diff --git a/doc/LectureNotes/DataFiles/cancer.dot b/doc/LectureNotes/DataFiles/cancer.dot
index 5b4b48a9b..5804ca671 100644
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diff --git a/doc/LectureNotes/DataFiles/cancer.png b/doc/LectureNotes/DataFiles/cancer.png
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Binary files a/doc/LectureNotes/DataFiles/cancer.png and b/doc/LectureNotes/DataFiles/cancer.png differ
diff --git a/doc/LectureNotes/Project3.ipynb b/doc/LectureNotes/Project3.ipynb
new file mode 100644
index 000000000..d45c511e8
--- /dev/null
+++ b/doc/LectureNotes/Project3.ipynb
@@ -0,0 +1,516 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "5831c36a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d774c8b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "# Project 3 on Machine Learning, deadline December 18 (midnight), 2023\n",
+ "**[Data Analysis and Machine Learning FYS-STK3155/FYS4155](http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html)**, Department of Physics, University of Oslo, Norway\n",
+ "\n",
+ "Date: **Nov 12, 2023**\n",
+ "\n",
+ "Copyright 1999-2023, [Data Analysis and Machine Learning FYS-STK3155/FYS4155](http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html). Released under CC Attribution-NonCommercial 4.0 license"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5cbde03b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "# Paths for project 3"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5170403b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Defining the data sets to analyze yourself\n",
+ "\n",
+ "For project 3, you can propose own data sets that relate to your research interests or just use existing data sets from say\n",
+ "1. [Kaggle](https://www.kaggle.com/datasets) \n",
+ "\n",
+ "2. The [University of California at Irvine (UCI) with its machine learning repository](https://archive.ics.uci.edu/ml/index.php).\n",
+ "\n",
+ "3. Or other sources.\n",
+ "\n",
+ "The approach to the analysis of these new data sets should follow to a large extent what you did in projects 1 and 2. That is:\n",
+ "1. Whether you end up with a regression or a classification problem, you should employ at least two of the methods we have discussed among **linear regression (including Ridge and Lasso)**, **Logistic Regression**, **Neural Networks**, **Convolution Neural Networks**, **Recurrent Neural Networks**, and **Decision Trees, Random Forests, Bagging and Boosting**. You could for example explore all of the approaches from decision trees, via bagging and voting classifiers, to random forests, boosting and finally XGboost. If you wish to venture into **convolutional neural networks** or **recurrent neural networks**, or extensions of neural networkds, feel free to do so. You can also study unsupervised methods, although we in this course have mainly paid attention to supervised learning. \n",
+ "\n",
+ "For Boosting, feel also free to write your own codes.\n",
+ "\n",
+ "1. For project 3, you should feel free to use your own codes from projects 1 and 2, eventually write your own for Decision trees/random forests/bagging/boosting' or use the available functionality of **Scikit-Learn**, **Tensorflow**, etc. \n",
+ "\n",
+ "2. The estimates you used and tested in projects 1 and 2 should also be included, that is the $R2$-score, **MSE**, confusion matrix, accuracy score, information gain, ROC and Cumulative gains curves and other, cross-validation and/or bootstrap if these are relevant.\n",
+ "\n",
+ "3. Similarly, feel free to explore various activations functions in deep learning and various approachs to stochastic gradient descent approaches.\n",
+ "\n",
+ "4. If possible, you should link the data sets with exisiting research and analyses thereof. Scientific articles which have used Machine Learning algorithms to analyze the data are highly welcome. Perhaps you can improve previous analyses and even publish a new article? \n",
+ "\n",
+ "5. A critical assessment of the methods with ditto perspectives and recommendations is also something you need to include.\n",
+ "\n",
+ "All in all, the report should follow the same pattern as the two previous ones, with abstract, introduction, methods, code, results, conclusions etc..\n",
+ "\n",
+ "We propose also an alternative to the above. This is a project on using machine learning methods (neural networks mainly) to the solution of ordinary differential equations and partial differential equations, with a final twist on how to diagonalize a symmetric matrix with neural networks.\n",
+ "\n",
+ "This is a field with a large interest recently, spanning from studies of turbulence in fluid mechanics and meteorology to the solution of quantum mechanical systems. As reading background you can use the slides [from week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week42.html) and/or the textbook by [Yadav et al](https://www.springer.com/gp/book/9789401798150)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e9fc8e8e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## The basic structure of your project\n",
+ "\n",
+ "Here follows a set up on how to structure your report and analyze the data you have opted for."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ce8b52a3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part a)\n",
+ "\n",
+ "The first part deals with structuring and reading the data, much along the same lines as done in projects 1 and 2. Explain how the data are produced and place them in a proper context."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "eb943f7a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part b)\n",
+ "\n",
+ "You need to include at least two central algorithms, or as an alternative explore methods from decisions tree to bagging, random forests and boosting. Explain the basics of the methods you have chosen to work with. This would be your theory part."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "17447dcc",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part c)\n",
+ "\n",
+ "Then describe your algorithm and its implementation and tests you have performed."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d9bea856",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part d)\n",
+ "\n",
+ "Then presents your results and findings, link with existing literature and more."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5b74282b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part e)\n",
+ "\n",
+ "Finally, here you should present a critical assessment of the methods you have studied and link your results with the existing literature."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "28f082d7",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Solving partial differential equations with neural networks\n",
+ "\n",
+ "For this variant of project 3, we will assume that you have some\n",
+ "background in the solution of partial differential equations using\n",
+ "finite difference schemes. We will study the solution of the diffusion\n",
+ "equation in one dimension using a standard explicit scheme and neural\n",
+ "networks to solve the same equations.\n",
+ "\n",
+ "For the explicit scheme, you can study for example chapter 10 of the lecture notes in [Computational Physics](https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Lectures/lectures2015.pdf) or alternative sources. For the solution of ordinary and partial differential equations using neural networks, the lectures by [Kristine Baluka Hein and included in the lectures of week 43](https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week43.html) at this course are highly recommended.\n",
+ "\n",
+ "For the machine learning part you can use your own code from project 2 or the functionality of for example **Tensorflow/Keras**.."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5e855e7f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part a), setting up the problem\n",
+ "\n",
+ "The physical problem can be that of the temperature gradient in a rod of length $L=1$ at $x=0$ and $x=1$.\n",
+ "We are looking at a one-dimensional\n",
+ "problem"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bc4be75f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\frac{\\partial^2 u(x,t)}{\\partial x^2} =\\frac{\\partial u(x,t)}{\\partial t}, t> 0, x\\in [0,L]\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "caa1eca4",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "or"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "98a2bd5b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx} = u_t,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "980b949d",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "with initial conditions, i.e., the conditions at $t=0$,"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7401e9ec",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(x,0)= \\sin{(\\pi x)} \\hspace{0.5cm} 0 < x < L,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e953662e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "with $L=1$ the length of the $x$-region of interest. The \n",
+ "boundary conditions are"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "52122eb7",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(0,t)= 0 \\hspace{0.5cm} t \\ge 0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ab71a383",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "and"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4f886681",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(L,t)= 0 \\hspace{0.5cm} t \\ge 0.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "83f8d12e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "The function $u(x,t)$ can be the temperature gradient of a rod.\n",
+ "As time increases, the velocity approaches a linear variation with $x$. \n",
+ "\n",
+ "We will limit ourselves to the so-called explicit forward Euler algorithm with discretized versions of time given by a forward formula and a centered difference in space resulting in"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4bcf494f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_t\\approx \\frac{u(x,t+\\Delta t)-u(x,t)}{\\Delta t}=\\frac{u(x_i,t_j+\\Delta t)-u(x_i,t_j)}{\\Delta t}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7f95982b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "and"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1ee81ac0",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx}\\approx \\frac{u(x+\\Delta x,t)-2u(x,t)+u(x-\\Delta x,t)}{\\Delta x^2},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bae7898a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "or"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5de23ea9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx}\\approx \\frac{u(x_i+\\Delta x,t_j)-2u(x_i,t_j)+u(x_i-\\Delta x,t_j)}{\\Delta x^2}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4d5d56e3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "Write down the algorithm and the equations you need to implement.\n",
+ "Find also the analytical solution to the problem."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e52523ef",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part b)\n",
+ "\n",
+ "Implement the explicit scheme algorithm and perform tests of the solution \n",
+ "for $\\Delta x=1/10$, $\\Delta x=1/100$ using $\\Delta t$ as dictated by the stability limit of the explicit scheme. The stability criterion for the explicit scheme requires that $\\Delta t/\\Delta x^2 \\leq 1/2$. \n",
+ "\n",
+ "Study the solutions at two time points $t_1$ and $t_2$ where $u(x,t_1)$ is smooth but still significantly curved\n",
+ "and $u(x,t_2)$ is almost linear, close to the stationary state."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "678607d0",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part c) Neural networks\n",
+ "\n",
+ "Study now the lecture notes on solving ODEs and PDEs with neural\n",
+ "network and use either your own code from project 2 or the\n",
+ "functionality of tensorflow/keras to solve the same equation as in\n",
+ "part b). Discuss your results and compare them with the standard\n",
+ "explicit scheme. Include also the analytical solution and compare with\n",
+ "that."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "61d51d93",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part d)\n",
+ "\n",
+ "Finally, present a critical assessment of the methods you have studied\n",
+ "and discuss the potential for the solving differential equations and\n",
+ "eigenvalue problems with machine learning methods."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d2163034",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Introduction to numerical projects\n",
+ "\n",
+ "Here follows a brief recipe and recommendation on how to write a report for each\n",
+ "project.\n",
+ "\n",
+ " * Give a short description of the nature of the problem and the eventual numerical methods you have used.\n",
+ "\n",
+ " * Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.\n",
+ "\n",
+ " * Include the source code of your program. Comment your program properly.\n",
+ "\n",
+ " * If possible, try to find analytic solutions, or known limits in order to test your program when developing the code.\n",
+ "\n",
+ " * Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes.\n",
+ "\n",
+ " * Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc.\n",
+ "\n",
+ " * Try to give an interpretation of you results in your answers to the problems.\n",
+ "\n",
+ " * Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts you've made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it.\n",
+ "\n",
+ " * Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don't properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e31290fd",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Format for electronic delivery of report and programs\n",
+ "\n",
+ "The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:\n",
+ "\n",
+ " * Use Canvas to hand in your projects, log in at with your normal UiO username and password.\n",
+ "\n",
+ " * Upload **only** the report file or the link to your GitHub/GitLab or similar typo of repos! For the source code file(s) you have developed please provide us with your link to your GitHub/GitLab or similar domain. The report file should include all of your discussions and a list of the codes you have developed. Do not include library files which are available at the course homepage, unless you have made specific changes to them.\n",
+ "\n",
+ " * In your GitHub/GitLab or similar repository, please include a folder which contains selected results. These can be in the form of output from your code for a selected set of runs and input parameters.\n",
+ "\n",
+ "Finally, \n",
+ "we encourage you to collaborate. Optimal working groups consist of \n",
+ "2-3 students. You can then hand in a common report."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "99e78b73",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Software and needed installations\n",
+ "\n",
+ "If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages, \n",
+ "we recommend that you install the following Python packages via **pip** as\n",
+ "1. pip install numpy scipy matplotlib ipython scikit-learn tensorflow sympy pandas pillow\n",
+ "\n",
+ "For Python3, replace **pip** with **pip3**.\n",
+ "\n",
+ "See below for a discussion of **tensorflow** and **scikit-learn**. \n",
+ "\n",
+ "For OSX users we recommend also, after having installed Xcode, to install **brew**. Brew allows \n",
+ "for a seamless installation of additional software via for example\n",
+ "1. brew install python3\n",
+ "\n",
+ "For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution\n",
+ "you can use **pip** as well and simply install Python as \n",
+ "1. sudo apt-get install python3 (or python for python2.7)\n",
+ "\n",
+ "etc etc. \n",
+ "\n",
+ "If you don't want to install various Python packages with their dependencies separately, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely\n",
+ "1. [Anaconda](https://docs.anaconda.com/) Anaconda is an open source distribution of the Python and R programming languages for large-scale data processing, predictive analytics, and scientific computing, that aims to simplify package management and deployment. Package versions are managed by the package management system **conda**\n",
+ "\n",
+ "2. [Enthought canopy](https://www.enthought.com/product/canopy/) is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.\n",
+ "\n",
+ "Popular software packages written in Python for ML are\n",
+ "\n",
+ "* [Scikit-learn](http://scikit-learn.org/stable/), \n",
+ "\n",
+ "* [Tensorflow](https://www.tensorflow.org/),\n",
+ "\n",
+ "* [PyTorch](http://pytorch.org/) and \n",
+ "\n",
+ "* [Keras](https://keras.io/).\n",
+ "\n",
+ "These are all freely available at their respective GitHub sites. They \n",
+ "encompass communities of developers in the thousands or more. And the number\n",
+ "of code developers and contributors keeps increasing."
