2.116753731999999888e-01 +-2.830637391999999974e-01 +-3.581341341000000011e-01 +-4.364624349999999819e-01 +-5.177783846000000301e-01 +-6.019067271000000385e-01 +-6.887363571000000295e-01 +-7.782028951999999666e-01 +-8.702784033999999558e-01 +-9.662013053894042969e-01 +-1.064232707023620605e+00 +-1.165069699287414551e+00 +-1.268005967140197754e+00 +-1.373028755187988281e+00 +-1.479670405387878418e+00 +-1.587635993957519531e+00 +-1.696464180946350098e+00 diff --git a/doc/pub/week43/ipynb/.ipynb_checkpoints/week43-checkpoint.ipynb b/doc/pub/week43/ipynb/.ipynb_checkpoints/week43-checkpoint.ipynb new file mode 100644 index 000000000..76a2342cd --- /dev/null +++ b/doc/pub/week43/ipynb/.ipynb_checkpoints/week43-checkpoint.ipynb @@ -0,0 +1,4634 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Week 43: Solving Differential Equations with Deep Learning and Dimensionality Reduction methods\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Oct 23, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "* Thursday: Wrapping up Recurrent Neural Networks and solving differential equations. [Video of Lecture October 22](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureOctober22.mp4?vrtx=view-as-webpage)\n", + "\n", + "* Friday: Principal Component Analysis and Dimensionality Reduction. [Video of Lecture October 23](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureOctober23.mp4?vrtx=view-as-webpage)\n", + "\n", + "We will also study the usage of [Autograd](https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola) in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from [week 40](https://compphysics.github.io/MachineLearning/doc/pub/week40/html/week40.html) and the [Autograd doucmentation](https://github.com/HIPS/autograd).\n", + "\n", + "## Recurrent Neural Networks\n", + "\n", + "[Overview video](https://www.youtube.com/watch?v=SEnXr6v2ifU&ab_channel=AlexanderAmini).\n", + "See also lecture on Thursday October 22 and examples from [week 42](https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week42.html).\n", + "\n", + "[IN5400 at UiO Lecture](https://www.uio.no/studier/emner/matnat/ifi/IN5400/v20/material/week10/in5400_2020_week10_recurrent_neural_network.pdf)\n", + "\n", + "[CS231 at Stanford Lecture](https://www.youtube.com/watch?v=6niqTuYFZLQ&list=PLzUTmXVwsnXod6WNdg57Yc3zFx_f-RYsq&index=10&ab_channel=StanfordUniversitySchoolofEngineering)\n", + "\n", + "## Solving ODEs 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", + "## Ordinary Differential Equations\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", + "## The trial solution\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", + "## Minimization process\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", + "## Minimizing the cost function using gradient descent and automatic differentiation\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", + "## The function to solve for\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", + "## The trial solution\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", + "## Setup of Network\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", + "## More technicalities\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", + "## More details\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", + "## A possible implementation of a neural network\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", + "## Technicalities\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": [ + "## Final technicalities I\n", + "\n", + "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": [ + "## Final technicalities II\n", + "\n", + "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", + "## Final technicalities III\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": [ + "## Final technicalities IV\n", + "\n", + "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", + "## Back propagation\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": {}, + "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": 2, + "metadata": {}, + "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", + "## Setting up the problem\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", + "## The trial solution\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", + "## The program using Autograd\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": {}, + "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": 4, + "metadata": {}, + "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": [ + "## Example: 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", + "## The specific equation to solve for\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": "markdown", + "metadata": {}, + "source": [ + "## Solving the equation using Autograd" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "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", + "## Setting up the code\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": {}, + "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", + "## 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": [ + "## More details\n", + "\n", + "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", + "## Defining the problem\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", + "## Setting up the network using Autograd\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": {}, + "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": [ + "## Setting up the network using Autograd; The trial solution\n", + "\n", + "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", + "## Why the jacobian?\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": {}, + "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": {}, + "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": [ + "## Example: 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", + "## The problem to solve for\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", + "## The trial solution\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", + "## The analytical solution\n", + "\n", + "The analytical solution for our specific problem, is\n", + "\n", + "$$\n", + "g(x,t) = \\sin(\\pi x)\\cos(\\pi t) - \\sin(\\pi x)\\sin(\\pi t)\n", + "$$\n", + "\n", + "## Solving the wave equation - the full program using Autograd" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "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)\n", + "\n", + "## Friday, Principal Component Analysis\n", + "\n", + "[Overview video](https://www.youtube.com/watch?v=fkf4IBRSeEc&ab_channel=SteveBrunton)\n", + "\n", + "## 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", + "## 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", + "## Correlation Function and Design/Feature Matrix\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": [ + "## Covariance Matrix Examples\n", + "\n", + "\n", + "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": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "-0.0295549375800962\n", + "3.790157415516731\n", + "[[ 1.14945017 3.28385419]\n", + " [ 3.28385419 10.22579788]]\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": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.08073726712724406\n", + "1.6145539590295142\n", + "[[1. 0.63404481]\n", + " [0.63404481 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", + "## Correlation Matrix with Pandas\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": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[ 0.01396941 0.49068974]\n", + " [ 0.54918099 2.04838299]\n", + " [-0.35991553 -1.16529785]\n", + " [ 0.74909071 1.11467729]\n", + " [ 1.10998316 4.0040917 ]\n", + " [-0.98934642 -2.16772616]\n", + " [ 0.25009971 0.75283979]\n", + " [-0.57918262 -1.70870953]\n", + " [-0.98545332 -3.90181134]\n", + " [ 0.24157391 0.53286336]]\n", + " 0 1\n", + "0 0.013969 0.490690\n", + "1 0.549181 2.048383\n", + "2 -0.359916 -1.165298\n", + "3 0.749091 1.114677\n", + "4 1.109983 4.004092\n", + "5 -0.989346 -2.167726\n", + "6 0.250100 0.752840\n", + "7 -0.579183 -1.708710\n", + "8 -0.985453 -3.901811\n", + "9 0.241574 0.532863\n", + " 0 1\n", + "0 1.000000 0.959994\n", + "1 0.959994 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.\n", + "\n", + "## Correlation Matrix with Pandas and the Franke function" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "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 0.072184 0.069825 0.069428 0.071162 0.072822 0.060421 0.061903 \n", + "2 0.0 0.069825 0.069297 0.065490 0.068155 0.070919 0.055665 0.057749 \n", + "3 0.0 0.069428 0.065490 0.071945 0.072269 0.072413 0.065947 0.066493 \n", + "4 0.0 0.071162 0.068155 0.072269 0.073368 0.074365 0.065102 0.066207 \n", + "5 0.0 0.072822 0.070919 0.072413 0.074365 0.076313 0.064012 0.065724 \n", + "6 0.0 0.060421 0.055665 0.065947 0.065102 0.064012 0.062745 0.062435 \n", + "7 0.0 0.061903 0.057749 0.066493 0.066207 0.065724 0.062435 0.062545 \n", + "8 0.0 0.063660 0.060176 0.067220 0.067552 0.067741 0.062209 0.062780 \n", + "9 0.0 0.065675 0.062949 0.068101 0.069119 0.070057 0.062036 0.063114 \n", + "10 0.0 0.052443 0.047291 0.059402 0.057799 0.055912 0.058081 0.057183 \n", + "11 0.0 0.053341 0.048621 0.059668 0.058469 0.057019 0.057756 0.057169 \n", + "12 0.0 0.054483 0.050229 0.060122 0.059366 0.058394 0.057550 0.057302 \n", + "13 0.0 0.055889 0.052145 0.060775 0.060506 0.060062 0.057464 0.057588 \n", + "14 0.0 0.057577 0.054396 0.061634 0.061905 0.062047 0.057497 0.058031 \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 0.063660 0.065675 0.052443 0.053341 0.054483 0.055889 0.057577 \n", + "2 0.060176 0.062949 0.047291 0.048621 0.050229 0.052145 0.054396 \n", + "3 0.067220 0.068101 0.059402 0.059668 0.060122 0.060775 0.061634 \n", + "4 0.067552 0.069119 0.057799 0.058469 0.059366 0.060506 0.061905 \n", + "5 0.067741 0.070057 0.055912 0.057019 0.058394 0.060062 0.062047 \n", + "6 0.062209 0.062036 0.058081 0.057756 0.057550 0.057464 0.057497 \n", + "7 0.062780 0.063114 0.057183 0.057169 0.057302 0.057588 0.058031 \n", + "8 0.063524 0.064419 0.056301 0.056626 0.057130 0.057825 0.058718 \n", + "9 0.064419 0.065935 0.055400 0.056095 0.057007 0.058150 0.059542 \n", + "10 0.056301 0.055400 0.054868 0.054129 0.053455 0.052843 0.052283 \n", + "11 0.056626 0.056095 0.054129 0.053624 0.053205 0.052870 0.052614 \n", + "12 0.057130 0.057007 0.053455 0.053205 0.053063 0.053031 0.053108 \n", + "13 0.057825 0.058150 0.052843 0.052870 0.053031 0.053332 0.053775 \n", + "14 0.058718 0.059542 0.052283 0.052614 0.053108 0.053775 0.054625 \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", + "## Rewriting the Covariance and/or Correlation Matrix\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": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}\\boldsymbol{X}^T= \\mathbb{E}[\\boldsymbol{X}\\boldsymbol{X}^T].\n", + "$$" + ] + }, + { + "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": [ + "$$\n", + "\\mathbb{E}[\\boldsymbol{X}\\boldsymbol{X}^T] = \\frac{1}{n}\\boldsymbol{X}\\boldsymbol{X}^T=\\begin{bmatrix}\n", + "x_{00}^2+x_{01}^2 & x_{00}x_{10}+x_{01}x_{11}\\\\\n", + "x_{10}x_{00}+x_{11}x_{01} & x_{10}^2+x_{11}^2\\\\\n", + "\\end{bmatrix},\n", + "$$" + ] + }, + { + "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", + "## 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": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}\\boldsymbol{X}^T= \\mathbb{E}[\\boldsymbol{X}\\boldsymbol{X}^T].\n", + "$$" + ] + }, + { + "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 $\\boldsymbol{S}\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S}^T=\\boldsymbol{C}[\\boldsymbol{y}]$ 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": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{y}] = \\mathbb{E}[\\boldsymbol{S}\\boldsymbol{X}\\boldsymbol{X}^T\\boldsymbol{S}^T]=\\boldsymbol{S}\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S}^T,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "since the matrix $\\boldsymbol{S}$ is not a data dependent matrix. Multiplying with $\\boldsymbol{S}^T$ from the left we have" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{S}^T\\boldsymbol{C}[\\boldsymbol{y}] = \\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S}^T,\n", + "$$" + ] + }, + { + "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": [ + "$$\n", + "\\boldsymbol{S}^T_i\\lambda_i = \\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{S}^T_i.\n", + "$$" + ] + }, + { + "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 $\\mathbb{E}[\\overline{\\boldsymbol{X}}\\overline{\\boldsymbol{X}}^T]$.\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:" + ] + }, + { + "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 $n = 1000$ points $X = \\{ x_1, \\ldots, x_N \\}$ from\n", + "this distribution, and store them in the $1000 \\times 2$ matrix $\\boldsymbol{X}$.\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": {}, + "outputs": [], + "source": [ + "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", + "### Compute the sample mean and center the data\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! Compare your code with the functionality from **Scikit-Learn** discussed above.\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": 16, + "metadata": {}, + "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", + "### Compute the sample covariance\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": 17, + "metadata": {}, + "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": 18, + "metadata": {}, + "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?\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": 19, + "metadata": {}, + "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", + "## 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", + "We assume also that we have an orthogonal transformation $\\boldsymbol{W}\\in {\\mathbb{R}}^{p\\times p}$. We define the reconstruction error (which is similar to the mean squared error we have seen before) as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "J(\\boldsymbol{W},\\boldsymbol{Z}) = \\frac{1}{n}\\sum_i (\\boldsymbol{x}_i - \\overline{\\boldsymbol{x}}_i)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "with $\\overline{\\boldsymbol{x}}_i = \\boldsymbol{W}\\boldsymbol{z}_i$, where $\\boldsymbol{z}_i$ is a row vector with dimension ${\\mathbb{R}}^{n}$ of the matrix\n", + "$\\boldsymbol{Z}\\in{\\mathbb{R}}^{p\\times n}$. When doing PCA we want to reduce this dimensionality. \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", + "## Proof of the PCA Theorem\n", + "\n", + "To show the PCA theorem let us start with the assumption that there is one vector $\\boldsymbol{w}_0$ 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" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)= \\frac{1}{n}\\sum_i (\\boldsymbol{x}_i - z_{i0}\\boldsymbol{w}_0)^2=\\frac{1}{n}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2\\boldsymbol{w}_0^T\\boldsymbol{w}_0),\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "which we can rewrite due to the orthogonality of $\\boldsymbol{w}_i$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)=\\frac{1}{n}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Minimizing $J$ with respect to the unknown parameters $z_{0i}$ we obtain that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "z_{i0}=\\boldsymbol{w}_0^T\\boldsymbol{x}_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where the vectors on the rhs are known. \n", + "\n", + "\n", + "## PCA Proof continued\n", + "\n", + "We have now found the unknown parameters $z_{i0}$. These correspond to the projected coordinates and we can write" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "J(\\boldsymbol{w}_0)= \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - z_{i0}^2)=\\mathrm{const}-\\frac{1}{n}\\sum_i z_{i0}^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can show that the variance of the projected coordinates defined by $\\boldsymbol{w}_0^T\\boldsymbol{x}_i$ are given by" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{var}[\\boldsymbol{w}_0^T\\boldsymbol{x}_i] = \\frac{1}{n}\\sum_i z_{i0}^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "since the expectation value of" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathbb{E}[\\boldsymbol{w}_0^T\\boldsymbol{x}_i] = \\mathbb{E}[z_{i0}]= \\boldsymbol{w}_0^T\\mathbb{E}[\\boldsymbol{x}_i]=0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where we have used the fact that our data are centered.\n", + "\n", + "Recalling our definition of the covariance as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}\\boldsymbol{X}^T=\\mathbb{E}[\\boldsymbol{X}\\boldsymbol{X}^T],\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "we have thus that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\mathrm{var}[\\boldsymbol{w}_0^T\\boldsymbol{x}_i] = \\frac{1}{n}\\sum_i z_{i0}^2=\\boldsymbol{w}_0^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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", + "## The final step\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", + "## 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\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": 20, + "metadata": {}, + "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": 21, + "metadata": {}, + "outputs": [], + "source": [ + "W2 = V.T[:, :2]\n", + "X2D = X_centered.dot(W2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## 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": 22, + "metadata": {}, + "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": 23, + "metadata": {}, + "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": 24, + "metadata": {}, + "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", + "## More on the PCA\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": 25, + "metadata": {}, + "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": 26, + "metadata": {}, + "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", + "## 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", + "\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": 27, + "metadata": {}, + "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": [ + "## LLE\n", + "\n", + "Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction\n", + "(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous\n", + "algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its\n", + "closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where\n", + "these local relationships are best preserved (more details shortly). \n", + "\n", + "\n", + "\n", + "## 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": { + "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/pub/week44/html/._week44-bs043.html b/doc/pub/week44/html/._week44-bs043.html new file mode 100644 index 000000000..e6d28bbc0 --- /dev/null +++ b/doc/pub/week44/html/._week44-bs043.html @@ -0,0 +1,306 @@ + + + + + + + + +Week 44: From Decision Trees to Bagging methods + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Week 44: From Decision Trees to Bagging methods +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Using the Voting Classifier

+

+ + +

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))
+
+

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week44/html/._week44-bs044.html b/doc/pub/week44/html/._week44-bs044.html new file mode 100644 index 000000000..22c872b5e --- /dev/null +++ b/doc/pub/week44/html/._week44-bs044.html @@ -0,0 +1,313 @@ + + + + + + + + +Week 44: From Decision Trees to Bagging methods + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Week 44: From Decision Trees to Bagging methods +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Please, not the moons again! Voting and Bagging

+ +

+ + +

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))
+
+

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week44/html/._week44-bs045.html b/doc/pub/week44/html/._week44-bs045.html new file mode 100644 index 000000000..da656dec3 --- /dev/null +++ b/doc/pub/week44/html/._week44-bs045.html @@ -0,0 +1,315 @@ + + + + + + + + +Week 44: From Decision Trees to Bagging methods + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Week 44: From Decision Trees to Bagging methods +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

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))
+
+

+ + +

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()
+
+

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week44/html/._week44-bs046.html b/doc/pub/week44/html/._week44-bs046.html new file mode 100644 index 000000000..f4379b151 --- /dev/null +++ b/doc/pub/week44/html/._week44-bs046.html @@ -0,0 +1,321 @@ + + + + + + + + +Week 44: From Decision Trees to Bagging methods + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Week 44: From Decision Trees to Bagging methods +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