+ ]
+ }
+ ],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
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+
+
+
+
+
+
+
+ Project 3 on Machine Learning, deadline December 18 (midnight), 2023 — Applied Data Analysis and Machine Learning
+
+
+
+
+
+
+
+
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+
+
+
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+
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+
+
The approach to the analysis of these new data sets should follow to a large extent what you did in projects 1 and 2. That is:
+
+
Whether you end up with a regression or a classification problem, you should employ at least two of the methods we have discussed among linear regression (including Ridge and Lasso), Logistic Regression, Neural Networks, Convolution Neural Networks, Recurrent Neural Networks, and Decision Trees, Random Forests, Bagging and Boosting. You could for example explore all of the approaches from decision trees, via bagging and voting classifiers, to random forests, boosting and finally XGboost. If you wish to venture into convolutional neural networks or recurrent neural networks, or extensions of neural networkds, feel free to do so. You can also study unsupervised methods, although we in this course have mainly paid attention to supervised learning.
+
+
For Boosting, feel also free to write your own codes.
+
+
For project 3, you should feel free to use your own codes from projects 1 and 2, eventually write your own for Decision trees/random forests/bagging/boosting’ or use the available functionality of Scikit-Learn, Tensorflow, etc.
+
The estimates you used and tested in projects 1 and 2 should also be included, that is the \(R2\)-score, MSE, confusion matrix, accuracy score, information gain, ROC and Cumulative gains curves and other, cross-validation and/or bootstrap if these are relevant.
+
Similarly, feel free to explore various activations functions in deep learning and various approachs to stochastic gradient descent approaches.
+
If possible, you should link the data sets with exisiting research and analyses thereof. Scientific articles which have used Machine Learning algorithms to analyze the data are highly welcome. Perhaps you can improve previous analyses and even publish a new article?
+
A critical assessment of the methods with ditto perspectives and recommendations is also something you need to include.
+
+
All in all, the report should follow the same pattern as the two previous ones, with abstract, introduction, methods, code, results, conclusions etc..
+
We propose also an alternative to the above. This is a project on using machine learning methods (neural networks mainly) to the solution of ordinary differential equations and partial differential equations, with a final twist on how to diagonalize a symmetric matrix with neural networks.
+
This is a field with a large interest recently, spanning from studies of turbulence in fluid mechanics and meteorology to the solution of quantum mechanical systems. As reading background you can use the slides from week 43 and/or the textbook by Yadav et al.
The first part deals with structuring and reading the data, much along the same lines as done in projects 1 and 2. Explain how the data are produced and place them in a proper context.
You need to include at least two central algorithms, or as an alternative explore methods from decisions tree to bagging, random forests and boosting. Explain the basics of the methods you have chosen to work with. This would be your theory part.
Finally, here you should present a critical assessment of the methods you have studied and link your results with the existing literature.
+
+
+
+
Solving partial differential equations with neural networks¶
+
For this variant of project 3, we will assume that you have some
+background in the solution of partial differential equations using
+finite difference schemes. We will study the solution of the diffusion
+equation in one dimension using a standard explicit scheme and neural
+networks to solve the same equations.
+
For the explicit scheme, you can study for example chapter 10 of the lecture notes in Computational Physics or alternative sources. For the solution of ordinary and partial differential equations using neural networks, the lectures by Kristine Baluka Hein and included in the lectures of week 43 at this course are highly recommended.
+
For the machine learning part you can use your own code from project 2 or the functionality of for example Tensorflow/Keras..
The physical problem can be that of the temperature gradient in a rod of length \(L=1\) at \(x=0\) and \(x=1\).
+We are looking at a one-dimensional
+problem
with \(L=1\) the length of the \(x\)-region of interest. The
+boundary conditions are
+
+\[
+u(0,t)= 0 \hspace{0.5cm} t \ge 0,
+\]
+
and
+
+\[
+u(L,t)= 0 \hspace{0.5cm} t \ge 0.
+\]
+
The function \(u(x,t)\) can be the temperature gradient of a rod.
+As time increases, the velocity approaches a linear variation with \(x\).
+
We will limit ourselves to the so-called explicit forward Euler algorithm with discretized versions of time given by a forward formula and a centered difference in space resulting in
Implement the explicit scheme algorithm and perform tests of the solution
+for \(\Delta x=1/10\), \(\Delta x=1/100\) using \(\Delta t\) as dictated by the stability limit of the explicit scheme. The stability criterion for the explicit scheme requires that \(\Delta t/\Delta x^2 \leq 1/2\).
+
Study the solutions at two time points \(t_1\) and \(t_2\) where \(u(x,t_1)\) is smooth but still significantly curved
+and \(u(x,t_2)\) is almost linear, close to the stationary state.
Study now the lecture notes on solving ODEs and PDEs with neural
+network and use either your own code from project 2 or the
+functionality of tensorflow/keras to solve the same equation as in
+part b). Discuss your results and compare them with the standard
+explicit scheme. Include also the analytical solution and compare with
+that.
Finally, present a critical assessment of the methods you have studied
+and discuss the potential for the solving differential equations and
+eigenvalue problems with machine learning methods.
Here follows a brief recipe and recommendation on how to write a report for each
+project.
+
+
Give a short description of the nature of the problem and the eventual numerical methods you have used.
+
Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.
+
Include the source code of your program. Comment your program properly.
+
If possible, try to find analytic solutions, or known limits in order to test your program when developing the code.
+
Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes.
+
Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc.
+
Try to give an interpretation of you results in your answers to the problems.
+
Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts you’ve made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it.
+
Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don’t properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning.
+
+
+
+
Format for electronic delivery of report and programs¶
+
The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:
Upload only the report file or the link to your GitHub/GitLab or similar typo of repos! For the source code file(s) you have developed please provide us with your link to your GitHub/GitLab or similar domain. The report file should include all of your discussions and a list of the codes you have developed. Do not include library files which are available at the course homepage, unless you have made specific changes to them.
+
In your GitHub/GitLab or similar repository, please include a folder which contains selected results. These can be in the form of output from your code for a selected set of runs and input parameters.
+
+
Finally,
+we encourage you to collaborate. Optimal working groups consist of
+2-3 students. You can then hand in a common report.
If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages,
+we recommend that you install the following Python packages via pip as
See below for a discussion of tensorflow and scikit-learn.
+
For OSX users we recommend also, after having installed Xcode, to install brew. Brew allows
+for a seamless installation of additional software via for example
+
+
brew install python3
+
+
For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution
+you can use pip as well and simply install Python as
+
+
sudo apt-get install python3 (or python for python2.7)
+
+
etc etc.
+
If you don’t want to install various Python packages with their dependencies separately, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely
+
+
Anaconda Anaconda is an open source distribution of the Python and R programming languages for large-scale data processing, predictive analytics, and scientific computing, that aims to simplify package management and deployment. Package versions are managed by the package management system conda
+
Enthought canopy is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.
+
+
Popular software packages written in Python for ML are
These are all freely available at their respective GitHub sites. They
+encompass communities of developers in the thousands or more. And the number
+of code developers and contributors keeps increasing.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
\ No newline at end of file
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@@ -0,0 +1,516 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "5831c36a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d774c8b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "# Project 3 on Machine Learning, deadline December 18 (midnight), 2023\n",
+ "**[Data Analysis and Machine Learning FYS-STK3155/FYS4155](http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html)**, Department of Physics, University of Oslo, Norway\n",
+ "\n",
+ "Date: **Nov 12, 2023**\n",
+ "\n",
+ "Copyright 1999-2023, [Data Analysis and Machine Learning FYS-STK3155/FYS4155](http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html). Released under CC Attribution-NonCommercial 4.0 license"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5cbde03b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "# Paths for project 3"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5170403b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Defining the data sets to analyze yourself\n",
+ "\n",
+ "For project 3, you can propose own data sets that relate to your research interests or just use existing data sets from say\n",
+ "1. [Kaggle](https://www.kaggle.com/datasets) \n",
+ "\n",
+ "2. The [University of California at Irvine (UCI) with its machine learning repository](https://archive.ics.uci.edu/ml/index.php).\n",
+ "\n",
+ "3. Or other sources.\n",
+ "\n",
+ "The approach to the analysis of these new data sets should follow to a large extent what you did in projects 1 and 2. That is:\n",
+ "1. Whether you end up with a regression or a classification problem, you should employ at least two of the methods we have discussed among **linear regression (including Ridge and Lasso)**, **Logistic Regression**, **Neural Networks**, **Convolution Neural Networks**, **Recurrent Neural Networks**, and **Decision Trees, Random Forests, Bagging and Boosting**. You could for example explore all of the approaches from decision trees, via bagging and voting classifiers, to random forests, boosting and finally XGboost. If you wish to venture into **convolutional neural networks** or **recurrent neural networks**, or extensions of neural networkds, feel free to do so. You can also study unsupervised methods, although we in this course have mainly paid attention to supervised learning. \n",
+ "\n",
+ "For Boosting, feel also free to write your own codes.\n",
+ "\n",
+ "1. For project 3, you should feel free to use your own codes from projects 1 and 2, eventually write your own for Decision trees/random forests/bagging/boosting' or use the available functionality of **Scikit-Learn**, **Tensorflow**, etc. \n",
+ "\n",
+ "2. The estimates you used and tested in projects 1 and 2 should also be included, that is the $R2$-score, **MSE**, confusion matrix, accuracy score, information gain, ROC and Cumulative gains curves and other, cross-validation and/or bootstrap if these are relevant.\n",
+ "\n",
+ "3. Similarly, feel free to explore various activations functions in deep learning and various approachs to stochastic gradient descent approaches.\n",
+ "\n",
+ "4. If possible, you should link the data sets with exisiting research and analyses thereof. Scientific articles which have used Machine Learning algorithms to analyze the data are highly welcome. Perhaps you can improve previous analyses and even publish a new article? \n",
+ "\n",
+ "5. A critical assessment of the methods with ditto perspectives and recommendations is also something you need to include.\n",
+ "\n",
+ "All in all, the report should follow the same pattern as the two previous ones, with abstract, introduction, methods, code, results, conclusions etc..\n",
+ "\n",
+ "We propose also an alternative to the above. This is a project on using machine learning methods (neural networks mainly) to the solution of ordinary differential equations and partial differential equations, with a final twist on how to diagonalize a symmetric matrix with neural networks.\n",
+ "\n",
+ "This is a field with a large interest recently, spanning from studies of turbulence in fluid mechanics and meteorology to the solution of quantum mechanical systems. As reading background you can use the slides [from week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week42.html) and/or the textbook by [Yadav et al](https://www.springer.com/gp/book/9789401798150)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e9fc8e8e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## The basic structure of your project\n",
+ "\n",
+ "Here follows a set up on how to structure your report and analyze the data you have opted for."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ce8b52a3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part a)\n",
+ "\n",
+ "The first part deals with structuring and reading the data, much along the same lines as done in projects 1 and 2. Explain how the data are produced and place them in a proper context."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "eb943f7a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part b)\n",
+ "\n",
+ "You need to include at least two central algorithms, or as an alternative explore methods from decisions tree to bagging, random forests and boosting. Explain the basics of the methods you have chosen to work with. This would be your theory part."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "17447dcc",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part c)\n",
+ "\n",
+ "Then describe your algorithm and its implementation and tests you have performed."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d9bea856",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part d)\n",
+ "\n",
+ "Then presents your results and findings, link with existing literature and more."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5b74282b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part e)\n",
+ "\n",
+ "Finally, here you should present a critical assessment of the methods you have studied and link your results with the existing literature."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "28f082d7",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Solving partial differential equations with neural networks\n",
+ "\n",
+ "For this variant of project 3, we will assume that you have some\n",
+ "background in the solution of partial differential equations using\n",
+ "finite difference schemes. We will study the solution of the diffusion\n",
+ "equation in one dimension using a standard explicit scheme and neural\n",
+ "networks to solve the same equations.\n",
+ "\n",
+ "For the explicit scheme, you can study for example chapter 10 of the lecture notes in [Computational Physics](https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Lectures/lectures2015.pdf) or alternative sources. For the solution of ordinary and partial differential equations using neural networks, the lectures by [Kristine Baluka Hein and included in the lectures of week 43](https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week43.html) at this course are highly recommended.\n",
+ "\n",
+ "For the machine learning part you can use your own code from project 2 or the functionality of for example **Tensorflow/Keras**.."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5e855e7f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part a), setting up the problem\n",
+ "\n",
+ "The physical problem can be that of the temperature gradient in a rod of length $L=1$ at $x=0$ and $x=1$.\n",
+ "We are looking at a one-dimensional\n",
+ "problem"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bc4be75f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\frac{\\partial^2 u(x,t)}{\\partial x^2} =\\frac{\\partial u(x,t)}{\\partial t}, t> 0, x\\in [0,L]\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "caa1eca4",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "or"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "98a2bd5b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx} = u_t,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "980b949d",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "with initial conditions, i.e., the conditions at $t=0$,"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7401e9ec",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(x,0)= \\sin{(\\pi x)} \\hspace{0.5cm} 0 < x < L,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e953662e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "with $L=1$ the length of the $x$-region of interest. The \n",
+ "boundary conditions are"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "52122eb7",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(0,t)= 0 \\hspace{0.5cm} t \\ge 0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ab71a383",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "and"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4f886681",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(L,t)= 0 \\hspace{0.5cm} t \\ge 0.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "83f8d12e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "The function $u(x,t)$ can be the temperature gradient of a rod.\n",
+ "As time increases, the velocity approaches a linear variation with $x$. \n",
+ "\n",
+ "We will limit ourselves to the so-called explicit forward Euler algorithm with discretized versions of time given by a forward formula and a centered difference in space resulting in"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4bcf494f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_t\\approx \\frac{u(x,t+\\Delta t)-u(x,t)}{\\Delta t}=\\frac{u(x_i,t_j+\\Delta t)-u(x_i,t_j)}{\\Delta t}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7f95982b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "and"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1ee81ac0",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx}\\approx \\frac{u(x+\\Delta x,t)-2u(x,t)+u(x-\\Delta x,t)}{\\Delta x^2},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bae7898a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "or"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5de23ea9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx}\\approx \\frac{u(x_i+\\Delta x,t_j)-2u(x_i,t_j)+u(x_i-\\Delta x,t_j)}{\\Delta x^2}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4d5d56e3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "Write down the algorithm and the equations you need to implement.\n",
+ "Find also the analytical solution to the problem."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e52523ef",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part b)\n",
+ "\n",
+ "Implement the explicit scheme algorithm and perform tests of the solution \n",
+ "for $\\Delta x=1/10$, $\\Delta x=1/100$ using $\\Delta t$ as dictated by the stability limit of the explicit scheme. The stability criterion for the explicit scheme requires that $\\Delta t/\\Delta x^2 \\leq 1/2$. \n",
+ "\n",
+ "Study the solutions at two time points $t_1$ and $t_2$ where $u(x,t_1)$ is smooth but still significantly curved\n",
+ "and $u(x,t_2)$ is almost linear, close to the stationary state."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "678607d0",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part c) Neural networks\n",
+ "\n",
+ "Study now the lecture notes on solving ODEs and PDEs with neural\n",
+ "network and use either your own code from project 2 or the\n",
+ "functionality of tensorflow/keras to solve the same equation as in\n",
+ "part b). Discuss your results and compare them with the standard\n",
+ "explicit scheme. Include also the analytical solution and compare with\n",
+ "that."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "61d51d93",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part d)\n",
+ "\n",
+ "Finally, present a critical assessment of the methods you have studied\n",
+ "and discuss the potential for the solving differential equations and\n",
+ "eigenvalue problems with machine learning methods."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d2163034",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Introduction to numerical projects\n",
+ "\n",
+ "Here follows a brief recipe and recommendation on how to write a report for each\n",
+ "project.\n",
+ "\n",
+ " * Give a short description of the nature of the problem and the eventual numerical methods you have used.\n",
+ "\n",
+ " * Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.\n",
+ "\n",
+ " * Include the source code of your program. Comment your program properly.\n",
+ "\n",
+ " * If possible, try to find analytic solutions, or known limits in order to test your program when developing the code.\n",
+ "\n",
+ " * Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes.\n",
+ "\n",
+ " * Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc.\n",
+ "\n",
+ " * Try to give an interpretation of you results in your answers to the problems.\n",
+ "\n",
+ " * Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts you've made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it.\n",
+ "\n",
+ " * Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don't properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e31290fd",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Format for electronic delivery of report and programs\n",
+ "\n",
+ "The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:\n",
+ "\n",
+ " * Use Canvas to hand in your projects, log in at with your normal UiO username and password.\n",
+ "\n",
+ " * Upload **only** the report file or the link to your GitHub/GitLab or similar typo of repos! For the source code file(s) you have developed please provide us with your link to your GitHub/GitLab or similar domain. The report file should include all of your discussions and a list of the codes you have developed. Do not include library files which are available at the course homepage, unless you have made specific changes to them.\n",
+ "\n",
+ " * In your GitHub/GitLab or similar repository, please include a folder which contains selected results. These can be in the form of output from your code for a selected set of runs and input parameters.\n",
+ "\n",
+ "Finally, \n",
+ "we encourage you to collaborate. Optimal working groups consist of \n",
+ "2-3 students. You can then hand in a common report."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "99e78b73",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Software and needed installations\n",
+ "\n",
+ "If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages, \n",
+ "we recommend that you install the following Python packages via **pip** as\n",
+ "1. pip install numpy scipy matplotlib ipython scikit-learn tensorflow sympy pandas pillow\n",
+ "\n",
+ "For Python3, replace **pip** with **pip3**.\n",
+ "\n",
+ "See below for a discussion of **tensorflow** and **scikit-learn**. \n",
+ "\n",
+ "For OSX users we recommend also, after having installed Xcode, to install **brew**. Brew allows \n",
+ "for a seamless installation of additional software via for example\n",
+ "1. brew install python3\n",
+ "\n",
+ "For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution\n",
+ "you can use **pip** as well and simply install Python as \n",
+ "1. sudo apt-get install python3 (or python for python2.7)\n",
+ "\n",
+ "etc etc. \n",
+ "\n",
+ "If you don't want to install various Python packages with their dependencies separately, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely\n",
+ "1. [Anaconda](https://docs.anaconda.com/) Anaconda is an open source distribution of the Python and R programming languages for large-scale data processing, predictive analytics, and scientific computing, that aims to simplify package management and deployment. Package versions are managed by the package management system **conda**\n",
+ "\n",
+ "2. [Enthought canopy](https://www.enthought.com/product/canopy/) is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.\n",
+ "\n",
+ "Popular software packages written in Python for ML are\n",
+ "\n",
+ "* [Scikit-learn](http://scikit-learn.org/stable/), \n",
+ "\n",
+ "* [Tensorflow](https://www.tensorflow.org/),\n",
+ "\n",
+ "* [PyTorch](http://pytorch.org/) and \n",
+ "\n",
+ "* [Keras](https://keras.io/).\n",
+ "\n",
+ "These are all freely available at their respective GitHub sites. They \n",
+ "encompass communities of developers in the thousands or more. And the number\n",
+ "of code developers and contributors keeps increasing."