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()
+
+

+ +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week44/ipynb/.ipynb_checkpoints/week44-checkpoint.ipynb b/doc/pub/week44/ipynb/.ipynb_checkpoints/week44-checkpoint.ipynb new file mode 100644 index 000000000..29b761332 --- /dev/null +++ b/doc/pub/week44/ipynb/.ipynb_checkpoints/week44-checkpoint.ipynb @@ -0,0 +1,1972 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# week 44: From Decision Trees to Bagging methods\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Oct 30, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "\n", + "## Overview of week 44\n", + "\n", + "* [Thursday: Wrapping up PCA from last week and basics of decision trees, classification and regression algorithms with video of lecture](https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureOctober29.mp4?vrtx=view-as-webpage) \n", + "\n", + "* Friday: Decision trees, voting models and bagging\n", + "\n", + "Geron's chapter 6 covers decision trees while ensemble models, voting and bagging are discussed in chapter 7. See also lecture from [STK-IN4300, lecture 7](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_7.pdf). Chapter 9.2 of Hastie et al contains also a good discussion.\n", + "\n", + "\n", + "## Thursday\n", + "\n", + "\n", + "\n", + "## 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", + "## A Sketch of a Tree, Regression problem\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## A Sketch of a Tree, Classification problem\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## A typical Decision Tree with its pertinent Jargon, Classification Problem\n", + "\n", + "\n", + "\n", + "\n", + "

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "This tree was produced using the Wisconsin cancer data (discussed here as well, see code examples below) using **Scikit-Learn**'s decision tree classifier. Here we have used the so-called **gini** index (see below) to split the various branches.\n", + "\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", + "## How do we set it up?\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!\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Decision trees and Regression" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "2nd degree coefficients:\n", + "zero power: 3.6463116984065373\n", + "first power: -0.13976367577641613\n", + "second power: 0.0006331574116245765\n" + ] + }, + { + "data": { + "image/png": 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\n", 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "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\n", + "## Pruning the tree\n", + "\n", + "The above procedure is rather straightforward, but leads often to\n", + "overfitting and unnecessarily large and complicated trees. The basic\n", + "idea is to grow a large tree $T_0$ and then prune it back in order to\n", + "obtain a subtree. A smaller tree with fewer splits (fewer regions) can\n", + "lead to smaller variance and better interpretation at the cost of a\n", + "little more bias.\n", + "\n", + "The so-called Cost complexity pruning algorithm gives us a\n", + "way to do just this. Rather than considering every possible subtree,\n", + "we consider a sequence of trees indexed by a nonnegative tuning\n", + "parameter $\\alpha$.\n", + "\n", + "Read more at the following [Scikit-Learn link on pruning](https://scikit-learn.org/stable/auto_examples/tree/plot_cost_complexity_pruning.html#sphx-glr-auto-examples-tree-plot-cost-complexity-pruning-py).\n", + "\n", + "## Cost complexity pruning\n", + "\n", + "For each value of $\\alpha$ there corresponds a subtree $T \\in T_0$ such that" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\sum_{m=1}^{\\overline{T}}\\sum_{i:x_i\\in R_m}(y_i-\\overline{y}_{R_m})^2+\\alpha\\overline{T},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "is as small as possible. Here $\\overline{T}$ is \n", + "the number of terminal nodes of the tree $T$ , $R_m$ is the\n", + "rectangle (i.e. the subset of predictor space) corresponding to the $m$-th terminal node.\n", + "\n", + "The tuning parameter $\\alpha$ controls a trade-off between the subtree’s\n", + "complexity and its fit to the training data. When $\\alpha = 0$, then the\n", + "subtree $T$ will simply equal $T_0$, \n", + "because then the above equation just measures the\n", + "training error. \n", + "However, as $\\alpha$ increases, there is a price to pay for\n", + "having a tree with many terminal nodes. The above equation will\n", + "tend to be minimized for a smaller subtree. \n", + "\n", + "\n", + "It turns out that as we increase $\\alpha$ from zero\n", + "branches get pruned from the tree in a nested and predictable fashion,\n", + "so obtaining the whole sequence of subtrees as a function of $\\alpha$ is\n", + "easy. We can select a value of $\\alpha$ using a validation set or using\n", + "cross-validation. We then return to the full data set and obtain the\n", + "subtree corresponding to $\\alpha$. \n", + "\n", + "\n", + "## Schematic Regression Procedure\n", + "\n", + "**Building a Regression Tree.**\n", + "\n", + "\n", + "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.\n", + "\n", + "2. Apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of $\\alpha$.\n", + "\n", + "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: \n", + "\n", + " * repeat steps 1 and 2 on all but the $k$-th fold of the training data. \n", + "\n", + " * Then we valuate the mean squared prediction error on the data in the left-out $k$-th fold, as a function of $\\alpha$.\n", + "\n", + " * Finally we average the results for each value of $\\alpha$, and pick $\\alpha$ to minimize the average error.\n", + "\n", + "\n", + "4. Return the subtree from Step 2 that corresponds to the chosen value of $\\alpha$.\n", + "\n", + "\n", + "\n", + "\n", + "## A Classification Tree\n", + "\n", + "A classification tree is very similar to a regression tree, except\n", + "that it is used to predict a qualitative response rather than a\n", + "quantitative one. Recall that for a regression tree, the predicted\n", + "response for an observation is given by the mean response of the\n", + "training observations that belong to the same terminal node. In\n", + "contrast, for a classification tree, we predict that each observation\n", + "belongs to the most commonly occurring class of training observations\n", + "in the region to which it belongs. In interpreting the results of a\n", + "classification tree, we are often interested not only in the class\n", + "prediction corresponding to a particular terminal node region, but\n", + "also in the class proportions among the training observations that\n", + "fall into that region. \n", + "\n", + "## Growing a classification tree\n", + "\n", + "The task of growing a\n", + "classification tree is quite similar to the task of growing a\n", + "regression tree. Just as in the regression setting, we use recursive\n", + "binary splitting to grow a classification tree. However, in the\n", + "classification setting, the MSE cannot be used as a criterion for making\n", + "the binary splits. A natural alternative to MSE is the **classification\n", + "error rate**. Since we plan to assign an observation in a given region\n", + "to the most commonly occurring error rate class of training\n", + "observations in that region, the classification error rate is simply\n", + "the fraction of the training observations in that region that do not\n", + "belong to the most common class. \n", + "\n", + "When building a classification tree, either the Gini index or the\n", + "entropy are typically used to evaluate the quality of a particular\n", + "split, since these two approaches are more sensitive to node purity\n", + "than is the classification error rate. \n", + "\n", + "\n", + "## Classification tree, how to split nodes\n", + "\n", + "If our targets are the outcome of a classification process that takes\n", + "for example $k=1,2,\\dots,K$ values, the only thing we need to think of\n", + "is to set up the splitting criteria for each node.\n", + "\n", + "We define a PDF $p_{mk}$ that represents the number of observations of\n", + "a class $k$ in a region $R_m$ with $N_m$ observations. We represent\n", + "this likelihood function in terms of the proportion $I(y_i=k)$ of\n", + "observations of this class in the region $R_m$ as" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p_{mk} = \\frac{1}{N_m}\\sum_{x_i\\in R_m}I(y_i=k).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We let $p_{mk}$ represent the majority class of observations in region\n", + "$m$. The three most common ways of splitting a node are given by\n", + "\n", + "* Misclassification error" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "p_{mk} = \\frac{1}{N_m}\\sum_{x_i\\in R_m}I(y_i\\ne k) = 1-p_{mk}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "* Gini index $g$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "g = \\sum_{k=1}^K p_{mk}(1-p_{mk}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "* Information entropy or just entropy $s$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "s = -\\sum_{k=1}^K p_{mk}\\log{p_{mk}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Visualizing the Tree, Classification" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " mean radius mean texture mean perimeter mean area mean smoothness \\\n", + "0 17.99 10.38 122.80 1001.0 0.11840 \n", + "1 20.57 17.77 132.90 1326.0 0.08474 \n", + "2 19.69 21.25 130.00 1203.0 0.10960 \n", + "3 11.42 20.38 77.58 386.1 0.14250 \n", + "4 20.29 14.34 135.10 1297.0 0.10030 \n", + ".. ... ... ... ... ... \n", + "564 21.56 22.39 142.00 1479.0 0.11100 \n", + "565 20.13 28.25 131.20 1261.0 0.09780 \n", + "566 16.60 28.08 108.30 858.1 0.08455 \n", + "567 20.60 29.33 140.10 1265.0 0.11780 \n", + "568 7.76 24.54 47.92 181.0 0.05263 \n", + "\n", + " mean compactness mean concavity mean concave points mean symmetry \\\n", + "0 0.27760 0.30010 0.14710 0.2419 \n", + "1 0.07864 0.08690 0.07017 0.1812 \n", + "2 0.15990 0.19740 0.12790 0.2069 \n", + "3 0.28390 0.24140 0.10520 0.2597 \n", + "4 0.13280 0.19800 0.10430 0.1809 \n", + ".. ... ... ... ... \n", + "564 0.11590 0.24390 0.13890 0.1726 \n", + "565 0.10340 0.14400 0.09791 0.1752 \n", + "566 0.10230 0.09251 0.05302 0.1590 \n", + "567 0.27700 0.35140 0.15200 0.2397 \n", + "568 0.04362 0.00000 0.00000 0.1587 \n", + "\n", + " mean fractal dimension ... worst radius worst texture \\\n", + "0 0.07871 ... 25.380 17.33 \n", + "1 0.05667 ... 24.990 23.41 \n", + "2 0.05999 ... 23.570 25.53 \n", + "3 0.09744 ... 14.910 26.50 \n", + "4 0.05883 ... 22.540 16.67 \n", + ".. ... ... ... ... \n", + "564 0.05623 ... 25.450 26.40 \n", + "565 0.05533 ... 23.690 38.25 \n", + "566 0.05648 ... 18.980 34.12 \n", + "567 0.07016 ... 25.740 39.42 \n", + "568 0.05884 ... 9.456 30.37 \n", + "\n", + " worst perimeter worst area worst smoothness worst compactness \\\n", + "0 184.60 2019.0 0.16220 0.66560 \n", + "1 158.80 1956.0 0.12380 0.18660 \n", + "2 152.50 1709.0 0.14440 0.42450 \n", + "3 98.87 567.7 0.20980 0.86630 \n", + "4 152.20 1575.0 0.13740 0.20500 \n", + ".. ... ... ... ... \n", + "564 166.10 2027.0 0.14100 0.21130 \n", + "565 155.00 1731.0 0.11660 0.19220 \n", + "566 126.70 1124.0 0.11390 0.30940 \n", + "567 184.60 1821.0 0.16500 0.86810 \n", + "568 59.16 268.6 0.08996 0.06444 \n", + "\n", + " worst concavity worst concave points worst symmetry \\\n", + "0 0.7119 0.2654 0.4601 \n", + "1 0.2416 0.1860 0.2750 \n", + "2 0.4504 0.2430 0.3613 \n", + "3 0.6869 0.2575 0.6638 \n", + "4 0.4000 0.1625 0.2364 \n", + ".. ... ... ... \n", + "564 0.4107 0.2216 0.2060 \n", + "565 0.3215 0.1628 0.2572 \n", + "566 0.3403 0.1418 0.2218 \n", + "567 0.9387 0.2650 0.4087 \n", + "568 0.0000 0.0000 0.2871 \n", + "\n", + " worst fractal dimension \n", + "0 0.11890 \n", + "1 0.08902 \n", + "2 0.08758 \n", + "3 0.17300 \n", + "4 0.07678 \n", + ".. ... \n", + "564 0.07115 \n", + "565 0.06637 \n", + "566 0.07820 \n", + "567 0.12400 \n", + "568 0.07039 \n", + "\n", + "[569 rows x 30 columns]\n", + " malignant benign\n", + "0 1 0\n", + "1 1 0\n", + "2 1 0\n", + "3 1 0\n", + "4 1 0\n", + ".. ... ...\n", + "564 1 0\n", + "565 1 0\n", + "566 1 0\n", + "567 1 0\n", + "568 0 1\n", + "\n", + "[569 rows x 2 columns]\n" + ] + }, + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "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": "markdown", + "metadata": {}, + "source": [ + "## Visualizing the Tree, The Moons" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "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": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[Text(167.4, 199.32, 'X[2] <= 2.45\\ngini = 0.667\\nsamples 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "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": [ + "## Printing out as text\n", + "\n", + "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": 9, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "|--- petal width (cm) <= 0.80\n", + "| |--- class: 0\n", + "|--- petal width (cm) > 0.80\n", + "| |--- petal width (cm) <= 1.75\n", + "| | |--- class: 1\n", + "| |--- petal width (cm) > 1.75\n", + "| | |--- class: 2\n", + "\n" + ] + } + ], + "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", + "## 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", + "## 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": 6, + "metadata": {}, + "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": [ + "## Computing the Gini Factor\n", + "\n", + "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": 7, + "metadata": {}, + "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": 8, + "metadata": {}, + "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": 9, + "metadata": {}, + "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": "markdown", + "metadata": {}, + "source": [ + "## Playing around with regions" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "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": "markdown", + "metadata": {}, + "source": [ + "## Regression trees" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "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": 12, + "metadata": {}, + "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": "markdown", + "metadata": {}, + "source": [ + "## Final regressor code" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "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": 14, + "metadata": {}, + "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.\n", + "\n", + "\n", + "## Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods\n", + "\n", + "As stated above and seen in many of the examples discussed here 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", + "\n", + "## An Overview of Ensemble Methods\n", + "\n", + "\n", + "\n", + "\n", + "

\n", + "\n", + "\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", + "## More bagging\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.\n", + "\n", + "## Simple Voting Example, head or tail" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "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": "markdown", + "metadata": {}, + "source": [ + "## Using the Voting Classifier" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "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": "markdown", + "metadata": {}, + "source": [ + "## Please, not the moons again! Voting and Bagging" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "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": 18, + "metadata": {}, + "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": 19, + "metadata": {}, + "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": 20, + "metadata": {}, + "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": 21, + "metadata": {}, + "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": 22, + "metadata": {}, + "outputs": [], + "source": [ + "from sklearn.metrics import accuracy_score\n", + "print(accuracy_score(y_test, y_pred))" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "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": 24, + "metadata": {}, + "outputs": [], + "source": [ + "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": 25, + "metadata": {}, + "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", + "plt.xlim(1,maxdepth)\n", + "plt.plot(polydegree, error, label='MSE simple tree')\n", + "plt.plot(polydegree, mse_simpletree, label='MSE for Bootstrap')\n", + "plt.plot(polydegree, bias, label='bias')\n", + "plt.plot(polydegree, variance, label='Variance')\n", + "plt.legend()\n", + "save_fig(\"baggingboot\")\n", + "plt.show()" + ] + } + ], + "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/pub/week44/ipynb/Datafiles/moons.dot b/doc/pub/week44/ipynb/Datafiles/moons.dot new file mode 100644 index 000000000..05d91c44c --- /dev/null +++ b/doc/pub/week44/ipynb/Datafiles/moons.dot @@ -0,0 +1,45 @@ +digraph Tree { +node [shape=box, style="filled, rounded", color="black", fontname=helvetica] ; 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+15 -> 19 ; +20 [label="gini = 0.0\nsamples = 5\nvalue = [0, 5]", fillcolor="#399de5"] ; +12 -> 20 ; +} \ No newline at end of file diff --git a/doc/pub/week44/ipynb/Datafiles/moons.png b/doc/pub/week44/ipynb/Datafiles/moons.png new file mode 100644 index 000000000..43ab76969 Binary files /dev/null and b/doc/pub/week44/ipynb/Datafiles/moons.png differ diff --git a/doc/pub/week44/ipynb/Results/FigureFiles/baggingtree.png b/doc/pub/week44/ipynb/Results/FigureFiles/baggingtree.png new file mode 100644 index 000000000..df249981e Binary files /dev/null and b/doc/pub/week44/ipynb/Results/FigureFiles/baggingtree.png differ diff --git a/doc/pub/week44/ipynb/Results/FigureFiles/votingsimple.png b/doc/pub/week44/ipynb/Results/FigureFiles/votingsimple.png new file mode 100644 index 000000000..30ec67491 Binary files /dev/null and b/doc/pub/week44/ipynb/Results/FigureFiles/votingsimple.png differ diff --git a/doc/pub/week45/html/._week45-bs033.html b/doc/pub/week45/html/._week45-bs033.html new file mode 100644 index 000000000..3af6f74a3 --- /dev/null +++ b/doc/pub/week45/html/._week45-bs033.html @@ -0,0 +1,294 @@ + + + + + + + + +Week 45: Random Forests and Boosting + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Week 45: Random Forests and Boosting +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Xgboost on the Cancer Data