+ ]
+ }
+ ],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/doc/LectureNotes/_build/html/_sources/week46.ipynb b/doc/LectureNotes/_build/html/_sources/week46.ipynb
new file mode 100644
index 000000000..5aaeebdc2
--- /dev/null
+++ b/doc/LectureNotes/_build/html/_sources/week46.ipynb
@@ -0,0 +1,3156 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "3ae913f7",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bde4c6f5",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "# Week 46: Decision Trees, Ensemble methods and Random Forests\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: **Week 46, November 13-17**"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9e846d96",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Plan for week 46\n",
+ "\n",
+ "**Active learning sessions on Tuesday and Wednesday.**\n",
+ "\n",
+ " * Work and Discussion of project 2\n",
+ "\n",
+ " * Discussion of project 3 as well\n",
+ "\n",
+ " \n",
+ "\n",
+ "**Material for the lecture on Thursday November 16, 2023.**\n",
+ "\n",
+ " * Thursday: Basics of decision trees, classification and regression algorithms and ensemble models \n",
+ "\n",
+ " * Readings and Videos:\n",
+ "\n",
+ " * These lecture notes\n",
+ "\n",
+ " * [Video of lecture to be added](https://youtu.be/)\n",
+ "\n",
+ " * [Video on Decision trees](https://www.youtube.com/watch?v=RmajweUFKvM&ab_channel=Simplilearn)\n",
+ "\n",
+ " * Decision Trees: 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b181bac8",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Decision trees, overarching aims\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",
+ "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**."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5c065b15",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0685a1f3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## A Sketch of a Tree, Regression problem\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9deb4609",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## A Sketch of a Tree, Classification problem\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "971fbe6d",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## A typical Decision Tree with its pertinent Jargon, Classification Problem\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "
Figure 1:
\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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3c3fee0c",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5d7a417e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## How do we set it up?\n",
+ "\n",
+ "In simplified terms, the process of training a decision tree and\n",
+ "predicting the target features of query instances is as follows:\n",
+ "\n",
+ "1. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature\n",
+ "\n",
+ "2. Train the decision tree model by continuously splitting the target feature along the values of the descriptive features using a measure of information gain during the training process\n",
+ "\n",
+ "3. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the *predictions* we want to make for new query instances\n",
+ "\n",
+ "4. Show query instances to the tree and run down the tree until we arrive at leaf nodes\n",
+ "\n",
+ "Then we are essentially done!"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "98739ae8",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Decision trees and Regression"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "2e75ab9e",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "%matplotlib inline\n",
+ "\n",
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.preprocessing import PolynomialFeatures\n",
+ "from sklearn.linear_model import LinearRegression\n",
+ "\n",
+ "steps=250\n",
+ "\n",
+ "distance=0\n",
+ "x=0\n",
+ "distance_list=[]\n",
+ "steps_list=[]\n",
+ "while x\n",
+ "\n",
+ "
Grade Trend
Hours slept
Hours Studied
Grade
\n",
+ "\n",
+ "\n",
+ "
Above
Low
High
Above
\n",
+ "
Below
High
Low
Below
\n",
+ "
Above
Low
High
Above
\n",
+ "
Above
High
High
Above
\n",
+ "
Below
Low
High
Below
\n",
+ "
Above
Low
Low
Below
\n",
+ "
Below
High
High
Below
\n",
+ "
Below
Low
High
Below
\n",
+ "
Above
Low
Low
Below
\n",
+ "
Above
High
High
Above
\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8ba76cfd",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Computing the various Gini Indices\n",
+ "\n",
+ "In computations we will translate all classes into numbers. Being\n",
+ "these binary classes, they can easily be split into ones and zeros.\n",
+ "\n",
+ "**Gini index for Average trend.**\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a7a4d204",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Computing the various Gini Indices, Hours slept\n",
+ "\n",
+ "**Gini index for hour slept.**\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "51de5c69",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Computing the various Gini Indices, Hours studied\n",
+ "\n",
+ "**Gini index for hour studied.**\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)\n",
+ "\n",
+ "For final tree, see the above handwritten notes"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9f4bf856",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## A possible code using Scikit-Learn"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "id": "020402b2",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Common imports\n",
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.tree import DecisionTreeClassifier\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.tree import export_graphviz\n",
+ "from sklearn.preprocessing import StandardScaler, OneHotEncoder\n",
+ "from sklearn.compose import ColumnTransformer\n",
+ "from IPython.display import Image \n",
+ "from pydot import graph_from_dot_data\n",
+ "import os\n",
+ "\n",
+ "# Where to save the figures and data files\n",
+ "PROJECT_ROOT_DIR = \"Results\"\n",
+ "FIGURE_ID = \"Results/FigureFiles\"\n",
+ "DATA_ID = \"DataFiles/\"\n",
+ "\n",
+ "if not os.path.exists(PROJECT_ROOT_DIR):\n",
+ " os.mkdir(PROJECT_ROOT_DIR)\n",
+ "\n",
+ "if not os.path.exists(FIGURE_ID):\n",
+ " os.makedirs(FIGURE_ID)\n",
+ "\n",
+ "if not os.path.exists(DATA_ID):\n",
+ " os.makedirs(DATA_ID)\n",
+ "\n",
+ "def image_path(fig_id):\n",
+ " return os.path.join(FIGURE_ID, fig_id)\n",
+ "\n",
+ "def data_path(dat_id):\n",
+ " return os.path.join(DATA_ID, dat_id)\n",
+ "\n",
+ "def save_fig(fig_id):\n",
+ " plt.savefig(image_path(fig_id) + \".png\", format='png')\n",
+ "\n",
+ "infile = open(data_path(\"grades.csv\"),'r')\n",
+ "\n",
+ "# Read the experimental data with Pandas\n",
+ "from IPython.display import display\n",
+ "grades = pd.read_csv(infile)\n",
+ "grades = pd.DataFrame(grades)\n",
+ "display(grades)\n",
+ "# Features and targets\n",
+ "X = grades.loc[:, grades.columns != 'Grade'].values\n",
+ "y = grades.loc[:, grades.columns == 'Grade'].values\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/grade.dot\",\n",
+ " rounded=True,\n",
+ " filled=True\n",
+ ")\n",
+ "cmd = 'dot -Tpng DataFiles/grade.dot -o DataFiles/grades.png'\n",
+ "os.system(cmd)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "082c7f06",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Further example: Computing the Gini index\n",
+ "\n",
+ "The next 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",
+ "
Day
Outlook
Temperature
Humidity
Wind
Ride
\n",
+ "\n",
+ "\n",
+ "
1
Sunny
Hot
High
Weak
0
\n",
+ "
2
Sunny
Hot
High
Strong
1
\n",
+ "
3
Overcast
Hot
High
Weak
1
\n",
+ "
4
Rain
Mild
High
Weak
1
\n",
+ "
5
Rain
Cool
Normal
Weak
1
\n",
+ "
6
Rain
Cool
Normal
Strong
0
\n",
+ "
7
Overcast
Cool
Normal
Strong
1
\n",
+ "
8
Sunny
Mild
High
Weak
0
\n",
+ "
9
Sunny
Cool
Normal
Weak
1
\n",
+ "
10
Rain
Mild
Normal
Weak
1
\n",
+ "
11
Sunny
Mild
Normal
Strong
1
\n",
+ "
12
Overcast
Mild
High
Strong
1
\n",
+ "
13
Overcast
Hot
Normal
Weak
1
\n",
+ "
14
Rain
Mild
High
Strong
0
\n",
+ "\n",
+ "
"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b742d7cc",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Simple Python Code to read in Data and perform Classification"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "1f114a5a",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Common imports\n",
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.tree import DecisionTreeClassifier\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.tree import export_graphviz\n",
+ "from sklearn.preprocessing import StandardScaler, OneHotEncoder\n",
+ "from sklearn.compose import ColumnTransformer\n",
+ "from IPython.display import Image \n",
+ "from pydot import graph_from_dot_data\n",
+ "import os\n",
+ "\n",
+ "# Where to save the figures and data files\n",
+ "PROJECT_ROOT_DIR = \"Results\"\n",
+ "FIGURE_ID = \"Results/FigureFiles\"\n",
+ "DATA_ID = \"DataFiles/\"\n",
+ "\n",
+ "if not os.path.exists(PROJECT_ROOT_DIR):\n",
+ " os.mkdir(PROJECT_ROOT_DIR)\n",
+ "\n",
+ "if not os.path.exists(FIGURE_ID):\n",
+ " os.makedirs(FIGURE_ID)\n",
+ "\n",
+ "if not os.path.exists(DATA_ID):\n",
+ " os.makedirs(DATA_ID)\n",
+ "\n",
+ "def image_path(fig_id):\n",
+ " return os.path.join(FIGURE_ID, fig_id)\n",
+ "\n",
+ "def data_path(dat_id):\n",
+ " return os.path.join(DATA_ID, dat_id)\n",
+ "\n",
+ "def save_fig(fig_id):\n",
+ " plt.savefig(image_path(fig_id) + \".png\", format='png')\n",
+ "\n",
+ "infile = open(data_path(\"rideclass.csv\"),'r')\n",
+ "\n",
+ "# Read the experimental data with Pandas\n",
+ "from IPython.display import display\n",
+ "ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))\n",
+ "ridedata = pd.DataFrame(ridedata)\n",
+ "\n",
+ "# Features and targets\n",
+ "X = ridedata.loc[:, ridedata.columns != 'Ride'].values\n",
+ "y = ridedata.loc[:, ridedata.columns == 'Ride'].values\n",
+ "\n",
+ "# Create the encoder.\n",
+ "encoder = OneHotEncoder(handle_unknown=\"ignore\")\n",
+ "# Assume for simplicity all features are categorical.\n",
+ "encoder.fit(X) \n",
+ "# Apply the encoder.\n",
+ "X = encoder.transform(X)\n",
+ "print(X)\n",
+ "# Then do a Classification tree\n",
+ "tree_clf = DecisionTreeClassifier(max_depth=2)\n",
+ "tree_clf.fit(X, y)\n",
+ "print(\"Train set accuracy with Decision Tree: {:.2f}\".format(tree_clf.score(X,y)))\n",
+ "#transfer to a decision tree graph\n",
+ "export_graphviz(\n",
+ " tree_clf,\n",
+ " out_file=\"DataFiles/ride.dot\",\n",
+ " rounded=True,\n",
+ " filled=True\n",
+ ")\n",
+ "cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'\n",
+ "os.system(cmd)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3ae90db1",
+ "metadata": {
+ "editable": true
+ },
+ "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": 8,
+ "id": "22320738",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Split a dataset based on an attribute and an attribute value\n",
+ "def test_split(index, value, dataset):\n",
+ "\tleft, right = list(), list()\n",
+ "\tfor row in dataset:\n",
+ "\t\tif row[index] < value:\n",
+ "\t\t\tleft.append(row)\n",
+ "\t\telse:\n",
+ "\t\t\tright.append(row)\n",
+ "\treturn left, right\n",
+ " \n",
+ "# Calculate the Gini index for a split dataset\n",
+ "def gini_index(groups, classes):\n",
+ "\t# count all samples at split point\n",
+ "\tn_instances = float(sum([len(group) for group in groups]))\n",
+ "\t# sum weighted Gini index for each group\n",
+ "\tgini = 0.0\n",
+ "\tfor group in groups:\n",
+ "\t\tsize = float(len(group))\n",
+ "\t\t# avoid divide by zero\n",
+ "\t\tif size == 0:\n",
+ "\t\t\tcontinue\n",
+ "\t\tscore = 0.0\n",
+ "\t\t# score the group based on the score for each class\n",
+ "\t\tfor class_val in classes:\n",
+ "\t\t\tp = [row[-1] for row in group].count(class_val) / size\n",
+ "\t\t\tscore += p * p\n",
+ "\t\t# weight the group score by its relative size\n",
+ "\t\tgini += (1.0 - score) * (size / n_instances)\n",
+ "\treturn gini\n",
+ "\n",
+ "# Select the best split point for a dataset\n",
+ "def get_split(dataset):\n",
+ "\tclass_values = list(set(row[-1] for row in dataset))\n",
+ "\tb_index, b_value, b_score, b_groups = 999, 999, 999, None\n",
+ "\tfor index in range(len(dataset[0])-1):\n",
+ "\t\tfor row in dataset:\n",
+ "\t\t\tgroups = test_split(index, row[index], dataset)\n",
+ "\t\t\tgini = gini_index(groups, class_values)\n",
+ "\t\t\tprint('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))\n",
+ "\t\t\tif gini < b_score:\n",
+ "\t\t\t\tb_index, b_value, b_score, b_groups = index, row[index], gini, groups\n",
+ "\treturn {'index':b_index, 'value':b_value, 'groups':b_groups}\n",
+ " \n",
+ "dataset = [[0,0,0,0,0],\n",
+ " [0,0,0,1,1],\n",
+ " [1,0,0,0,1],\n",
+ " [2,1,0,0,1],\n",
+ " [2,2,1,0,1],\n",
+ " [2,2,1,1,0],\n",
+ " [1,2,1,1,1],\n",
+ " [0,1,0,0,0],\n",
+ " [0,2,1,0,1],\n",
+ " [2,1,1,0,1],\n",
+ " [0,1,1,1,1],\n",
+ " [1,1,0,1,1],\n",
+ " [1,0,1,0,1],\n",
+ " [2,1,0,1,0]]\n",
+ "\n",
+ "split = get_split(dataset)\n",
+ "print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ebd0ac9f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Regression trees"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "id": "bfdc5109",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Quadratic training set + noise\n",
+ "np.random.seed(42)\n",
+ "m = 200\n",
+ "X = np.random.rand(m, 1)\n",
+ "y = 4 * (X - 0.5) ** 2\n",
+ "y = y + np.random.randn(m, 1) / 10"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "id": "ab8bb0be",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.tree import DecisionTreeRegressor\n",
+ "\n",
+ "tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)\n",
+ "tree_reg.fit(X, y)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b102de57",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Final regressor code"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "id": "3aeec95d",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.tree import DecisionTreeRegressor\n",
+ "\n",
+ "tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)\n",
+ "tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)\n",
+ "tree_reg1.fit(X, y)\n",
+ "tree_reg2.fit(X, y)\n",
+ "\n",
+ "def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel=\"$y$\"):\n",
+ " x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)\n",
+ " y_pred = tree_reg.predict(x1)\n",
+ " plt.axis(axes)\n",
+ " plt.xlabel(\"$x_1$\", fontsize=18)\n",
+ " if ylabel:\n",
+ " plt.ylabel(ylabel, fontsize=18, rotation=0)\n",
+ " plt.plot(X, y, \"b.\")\n",
+ " plt.plot(x1, y_pred, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n",
+ "\n",
+ "plt.figure(figsize=(11, 4))\n",
+ "plt.subplot(121)\n",
+ "plot_regression_predictions(tree_reg1, X, y)\n",
+ "for split, style in ((0.1973, \"k-\"), (0.0917, \"k--\"), (0.7718, \"k--\")):\n",
+ " plt.plot([split, split], [-0.2, 1], style, linewidth=2)\n",
+ "plt.text(0.21, 0.65, \"Depth=0\", fontsize=15)\n",
+ "plt.text(0.01, 0.2, \"Depth=1\", fontsize=13)\n",
+ "plt.text(0.65, 0.8, \"Depth=1\", fontsize=13)\n",
+ "plt.legend(loc=\"upper center\", fontsize=18)\n",
+ "plt.title(\"max_depth=2\", fontsize=14)\n",
+ "\n",
+ "plt.subplot(122)\n",
+ "plot_regression_predictions(tree_reg2, X, y, ylabel=None)\n",
+ "for split, style in ((0.1973, \"k-\"), (0.0917, \"k--\"), (0.7718, \"k--\")):\n",
+ " plt.plot([split, split], [-0.2, 1], style, linewidth=2)\n",
+ "for split in (0.0458, 0.1298, 0.2873, 0.9040):\n",
+ " plt.plot([split, split], [-0.2, 1], \"k:\", linewidth=1)\n",
+ "plt.text(0.3, 0.5, \"Depth=2\", fontsize=13)\n",
+ "plt.title(\"max_depth=3\", fontsize=14)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "id": "638f8ac8",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "tree_reg1 = DecisionTreeRegressor(random_state=42)\n",
+ "tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)\n",
+ "tree_reg1.fit(X, y)\n",
+ "tree_reg2.fit(X, y)\n",
+ "\n",
+ "x1 = np.linspace(0, 1, 500).reshape(-1, 1)\n",
+ "y_pred1 = tree_reg1.predict(x1)\n",
+ "y_pred2 = tree_reg2.predict(x1)\n",
+ "\n",
+ "plt.figure(figsize=(11, 4))\n",
+ "\n",
+ "plt.subplot(121)\n",
+ "plt.plot(X, y, \"b.\")\n",
+ "plt.plot(x1, y_pred1, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n",
+ "plt.axis([0, 1, -0.2, 1.1])\n",
+ "plt.xlabel(\"$x_1$\", fontsize=18)\n",
+ "plt.ylabel(\"$y$\", fontsize=18, rotation=0)\n",
+ "plt.legend(loc=\"upper center\", fontsize=18)\n",
+ "plt.title(\"No restrictions\", fontsize=14)\n",
+ "\n",
+ "plt.subplot(122)\n",
+ "plt.plot(X, y, \"b.\")\n",
+ "plt.plot(x1, y_pred2, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n",
+ "plt.axis([0, 1, -0.2, 1.1])\n",
+ "plt.xlabel(\"$x_1$\", fontsize=18)\n",
+ "plt.title(\"min_samples_leaf={}\".format(tree_reg2.min_samples_leaf), fontsize=14)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "939c2f5c",
+ "metadata": {
+ "editable": true
+ },
+ "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)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "121bdc13",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f58b9924",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3369dc37",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## An Overview of Ensemble Methods\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "
Figure 1:
\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "247d8e1f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "418dee56",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Tossing coins\n",
+ "\n",
+ "The simplest case is a so-called voting ensemble. To illustrate this,\n",
+ "think of yourself tossing coins with a biased outcome of 51 per cent\n",
+ "for heads and 49% for tails. With only few tosses,\n",
+ "you may not clearly see this distribution for heads and tails. However, after some\n",
+ "thousands of tosses, there will be a clear majority of heads. With 2000 tosses\n",
+ "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\n",
+ "you do this ten thousand times, it is easy to see that there is a 97%\n",
+ "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,\n",
+ "clealry demostrating that after some tosses we see the [law of large](https://en.wikipedia.org/wiki/Law_of_large_numbers)\n",
+ "numbers kicking in."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "81c7b4f6",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Standard imports first"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "id": "c8635354",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Common imports\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",
+ "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')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6945cdc1",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Simple Voting Example, head or tail"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "id": "dbe486ef",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "\n",
+ "# Common imports\n",
+ "import numpy as np\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",
+ "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",
+ "id": "31814f98",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Using the Voting Classifier\n",
+ "\n",
+ "We can use the voting classifier on other data sets, here the exciting binary case of two distinct objects using the make moons functionality of **Scikit-Learn**."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "id": "20858a25",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.datasets import make_moons\n",
+ "\n",
+ "X, y = make_moons(n_samples=500, noise=0.30, random_state=42)\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)\n",
+ "\n",
+ "from sklearn.ensemble import RandomForestClassifier\n",
+ "from sklearn.ensemble import VotingClassifier\n",
+ "from sklearn.linear_model import LogisticRegression\n",
+ "from sklearn.svm import SVC\n",
+ "\n",
+ "log_clf = LogisticRegression(solver=\"liblinear\", random_state=42)\n",
+ "rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)\n",
+ "svm_clf = SVC(gamma=\"auto\", random_state=42)\n",
+ "\n",
+ "voting_clf = VotingClassifier(\n",
+ " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n",
+ " voting='hard')\n",
+ "\n",
+ "voting_clf.fit(X_train, y_train)\n",
+ "\n",
+ "from sklearn.metrics import accuracy_score\n",
+ "\n",
+ "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n",
+ " clf.fit(X_train, y_train)\n",
+ " y_pred = clf.predict(X_test)\n",
+ " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))\n",
+ "\n",
+ "log_clf = LogisticRegression(solver=\"liblinear\", random_state=42)\n",
+ "rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)\n",
+ "svm_clf = SVC(gamma=\"auto\", probability=True, random_state=42)\n",
+ "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",
+ "id": "69bde19e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Voting and Bagging"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "579fcaba",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.datasets import make_moons\n",
+ "\n",
+ "X, y = make_moons(n_samples=500, noise=0.30, random_state=42)\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)\n",
+ "from sklearn.ensemble import RandomForestClassifier\n",
+ "from sklearn.ensemble import VotingClassifier\n",
+ "from sklearn.linear_model import LogisticRegression\n",
+ "from sklearn.svm import SVC\n",
+ "\n",
+ "log_clf = LogisticRegression(random_state=42)\n",
+ "rnd_clf = RandomForestClassifier(random_state=42)\n",
+ "svm_clf = SVC(random_state=42)\n",
+ "\n",
+ "voting_clf = VotingClassifier(\n",
+ " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n",
+ " voting='hard')\n",
+ "voting_clf.fit(X_train, y_train)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "id": "e19aed80",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.metrics import accuracy_score\n",
+ "\n",
+ "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n",
+ " clf.fit(X_train, y_train)\n",
+ " y_pred = clf.predict(X_test)\n",
+ " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "id": "5e7fff36",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "log_clf = LogisticRegression(random_state=42)\n",
+ "rnd_clf = RandomForestClassifier(random_state=42)\n",
+ "svm_clf = SVC(probability=True, random_state=42)\n",
+ "\n",
+ "voting_clf = VotingClassifier(\n",
+ " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n",
+ " voting='soft')\n",
+ "voting_clf.fit(X_train, y_train)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "id": "12a367f0",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.metrics import accuracy_score\n",