+ +

+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()
+
+

+ +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week45/ipynb/.ipynb_checkpoints/week45-checkpoint.ipynb b/doc/pub/week45/ipynb/.ipynb_checkpoints/week45-checkpoint.ipynb new file mode 100644 index 000000000..c0d1953be --- /dev/null +++ b/doc/pub/week45/ipynb/.ipynb_checkpoints/week45-checkpoint.ipynb @@ -0,0 +1,1435 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Week 45: Random Forests and Boosting\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Nov 2, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "## Overview of week 45\n", + "\n", + "* \"Thursday: Wrapping up from last week. Bagging and Random forests. Boosting methods.\n", + "\n", + "* \"Friday: Boosting and gradient boosting\n", + "\n", + "Geron's chapter 7. See also lecture from [STK-IN4300, lecture 9](https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_9.pdf). Chapter 10 (sections 10.1-10.10 are the most relevant ones) of Hastie et al contains also a good discussion.\n", + "\n", + "\n", + "## Thursday\n", + "\n", + "Bagging, voting and random forests.\n", + "The material on bagging and voting is a repeat from last week and can be found in the slides from week 44.\n", + "We repeat here the voting approach since this will serve as a motivation for boosting methods later.\n", + "\n", + "## Why Voting?\n", + "\n", + "The idea behind boosting, and voting as well can be phrased as follows:\n", + "**Can a group of people somehow arrive at highly\n", + "reasoned decisions, despite the weak judgement of the individual\n", + "members?**\n", + "\n", + "The aim is to create a good classifier by combining several weak classifiers.\n", + "**A weak classifier is a classifier which is able to produce results that are only slightly better than guessing at random.**\n", + "\n", + "The basic approach is to apply repeatedly (in boosting this is done in an iterative way) a weak classifier to modifications of the data.\n", + "In voting we simply apply the law of large numbers while in boosting we give more weight to misclassified data in\n", + "each iteration. \n", + "\n", + "Decision trees play an important role as our weak classifier. They serve as the basic method. \n", + "\n", + "## Tossing coins\n", + "The simplest case is a so-called voting ensemble. To illustrate this, Think of you tossing coins with a biased outcome of 51 per cent for heads and 49% for tails.\n", + "With only few tosses, you may not clearly see this distribution. However, after some thousands of tosses (sounds like you may have some spare time problems), there will be a clear majority of heads.\n", + "With 2000 tosses you should see approximately 1020 heads and 980 tails.\n", + "\n", + "We can then state that the outcome is a clear majority of heads. If you do this ten thousand times, it is easy to see that there is a 97% likelihood of a majority of heads.\n", + "\n", + "Another example would be to collect all polls before an\n", + "election. Different polls may show different likelihoods for a\n", + "candidate winning with say a majority of the popular vote. The majority vote\n", + "would then consist in many polls indicating that this candidate will\n", + "actually win.\n", + "\n", + "The example here shows how we can implement the coin tossing case, clealry demostrating that after some tosses we see the law of large numbers kicking in.\n", + "\n", + "## Simple Voting Example, head or tail" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "ename": "NameError", + "evalue": "name 'plt' 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 3\u001b[0m \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[1;32m 4\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 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\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[1;32m 7\u001b[0m \u001b[0mplt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m10000\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;36m0.51\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0.51\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m\"k--\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlinewidth\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlabel\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m\"51%\"\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 'plt' is not defined" + ] + } + ], + "source": [ + "import numpy as np\n", + "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": "markdown", + "metadata": {}, + "source": [ + "## Using the Voting Classifier\n", + "\n", + "We can use the voting classifier on other data sets, here the excting binary case of two distinct objects using the make moons functionality of -Scikit-Learn-." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "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", + "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": "markdown", + "metadata": {}, + "source": [ + "## Please, not the moons again! Voting and Bagging" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "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": 4, + "metadata": {}, + "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": 5, + "metadata": {}, + "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": 6, + "metadata": {}, + "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": [ + "## 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", + "## Random Forest Algorithm\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 of 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 either the CART algorithm or the ID3 for classification 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. \n", + "\n", + "## Random Forests Compared with other Methods on the Cancer Data" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\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.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)))\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": [ + "## Compare Bagging on Trees with Random Forests" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "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": 9, + "metadata": {}, + "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", + "## What is boosting? Additive Modelling/Iterative Fitting\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", + "## Squared-Error Example and Iterative Fitting\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": [ + "## Adaptive Boosting, AdaBoost\n", + "\n", + "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", + "## Building up AdaBoost\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", + "## Basic Steps of AdaBoost\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", + "## AdaBoost Examples\n", + "\n", + "Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "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": [ + "## AdaBoost for Regression\n", + "\n", + "Here we present [Drucker's AdaBoost](https://pdfs.semanticscholar.org/8d49/e2dedb817f2c3330e74b63c5fc86d2399ce3.pdf) tailored for regression.\n", + "\n", + "In bagging, each training example is equally likely to be\n", + "picked. In boosting, the probability of a particular\n", + "example being in the training set of a particular machine\n", + "depends on the performance of the prior machines on\n", + "that example. The following is a modification of\n", + "Adaboost by Drucker.\n", + "\n", + "Start by selecting a set of training data $n$ and assign to each entry a weight $w_i=1$ for $i=1,2,\\dots,n$. As we have done earlier, we could pick say $80\\%$ of the data set for training. The algorithm runs as follows:\n", + "1. We define the probability that the training sample $i$ is in the set by $p_i = w_i/\\sum_iw_i$. We pick $n$ samples (with replacement) to form our training set. We pick a number uniformly in the range $[0,\\sum_iw_i]$.\n", + "\n", + "2. We choose then a regression machine (for example plain linear regression or a simple decision tree). A given regression machine makes then a hypothesis.\n", + "\n", + "3. Using every member of the training set with the chosen regression machine we obtain then a prediction $\\tilde{y}_i$.\n", + "\n", + "4. We calculate then the loss function $L_i$ for each training sample. We can use various types of loss function as long as we have a value\n", + "\n", + "$L_i\\in [0,1]$. \n", + "\n", + "## Gradient boosting: Basics with Steepest 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", + "## The Squared-Error again! Steepest Descent\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": [ + "## Steepest Descent Example\n", + "\n", + "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", + "## Gradient Boosting, algorithm\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 $f_m(x) = f_{m-1}(x)+\\nu h_m(u_m,x)$;\n", + "\n", + "\n", + "4. The final estimate is then $f_M(x) = \\sum_{m=1}^M\\nu h_m(u_m,x)$.\n", + "\n", + "## Gradient Boosting Example, Regression\n", + "\n", + "We discuss here the difference between the steepest descent approach and gradient boosting by repeating our simple regression example above. \n", + "\n", + "\n", + "## Gradient Boosting, Examples of Regression" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "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": 12, + "metadata": {}, + "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": 13, + "metadata": {}, + "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": [ + "## Xgboost on the Cancer Data\n", + "\n", + "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": 14, + "metadata": {}, + "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": { + "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": "3.8.3" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/doc/pub/week46/html/._week46-bs029.html b/doc/pub/week46/html/._week46-bs029.html new file mode 100644 index 000000000..22e4fd44f --- /dev/null +++ b/doc/pub/week46/html/._week46-bs029.html @@ -0,0 +1,268 @@ + + + + + + + + +Week 46: Gradient Boosting Summary and Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Week 46: Gradient Boosting Summary and Support Vector Machines +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

A simple example

+ +

+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’]
+
+

+

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week46/html/._week46-bs030.html b/doc/pub/week46/html/._week46-bs030.html new file mode 100644 index 000000000..f67b2490c --- /dev/null +++ b/doc/pub/week46/html/._week46-bs030.html @@ -0,0 +1,222 @@ + + + + + + + + +Week 46: Gradient Boosting Summary and Support Vector Machines + + + + + + + + + + + + + + + + + + + + + + + +
+
+ + Week 46: Gradient Boosting Summary and Support Vector Machines +
+ +
+ +
+
+ + +
+ +

 

 

 

+ + + + +

Back to the more realistic cases

+ +

+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 \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). 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 + +

+ +

+ +

+ + +
+ + + + + + + +
+ +
+ + + + + + diff --git a/doc/pub/week48/html/reveal.js/.gitignore b/doc/pub/week48/html/reveal.js/.gitignore new file mode 100644 index 000000000..a5df3133d --- /dev/null +++ b/doc/pub/week48/html/reveal.js/.gitignore @@ -0,0 +1,8 @@ +.DS_Store +.svn +log/*.log +tmp/** +node_modules/ +.sass-cache +css/reveal.min.css +js/reveal.min.js diff --git a/doc/pub/week48/html/reveal.js/.travis.yml b/doc/pub/week48/html/reveal.js/.travis.yml new file mode 100644 index 000000000..165d9ae9f --- /dev/null +++ b/doc/pub/week48/html/reveal.js/.travis.yml @@ -0,0 +1,5 @@ +language: node_js +node_js: + - 0.10 +before_script: + - npm install -g grunt-cli \ No newline at end of file diff --git a/doc/pub/week48/html/reveal.js/CONTRIBUTING.md b/doc/pub/week48/html/reveal.js/CONTRIBUTING.md new file mode 100644 index 000000000..c2091e88f --- /dev/null +++ b/doc/pub/week48/html/reveal.js/CONTRIBUTING.md @@ -0,0 +1,23 @@ +## Contributing + +Please keep the [issue tracker](http://github.com/hakimel/reveal.js/issues) limited to **bug reports**, **feature requests** and **pull requests**. + + +### Personal Support +If you have personal support or setup questions the best place to ask those are [StackOverflow](http://stackoverflow.com/questions/tagged/reveal.js). + + +### Bug Reports +When reporting a bug make sure to include information about which browser and operating system you are on as well as the necessary steps to reproduce the issue. If possible please include a link to a sample presentation where the bug can be tested. + + +### Pull Requests +- Should follow the coding style of the file you work in, most importantly: + - Tabs to indent + - Single-quoted strings +- Should be made towards the **dev branch** +- Should be submitted from a feature/topic branch (not your master) + + +### Plugins +Please do not submit plugins as pull requests. They should be maintained in their own separate repository. More information here: https://github.com/hakimel/reveal.js/wiki/Plugin-Guidelines diff --git a/doc/pub/week48/html/reveal.js/Gruntfile.js b/doc/pub/week48/html/reveal.js/Gruntfile.js new file mode 100644 index 000000000..b257e8f32 --- /dev/null +++ b/doc/pub/week48/html/reveal.js/Gruntfile.js @@ -0,0 +1,140 @@ +/* global module:false */ +module.exports = function(grunt) { + var port = grunt.option('port') || 8000; + // Project configuration + grunt.initConfig({ + pkg: grunt.file.readJSON('package.json'), + meta: { + banner: + '/*!\n' + + ' * reveal.js <%= pkg.version %> (<%= grunt.template.today("yyyy-mm-dd, HH:MM") %>)\n' + + ' * http://lab.hakim.se/reveal-js\n' + + ' * MIT licensed\n' + + ' *\n' + + ' * Copyright (C) 2014 Hakim El Hattab, http://hakim.se\n' + + ' */' + }, + + qunit: { + files: [ 'test/*.html' ] + }, + + uglify: { + options: { + banner: '<%= meta.banner %>\n' + }, + build: { + src: 'js/reveal.js', + dest: 'js/reveal.min.js' + } + }, + + cssmin: { + compress: { + files: { + 'css/reveal.min.css': [ 'css/reveal.css' ] + } + } + }, + + sass: { + main: { + files: { + 'css/theme/darkgray.css': 'css/theme/source/darkgray.scss', + 'css/theme/beigesmall.css': 'css/theme/source/beigesmall.scss', + 'css/theme/cbc.css': 'css/theme/source/cbc.scss', + 'css/theme/default.css': 'css/theme/source/default.scss', + 'css/theme/beige.css': 'css/theme/source/beige.scss', + 'css/theme/night.css': 'css/theme/source/night.scss', + 'css/theme/serif.css': 'css/theme/source/serif.scss', + 'css/theme/simple.css': 'css/theme/source/simple.scss', + 'css/theme/sky.css': 'css/theme/source/sky.scss', + 'css/theme/moon.css': 'css/theme/source/moon.scss', + 'css/theme/solarized.css': 'css/theme/source/solarized.scss', + 'css/theme/blood.css': 'css/theme/source/blood.scss' + } + } + }, + + jshint: { + options: { + curly: false, + eqeqeq: true, + immed: true, + latedef: true, + newcap: true, + noarg: true, + sub: true, + undef: true, + eqnull: true, + browser: true, + expr: true, + globals: { + head: false, + module: false, + console: false, + unescape: false + } + }, + files: [ 'Gruntfile.js', 'js/reveal.js' ] + }, + + connect: { + server: { + options: { + port: port, + base: '.' + } + } + }, + + zip: { + 'reveal-js-presentation.zip': [ + 'index.html', + 'css/**', + 'js/**', + 'lib/**', + 'images/**', + 'plugin/**' + ] + }, + + watch: { + main: { + files: [ 'Gruntfile.js', 'js/reveal.js', 'css/reveal.css' ], + tasks: 'default' + }, + theme: { + files: [ 'css/theme/source/*.scss', 'css/theme/template/*.scss' ], + tasks: 'themes' + } + } + + }); + + // Dependencies + grunt.loadNpmTasks( 'grunt-contrib-qunit' ); + grunt.loadNpmTasks( 'grunt-contrib-jshint' ); + grunt.loadNpmTasks( 'grunt-contrib-cssmin' ); + grunt.loadNpmTasks( 'grunt-contrib-uglify' ); + grunt.loadNpmTasks( 'grunt-contrib-watch' ); + grunt.loadNpmTasks( 'grunt-contrib-sass' ); + grunt.loadNpmTasks( 'grunt-contrib-connect' ); + grunt.loadNpmTasks( 'grunt-zip' ); + + // Default task + grunt.registerTask( 'default', [ 'jshint', 'cssmin', 'uglify', 'qunit' ] ); + + // Theme task + grunt.registerTask( 'themes', [ 'sass' ] ); + + // Package presentation to archive + grunt.registerTask( 'package', [ 'default', 'zip' ] ); + + // Serve presentation locally + grunt.registerTask( 'serve', [ 'connect', 'watch' ] ); + + // Run tests + grunt.registerTask( 'test', [ 'jshint', 'qunit' ] ); + +}; diff --git a/doc/pub/week48/html/reveal.js/LICENSE b/doc/pub/week48/html/reveal.js/LICENSE new file mode 100644 index 000000000..09623076f --- /dev/null +++ b/doc/pub/week48/html/reveal.js/LICENSE @@ -0,0 +1,19 @@ +Copyright (C) 2015 Hakim El Hattab, http://hakim.se + +Permission is hereby granted, free of charge, to any person obtaining a copy +of this software and associated documentation files (the "Software"), to deal +in the Software without restriction, including without limitation the rights +to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +copies of the Software, and to permit persons to whom the Software is +furnished to do so, subject to the following conditions: + +The above copyright notice and this permission notice shall be included in +all copies or substantial portions of the Software. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN +THE SOFTWARE. \ No newline at end of file diff --git a/doc/pub/week48/html/reveal.js/README.md b/doc/pub/week48/html/reveal.js/README.md new file mode 100644 index 000000000..573b19597 --- /dev/null +++ b/doc/pub/week48/html/reveal.js/README.md @@ -0,0 +1,1052 @@ +# reveal.js [![Build Status](https://travis-ci.org/hakimel/reveal.js.svg?branch=master)](https://travis-ci.org/hakimel/reveal.js) + +A framework for easily creating beautiful presentations using HTML. [Check out the live demo](http://lab.hakim.se/reveal-js/). + +reveal.js comes with a broad range of features including [nested slides](https://github.com/hakimel/reveal.js#markup), [Markdown contents](https://github.com/hakimel/reveal.js#markdown), [PDF export](https://github.com/hakimel/reveal.js#pdf-export), [speaker notes](https://github.com/hakimel/reveal.js#speaker-notes) and a [JavaScript API](https://github.com/hakimel/reveal.js#api). It's best viewed in a modern browser but [fallbacks](https://github.com/hakimel/reveal.js/wiki/Browser-Support) are available to make sure your presentation can still be viewed elsewhere. + + +#### More reading: +- [Installation](#installation): Step-by-step instructions for getting reveal.js running on your computer. +- [Changelog](https://github.com/hakimel/reveal.js/releases): Up-to-date version history. +- [Examples](https://github.com/hakimel/reveal.js/wiki/Example-Presentations): Presentations created with reveal.js, add your own! +- [Browser Support](https://github.com/hakimel/reveal.js/wiki/Browser-Support): Explanation of browser support and fallbacks. +- [Plugins](https://github.com/hakimel/reveal.js/wiki/Plugins,-Tools-and-Hardware): A list of plugins that can be used to extend reveal.js. + +## Online Editor + +Presentations are written using HTML or Markdown but there's also an online editor for those of you who prefer a graphical interface. Give it a try at [http://slides.com](http://slides.com). + + +## Instructions + +### Markup + +Markup hierarchy needs to be ``
`` where the ``
`` represents one slide and can be repeated indefinitely. If you place multiple ``
``'s inside of another ``
`` they will be shown as vertical slides. The first of the vertical slides is the "root" of the others (at the top), and it will be included in the horizontal sequence. For example: + +```html +
+
+
Single Horizontal Slide
+
+
Vertical Slide 1
+
Vertical Slide 2
+
+
+
+``` + +### Markdown + +It's possible to write your slides using Markdown. To enable Markdown, add the ```data-markdown``` attribute to your ```
``` elements and wrap the contents in a ``` +
+``` + +#### External Markdown + +You can write your content as a separate file and have reveal.js load it at runtime. Note the separator arguments which determine how slides are delimited in the external file. The ```data-charset``` attribute is optional and specifies which charset to use when loading the external file. + +When used locally, this feature requires that reveal.js [runs from a local web server](#full-setup). + +```html +
+
+``` + +#### Element Attributes + +Special syntax (in html comment) is available for adding attributes to Markdown elements. This is useful for fragments, amongst other things. + +```html +
+ +
+``` + +#### Slide Attributes + +Special syntax (in html comment) is available for adding attributes to the slide `
` elements generated by your Markdown. + +```html +
+ +
+``` + + +### Configuration + +At the end of your page you need to initialize reveal by running the following code. Note that all config values are optional and will default as specified below. + +```javascript +Reveal.initialize({ + + // Display controls in the bottom right corner + controls: true, + + // Display a presentation progress bar + progress: true, + + // Display the page number of the current slide + slideNumber: false, + + // Push each slide change to the browser history + history: false, + + // Enable keyboard shortcuts for navigation + keyboard: true, + + // Enable the slide overview mode + overview: true, + + // Vertical centering of slides + center: true, + + // Enables touch navigation on devices with touch input + touch: true, + + // Loop the presentation + loop: false, + + // Change the presentation direction to be RTL + rtl: false, + + // Turns fragments on and off globally + fragments: true, + + // Flags if the presentation is running in an embedded mode, + // i.e. contained within a limited portion of the screen + embedded: false, + + // Flags if we should show a help overlay when the questionmark + // key is pressed + help: true, + + // Number of milliseconds between automatically proceeding to the + // next slide, disabled when set to 0, this value can be overwritten + // by using a data-autoslide attribute on your slides + autoSlide: 0, + + // Stop auto-sliding after user input + autoSlideStoppable: true, + + // Enable slide navigation via mouse wheel + mouseWheel: false, + + // Hides the address bar on mobile devices + hideAddressBar: true, + + // Opens links in an iframe preview overlay + previewLinks: false, + + // Transition style + transition: 'default', // none/fade/slide/convex/concave/zoom + + // Transition speed + transitionSpeed: 'default', // default/fast/slow + + // Transition style for full page slide backgrounds + backgroundTransition: 'default', // none/fade/slide/convex/concave/zoom + + // Number of slides away from the current that are visible + viewDistance: 3, + + // Parallax background image + parallaxBackgroundImage: '', // e.g. "'https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg'" + + // Parallax background size + parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" + + // Amount to move parallax background (horizontal and vertical) on slide change + // Number, e.g. 100 + parallaxBackgroundHorizontal: '', + parallaxBackgroundVertical: '' + +}); +``` + + +The configuration can be updated after initialization using the ```configure``` method: + +```javascript +// Turn autoSlide off +Reveal.configure({ autoSlide: 0 }); + +// Start auto-sliding every 5s +Reveal.configure({ autoSlide: 5000 }); +``` + + +### Dependencies + +Reveal.js doesn't _rely_ on any third party scripts to work but a few optional libraries are included by default. These libraries are loaded as dependencies in the order they appear, for example: + +```javascript +Reveal.initialize({ + dependencies: [ + // Cross-browser shim that fully implements classList - https://github.com/eligrey/classList.js/ + { src: 'lib/js/classList.js', condition: function() { return !document.body.classList; } }, + + // Interpret Markdown in
elements + { src: 'plugin/markdown/marked.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } }, + { src: 'plugin/markdown/markdown.js', condition: function() { return !!document.querySelector( '[data-markdown]' ); } }, + + // Syntax highlight for elements + { src: 'plugin/highlight/highlight.js', async: true, callback: function() { hljs.initHighlightingOnLoad(); } }, + + // Zoom in and out with Alt+click + { src: 'plugin/zoom-js/zoom.js', async: true }, + + // Speaker notes + { src: 'plugin/notes/notes.js', async: true }, + + // Remote control your reveal.js presentation using a touch device + { src: 'plugin/remotes/remotes.js', async: true }, + + // MathJax + { src: 'plugin/math/math.js', async: true } + ] +}); +``` + +You can add your own extensions using the same syntax. The following properties are available for each dependency object: +- **src**: Path to the script to load +- **async**: [optional] Flags if the script should load after reveal.js has started, defaults to false +- **callback**: [optional] Function to execute when the script has loaded +- **condition**: [optional] Function which must return true for the script to be loaded + + +### Ready Event + +A 'ready' event is fired when reveal.js has loaded all non-async dependencies and is ready to start navigating. To check if reveal.js is already 'ready' you can call `Reveal.isReady()`. + +```javascript +Reveal.addEventListener( 'ready', function( event ) { + // event.currentSlide, event.indexh, event.indexv +} ); +``` + + +### Presentation Size + +All presentations have a normal size, that is the resolution at which they are authored. The framework will automatically scale presentations uniformly based on this size to ensure that everything fits on any given display or viewport. + +See below for a list of configuration options related to sizing, including default values: + +```javascript +Reveal.initialize({ + + ... + + // The "normal" size of the presentation, aspect ratio will be preserved + // when the presentation is scaled to fit different resolutions. Can be + // specified using percentage units. + width: 960, + height: 700, + + // Factor of the display size that should remain empty around the content + margin: 0.1, + + // Bounds for smallest/largest possible scale to apply to content + minScale: 0.2, + maxScale: 1.5 + +}); +``` + + +### Auto-sliding + +Presentations can be configured to progress through slides automatically, without any user input. To enable this you will need to tell the framework how many milliseconds it should wait between slides: + +```javascript +// Slide every five seconds +Reveal.configure({ + autoSlide: 5000 +}); +``` +When this is turned on a control element will appear that enables users to pause and resume auto-sliding. Alternatively, sliding can be paused or resumed by pressing »a« on the keyboard. Sliding is paused automatically as soon as the user starts navigating. You can disable these controls by specifying ```autoSlideStoppable: false``` in your reveal.js config. + +You can also override the slide duration for individual slides and fragments by using the ```data-autoslide``` attribute: + +```html +
+

After 2 seconds the first fragment will be shown.

+

After 10 seconds the next fragment will be shown.

+

Now, the fragment is displayed for 2 seconds before the next slide is shown.

+
+``` + +Whenever the auto-slide mode is resumed or paused the ```autoslideresumed``` and ```autoslidepaused``` events are fired. + + +### Keyboard Bindings + +If you're unhappy with any of the default keyboard bindings you can override them using the ```keyboard``` config option: + +```javascript +Reveal.configure({ + keyboard: { + 13: 'next', // go to the next slide when the ENTER key is pressed + 27: function() {}, // do something custom when ESC is pressed + 32: null // don't do anything when SPACE is pressed (i.e. disable a reveal.js default binding) + } +}); +``` + +### Lazy Loading + +When working on presentation with a lot of media or iframe content it's important to load lazily. Lazy loading means that reveal.js will only load content for the few slides nearest to the current slide. The number of slides that are preloaded is determined by the `viewDistance` configuration option. + +To enable lazy loading all you need to do is change your "src" attributes to "data-src" as shown below. This is supported for image, video, audio and iframe elements. Lazy loaded iframes will also unload when the containing slide is no longer visible. + +```html +
+ + + +
+``` + + +### API + +The ``Reveal`` object exposes a JavaScript API for controlling navigation and reading state: + +```javascript +// Navigation +Reveal.slide( indexh, indexv, indexf ); +Reveal.left(); +Reveal.right(); +Reveal.up(); +Reveal.down(); +Reveal.prev(); +Reveal.next(); +Reveal.prevFragment(); +Reveal.nextFragment(); + +// Toggle presentation states, optionally pass true/false to force on/off +Reveal.toggleOverview(); +Reveal.togglePause(); +Reveal.toggleAutoSlide(); + +// Change a config value at runtime +Reveal.configure({ controls: true }); + +// Returns the present configuration options +Reveal.getConfig(); + +// Fetch the current scale of the presentation +Reveal.getScale(); + +// Retrieves the previous and current slide elements +Reveal.getPreviousSlide(); +Reveal.getCurrentSlide(); + +Reveal.getIndices(); // { h: 0, v: 0 } } +Reveal.getProgress(); // 0-1 +Reveal.getTotalSlides(); + +// State checks +Reveal.isFirstSlide(); +Reveal.isLastSlide(); +Reveal.isOverview(); +Reveal.isPaused(); +Reveal.isAutoSliding(); +``` + +### Slide Changed Event + +A 'slidechanged' event is fired each time the slide is changed (regardless of state). The event object holds the index values of the current slide as well as a reference to the previous and current slide HTML nodes. + +Some libraries, like MathJax (see [#226](https://github.com/hakimel/reveal.js/issues/226#issuecomment-10261609)), get confused by the transforms and display states of slides. Often times, this can be fixed by calling their update or render function from this callback. + +```javascript +Reveal.addEventListener( 'slidechanged', function( event ) { + // event.previousSlide, event.currentSlide, event.indexh, event.indexv +} ); +``` + +### Presentation State + +The presentation's current state can be fetched by using the `getState` method. A state object contains all of the information required to put the presentation back as it was when `getState` was first called. Sort of like a snapshot. It's a simple object that can easily be stringified and persisted or sent over the wire. + +```javascript +Reveal.slide( 1 ); +// we're on slide 1 + +var state = Reveal.getState(); + +Reveal.slide( 3 ); +// we're on slide 3 + +Reveal.setState( state ); +// we're back on slide 1 +``` + +### Slide States + +If you set ``data-state="somestate"`` on a slide ``
``, "somestate" will be applied as a class on the document element when that slide is opened. This allows you to apply broad style changes to the page based on the active slide. + +Furthermore you can also listen to these changes in state via JavaScript: + +```javascript +Reveal.addEventListener( 'somestate', function() { + // TODO: Sprinkle magic +}, false ); +``` + +### Slide Backgrounds + +Slides are contained within a limited portion of the screen by default to allow them to fit any display and scale uniformly. You can apply full page backgrounds outside of the slide area by adding a ```data-background``` attribute to your ```
``` elements. Four different types of backgrounds are supported: color, image, video and iframe. Below are a few examples. + +```html +
+