+ "\n",
+ "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n",
+ " clf.fit(X_train, y_train)\n",
+ " y_pred = clf.predict(X_test)\n",
+ " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "70753458",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d37bbe68",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "498956cc",
+ "metadata": {
+ "editable": true
+ },
+ "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": 20,
+ "id": "452de3bb",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "\n",
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.pipeline import make_pipeline\n",
+ "from sklearn.utils import resample\n",
+ "from sklearn.tree import DecisionTreeRegressor\n",
+ "\n",
+ "n = 100\n",
+ "n_boostraps = 100\n",
+ "maxdepth = 8\n",
+ "\n",
+ "# Make data set.\n",
+ "x = np.linspace(-3, 3, n).reshape(-1, 1)\n",
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n",
+ "error = np.zeros(maxdepth)\n",
+ "bias = np.zeros(maxdepth)\n",
+ "variance = np.zeros(maxdepth)\n",
+ "polydegree = np.zeros(maxdepth)\n",
+ "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n",
+ "\n",
+ "from sklearn.preprocessing import StandardScaler\n",
+ "scaler = StandardScaler()\n",
+ "scaler.fit(X_train)\n",
+ "X_train_scaled = scaler.transform(X_train)\n",
+ "X_test_scaled = scaler.transform(X_test)\n",
+ "\n",
+ "# we produce a simple tree first as benchmark\n",
+ "simpletree = DecisionTreeRegressor(max_depth=3) \n",
+ "simpletree.fit(X_train_scaled, y_train)\n",
+ "simpleprediction = simpletree.predict(X_test_scaled)\n",
+ "for degree in range(1,maxdepth):\n",
+ " model = DecisionTreeRegressor(max_depth=degree) \n",
+ " y_pred = np.empty((y_test.shape[0], n_boostraps))\n",
+ " for i in range(n_boostraps):\n",
+ " x_, y_ = resample(X_train_scaled, y_train)\n",
+ " model.fit(x_, y_)\n",
+ " y_pred[:, i] = model.predict(X_test_scaled)#.ravel()\n",
+ "\n",
+ " polydegree[degree] = degree\n",
+ " error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n",
+ " bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n",
+ " variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n",
+ " print('Polynomial degree:', degree)\n",
+ " print('Error:', error[degree])\n",
+ " print('Bias^2:', bias[degree])\n",
+ " print('Var:', variance[degree])\n",
+ " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n",
+ " \n",
+ "mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2))\n",
+ "print(\"Simple tree:\",mse_simpletree)\n",
+ "plt.xlim(1,maxdepth)\n",
+ "plt.plot(polydegree, error, label='MSE')\n",
+ "plt.plot(polydegree, bias, label='bias')\n",
+ "plt.plot(polydegree, variance, label='Variance')\n",
+ "plt.legend()\n",
+ "save_fig(\"baggingboot\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b7a39b9f",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "d39a6e13",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "m\\approx \\sqrt{p}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "42f8693b",
+ "metadata": {
+ "editable": true
+ },
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a25fd53e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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 from the training data organized in our $\\boldsymbol{X}$ matrix.\n",
+ "\n",
+ " * We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached\n",
+ "\n",
+ "1. we select $m \\le p$ variables at random from the $p$ predictors/features\n",
+ "\n",
+ "2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node\n",
+ "\n",
+ "3. split the node into daughter nodes\n",
+ "\n",
+ "4. Output then the ensemble of trees $\\{T_b\\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c72f80ca",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Random Forests Compared with other Methods on the Cancer Data"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "id": "9b20111b",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np\n",
+ "from sklearn.model_selection import train_test_split \n",
+ "from sklearn.datasets import load_breast_cancer\n",
+ "from sklearn.svm import SVC\n",
+ "from sklearn.linear_model import LogisticRegression\n",
+ "from sklearn.tree import DecisionTreeClassifier\n",
+ "from sklearn.ensemble import BaggingClassifier\n",
+ "\n",
+ "# Load the data\n",
+ "cancer = load_breast_cancer()\n",
+ "\n",
+ "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n",
+ "print(X_train.shape)\n",
+ "print(X_test.shape)\n",
+ "#define methods\n",
+ "# Logistic Regression\n",
+ "logreg = LogisticRegression(solver='lbfgs')\n",
+ "# Support vector machine\n",
+ "svm = SVC(gamma='auto', C=100)\n",
+ "# Decision Trees\n",
+ "deep_tree_clf = DecisionTreeClassifier(max_depth=None)\n",
+ "#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",
+ "id": "a2a69156",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "Recall that the cumulative gains curve shows the percentage of the\n",
+ "overall number of cases in a given category *gained* by targeting a\n",
+ "percentage of the total number of cases.\n",
+ "\n",
+ "Similarly, the receiver operating characteristic curve, or ROC curve,\n",
+ "displays the diagnostic ability of a binary classifier system as its\n",
+ "discrimination threshold is varied. It plots the true positive rate against the false positive rate."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "935e5ca1",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Compare Bagging on Trees with Random Forests"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "id": "2b6604c5",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "bag_clf = BaggingClassifier(\n",
+ " DecisionTreeClassifier(splitter=\"random\", max_leaf_nodes=16, random_state=42),\n",
+ " n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 23,
+ "id": "b6412285",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "bag_clf.fit(X_train, y_train)\n",
+ "y_pred = bag_clf.predict(X_test)\n",
+ "from sklearn.ensemble import RandomForestClassifier\n",
+ "rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)\n",
+ "rnd_clf.fit(X_train, y_train)\n",
+ "y_pred_rf = rnd_clf.predict(X_test)\n",
+ "np.sum(y_pred == y_pred_rf) / len(y_pred)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c8adc4f4",
+ "metadata": {
+ "editable": true
+ },
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e0bd3a34",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "b93c398e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7fc3ae93",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "e5c67b21",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\sigma(t) = \\frac{1}{1+\\exp{(-t)}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d573725",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "f8cf15d3",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "e734d8d4",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "0585b995",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\beta}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b983e555",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\\beta_m$ and $\\gamma_m$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "203a7a1c",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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 $C(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ 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",
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6d391d0c",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "3b98bb46",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "3d2a99d8",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "3af9f679",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "368371bc",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "and"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ebc23cc6",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "3285cd00",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "420c5c35",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\gamma \\boldsymbol{w}^T(\\boldsymbol{y}-\\beta\\gamma \\boldsymbol{w})=0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2a386bd3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "which gives us $\\beta = \\boldsymbol{w}^T\\boldsymbol{y}/(\\boldsymbol{w}^T\\boldsymbol{w})$. Similarly we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cb75d383",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\beta\\gamma \\boldsymbol{x}^T(\\boldsymbol{y}-\\beta(1+\\gamma \\boldsymbol{x}))=0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d7aac8e6",
+ "metadata": {
+ "editable": true
+ },
+ "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 $f$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8211e34f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "0759bace",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\mathrm{\\overline{err}}=\\frac{1}{n} \\sum_{i=0}^{n-1} I(y_i\\ne G(x_i)).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6a08f8a3",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "ea71963f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "42c5df3b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "will be a function of"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "237180f9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "G_M(x) = \\mathrm{sign} \\sum_{i=1}^M \\alpha_m G_m(x).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2374907b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Adaptive Boosting, AdaBoost\n",
+ "\n",
+ "In our iterative procedure we define thus"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "04048caf",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "f_m(x) = f_{m-1}(x)+\\beta_mG_m(x).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1a77afe0",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "981883bd",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "3d8d3830",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "fe3d598c",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "762ac6b8",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "where we have defined $w_i^m= \\exp{(-y_if_{m-1}(x_i))}$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "396805d8",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Building up AdaBoost\n",
+ "\n",