All CSS color formats are supported, like rgba() or hsl().

+
+
+

This slide will have a full-size background image.

+
+
+

This background image will be sized to 100px and repeated.

+
+
+

Video. Multiple sources can be defined using a comma separated list. Video will loop when the data-background-video-loop attribute is provided.

+
+
+

Embeds a web page as a background. Note that the page won't be interactive.

+
+``` + +Backgrounds transition using a fade animation by default. This can be changed to a linear sliding transition by passing ```backgroundTransition: 'slide'``` to the ```Reveal.initialize()``` call. Alternatively you can set ```data-background-transition``` on any section with a background to override that specific transition. + + +### Parallax Background + +If you want to use a parallax scrolling background, set the first two config properties below when initializing reveal.js (the other two are optional). + +```javascript +Reveal.initialize({ + + // Parallax background image + parallaxBackgroundImage: '', // e.g. "https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg" + + // Parallax background size + parallaxBackgroundSize: '', // CSS syntax, e.g. "2100px 900px" - currently only pixels are supported (don't use % or auto) + + // Amount of pixels to move the parallax background per slide step, + // a value of 0 disables movement along the given axis + // These are optional, if they aren't specified they'll be calculated automatically + parallaxBackgroundHorizontal: 200, + parallaxBackgroundVertical: 50 + +}); +``` + +Make sure that the background size is much bigger than screen size to allow for some scrolling. [View example](http://lab.hakim.se/reveal-js/?parallaxBackgroundImage=https%3A%2F%2Fs3.amazonaws.com%2Fhakim-static%2Freveal-js%2Freveal-parallax-1.jpg¶llaxBackgroundSize=2100px%20900px). + + + +### Slide Transitions +The global presentation transition is set using the ```transition``` config value. You can override the global transition for a specific slide by using the ```data-transition``` attribute: + +```html +
+

This slide will override the presentation transition and zoom!

+
+ +
+

Choose from three transition speeds: default, fast or slow!

+
+``` + +You can also use different in and out transitions for the same slide: + +```html +
+ The train goes on … +
+
+ and on … +
+
+ and stops. +
+
+ (Passengers entering and leaving) +
+
+ And it starts again. +
+``` + + +Note that this does not work with the page and cube transitions. + + +### Internal links + +It's easy to link between slides. The first example below targets the index of another slide whereas the second targets a slide with an ID attribute (```
```): + +```html +Link +Link +``` + +You can also add relative navigation links, similar to the built in reveal.js controls, by appending one of the following classes on any element. Note that each element is automatically given an ```enabled``` class when it's a valid navigation route based on the current slide. + +```html + + + + + + +``` + + +### Fragments +Fragments are used to highlight individual elements on a slide. Every element with the class ```fragment``` will be stepped through before moving on to the next slide. Here's an example: http://lab.hakim.se/reveal-js/#/fragments + +The default fragment style is to start out invisible and fade in. This style can be changed by appending a different class to the fragment: + +```html +
+

grow

+

shrink

+

fade-out

+

visible only once

+

blue only once

+

highlight-red

+

highlight-green

+

highlight-blue

+
+``` + +Multiple fragments can be applied to the same element sequentially by wrapping it, this will fade in the text on the first step and fade it back out on the second. + +```html +
+ + I'll fade in, then out + +
+``` + +The display order of fragments can be controlled using the ```data-fragment-index``` attribute. + +```html +
+

Appears last

+

Appears first

+

Appears second

+
+``` + +### Fragment events + +When a slide fragment is either shown or hidden reveal.js will dispatch an event. + +Some libraries, like MathJax (see #505), get confused by the initially hidden fragment elements. Often times this can be fixed by calling their update or render function from this callback. + +```javascript +Reveal.addEventListener( 'fragmentshown', function( event ) { + // event.fragment = the fragment DOM element +} ); +Reveal.addEventListener( 'fragmenthidden', function( event ) { + // event.fragment = the fragment DOM element +} ); +``` + +### Code syntax highlighting + +By default, Reveal is configured with [highlight.js](http://softwaremaniacs.org/soft/highlight/en/) for code syntax highlighting. Below is an example with clojure code that will be syntax highlighted. When the `data-trim` attribute is present surrounding whitespace is automatically removed. + +```html +
+ 

+(def lazy-fib
+  (concat
+   [0 1]
+   ((fn rfib [a b]
+        (lazy-cons (+ a b) (rfib b (+ a b)))) 0 1)))
+
+
+``` + +### Slide number +If you would like to display the page number of the current slide you can do so using the ```slideNumber``` configuration value. + +```javascript +// Shows the slide number using default formatting +Reveal.configure({ slideNumber: true }); + +// Slide number formatting can be configured using these variables: +// h: current slide's horizontal index +// v: current slide's vertical index +// c: current slide index (flattened) +// t: total number of slides (flattened) +Reveal.configure({ slideNumber: 'c / t' }); + +``` + + +### Overview mode + +Press "Esc" or "o" keys to toggle the overview mode on and off. While you're in this mode, you can still navigate between slides, +as if you were at 1,000 feet above your presentation. The overview mode comes with a few API hooks: + +```javascript +Reveal.addEventListener( 'overviewshown', function( event ) { /* ... */ } ); +Reveal.addEventListener( 'overviewhidden', function( event ) { /* ... */ } ); + +// Toggle the overview mode programmatically +Reveal.toggleOverview(); +``` + +### Fullscreen mode +Just press »F« on your keyboard to show your presentation in fullscreen mode. Press the »ESC« key to exit fullscreen mode. + + +### Embedded media +Embedded HTML5 `
+ +
+ +

 

 

 

+ + + + + + +
+

Week 48: Support Vector Machines and Summary of course

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Nov 22, 2020

+
+

+ + +

Read »

+ + +
+ +

+ +

+ + +
+ + + + + + + +
+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/week48/html/week48-reveal.html b/doc/pub/week48/html/week48-reveal.html new file mode 100644 index 000000000..89f6e67d9 --- /dev/null +++ b/doc/pub/week48/html/week48-reveal.html @@ -0,0 +1,1516 @@ + + + + + + + +Week 48: Support Vector Machines and Summary of course + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + +
+ + + + + + + + + + + + + + +
+ + + + +

Week 48: Support Vector Machines and Summary of course

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

 
+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

 
+

Nov 22, 2020

+
+

+ +

+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+
+ + +
+

Overview of week 48

+ +
    +

  • Thursday: Support Vector Machines, Kernels, Classification and Regression
    • +

  • Friday: Summary of course with perspectives for future studies
    • +
+

+ +Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion. +

+ + +
+

Thursday

+ +

+We finalize our discussion on Support Vector Machines with an emphasis on kernel transformations and applications to regression. The following video attempts at giving an overview on this part. See also the follow-up video. +

+ + +
+

Friday

+ +

+Friday's lecture is split in two parts. It starts with a summary of +what we have done this semester and continues with perspectives for future studies and +modern research projects in machine learning. +

+ + +
+

Support Vector Machines, overarching aims

+ +

+As discussed last week, +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. + +

+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. I recommend you take a look at the lectures from last week on the binary classification problem. +

+ + +
+

Kernels and non-linearity

+ +

+The cases we studied last week 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()
+
+
+ + +
+

The equations

+ +

+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. +

+ + +
+

The problem to solve

+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. +

  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. +

+ + +
+

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. +

+ + +
+

How do we solve these problems?

+ +

+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. +

+ + +
+

A simple example

+ +

+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’]
+
+
+ + +
+

Back to the more realistic cases

+ +

+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 \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). 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 \( 0\leq \lambda_i \leq C \). +

+ + +
+

Summary of course

+
+ + +
+

What? Me worry? No final exam in this course!

+



+
+ + +
+

Topics we have covered this year

+ +

+The course has two central parts + +

    +

  1. Statistical analysis and optimization of data
    2. +

  2. Machine learning
    3. +
+
+ + +
+

Statistical analysis and optimization of data

+ +

+The following topics be covered + +

    +

  1. Basic concepts, expectation values, variance, covariance, correlation functions and errors;
    2. +

  2. Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;
    3. +

  3. Central elements from linear algebra
    4. +

  4. Gradient methods for data optimization
    5. +

  5. Estimation of errors using cross-validation, bootstrapping and jackknife methods;
    6. +

  6. Practical optimization using Singular-value decomposition and least squares for parameterizing data.
    7. +

  7. Principal Component Analysis.
    8. +
+
+ + +
+

Machine learning

+ +

+The following topics will be covered + +

    +

  1. Linear methods for regression and classification: + +
      +

    1. Ordinary Least Squares
      2. +

    2. Ridge regression
      3. +

    3. Lasso regression
      4. +

    4. Logistic regression
      5. +
    +

  2. Neural networks and deep learning: + +
      +

    1. Feed Forward Neural Networks
      2. +

    2. Convolutional Neural Networks
      3. +

    3. Recurrent Neural Networks
      4. +
    +

  3. Decisions trees and ensemble methods: + +
      +

    1. Decision trees
      2. +

    2. Bagging and voting
      3. +

    3. Random forests
      4. +

    4. Boosting and gradient boosting
      5. +
    +

  4. Support vector machines + +
      +

    1. Binary classification and multiclass classification
      2. +

    2. Kernel methods
      3. +

    3. Regression
      4. +
    +

    +

+
+ + +
+

Learning outcomes and overarching aims of this course

+ +

+The course introduces a variety of central algorithms and methods +essential for studies of data analysis and machine learning. The +course is project based and through the various 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. The students will learn to develop and structure large codes +for studying these systems, get acquainted with computing facilities +and learn to handle large scientific projects. A good scientific and +ethical conduct is emphasized throughout the course. + +

    +

  • Understand linear methods for regression and classification;
    • +

  • Learn about neural network;
    • +

  • Learn about baggin, boosting and trees
    • +

  • Support vector machines
    • +

  • Learn about basic data analysis;
    • +

  • 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;
    • +

  • 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++.
    • +
+
+ + +
+

Perspective on Machine Learning

+ +
    +

  1. Rapidly emerging application area
    2. +

  2. Experiment AND theory are evolving in many many fields. Still many low-hanging fruits.
    3. +

  3. Requires education/retraining for more widespread adoption
    4. +

  4. A lot of “word-of-mouth” development methods
    5. +
+

+ +Huge amounts of data sets require automation, classical analysis tools often inadequate. +High energy physics hit this wall in the 90’s. +In 2009 single top quark production was determined via Boosted decision trees, Bayesian +Neural Networks, etc. +

+ + +
+

Machine Learning Research

+ +

+Where to find recent results: + +

    +

  1. Conference proceedings, arXiv and blog posts!
    2. +

  2. NIPS: Neural Information Processing Systems
    3. +

  3. ICLR: International Conference on Learning Representations
    4. +

  4. ICML: International Conference on Machine Learning
    5. +

  5. Journal of Machine Learning Research
    6. +
+
+ + +
+

Starting your Machine Learning Project

+ +
    +

  1. Identify problem type: classification, generation, regression
    2. +

  2. Consider your data carefully
    3. +

  3. Choose a simple model that fits 1. and 2.
    4. +

  4. Consider your data carefully again… data representation
    5. +

  5. Based on results, feedback loop to earliest possible point
    6. +
+
+ + +
+

Choose a Model and Algorithm

+ +
    +

  1. Supervised?
    2. +

  2. Start with the simplest model that fits your problem
    3. +

  3. Start with minimal processing of data
    4. +
+
+ + +
+

Preparing Your Data

+ +
    +

  1. Shuffle your data
    2. +

  2. Mean center your data
    3. + +
      + +

    • Why?
      • +
    +

  3. Normalize the variance
    4. + +
      + +

    • Why?
      • +
    +

  4. Whitening
    5. + +
      + +

    • Decorrelates data
      • + +

    • Can be hit or miss
      • +
    +

  5. When to do train/test split?
    6. +
+
+ + +
+

Which Activation and Weights to Choose in Neural Networks

+ +
    +

  1. RELU? ELU?
    2. +

  2. Sigmoid or Tanh?
    3. +

  3. Set all weights to 0?
    4. + +
      + +

    • Terrible idea
      • +
    +

  4. Set all weights to random values?
    5. + +
      + +

    • Small random values
      • +
    +

    +

+
+ + +
+

Optimization Methods and Hyperparameters

+ +
    +

  1. Stochastic gradient descent + +
      +

    1. Stochastic gradient descent + momentum
      2. +
    +

  2. State-of-the-art approaches:
    3. + +
      + +

    • RMSProp
      • + +

    • Adam
      • +
    +

    +

+

+ +Which regularization and hyperparameters? \( L_1 \) or \( L_2 \), soft classifiers, depths of trees and many other. Need to explore a large set of hyperparameters and regularization methods. +

+ + +
+

Resampling

+ +

+When do we resample? + +

    +

  1. Bootstrap
    2. +

  2. Cross-validation
    3. +

  3. Jackknife and many other
    4. +
+
+ + +
+

Other courses on Data science and Machine Learning at UiO

+ +

+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. + +

    +

  1. STK2100 Machine learning and statistical methods for prediction and classification.
    2. +

  2. IN3050/IN4050 Introduction to Artificial Intelligence and Machine Learning. Introductory course in machine learning and AI with an algorithmic approach.
    3. +

  3. STK-INF3000/4000 Selected Topics in Data Science. The course provides insight into selected contemporary relevant topics within Data Science.
    4. +

  4. IN4080 Natural Language Processing. Probabilistic and machine learning techniques applied to natural language processing.
    5. +

  5. STK-IN4300 – Statistical learning methods in Data Science. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.
    6. +

  6. IN-STK5000 Adaptive Methods for Data-Based Decision Making. Methods for adaptive collection and processing of data based on machine learning techniques.
    7. +

  7. IN5400/INF5860 – Machine Learning for Image Analysis. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.
    8. +

  8. TEK5040 – Dyp læring for autonome systemer. 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.
    9. +
+
+ + +
+

Additional courses of interest

+ +
    +

  1. STK4051 Computational Statistics
    2. +

  2. STK4021 Applied Bayesian Analysis and Numerical Methods
    3. +
+
+ + +
+

What's the future like?

+ +

+Based on multi-layer nonlinear neural networks, deep learning can +learn directly from raw data, automatically extract and abstract +features from layer to layer, and then achieve the goal of regression, +classification, or ranking. Deep learning has made breakthroughs in +computer vision, speech processing and natural language, and reached +or even surpassed human level. The success of deep learning is mainly +due to the three factors: big data, big model, and big computing. + +

+In the past few decades, many different architectures of deep neural +networks have been proposed, such as + +

    +

  1. Convolutional neural networks, which are mostly used in image and video data processing, and have also been applied to sequential data such as text processing;
    2. +

  2. Recurrent neural networks, which can process sequential data of variable length and have been widely used in natural language understanding and speech processing;
    3. +

  3. Encoder-decoder framework, which is mostly used for image or sequence generation, such as machine translation, text summarization, and image captioning.
    4. +
+
+ + +
+

Bayesian Machine Learning

+ +

+This is an important topic if we aim at extracting a probability +distribution. This gives us also a confidence interval and error +estimates. + +

+Bayesian machine learning allows us to encode our prior beliefs about +what those models should look like, independent of what the data tells +us. This is especially useful when we don’t have a ton of data to +confidently learn our model. +

+ + +
+

Reinforcement Learning

+ +

+Reinforcement learning is a sub-area of machine learning. It studies +how agents take actions based on trial and error, so as to maximize +some notion of cumulative reward in a dynamic system or +environment. Due to its generality, the problem has also been studied +in many other disciplines, such as game theory, control theory, +operations research, information theory, multi-agent systems, swarm +intelligence, statistics, and genetic algorithms. + +

+In March 2016, AlphaGo, a computer program that plays the board game +Go, beat Lee Sedol in a five-game match. This was the first time a +computer Go program had beaten a 9-dan (highest rank) professional +without handicaps. AlphaGo is based on deep convolutional neural +networks and reinforcement learning. AlphaGo’s victory was a major +milestone in artificial intelligence and it has also made +reinforcement learning a hot research area in the field of machine +learning. +

+ + +
+

Transfer learning

+ +

+The goal of transfer learning is to transfer the model or knowledge +obtained from a source task to the target task, in order to resolve +the issues of insufficient training data in the target task. The +rationality of doing so lies in that usually the source and target +tasks have inter-correlations, and therefore either the features, +samples, or models in the source task might provide useful information +for us to better solve the target task. Transfer learning is a hot +research topic in recent years, with many problems still waiting to be +solved in this space. +

+ + +
+

Adversarial learning

+ +

+The conventional deep generative model has a potential problem: the +model tends to generate extreme instances to maximize the +probabilistic likelihood, which will hurt its performance. Adversarial +learning utilizes the adversarial behaviors (e.g., generating +adversarial instances or training an adversarial model) to enhance the +robustness of the model and improve the quality of the generated +data. In recent years, one of the most promising unsupervised learning +technologies, generative adversarial networks (GAN), has already been +successfully applied to image, speech, and text. +

+ + +
+

Dual learning

+ +

+Dual learning is a new learning paradigm, the basic idea of which is +to use the primal-dual structure between machine learning tasks to +obtain effective feedback/regularization, and guide and strengthen the +learning process, thus reducing the requirement of large-scale labeled +data for deep learning. The idea of dual learning has been applied to +many problems in machine learning, including machine translation, +image style conversion, question answering and generation, image +classification and generation, text classification and generation, +image-to-text, and text-to-image. +

+ + +
+

Distributed machine learning

+ +

+Distributed computation will speed up machine learning algorithms, +significantly improve their efficiency, and thus enlarge their +application. When distributed meets machine learning, more than just +implementing the machine learning algorithms in parallel is required. +

+ + +
+

Meta learning

+ +

+Meta learning is an emerging research direction in machine +learning. Roughly speaking, meta learning concerns learning how to +learn, and focuses on the understanding and adaptation of the learning +itself, instead of just completing a specific learning task. That is, +a meta learner needs to be able to evaluate its own learning methods +and adjust its own learning methods according to specific learning +tasks. +

+ + +
+

The Challenges Facing Machine Learning

+ +

+While there has been much progress in machine learning, there are also challenges. + +

+For example, the mainstream machine learning technologies are +black-box approaches, making us concerned about their potential +risks. To tackle this challenge, we may want to make machine learning +more explainable and controllable. As another example, the +computational complexity of machine learning algorithms is usually +very high and we may want to invent lightweight algorithms or +implementations. Furthermore, in many domains such as physics, +chemistry, biology, and social sciences, people usually seek elegantly +simple equations (e.g., the Schrödinger equation) to uncover the +underlying laws behind various phenomena. In the field of machine +learning, can we reveal simple laws instead of designing more complex +models for data fitting? Although there are many challenges, we are +still very optimistic about the future of machine learning. As we look +forward to the future, here are what we think the research hotspots in +the next ten years will be. +

+ + +
+

Explainable machine learning

+ +

+Machine learning, especially deep learning, evolves rapidly. The +ability gap between machine and human on many complex cognitive tasks +becomes narrower and narrower. However, we are still in the very early +stage in terms of explaining why those effective models work and how +they work. + +

+What is missing: the gap between correlation and causation Most +machine learning techniques, especially the statistical ones, depend +highly on data correlation to make predictions and analyses. In +contrast, rational humans tend to reply on clear and trustworthy +causality relations obtained via logical reasoning on real and clear +facts. It is one of the core goals of explainable machine learning to +transition from solving problems by data correlation to solving +problems by logical reasoning. +

+ + +
+

Quantum machine learning

+ +

+Quantum machine learning is an emerging interdisciplinary research +area at the intersection of quantum computing and machine learning. + +

+Quantum computers use effects such as quantum coherence and quantum +entanglement to process information, which is fundamentally different +from classical computers. Quantum algorithms have surpassed the best +classical algorithms in several problems (e.g., searching for an +unsorted database, inverting a sparse matrix), which we call quantum +acceleration. + +

+When quantum computing meets machine learning, it can be a mutually +beneficial and reinforcing process, as it allows us to take advantage +of quantum computing to improve the performance of classical machine +learning algorithms. In addition, we can also use the machine learning +algorithms (on classic computers) to analyze and improve quantum +computing systems. +

+ + +
+

Quantum machine learning algorithms based on linear algebra

+ +

+Many quantum machine learning algorithms are based on variants of +quantum algorithms for solving linear equations, which can efficiently +solve N-variable linear equations with complexity of O(log2 N) under +certain conditions. The quantum matrix inversion algorithm can +accelerate many machine learning methods, such as least square linear +regression, least square version of support vector machine, Gaussian +process, and more. The training of these algorithms can be simplified +to solve linear equations. The key bottleneck of this type of quantum +machine learning algorithms is data input—that is, how to initialize +the quantum system with the entire data set. Although efficient +data-input algorithms exist for certain situations, how to efficiently +input data into a quantum system is as yet unknown for most cases. +