+ "First, for any $\\beta > 0$, we optimize $G$ by setting"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "19ca93fe",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "5b78c347",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "742c335a",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "766b1fdd",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "which can be rewritten as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d9698851",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "8b9dc891",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "which leads to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "32e16984",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\beta_m = \\frac{1}{2}\\log{\\frac{1-\\mathrm{\\overline{err}}}{\\mathrm{\\overline{err}}}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "85898ea6",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "where we have redefined the error as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e9c70102",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "a00653d9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "which leads to an update of"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e041fdeb",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "f_m(x) = f_{m-1}(x) +\\beta_m G_m(x).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5f07c0b1",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "This leads to the new weights"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d13323ee",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "w_i^{m+1} = w_i^m \\exp{(-y_i\\beta_m G_m(x_i))}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c7237587",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "5eabf8df",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\mathrm{err}=\\frac{1}{n}\\sum_{i=0}^{n-1}I(y_i\\ne G(x_i)),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "309fe485",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "where the function $I()$ is one if we misclassify and zero if we classify correctly."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "344c2fc2",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "0af051ff",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "ebf99e70",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "38533606",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## AdaBoost Examples\n",
+ "\n",
+ "Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "id": "58b8da05",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.ensemble import AdaBoostClassifier\n",
+ "\n",
+ "ada_clf = AdaBoostClassifier(\n",
+ " DecisionTreeClassifier(max_depth=1), n_estimators=200,\n",
+ " algorithm=\"SAMME.R\", learning_rate=0.5, random_state=42)\n",
+ "ada_clf.fit(X_train, y_train)\n",
+ "\n",
+ "from sklearn.ensemble import AdaBoostClassifier\n",
+ "\n",
+ "ada_clf = AdaBoostClassifier(\n",
+ " DecisionTreeClassifier(max_depth=1), n_estimators=200,\n",
+ " algorithm=\"SAMME.R\", learning_rate=0.5, random_state=42)\n",
+ "ada_clf.fit(X_train_scaled, y_train)\n",
+ "y_pred = ada_clf.predict(X_test_scaled)\n",
+ "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n",
+ "plt.show()\n",
+ "y_probas = ada_clf.predict_proba(X_test_scaled)\n",
+ "skplt.metrics.plot_roc(y_test, y_probas)\n",
+ "plt.show()\n",
+ "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n",
+ "plt.show()"
+ ]
+ }
+ ],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/doc/LectureNotes/_build/html/genindex.html b/doc/LectureNotes/_build/html/genindex.html
index 5ce28b896..7bf64532e 100644
--- a/doc/LectureNotes/_build/html/genindex.html
+++ b/doc/LectureNotes/_build/html/genindex.html
@@ -349,6 +349,11 @@ const thebe_selector_output = ".output, .cell_output"
Week 45, Recurrent Neural Networks
+
diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js
index 1dd30796e..2ac07f344 100644
--- a/doc/LectureNotes/_build/html/searchindex.js
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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Exercises week 38","Exercises week 39","Exercises week 41","Exercises week 42","Exercises weeks 43 and 44","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 9 (midnight), 2023","Project 2 on Machine Learning, deadline November 13 (Midnight)","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling techniques","Week 37: Statistical interpretations and Resampling Methods","Week 38: Logistic Regression and Optimization","Week 39: Optimization and Gradient Methods","Week 40: Gradient descent methods (continued) and start Neural networks","Week 41 Neural networks and constructing a neural network code","Week 42 Constructing a Neural Network code with introduction to Tensor flow","Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations","Week 44, Convolutional Neural Networks (CNN)","Week 45, Recurrent Neural 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3 on Machine Learning, deadline December 18 (midnight), 2023","3. Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Exercises week 37","Exercises week 38","Exercises week 39","Exercises week 41","Exercises week 42","Exercises weeks 43 and 44","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Project 1 on Machine Learning, deadline October 9 (midnight), 2023","Project 2 on Machine Learning, deadline November 13 (Midnight)","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling techniques","Week 37: Statistical interpretations and Resampling Methods","Week 38: Logistic Regression and Optimization","Week 39: Optimization and Gradient Methods","Week 40: Gradient descent methods (continued) and start Neural networks","Week 41 Neural networks and constructing a neural network code","Week 42 Constructing a Neural Network code with introduction to Tensor flow","Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations","Week 44, Convolutional Neural Networks (CNN)","Week 45, Recurrent Neural Networks","Week 46: Decision Trees, Ensemble methods and Random 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+ Week 46: Decision Trees, Ensemble methods and Random Forests — Applied Data Analysis and Machine Learning
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Week 46: Decision Trees, Ensemble methods and Random Forests
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Week 46: Decision Trees, Ensemble methods and Random Forests¶
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Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
Decision Trees: 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. Chapter 9.2 of Hastie et al contains also a good discussion.
We start here with the most basic algorithm, the so-called decision
+tree. With this basic algorithm we can in turn build more complex
+networks, spanning from homogeneous and heterogenous forests (bagging,
+random forests and more) to one of the most popular supervised
+algorithms nowadays, the extreme gradient boosting, or just
+XGBoost. But let us start with the simplest possible ingredient.
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Decision trees are supervised learning algorithms used for both,
+classification and regression tasks.
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The main idea of decision trees
+is to find those descriptive features which contain the most
+information regarding the target feature and then split the dataset
+along the values of these features such that the target feature values
+for the resulting underlying datasets are as pure as possible.
+
The descriptive features which reproduce best the target/output features are normally said
+to be the most informative ones. The process of finding the most
+informative feature is done until we accomplish a stopping criteria
+where we then finally end up in so called leaf nodes.
A decision tree is typically divided into a root node, the interior nodes,
+and the final leaf nodes or just leaves. These entities are then connected by so-called branches.
+
The leaf nodes
+contain the predictions we will make for new query instances presented
+to our trained model. This is possible since the model has
+learned the underlying structure of the training data and hence can,
+given some assumptions, make predictions about the target feature value
+(class) of unseen query instances.
A typical Decision Tree with its pertinent Jargon, Classification Problem¶
+
+
+
Figure 1:
+
+
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.
The overarching approach to decision trees is a top-down approach.
+
+
A leaf provides the classification of a given instance.
+
A node specifies a test of some attribute of the instance.
+
A branch corresponds to a possible values of an attribute.
+
An instance is classified by starting at the root node of the tree, testing the attribute specified by this node, then moving down the tree branch corresponding to the value of the attribute in the given example.
+
+
This process is then repeated for the subtree rooted at the new
+node.
In simplified terms, the process of training a decision tree and
+predicting the target features of query instances is as follows:
+
+
Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature
+
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
+
Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the predictions we want to make for new query instances
+
Show query instances to the tree and run down the tree until we arrive at leaf nodes
We split the predictor space (the set of possible values \(x_1,x_2,\dots, x_p\)) into \(J\) distinct and non-non-overlapping regions, \(R_1,R_2,\dots,R_J\).
+
For every observation that falls into the region \(R_j\) , we make the same prediction, which is simply the mean of the response values for the training observations in \(R_j\).
+
+
How do we construct the regions \(R_1,\dots,R_J\)? In theory, the
+regions could have any shape. However, we choose to divide the
+predictor space into high-dimensional rectangles, or boxes, for
+simplicity and for ease of interpretation of the resulting predictive
+model. The goal is to find boxes \(R_1,\dots,R_J\) that minimize the
+MSE, given by
Unfortunately, it is computationally infeasible to consider every
+possible partition of the feature space into \(J\) boxes. The common
+strategy is to take a top-down approach
+
The approach is top-down because it begins at the top of the tree (all
+observations belong to a single region) and then successively splits
+the predictor space; each split is indicated via two new branches
+further down on the tree. It is greedy because at each step of the
+tree-building process, the best split is made at that particular step,
+rather than looking ahead and picking a split that will lead to a
+better tree in some future step.
In order to implement the recursive binary splitting we start by selecting
+the predictor \(x_j\) and a cutpoint \(s\) that splits the predictor space into two regions \(R_1\) and \(R_2\)
which we want to minimize by considering all predictors
+\(x_1,x_2,\dots,x_p\). We consider also all possible values of \(s\) for
+each predictor. These values could be determined by randomly assigned
+numbers or by starting at the midpoint and then proceed till we find
+an optimal value.
+
For any \(j\) and \(s\), we define the pair of half-planes where
+\(\overline{y}_{R_1}\) is the mean response for the training
+observations in \(R_1(j,s)\), and \(\overline{y}_{R_2}\) is the mean
+response for the training observations in \(R_2(j,s)\).
+
Finding the values of \(j\) and \(s\) that minimize the above equation can be
+done quite quickly, especially when the number of features \(p\) is not
+too large.
+
Next, we repeat the process, looking
+for the best predictor and best cutpoint in order to split the data
+further so as to minimize the MSE within each of the resulting
+regions. However, this time, instead of splitting the entire predictor
+space, we split one of the two previously identified regions. We now
+have three regions. Again, we look to split one of these three regions
+further, so as to minimize the MSE. The process continues until a
+stopping criterion is reached; for instance, we may continue until no
+region contains more than five observations.