+ + +
+

Quantum reinforcement learning

+ +

+In quantum reinforcement learning, a quantum agent interacts with the +classical environment to obtain rewards from the environment, so as to +adjust and improve its behavioral strategies. In some cases, it +achieves quantum acceleration by the quantum processing capabilities +of the agent or the possibility of exploring the environment through +quantum superposition. Such algorithms have been proposed in +superconducting circuits and systems of trapped ions. +

+ + +
+

Quantum deep learning

+ +

+Dedicated quantum information processors, such as quantum annealers +and programmable photonic circuits, are well suited for building deep +quantum networks. The simplest deep quantum network is the Boltzmann +machine. The classical Boltzmann machine consists of bits with tunable +interactions and is trained by adjusting the interaction of these bits +so that the distribution of its expression conforms to the statistics +of the data. To quantize the Boltzmann machine, the neural network can +simply be represented as a set of interacting quantum spins that +correspond to an adjustable Ising model. Then, by initializing the +input neurons in the Boltzmann machine to a fixed state and allowing +the system to heat up, we can read out the output qubits to get the +result. +

+ + +
+

Social machine learning

+ +

+Machine learning aims to imitate how humans +learn. While we have developed successful machine learning algorithms, +until now we have ignored one important fact: humans are social. Each +of us is one part of the total society and it is difficult for us to +live, learn, and improve ourselves, alone and isolated. Therefore, we +should design machines with social properties. Can we let machines +evolve by imitating human society so as to achieve more effective, +intelligent, interpretable “social machine learning”? + +

+And much more. +

+ + +
+

The last words?

+ +

+Early computer scientist Alan Kay said, The best way to predict the +future is to create it. Therefore, all machine learning +practitioners, whether scholars or engineers, professors or students, +need to work together to advance these important research +topics. Together, we will not just predict the future, but create it. +

+ + +
+

Best wishes to you all and thanks so much for your heroic efforts this semester

+ +

+



+
+ + + +
+
+ + + + + + + + + + + + diff --git a/doc/pub/week48/html/week48-solarized.html b/doc/pub/week48/html/week48-solarized.html new file mode 100644 index 000000000..ee25ccd86 --- /dev/null +++ b/doc/pub/week48/html/week48-solarized.html @@ -0,0 +1,1321 @@ + + + + + + + + +Week 48: Support Vector Machines and Summary of course + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Week 48: Support Vector Machines and Summary of course

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Nov 22, 2020

+
+

+









+ +

Overview of week 48

+ +
    +
  • Thursday: Support Vector Machines, Kernels, Classification and Regression
    • +
  • Friday: Summary of course with perspectives for future studies
    • +
+ +Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion. + +

+









+ +

Thursday

+ +

+We finalize our discussion on Support Vector Machines with an emphasis on kernel transformations and applications to regression. The following video attempts at giving an overview on this part. See also the follow-up video. + +

+









+ +

Friday

+ +

+Friday's lecture is split in two parts. It starts with a summary of +what we have done this semester and continues with perspectives for future studies and +modern research projects in machine learning. + +

+









+ +

Support Vector Machines, overarching aims

+ +

+As discussed last week, +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. + +

+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. I recommend you take a look at the lectures from last week on the binary classification problem. + +

+









+ +

Kernels and non-linearity

+ +

+The cases we studied last week 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()
+
+

+









+ +

The equations

+ +

+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. + +

+









+ +

The problem to solve

+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. +
  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. + +

+









+ +

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. + +

+









+ +

How do we solve these problems?

+ +

+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. + +

+









+ +

A simple example

+ +

+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’]
+
+

+









+ +

Back to the more realistic cases

+ +

+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 \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). 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 \( 0\leq \lambda_i \leq C \). + +

+









+ +

Summary of course

+ +

+









+ +

What? Me worry? No final exam in this course!

+



+ +

+









+ +

Topics we have covered this year

+ +

+The course has two central parts + +

    +
  1. Statistical analysis and optimization of data
    2. +
  2. Machine learning
    3. +
+ +









+ +

Statistical analysis and optimization of data

+ +

+The following topics be covered + +

    +
  1. Basic concepts, expectation values, variance, covariance, correlation functions and errors;
    2. +
  2. Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;
    3. +
  3. Central elements from linear algebra
    4. +
  4. Gradient methods for data optimization
    5. +
  5. Estimation of errors using cross-validation, bootstrapping and jackknife methods;
    6. +
  6. Practical optimization using Singular-value decomposition and least squares for parameterizing data.
    7. +
  7. Principal Component Analysis.
    8. +
+ +









+ +

Machine learning

+ +

+The following topics will be covered + +

    +
  1. Linear methods for regression and classification: + +
      +
    1. Ordinary Least Squares
      2. +
    2. Ridge regression
      3. +
    3. Lasso regression
      4. +
    4. Logistic regression
      5. +
    + +
  2. Neural networks and deep learning: + +
      +
    1. Feed Forward Neural Networks
      2. +
    2. Convolutional Neural Networks
      3. +
    3. Recurrent Neural Networks
      4. +
    + +
  3. Decisions trees and ensemble methods: + +
      +
    1. Decision trees
      2. +
    2. Bagging and voting
      3. +
    3. Random forests
      4. +
    4. Boosting and gradient boosting
      5. +
    + +
  4. Support vector machines + +
      +
    1. Binary classification and multiclass classification
      2. +
    2. Kernel methods
      3. +
    3. Regression
      4. +
    + +
+ +









+ +

Learning outcomes and overarching aims of this course

+ +

+The course introduces a variety of central algorithms and methods +essential for studies of data analysis and machine learning. The +course is project based and through the various 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. The students will learn to develop and structure large codes +for studying these systems, get acquainted with computing facilities +and learn to handle large scientific projects. A good scientific and +ethical conduct is emphasized throughout the course. + +

    +
  • Understand linear methods for regression and classification;
    • +
  • Learn about neural network;
    • +
  • Learn about baggin, boosting and trees
    • +
  • Support vector machines
    • +
  • Learn about basic data analysis;
    • +
  • 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;
    • +
  • 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++.
    • +
+ +









+ +

Perspective on Machine Learning

+ +
    +
  1. Rapidly emerging application area
    2. +
  2. Experiment AND theory are evolving in many many fields. Still many low-hanging fruits.
    3. +
  3. Requires education/retraining for more widespread adoption
    4. +
  4. A lot of “word-of-mouth” development methods
    5. +
+ +Huge amounts of data sets require automation, classical analysis tools often inadequate. +High energy physics hit this wall in the 90’s. +In 2009 single top quark production was determined via Boosted decision trees, Bayesian +Neural Networks, etc. + +

+









+ +

Machine Learning Research

+ +

+Where to find recent results: + +

    +
  1. Conference proceedings, arXiv and blog posts!
    2. +
  2. NIPS: Neural Information Processing Systems
    3. +
  3. ICLR: International Conference on Learning Representations
    4. +
  4. ICML: International Conference on Machine Learning
    5. +
  5. Journal of Machine Learning Research
    6. +
+ +









+ +

Starting your Machine Learning Project

+ +
    +
  1. Identify problem type: classification, generation, regression
    2. +
  2. Consider your data carefully
    3. +
  3. Choose a simple model that fits 1. and 2.
    4. +
  4. Consider your data carefully again… data representation
    5. +
  5. Based on results, feedback loop to earliest possible point
    6. +
+ +









+ +

Choose a Model and Algorithm

+ +
    +
  1. Supervised?
    2. +
  2. Start with the simplest model that fits your problem
    3. +
  3. Start with minimal processing of data
    4. +
+ +









+ +

Preparing Your Data

+ +
    +
  1. Shuffle your data
    2. +
  2. Mean center your data
    3. + +
      +
    • Why?
      • +
    + +
  3. Normalize the variance
    4. + +
      +
    • Why?
      • +
    + +
  4. Whitening
    5. + +
      +
    • Decorrelates data
      • +
    • Can be hit or miss
      • +
    + +
  5. When to do train/test split?
    6. +
+ +









+ +

Which Activation and Weights to Choose in Neural Networks

+ +
    +
  1. RELU? ELU?
    2. +
  2. Sigmoid or Tanh?
    3. +
  3. Set all weights to 0?
    4. + +
      +
    • Terrible idea
      • +
    + +
  4. Set all weights to random values?
    5. + +
      +
    • Small random values
      • +
    + +
+ +









+ +

Optimization Methods and Hyperparameters

+ +
    +
  1. Stochastic gradient descent + +
      +
    1. Stochastic gradient descent + momentum
      2. +
    + +
  2. State-of-the-art approaches:
    3. + +
      +
    • RMSProp
      • +
    • Adam
      • +
    + +
+ +Which regularization and hyperparameters? \( L_1 \) or \( L_2 \), soft classifiers, depths of trees and many other. Need to explore a large set of hyperparameters and regularization methods. + +

+









+ +

Resampling

+ +

+When do we resample? + +

    +
  1. Bootstrap
    2. +
  2. Cross-validation
    3. +
  3. Jackknife and many other
    4. +
+ +









+ +

Other courses on Data science and Machine Learning at UiO

+ +

+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. + +

    +
  1. STK2100 Machine learning and statistical methods for prediction and classification.
    2. +
  2. IN3050/IN4050 Introduction to Artificial Intelligence and Machine Learning. Introductory course in machine learning and AI with an algorithmic approach.
    3. +
  3. STK-INF3000/4000 Selected Topics in Data Science. The course provides insight into selected contemporary relevant topics within Data Science.
    4. +
  4. IN4080 Natural Language Processing. Probabilistic and machine learning techniques applied to natural language processing.
    5. +
  5. STK-IN4300 – Statistical learning methods in Data Science. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.
    6. +
  6. IN-STK5000 Adaptive Methods for Data-Based Decision Making. Methods for adaptive collection and processing of data based on machine learning techniques.
    7. +
  7. IN5400/INF5860 – Machine Learning for Image Analysis. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.
    8. +
  8. TEK5040 – Dyp læring for autonome systemer. 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.
    9. +
+ +









+ +

Additional courses of interest

+ +
    +
  1. STK4051 Computational Statistics
    2. +
  2. STK4021 Applied Bayesian Analysis and Numerical Methods
    3. +
+ +









+ +

What's the future like?

+ +

+Based on multi-layer nonlinear neural networks, deep learning can +learn directly from raw data, automatically extract and abstract +features from layer to layer, and then achieve the goal of regression, +classification, or ranking. Deep learning has made breakthroughs in +computer vision, speech processing and natural language, and reached +or even surpassed human level. The success of deep learning is mainly +due to the three factors: big data, big model, and big computing. + +

+In the past few decades, many different architectures of deep neural +networks have been proposed, such as + +

    +
  1. Convolutional neural networks, which are mostly used in image and video data processing, and have also been applied to sequential data such as text processing;
    2. +
  2. Recurrent neural networks, which can process sequential data of variable length and have been widely used in natural language understanding and speech processing;
    3. +
  3. Encoder-decoder framework, which is mostly used for image or sequence generation, such as machine translation, text summarization, and image captioning.
    4. +
+ +









+ +

Bayesian Machine Learning

+ +

+This is an important topic if we aim at extracting a probability +distribution. This gives us also a confidence interval and error +estimates. + +

+Bayesian machine learning allows us to encode our prior beliefs about +what those models should look like, independent of what the data tells +us. This is especially useful when we don’t have a ton of data to +confidently learn our model. + +

+









+ +

Reinforcement Learning

+ +

+Reinforcement learning is a sub-area of machine learning. It studies +how agents take actions based on trial and error, so as to maximize +some notion of cumulative reward in a dynamic system or +environment. Due to its generality, the problem has also been studied +in many other disciplines, such as game theory, control theory, +operations research, information theory, multi-agent systems, swarm +intelligence, statistics, and genetic algorithms. + +

+In March 2016, AlphaGo, a computer program that plays the board game +Go, beat Lee Sedol in a five-game match. This was the first time a +computer Go program had beaten a 9-dan (highest rank) professional +without handicaps. AlphaGo is based on deep convolutional neural +networks and reinforcement learning. AlphaGo’s victory was a major +milestone in artificial intelligence and it has also made +reinforcement learning a hot research area in the field of machine +learning. + +

+









+ +

Transfer learning

+ +

+The goal of transfer learning is to transfer the model or knowledge +obtained from a source task to the target task, in order to resolve +the issues of insufficient training data in the target task. The +rationality of doing so lies in that usually the source and target +tasks have inter-correlations, and therefore either the features, +samples, or models in the source task might provide useful information +for us to better solve the target task. Transfer learning is a hot +research topic in recent years, with many problems still waiting to be +solved in this space. + +

+









+ +

Adversarial learning

+ +

+The conventional deep generative model has a potential problem: the +model tends to generate extreme instances to maximize the +probabilistic likelihood, which will hurt its performance. Adversarial +learning utilizes the adversarial behaviors (e.g., generating +adversarial instances or training an adversarial model) to enhance the +robustness of the model and improve the quality of the generated +data. In recent years, one of the most promising unsupervised learning +technologies, generative adversarial networks (GAN), has already been +successfully applied to image, speech, and text. + +

+









+ +

Dual learning

+ +

+Dual learning is a new learning paradigm, the basic idea of which is +to use the primal-dual structure between machine learning tasks to +obtain effective feedback/regularization, and guide and strengthen the +learning process, thus reducing the requirement of large-scale labeled +data for deep learning. The idea of dual learning has been applied to +many problems in machine learning, including machine translation, +image style conversion, question answering and generation, image +classification and generation, text classification and generation, +image-to-text, and text-to-image. + +

+









+ +

Distributed machine learning

+ +

+Distributed computation will speed up machine learning algorithms, +significantly improve their efficiency, and thus enlarge their +application. When distributed meets machine learning, more than just +implementing the machine learning algorithms in parallel is required. + +

+









+ +

Meta learning

+ +

+Meta learning is an emerging research direction in machine +learning. Roughly speaking, meta learning concerns learning how to +learn, and focuses on the understanding and adaptation of the learning +itself, instead of just completing a specific learning task. That is, +a meta learner needs to be able to evaluate its own learning methods +and adjust its own learning methods according to specific learning +tasks. + +

+









+ +

The Challenges Facing Machine Learning

+ +

+While there has been much progress in machine learning, there are also challenges. + +

+For example, the mainstream machine learning technologies are +black-box approaches, making us concerned about their potential +risks. To tackle this challenge, we may want to make machine learning +more explainable and controllable. As another example, the +computational complexity of machine learning algorithms is usually +very high and we may want to invent lightweight algorithms or +implementations. Furthermore, in many domains such as physics, +chemistry, biology, and social sciences, people usually seek elegantly +simple equations (e.g., the Schrödinger equation) to uncover the +underlying laws behind various phenomena. In the field of machine +learning, can we reveal simple laws instead of designing more complex +models for data fitting? Although there are many challenges, we are +still very optimistic about the future of machine learning. As we look +forward to the future, here are what we think the research hotspots in +the next ten years will be. + +

+









+ +

Explainable machine learning

+ +

+Machine learning, especially deep learning, evolves rapidly. The +ability gap between machine and human on many complex cognitive tasks +becomes narrower and narrower. However, we are still in the very early +stage in terms of explaining why those effective models work and how +they work. + +

+What is missing: the gap between correlation and causation Most +machine learning techniques, especially the statistical ones, depend +highly on data correlation to make predictions and analyses. In +contrast, rational humans tend to reply on clear and trustworthy +causality relations obtained via logical reasoning on real and clear +facts. It is one of the core goals of explainable machine learning to +transition from solving problems by data correlation to solving +problems by logical reasoning. + +

+









+ +

Quantum machine learning

+ +

+Quantum machine learning is an emerging interdisciplinary research +area at the intersection of quantum computing and machine learning. + +

+Quantum computers use effects such as quantum coherence and quantum +entanglement to process information, which is fundamentally different +from classical computers. Quantum algorithms have surpassed the best +classical algorithms in several problems (e.g., searching for an +unsorted database, inverting a sparse matrix), which we call quantum +acceleration. + +

+When quantum computing meets machine learning, it can be a mutually +beneficial and reinforcing process, as it allows us to take advantage +of quantum computing to improve the performance of classical machine +learning algorithms. In addition, we can also use the machine learning +algorithms (on classic computers) to analyze and improve quantum +computing systems. + +

+









+ +

Quantum machine learning algorithms based on linear algebra

+ +

+Many quantum machine learning algorithms are based on variants of +quantum algorithms for solving linear equations, which can efficiently +solve N-variable linear equations with complexity of O(log2 N) under +certain conditions. The quantum matrix inversion algorithm can +accelerate many machine learning methods, such as least square linear +regression, least square version of support vector machine, Gaussian +process, and more. The training of these algorithms can be simplified +to solve linear equations. The key bottleneck of this type of quantum +machine learning algorithms is data input—that is, how to initialize +the quantum system with the entire data set. Although efficient +data-input algorithms exist for certain situations, how to efficiently +input data into a quantum system is as yet unknown for most cases. + +

+









+ +

Quantum reinforcement learning

+ +

+In quantum reinforcement learning, a quantum agent interacts with the +classical environment to obtain rewards from the environment, so as to +adjust and improve its behavioral strategies. In some cases, it +achieves quantum acceleration by the quantum processing capabilities +of the agent or the possibility of exploring the environment through +quantum superposition. Such algorithms have been proposed in +superconducting circuits and systems of trapped ions. + +

+









+ +

Quantum deep learning

+ +

+Dedicated quantum information processors, such as quantum annealers +and programmable photonic circuits, are well suited for building deep +quantum networks. The simplest deep quantum network is the Boltzmann +machine. The classical Boltzmann machine consists of bits with tunable +interactions and is trained by adjusting the interaction of these bits +so that the distribution of its expression conforms to the statistics +of the data. To quantize the Boltzmann machine, the neural network can +simply be represented as a set of interacting quantum spins that +correspond to an adjustable Ising model. Then, by initializing the +input neurons in the Boltzmann machine to a fixed state and allowing +the system to heat up, we can read out the output qubits to get the +result. + +

+









+ +

Social machine learning

+ +

+Machine learning aims to imitate how humans +learn. While we have developed successful machine learning algorithms, +until now we have ignored one important fact: humans are social. Each +of us is one part of the total society and it is difficult for us to +live, learn, and improve ourselves, alone and isolated. Therefore, we +should design machines with social properties. Can we let machines +evolve by imitating human society so as to achieve more effective, +intelligent, interpretable “social machine learning”? + +

+And much more. + +

+









+ +

The last words?

+ +

+Early computer scientist Alan Kay said, The best way to predict the +future is to create it. Therefore, all machine learning +practitioners, whether scholars or engineers, professors or students, +need to work together to advance these important research +topics. Together, we will not just predict the future, but create it. + +

+









+ +

Best wishes to you all and thanks so much for your heroic efforts this semester

+ +

+



+ + + + +
+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/week48/html/week48.html b/doc/pub/week48/html/week48.html new file mode 100644 index 000000000..ea4f4589c --- /dev/null +++ b/doc/pub/week48/html/week48.html @@ -0,0 +1,1326 @@ + + + + + + + + +Week 48: Support Vector Machines and Summary of course + + + + + + + + + + + + + + + + + + + + + + + +

Week 48: Support Vector Machines and Summary of course

+ +

+ + +

+Morten Hjorth-Jensen [1, 2] +
+ +

+ + +

[1] Department of Physics, University of Oslo
+
[2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+
+

+

Nov 22, 2020

+
+

+









+ +

Overview of week 48

+ +
    +
  • Thursday: Support Vector Machines, Kernels, Classification and Regression
    • +
  • Friday: Summary of course with perspectives for future studies
    • +
+ +Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion. + +

+









+ +

Thursday

+ +

+We finalize our discussion on Support Vector Machines with an emphasis on kernel transformations and applications to regression. The following video attempts at giving an overview on this part. See also the follow-up video. + +

+









+ +

Friday

+ +

+Friday's lecture is split in two parts. It starts with a summary of +what we have done this semester and continues with perspectives for future studies and +modern research projects in machine learning. + +

+









+ +

Support Vector Machines, overarching aims

+ +

+As discussed last week, +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. + +

+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. I recommend you take a look at the lectures from last week on the binary classification problem. + +

+









+ +

Kernels and non-linearity

+ +

+The cases we studied last week 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()
+
+

+









+ +

The equations

+ +

+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. + +

+









+ +

The problem to solve

+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. +
  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. + +

+









+ +

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. + +

+









+ +

How do we solve these problems?

+ +

+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. + +

+









+ +

A simple example

+ +

+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’]
+
+

+









+ +

Back to the more realistic cases

+ +

+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 \( \boldsymbol{y}^T\boldsymbol{\lambda}=0 \). 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 \( 0\leq \lambda_i \leq C \). + +

+









+ +

Summary of course

+ +

+









+ +

What? Me worry? No final exam in this course!

+



+ +

+









+ +

Topics we have covered this year

+ +

+The course has two central parts + +

    +
  1. Statistical analysis and optimization of data
    2. +
  2. Machine learning
    3. +
+ +









+ +

Statistical analysis and optimization of data

+ +

+The following topics be covered + +

    +
  1. Basic concepts, expectation values, variance, covariance, correlation functions and errors;
    2. +
  2. Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;
    3. +
  3. Central elements from linear algebra
    4. +
  4. Gradient methods for data optimization
    5. +
  5. Estimation of errors using cross-validation, bootstrapping and jackknife methods;
    6. +
  6. Practical optimization using Singular-value decomposition and least squares for parameterizing data.
    7. +
  7. Principal Component Analysis.
    8. +
+ +









+ +

Machine learning

+ +

+The following topics will be covered + +

    +
  1. Linear methods for regression and classification: + +
      +
    1. Ordinary Least Squares
      2. +
    2. Ridge regression
      3. +
    3. Lasso regression
      4. +
    4. Logistic regression
      5. +
    + +
  2. Neural networks and deep learning: + +
      +
    1. Feed Forward Neural Networks
      2. +
    2. Convolutional Neural Networks
      3. +
    3. Recurrent Neural Networks
      4. +
    + +
  3. Decisions trees and ensemble methods: + +
      +
    1. Decision trees
      2. +
    2. Bagging and voting
      3. +
    3. Random forests
      4. +
    4. Boosting and gradient boosting
      5. +
    + +
  4. Support vector machines + +
      +
    1. Binary classification and multiclass classification
      2. +
    2. Kernel methods
      3. +
    3. Regression
      4. +
    + +
+ +









+ +

Learning outcomes and overarching aims of this course

+ +

+The course introduces a variety of central algorithms and methods +essential for studies of data analysis and machine learning. The +course is project based and through the various 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. The students will learn to develop and structure large codes +for studying these systems, get acquainted with computing facilities +and learn to handle large scientific projects. A good scientific and +ethical conduct is emphasized throughout the course. + +

    +
  • Understand linear methods for regression and classification;
    • +
  • Learn about neural network;
    • +
  • Learn about baggin, boosting and trees
    • +
  • Support vector machines
    • +
  • Learn about basic data analysis;
    • +
  • 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;
    • +
  • 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++.
    • +
+ +









+ +

Perspective on Machine Learning

+ +
    +
  1. Rapidly emerging application area
    2. +
  2. Experiment AND theory are evolving in many many fields. Still many low-hanging fruits.
    3. +
  3. Requires education/retraining for more widespread adoption
    4. +
  4. A lot of “word-of-mouth” development methods
    5. +
+ +Huge amounts of data sets require automation, classical analysis tools often inadequate. +High energy physics hit this wall in the 90’s. +In 2009 single top quark production was determined via Boosted decision trees, Bayesian +Neural Networks, etc. + +

+









+ +

Machine Learning Research

+ +

+Where to find recent results: + +

    +
  1. Conference proceedings, arXiv and blog posts!
    2. +
  2. NIPS: Neural Information Processing Systems
    3. +
  3. ICLR: International Conference on Learning Representations
    4. +
  4. ICML: International Conference on Machine Learning
    5. +
  5. Journal of Machine Learning Research
    6. +
+ +









+ +

Starting your Machine Learning Project

+ +
    +
  1. Identify problem type: classification, generation, regression
    2. +
  2. Consider your data carefully
    3. +
  3. Choose a simple model that fits 1. and 2.
    4. +
  4. Consider your data carefully again… data representation
    5. +
  5. Based on results, feedback loop to earliest possible point
    6. +
+ +









+ +

Choose a Model and Algorithm

+ +
    +
  1. Supervised?
    2. +
  2. Start with the simplest model that fits your problem
    3. +
  3. Start with minimal processing of data
    4. +
+ +









+ +

Preparing Your Data

+ +
    +
  1. Shuffle your data
    2. +
  2. Mean center your data
    3. + +
      +
    • Why?
      • +
    + +
  3. Normalize the variance
    4. + +
      +
    • Why?
      • +
    + +
  4. Whitening
    5. + +
      +
    • Decorrelates data
      • +
    • Can be hit or miss
      • +
    + +
  5. When to do train/test split?
    6. +
+ +









+ +

Which Activation and Weights to Choose in Neural Networks

+ +
    +
  1. RELU? ELU?
    2. +
  2. Sigmoid or Tanh?
    3. +
  3. Set all weights to 0?
    4. + +
      +
    • Terrible idea
      • +
    + +
  4. Set all weights to random values?
    5. + +
      +
    • Small random values
      • +
    + +
+ +









+ +

Optimization Methods and Hyperparameters

+ +
    +
  1. Stochastic gradient descent + +
      +
    1. Stochastic gradient descent + momentum
      2. +
    + +
  2. State-of-the-art approaches:
    3. + +
      +
    • RMSProp
      • +
    • Adam
      • +
    + +
+ +Which regularization and hyperparameters? \( L_1 \) or \( L_2 \), soft classifiers, depths of trees and many other. Need to explore a large set of hyperparameters and regularization methods. + +

+









+ +

Resampling

+ +

+When do we resample? + +

    +
  1. Bootstrap
    2. +
  2. Cross-validation
    3. +
  3. Jackknife and many other
    4. +
+ +









+ +

Other courses on Data science and Machine Learning at UiO

+ +

+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. + +

    +
  1. STK2100 Machine learning and statistical methods for prediction and classification.
    2. +
  2. IN3050/IN4050 Introduction to Artificial Intelligence and Machine Learning. Introductory course in machine learning and AI with an algorithmic approach.
    3. +
  3. STK-INF3000/4000 Selected Topics in Data Science. The course provides insight into selected contemporary relevant topics within Data Science.
    4. +
  4. IN4080 Natural Language Processing. Probabilistic and machine learning techniques applied to natural language processing.
    5. +
  5. STK-IN4300 – Statistical learning methods in Data Science. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.
    6. +
  6. IN-STK5000 Adaptive Methods for Data-Based Decision Making. Methods for adaptive collection and processing of data based on machine learning techniques.
    7. +
  7. IN5400/INF5860 – Machine Learning for Image Analysis. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.
    8. +
  8. TEK5040 – Dyp læring for autonome systemer. 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.
    9. +
+ +









+ +

Additional courses of interest

+ +
    +
  1. STK4051 Computational Statistics
    2. +
  2. STK4021 Applied Bayesian Analysis and Numerical Methods
    3. +
+ +









+ +

What's the future like?

+ +

+Based on multi-layer nonlinear neural networks, deep learning can +learn directly from raw data, automatically extract and abstract +features from layer to layer, and then achieve the goal of regression, +classification, or ranking. Deep learning has made breakthroughs in +computer vision, speech processing and natural language, and reached +or even surpassed human level. The success of deep learning is mainly +due to the three factors: big data, big model, and big computing. + +

+In the past few decades, many different architectures of deep neural +networks have been proposed, such as + +

    +
  1. Convolutional neural networks, which are mostly used in image and video data processing, and have also been applied to sequential data such as text processing;
    2. +
  2. Recurrent neural networks, which can process sequential data of variable length and have been widely used in natural language understanding and speech processing;
    3. +
  3. Encoder-decoder framework, which is mostly used for image or sequence generation, such as machine translation, text summarization, and image captioning.
    4. +
+ +









+ +

Bayesian Machine Learning

+ +

+This is an important topic if we aim at extracting a probability +distribution. This gives us also a confidence interval and error +estimates. + +

+Bayesian machine learning allows us to encode our prior beliefs about +what those models should look like, independent of what the data tells +us. This is especially useful when we don’t have a ton of data to +confidently learn our model. + +

+









+ +

Reinforcement Learning

+ +

+Reinforcement learning is a sub-area of machine learning. It studies +how agents take actions based on trial and error, so as to maximize +some notion of cumulative reward in a dynamic system or +environment. Due to its generality, the problem has also been studied +in many other disciplines, such as game theory, control theory, +operations research, information theory, multi-agent systems, swarm +intelligence, statistics, and genetic algorithms. + +

+In March 2016, AlphaGo, a computer program that plays the board game +Go, beat Lee Sedol in a five-game match. This was the first time a +computer Go program had beaten a 9-dan (highest rank) professional +without handicaps. AlphaGo is based on deep convolutional neural +networks and reinforcement learning. AlphaGo’s victory was a major +milestone in artificial intelligence and it has also made +reinforcement learning a hot research area in the field of machine +learning. + +

+









+ +

Transfer learning

+ +

+The goal of transfer learning is to transfer the model or knowledge +obtained from a source task to the target task, in order to resolve +the issues of insufficient training data in the target task. The +rationality of doing so lies in that usually the source and target +tasks have inter-correlations, and therefore either the features, +samples, or models in the source task might provide useful information +for us to better solve the target task. Transfer learning is a hot +research topic in recent years, with many problems still waiting to be +solved in this space. + +

+









+ +

Adversarial learning

+ +

+The conventional deep generative model has a potential problem: the +model tends to generate extreme instances to maximize the +probabilistic likelihood, which will hurt its performance. Adversarial +learning utilizes the adversarial behaviors (e.g., generating +adversarial instances or training an adversarial model) to enhance the +robustness of the model and improve the quality of the generated +data. In recent years, one of the most promising unsupervised learning +technologies, generative adversarial networks (GAN), has already been +successfully applied to image, speech, and text. + +

+









+ +

Dual learning

+ +

+Dual learning is a new learning paradigm, the basic idea of which is +to use the primal-dual structure between machine learning tasks to +obtain effective feedback/regularization, and guide and strengthen the +learning process, thus reducing the requirement of large-scale labeled +data for deep learning. The idea of dual learning has been applied to +many problems in machine learning, including machine translation, +image style conversion, question answering and generation, image +classification and generation, text classification and generation, +image-to-text, and text-to-image. + +

+









+ +

Distributed machine learning

+ +

+Distributed computation will speed up machine learning algorithms, +significantly improve their efficiency, and thus enlarge their +application. When distributed meets machine learning, more than just +implementing the machine learning algorithms in parallel is required. + +

+









+ +

Meta learning

+ +

+Meta learning is an emerging research direction in machine +learning. Roughly speaking, meta learning concerns learning how to +learn, and focuses on the understanding and adaptation of the learning +itself, instead of just completing a specific learning task. That is, +a meta learner needs to be able to evaluate its own learning methods +and adjust its own learning methods according to specific learning +tasks. + +

+









+ +

The Challenges Facing Machine Learning

+ +

+While there has been much progress in machine learning, there are also challenges. + +

+For example, the mainstream machine learning technologies are +black-box approaches, making us concerned about their potential +risks. To tackle this challenge, we may want to make machine learning +more explainable and controllable. As another example, the +computational complexity of machine learning algorithms is usually +very high and we may want to invent lightweight algorithms or +implementations. Furthermore, in many domains such as physics, +chemistry, biology, and social sciences, people usually seek elegantly +simple equations (e.g., the Schrödinger equation) to uncover the +underlying laws behind various phenomena. In the field of machine +learning, can we reveal simple laws instead of designing more complex +models for data fitting? Although there are many challenges, we are +still very optimistic about the future of machine learning. As we look +forward to the future, here are what we think the research hotspots in +the next ten years will be. + +

+









+ +

Explainable machine learning

+ +

+Machine learning, especially deep learning, evolves rapidly. The +ability gap between machine and human on many complex cognitive tasks +becomes narrower and narrower. However, we are still in the very early +stage in terms of explaining why those effective models work and how +they work. + +

+What is missing: the gap between correlation and causation Most +machine learning techniques, especially the statistical ones, depend +highly on data correlation to make predictions and analyses. In +contrast, rational humans tend to reply on clear and trustworthy +causality relations obtained via logical reasoning on real and clear +facts. It is one of the core goals of explainable machine learning to +transition from solving problems by data correlation to solving +problems by logical reasoning. + +

+









+ +

Quantum machine learning

+ +

+Quantum machine learning is an emerging interdisciplinary research +area at the intersection of quantum computing and machine learning. + +

+Quantum computers use effects such as quantum coherence and quantum +entanglement to process information, which is fundamentally different +from classical computers. Quantum algorithms have surpassed the best +classical algorithms in several problems (e.g., searching for an +unsorted database, inverting a sparse matrix), which we call quantum +acceleration. + +

+When quantum computing meets machine learning, it can be a mutually +beneficial and reinforcing process, as it allows us to take advantage +of quantum computing to improve the performance of classical machine +learning algorithms. In addition, we can also use the machine learning +algorithms (on classic computers) to analyze and improve quantum +computing systems. + +

+









+ +

Quantum machine learning algorithms based on linear algebra

+ +

+Many quantum machine learning algorithms are based on variants of +quantum algorithms for solving linear equations, which can efficiently +solve N-variable linear equations with complexity of O(log2 N) under +certain conditions. The quantum matrix inversion algorithm can +accelerate many machine learning methods, such as least square linear +regression, least square version of support vector machine, Gaussian +process, and more. The training of these algorithms can be simplified +to solve linear equations. The key bottleneck of this type of quantum +machine learning algorithms is data input—that is, how to initialize +the quantum system with the entire data set. Although efficient +data-input algorithms exist for certain situations, how to efficiently +input data into a quantum system is as yet unknown for most cases. + +

+









+ +

Quantum reinforcement learning

+ +

+In quantum reinforcement learning, a quantum agent interacts with the +classical environment to obtain rewards from the environment, so as to +adjust and improve its behavioral strategies. In some cases, it +achieves quantum acceleration by the quantum processing capabilities +of the agent or the possibility of exploring the environment through +quantum superposition. Such algorithms have been proposed in +superconducting circuits and systems of trapped ions. + +

+









+ +

Quantum deep learning

+ +

+Dedicated quantum information processors, such as quantum annealers +and programmable photonic circuits, are well suited for building deep +quantum networks. The simplest deep quantum network is the Boltzmann +machine. The classical Boltzmann machine consists of bits with tunable +interactions and is trained by adjusting the interaction of these bits +so that the distribution of its expression conforms to the statistics +of the data. To quantize the Boltzmann machine, the neural network can +simply be represented as a set of interacting quantum spins that +correspond to an adjustable Ising model. Then, by initializing the +input neurons in the Boltzmann machine to a fixed state and allowing +the system to heat up, we can read out the output qubits to get the +result. + +

+









+ +

Social machine learning

+ +

+Machine learning aims to imitate how humans +learn. While we have developed successful machine learning algorithms, +until now we have ignored one important fact: humans are social. Each +of us is one part of the total society and it is difficult for us to +live, learn, and improve ourselves, alone and isolated. Therefore, we +should design machines with social properties. Can we let machines +evolve by imitating human society so as to achieve more effective, +intelligent, interpretable “social machine learning”? + +

+And much more. + +

+









+ +

The last words?

+ +

+Early computer scientist Alan Kay said, The best way to predict the +future is to create it. Therefore, all machine learning +practitioners, whether scholars or engineers, professors or students, +need to work together to advance these important research +topics. Together, we will not just predict the future, but create it. + +

+









+ +

Best wishes to you all and thanks so much for your heroic efforts this semester

+ +

+



+ + + + +
+ © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license +
+ + + + + + diff --git a/doc/pub/week48/ipynb/ipynb-week48-src.tar.gz b/doc/pub/week48/ipynb/ipynb-week48-src.tar.gz new file mode 100644 index 000000000..3df999677 Binary files /dev/null and b/doc/pub/week48/ipynb/ipynb-week48-src.tar.gz differ diff --git a/doc/pub/week48/ipynb/week48.ipynb b/doc/pub/week48/ipynb/week48.ipynb new file mode 100644 index 000000000..8c3836ea1 --- /dev/null +++ b/doc/pub/week48/ipynb/week48.ipynb @@ -0,0 +1,1274 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Week 48: Support Vector Machines and Summary of course\n", + "\n", + " \n", + "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", + "\n", + "Date: **Nov 22, 2020**\n", + "\n", + "Copyright 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n", + "\n", + "\n", + "\n", + "## Overview of week 48\n", + "\n", + "* **Thursday**: Support Vector Machines, Kernels, Classification and Regression\n", + "\n", + "* **Friday**: Summary of course with perspectives for future studies\n", + "\n", + "Geron's chapter 5. Chapter 12 (sections 12.1-12.3 are the most relevant ones) of Hastie et al contains also a good discussion.\n", + "\n", + "\n", + "## Thursday\n", + "\n", + "We finalize our discussion on Support Vector Machines with an emphasis on kernel transformations and applications to regression. The following [video attempts at giving an overview on this part](https://www.youtube.com/watch?v=Toet3EiSFcM&ab_channel=StatQuestwithJoshStarmer). See also the [follow-up video](https://www.youtube.com/watch?v=Qc5IyLW_hns&ab_channel=StatQuestwithJoshStarmer).\n", + "\n", + "## Friday\n", + "\n", + "Friday's lecture is split in two parts. It starts with a summary of\n", + "what we have done this semester and continues with perspectives for future studies and\n", + "modern research projects in machine learning.\n", + "\n", + "## Support Vector Machines, overarching aims\n", + "\n", + "As discussed last week, \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.\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. I recommend you take a look at the lectures from last week on the binary classification problem.\n", + "\n", + "## Kernels and non-linearity\n", + "\n", + "The cases we studied last week 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": 1, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "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": [ + "## The equations\n", + "\n", + "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", + "## The problem to solve\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", + "## 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": 2, + "metadata": { + "collapsed": false + }, + "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", + "## How do we solve these problems?\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": 3, + "metadata": { + "collapsed": false + }, + "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", + "## A simple example\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": 4, + "metadata": { + "collapsed": false + }, + "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": [ + "## Back to the more realistic cases\n", + "\n", + "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", + "\n", + "\n", + "## Summary of course\n", + "\n", + "## What? Me worry? No final exam in this course!\n", + "\n", + "\n", + "\n", + "

\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "\n", + "## Topics we have covered this year\n", + "\n", + "The course has two central parts\n", + "\n", + "1. Statistical analysis and optimization of data\n", + "\n", + "2. Machine learning\n", + "\n", + "## Statistical analysis and optimization of data\n", + "\n", + "The following topics be covered\n", + "1. Basic concepts, expectation values, variance, covariance, correlation functions and errors;\n", + "\n", + "2. Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;\n", + "\n", + "3. Central elements from linear algebra\n", + "\n", + "4. Gradient methods for data optimization\n", + "\n", + "5. Estimation of errors using cross-validation, bootstrapping and jackknife methods;\n", + "\n", + "6. Practical optimization using Singular-value decomposition and least squares for parameterizing data.\n", + "\n", + "7. Principal Component Analysis.\n", + "\n", + "## Machine learning\n", + "\n", + "The following topics will be covered\n", + "1. Linear methods for regression and classification:\n", + "\n", + "a. Ordinary Least Squares\n", + "\n", + "b. Ridge regression\n", + "\n", + "c. Lasso regression\n", + "\n", + "d. Logistic regression\n", + "\n", + "\n", + "5. Neural networks and deep learning:\n", + "\n", + "a. Feed Forward Neural Networks\n", + "\n", + "b. Convolutional Neural Networks\n", + "\n", + "c. Recurrent Neural Networks\n", + "\n", + "\n", + "4. Decisions trees and ensemble methods:\n", + "\n", + "a. Decision trees\n", + "\n", + "b. Bagging and voting\n", + "\n", + "c. Random forests\n", + "\n", + "d. Boosting and gradient boosting\n", + "\n", + "\n", + "5. Support vector machines\n", + "\n", + "a. Binary classification and multiclass classification\n", + "\n", + "b. Kernel methods\n", + "\n", + "c. Regression\n", + "\n", + "\n", + "## Learning outcomes and overarching aims of this course\n", + "\n", + "The course introduces a variety of central algorithms and methods\n", + "essential for studies of data analysis and machine learning. The\n", + "course is project based and through the various projects, normally\n", + "three, you will be exposed to fundamental research problems\n", + "in these fields, with the aim to reproduce state of the art scientific\n", + "results. The students will learn to develop and structure large codes\n", + "for studying these systems, get acquainted with computing facilities\n", + "and learn to handle large scientific projects. A good scientific and\n", + "ethical conduct is emphasized throughout the course. \n", + "\n", + "* Understand linear methods for regression and classification;\n", + "\n", + "* Learn about neural network;\n", + "\n", + "* Learn about baggin, boosting and trees\n", + "\n", + "* Support vector machines\n", + "\n", + "* Learn about basic data analysis;\n", + "\n", + "* Be capable of extending the acquired knowledge to other systems and cases;\n", + "\n", + "* Have an understanding of central algorithms used in data analysis and machine learning;\n", + "\n", + "* 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++.\n", + "\n", + "## Perspective on Machine Learning\n", + "\n", + "1. Rapidly emerging application area\n", + "\n", + "2. Experiment AND theory are evolving in many many fields. Still many low-hanging fruits.\n", + "\n", + "3. Requires education/retraining for more widespread adoption\n", + "\n", + "4. A lot of “word-of-mouth” development methods\n", + "\n", + "Huge amounts of data sets require automation, classical analysis tools often inadequate. \n", + "High energy physics hit this wall in the 90’s.\n", + "In 2009 single top quark production was determined via [Boosted decision trees, Bayesian\n", + "Neural Networks, etc.](https://arxiv.org/pdf/0903.0850.pdf)\n", + "\n", + "\n", + "## Machine Learning Research\n", + "\n", + "Where to find recent results:\n", + "1. Conference proceedings, arXiv and blog posts!\n", + "\n", + "2. **NIPS**: [Neural Information Processing Systems](https://papers.nips.cc)\n", + "\n", + "3. **ICLR**: [International Conference on Learning Representations](https://openreview.net/group?id=ICLR.cc/2018/Conference#accepted-oral-papers)\n", + "\n", + "4. **ICML**: International Conference on Machine Learning\n", + "\n", + "5. [Journal of Machine Learning Research](http://www.jmlr.org/papers/v19/) \n", + "\n", + "## Starting your Machine Learning Project\n", + "\n", + "1. Identify problem type: classification, generation, regression\n", + "\n", + "2. Consider your data carefully\n", + "\n", + "3. Choose a simple model that fits 1. and 2.\n", + "\n", + "4. Consider your data carefully again… data representation\n", + "\n", + "5. Based on results, feedback loop to earliest possible point\n", + "\n", + "## Choose a Model and Algorithm\n", + "\n", + "1. Supervised?\n", + "\n", + "2. Start with the simplest model that fits your problem\n", + "\n", + "3. Start with minimal processing of data\n", + "\n", + "## Preparing Your Data\n", + "\n", + "1. Shuffle your data\n", + "\n", + "2. Mean center your data\n", + "\n", + " * Why?\n", + "\n", + "\n", + "3. Normalize the variance\n", + "\n", + " * Why?\n", + "\n", + "\n", + "4. **Whitening**\n", + "\n", + " * Decorrelates data\n", + "\n", + " * Can be hit or miss\n", + "\n", + "\n", + "5. When to do train/test split?\n", + "\n", + "## Which Activation and Weights to Choose in Neural Networks\n", + "\n", + "1. RELU? ELU?\n", + "\n", + "2. Sigmoid or Tanh?\n", + "\n", + "3. Set all weights to 0?\n", + "\n", + " * Terrible idea\n", + "\n", + "\n", + "4. Set all weights to random values?\n", + "\n", + " * Small random values\n", + "\n", + "\n", + "## Optimization Methods and Hyperparameters\n", + "1. Stochastic gradient descent\n", + "\n", + "a. Stochastic gradient descent + momentum\n", + "\n", + "\n", + "2. State-of-the-art approaches:\n", + "\n", + " * RMSProp\n", + "\n", + " * Adam\n", + "\n", + "\n", + "Which regularization and hyperparameters? $L_1$ or $L_2$, soft classifiers, depths of trees and many other. Need to explore a large set of hyperparameters and regularization methods. \n", + "\n", + "\n", + "## Resampling\n", + "\n", + "When do we resample?\n", + "\n", + "1. Bootstrap\n", + "\n", + "2. Cross-validation\n", + "\n", + "3. Jackknife and many other\n", + "\n", + "## Other courses on Data science and Machine Learning at UiO\n", + "\n", + "The link here gives an excellent overview of courses on Machine learning at UiO.\n", + "\n", + "1. [STK2100 Machine learning and statistical methods for prediction and classification](http://www.uio.no/studier/emner/matnat/math/STK2100/index-eng.html). \n", + "\n", + "2. [IN3050/IN4050 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. \n", + "\n", + "3. [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. \n", + "\n", + "4. [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. \n", + "\n", + "5. [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.\n", + "\n", + "6. [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. \n", + "\n", + "7. [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.\n", + "\n", + "8. [TEK5040 – Dyp læring for autonome systemer](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.\n", + "\n", + "## Additional courses of interest\n", + "\n", + "1. [STK4051 Computational Statistics](https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html)\n", + "\n", + "2. [STK4021 Applied Bayesian Analysis and Numerical Methods](https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html)\n", + "\n", + "## What's the future like?\n", + "\n", + "Based on multi-layer nonlinear neural networks, deep learning can\n", + "learn directly from raw data, automatically extract and abstract\n", + "features from layer to layer, and then achieve the goal of regression,\n", + "classification, or ranking. Deep learning has made breakthroughs in\n", + "computer vision, speech processing and natural language, and reached\n", + "or even surpassed human level. The success of deep learning is mainly\n", + "due to the three factors: big data, big model, and big computing.\n", + "\n", + "In the past few decades, many different architectures of deep neural\n", + "networks have been proposed, such as\n", + "1. Convolutional neural networks, which are mostly used in image and video data processing, and have also been applied to sequential data such as text processing;\n", + "\n", + "2. Recurrent neural networks, which can process sequential data of variable length and have been widely used in natural language understanding and speech processing;\n", + "\n", + "3. Encoder-decoder framework, which is mostly used for image or sequence generation, such as machine translation, text summarization, and image captioning.\n", + "\n", + "## Bayesian Machine Learning\n", + "\n", + "This is an important topic if we aim at extracting a probability\n", + "distribution. This gives us also a confidence interval and error\n", + "estimates.\n", + "\n", + "Bayesian machine learning allows us to encode our prior beliefs about\n", + "what those models should look like, independent of what the data tells\n", + "us. This is especially useful when we don’t have a ton of data to\n", + "confidently learn our model.\n", + "\n", + "\n", + "\n", + "\n", + "## Reinforcement Learning\n", + "\n", + "Reinforcement learning is a sub-area of machine learning. It studies\n", + "how agents take actions based on trial and error, so as to maximize\n", + "some notion of cumulative reward in a dynamic system or\n", + "environment. Due to its generality, the problem has also been studied\n", + "in many other disciplines, such as game theory, control theory,\n", + "operations research, information theory, multi-agent systems, swarm\n", + "intelligence, statistics, and genetic algorithms.\n", + "\n", + "In March 2016, AlphaGo, a computer program that plays the board game\n", + "Go, beat Lee Sedol in a five-game match. This was the first time a\n", + "computer Go program had beaten a 9-dan (highest rank) professional\n", + "without handicaps. AlphaGo is based on deep convolutional neural\n", + "networks and reinforcement learning. AlphaGo’s victory was a major\n", + "milestone in artificial intelligence and it has also made\n", + "reinforcement learning a hot research area in the field of machine\n", + "learning.\n", + "\n", + "## Transfer learning\n", + "\n", + "The goal of transfer learning is to transfer the model or knowledge\n", + "obtained from a source task to the target task, in order to resolve\n", + "the issues of insufficient training data in the target task. The\n", + "rationality of doing so lies in that usually the source and target\n", + "tasks have inter-correlations, and therefore either the features,\n", + "samples, or models in the source task might provide useful information\n", + "for us to better solve the target task. Transfer learning is a hot\n", + "research topic in recent years, with many problems still waiting to be\n", + "solved in this space.\n", + "\n", + "\n", + "## Adversarial learning\n", + "\n", + "The conventional deep generative model has a potential problem: the\n", + "model tends to generate extreme instances to maximize the\n", + "probabilistic likelihood, which will hurt its performance. Adversarial\n", + "learning utilizes the adversarial behaviors (e.g., generating\n", + "adversarial instances or training an adversarial model) to enhance the\n", + "robustness of the model and improve the quality of the generated\n", + "data. In recent years, one of the most promising unsupervised learning\n", + "technologies, generative adversarial networks (GAN), has already been\n", + "successfully applied to image, speech, and text.\n", + "\n", + "## Dual learning\n", + "\n", + "Dual learning is a new learning paradigm, the basic idea of which is\n", + "to use the primal-dual structure between machine learning tasks to\n", + "obtain effective feedback/regularization, and guide and strengthen the\n", + "learning process, thus reducing the requirement of large-scale labeled\n", + "data for deep learning. The idea of dual learning has been applied to\n", + "many problems in machine learning, including machine translation,\n", + "image style conversion, question answering and generation, image\n", + "classification and generation, text classification and generation,\n", + "image-to-text, and text-to-image.\n", + "\n", + "## Distributed machine learning\n", + "\n", + "Distributed computation will speed up machine learning algorithms,\n", + "significantly improve their efficiency, and thus enlarge their\n", + "application. When distributed meets machine learning, more than just\n", + "implementing the machine learning algorithms in parallel is required.\n", + "\n", + "\n", + "## Meta learning\n", + "\n", + "Meta learning is an emerging research direction in machine\n", + "learning. Roughly speaking, meta learning concerns learning how to\n", + "learn, and focuses on the understanding and adaptation of the learning\n", + "itself, instead of just completing a specific learning task. That is,\n", + "a meta learner needs to be able to evaluate its own learning methods\n", + "and adjust its own learning methods according to specific learning\n", + "tasks.\n", + "\n", + "## The Challenges Facing Machine Learning\n", + "\n", + "While there has been much progress in machine learning, there are also challenges.\n", + "\n", + "For example, the mainstream machine learning technologies are\n", + "black-box approaches, making us concerned about their potential\n", + "risks. To tackle this challenge, we may want to make machine learning\n", + "more explainable and controllable. As another example, the\n", + "computational complexity of machine learning algorithms is usually\n", + "very high and we may want to invent lightweight algorithms or\n", + "implementations. Furthermore, in many domains such as physics,\n", + "chemistry, biology, and social sciences, people usually seek elegantly\n", + "simple equations (e.g., the Schrödinger equation) to uncover the\n", + "underlying laws behind various phenomena. In the field of machine\n", + "learning, can we reveal simple laws instead of designing more complex\n", + "models for data fitting? Although there are many challenges, we are\n", + "still very optimistic about the future of machine learning. As we look\n", + "forward to the future, here are what we think the research hotspots in\n", + "the next ten years will be.\n", + "\n", + "\n", + "## Explainable machine learning\n", + "\n", + "Machine learning, especially deep learning, evolves rapidly. The\n", + "ability gap between machine and human on many complex cognitive tasks\n", + "becomes narrower and narrower. However, we are still in the very early\n", + "stage in terms of explaining why those effective models work and how\n", + "they work.\n", + "\n", + "What is missing: the gap between correlation and causation Most\n", + "machine learning techniques, especially the statistical ones, depend\n", + "highly on data correlation to make predictions and analyses. In\n", + "contrast, rational humans tend to reply on clear and trustworthy\n", + "causality relations obtained via logical reasoning on real and clear\n", + "facts. It is one of the core goals of explainable machine learning to\n", + "transition from solving problems by data correlation to solving\n", + "problems by logical reasoning.\n", + "\n", + "## Quantum machine learning\n", + "\n", + "Quantum machine learning is an emerging interdisciplinary research\n", + "area at the intersection of quantum computing and machine learning.\n", + "\n", + "Quantum computers use effects such as quantum coherence and quantum\n", + "entanglement to process information, which is fundamentally different\n", + "from classical computers. Quantum algorithms have surpassed the best\n", + "classical algorithms in several problems (e.g., searching for an\n", + "unsorted database, inverting a sparse matrix), which we call quantum\n", + "acceleration.\n", + "\n", + "When quantum computing meets machine learning, it can be a mutually\n", + "beneficial and reinforcing process, as it allows us to take advantage\n", + "of quantum computing to improve the performance of classical machine\n", + "learning algorithms. In addition, we can also use the machine learning\n", + "algorithms (on classic computers) to analyze and improve quantum\n", + "computing systems.\n", + "\n", + "\n", + "## Quantum machine learning algorithms based on linear algebra\n", + "\n", + "Many quantum machine learning algorithms are based on variants of\n", + "quantum algorithms for solving linear equations, which can efficiently\n", + "solve N-variable linear equations with complexity of O(log2 N) under\n", + "certain conditions. The quantum matrix inversion algorithm can\n", + "accelerate many machine learning methods, such as least square linear\n", + "regression, least square version of support vector machine, Gaussian\n", + "process, and more. The training of these algorithms can be simplified\n", + "to solve linear equations. The key bottleneck of this type of quantum\n", + "machine learning algorithms is data input—that is, how to initialize\n", + "the quantum system with the entire data set. Although efficient\n", + "data-input algorithms exist for certain situations, how to efficiently\n", + "input data into a quantum system is as yet unknown for most cases.\n", + "\n", + "## Quantum reinforcement learning\n", + "\n", + "In quantum reinforcement learning, a quantum agent interacts with the\n", + "classical environment to obtain rewards from the environment, so as to\n", + "adjust and improve its behavioral strategies. In some cases, it\n", + "achieves quantum acceleration by the quantum processing capabilities\n", + "of the agent or the possibility of exploring the environment through\n", + "quantum superposition. Such algorithms have been proposed in\n", + "superconducting circuits and systems of trapped ions.\n", + "\n", + "## Quantum deep learning\n", + "\n", + "Dedicated quantum information processors, such as quantum annealers\n", + "and programmable photonic circuits, are well suited for building deep\n", + "quantum networks. The simplest deep quantum network is the Boltzmann\n", + "machine. The classical Boltzmann machine consists of bits with tunable\n", + "interactions and is trained by adjusting the interaction of these bits\n", + "so that the distribution of its expression conforms to the statistics\n", + "of the data. To quantize the Boltzmann machine, the neural network can\n", + "simply be represented as a set of interacting quantum spins that\n", + "correspond to an adjustable Ising model. Then, by initializing the\n", + "input neurons in the Boltzmann machine to a fixed state and allowing\n", + "the system to heat up, we can read out the output qubits to get the\n", + "result.\n", + "\n", + "\n", + "## Social machine learning\n", + "\n", + "Machine learning aims to imitate how humans\n", + "learn. While we have developed successful machine learning algorithms,\n", + "until now we have ignored one important fact: humans are social. Each\n", + "of us is one part of the total society and it is difficult for us to\n", + "live, learn, and improve ourselves, alone and isolated. Therefore, we\n", + "should design machines with social properties. Can we let machines\n", + "evolve by imitating human society so as to achieve more effective,\n", + "intelligent, interpretable “social machine learning”?\n", + "\n", + "And much more.\n", + "\n", + "## The last words?\n", + "\n", + "Early computer scientist Alan Kay said, **The best way to predict the\n", + "future is to create it**. Therefore, all machine learning\n", + "practitioners, whether scholars or engineers, professors or students,\n", + "need to work together to advance these important research\n", + "topics. Together, we will not just predict the future, but create it.\n", + "\n", + "\n", + "\n", + "\n", + "## Best wishes to you all and thanks so much for your heroic efforts this semester\n", + "\n", + "\n", + "\n", + "\n", + "

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