The above procedure is rather straightforward, but leads often to
+overfitting and unnecessarily large and complicated trees. The basic
+idea is to grow a large tree \(T_0\) and then prune it back in order to
+obtain a subtree. A smaller tree with fewer splits (fewer regions) can
+lead to smaller variance and better interpretation at the cost of a
+little more bias.
+
The so-called Cost complexity pruning algorithm gives us a
+way to do just this. Rather than considering every possible subtree,
+we consider a sequence of trees indexed by a nonnegative tuning
+parameter \(\alpha\).
is as small as possible. Here \(\overline{T}\) is
+the number of terminal nodes of the tree \(T\) , \(R_m\) is the
+rectangle (i.e. the subset of predictor space) corresponding to the \(m\)-th terminal node.
+
The tuning parameter \(\alpha\) controls a trade-off between the subtree’s
+complexity and its fit to the training data. When \(\alpha = 0\), then the
+subtree \(T\) will simply equal \(T_0\),
+because then the above equation just measures the
+training error.
+However, as \(\alpha\) increases, there is a price to pay for
+having a tree with many terminal nodes. The above equation will
+tend to be minimized for a smaller subtree.
+
It turns out that as we increase \(\alpha\) from zero
+branches get pruned from the tree in a nested and predictable fashion,
+so obtaining the whole sequence of subtrees as a function of \(\alpha\) is
+easy. We can select a value of \(\alpha\) using a validation set or using
+cross-validation. We then return to the full data set and obtain the
+subtree corresponding to \(\alpha\).
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.
+
Apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of \(\alpha\).
+
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:
+
+
+
repeat steps 1 and 2 on all but the \(k\)-th fold of the training data.
+
Then we valuate the mean squared prediction error on the data in the left-out \(k\)-th fold, as a function of \(\alpha\).
+
Finally we average the results for each value of \(\alpha\), and pick \(\alpha\) to minimize the average error.
+
+
+
Return the subtree from Step 2 that corresponds to the chosen value of \(\alpha\).
A classification tree is very similar to a regression tree, except
+that it is used to predict a qualitative response rather than a
+quantitative one. Recall that for a regression tree, the predicted
+response for an observation is given by the mean response of the
+training observations that belong to the same terminal node. In
+contrast, for a classification tree, we predict that each observation
+belongs to the most commonly occurring class of training observations
+in the region to which it belongs. In interpreting the results of a
+classification tree, we are often interested not only in the class
+prediction corresponding to a particular terminal node region, but
+also in the class proportions among the training observations that
+fall into that region.
The task of growing a
+classification tree is quite similar to the task of growing a
+regression tree. Just as in the regression setting, we use recursive
+binary splitting to grow a classification tree. However, in the
+classification setting, the MSE cannot be used as a criterion for making
+the binary splits. A natural alternative to MSE is the classification
+error rate. Since we plan to assign an observation in a given region
+to the most commonly occurring error rate class of training
+observations in that region, the classification error rate is simply
+the fraction of the training observations in that region that do not
+belong to the most common class.
+
When building a classification tree, either the Gini index or the
+entropy are typically used to evaluate the quality of a particular
+split, since these two approaches are more sensitive to node purity
+than is the classification error rate.
If our targets are the outcome of a classification process that takes
+for example \(k=1,2,\dots,K\) values, the only thing we need to think of
+is to set up the splitting criteria for each node.
+
We define a PDF \(p_{mk}\) that represents the number of observations of
+a class \(k\) in a region \(R_m\) with \(N_m\) observations. We represent
+this likelihood function in terms of the proportion \(I(y_i=k)\) of
+observations of this class in the region \(R_m\) as
Scikit-Learn has also another way to visualize the trees which is very useful, here with the Iris data.
+
+
+
fromsklearn.datasetsimportload_iris
+fromsklearnimporttree
+X,y=load_iris(return_X_y=True)
+tree_clf=tree.DecisionTreeClassifier()
+tree_clf=tree_clf.fit(X,y)
+# and then plot the tree
+tree.plot_tree(tree_clf)
+
Alternatively, the tree can also be exported in textual format with the function exporttext.
+This method doesn’t require the installation of external libraries and is more compact:
Two algorithms stand out in the set up of decision trees:
+
+
The CART (Classification And Regression Tree) algorithm for both classification and regression
+
The ID3 algorithm based on the computation of the information gain for classification
+
+
We discuss both algorithms with applications here. The popular library
+Scikit-Learn uses the CART algorithm. For classification problems
+you can use either the gini index or the entropy to split a tree
+in two branches.
For classification, the CART algorithm splits the data set in two subsets using a single feature \(k\) and a threshold \(t_k\).
+This could be for example a threshold set by a number below a certain circumference of a malign tumor.
+
How do we find these two quantities?
+We search for the pair \((k,t_k)\) that produces the purest subset using for example the gini factor \(G\).
+The cost function it tries to minimize is then
where \(G_{\mathrm{left/right}}\) measures the impurity of the left/right subset and \(m_{\mathrm{left/right}}\)
+is the number of instances in the left/right subset
+
Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets
+and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the
+\(max\_depth\) hyperparameter), or if it cannot find a split that will reduce impurity. A few other
+hyperparameters control additional stopping conditions such as the \(min\_samples\_split\),
+\(min\_samples\_leaf\), \(min\_weight\_fraction\_leaf\), and \(max\_leaf\_nodes\).
The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the
+training set in a way that minimizes say the gini or entropy impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now
It is custom to split to a tree uising binary splits. The reason is
+that multiway splits fragment the data too quickly, leaving
+insufficient data at the next level down. Multiway splits can be
+achieved by a series of binary split and this is normally preferred.
Consider the following example with attributes/features and two
+possible outcomes (classes) for each attribute. Assume we wish to find some
+correlations between the average grade of a student as function of the
+number of hours studied and hours slept. We want also to correlate the
+grade in a given course with the general trend, whether the students
+recently has gotten grades below average or above.
+
We have three features/attributes
+
+
Trend of average grades before present course, classified as either below or above the average grade of the whole class
+
The number of hours studies, classified again as either higher (more than 3 hours per day) or lower . Here we have used a standard for one \(ECTS\) which is scaled to 25-30 hours of work for a semester which lasts 18 weeks, with 15 weeks of lectures and 3 weeks for exams, assuming a total of 30 ECTS per semester.
+
The number of hours slept as high for more than \(8\) hours and below for less than 8 hours of sleep, classified again as either high or low
+
The final grade whether it is above or below average
# Common imports
+importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.treeimportDecisionTreeClassifier
+fromsklearn.model_selectionimporttrain_test_split
+fromsklearn.treeimportexport_graphviz
+fromsklearn.preprocessingimportStandardScaler,OneHotEncoder
+fromsklearn.composeimportColumnTransformer
+fromIPython.displayimportImage
+frompydotimportgraph_from_dot_data
+importos
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR="Results"
+FIGURE_ID="Results/FigureFiles"
+DATA_ID="DataFiles/"
+
+ifnotos.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+ifnotos.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+ifnotos.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+defimage_path(fig_id):
+ returnos.path.join(FIGURE_ID,fig_id)
+
+defdata_path(dat_id):
+ returnos.path.join(DATA_ID,dat_id)
+
+defsave_fig(fig_id):
+ plt.savefig(image_path(fig_id)+".png",format='png')
+
+infile=open(data_path("grades.csv"),'r')
+
+# Read the experimental data with Pandas
+fromIPython.displayimportdisplay
+grades=pd.read_csv(infile)
+grades=pd.DataFrame(grades)
+display(grades)
+# Features and targets
+X=grades.loc[:,grades.columns!='Grade'].values
+y=grades.loc[:,grades.columns=='Grade'].values
+print(X)
+# Then do a Classification tree
+tree_clf=DecisionTreeClassifier(max_depth=2)
+tree_clf.fit(X,y)
+print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
+#transfer to a decision tree graph
+export_graphviz(
+ tree_clf,
+ out_file="DataFiles/grade.dot",
+ rounded=True,
+ filled=True
+)
+cmd='dot -Tpng DataFiles/grade.dot -o DataFiles/grades.png'
+os.system(cmd)
+
+
+
+
+
---------------------------------------------------------------------------
+FileNotFoundErrorTraceback (most recent call last)
+Input In [6], in <cell line: 37>()
+34defsave_fig(fig_id):
+35plt.savefig(image_path(fig_id)+".png",format='png')
+---> 37infile=open(data_path("grades.csv"),'r')
+39# Read the experimental data with Pandas
+40fromIPython.displayimportdisplay
+
+FileNotFoundError: [Errno 2] No such file or directory: 'DataFiles/grades.csv'
+
The next example we will look at is a classical one in many Machine
+Learning applications. Based on various meteorological features, we
+have several so-called attributes which decide whether we at the end
+will do some outdoor activity like skiing, going for a bike ride etc
+etc. The table here contains the feautures outlook, temperature,
+humidity and wind. The target or output is whether we ride
+(True=1) or whether we do something else that day (False=0). The
+attributes for each feature are then sunny, overcast and rain for the
+outlook, hot, cold and mild for temperature, high and normal for
+humidity and weak and strong for wind.
+
The table here summarizes the various attributes and
+
+
+
Day
Outlook
Temperature
Humidity
Wind
Ride
+
+
+
1
Sunny
Hot
High
Weak
0
+
2
Sunny
Hot
High
Strong
1
+
3
Overcast
Hot
High
Weak
1
+
4
Rain
Mild
High
Weak
1
+
5
Rain
Cool
Normal
Weak
1
+
6
Rain
Cool
Normal
Strong
0
+
7
Overcast
Cool
Normal
Strong
1
+
8
Sunny
Mild
High
Weak
0
+
9
Sunny
Cool
Normal
Weak
1
+
10
Rain
Mild
Normal
Weak
1
+
11
Sunny
Mild
Normal
Strong
1
+
12
Overcast
Mild
High
Strong
1
+
13
Overcast
Hot
Normal
Weak
1
+
14
Rain
Mild
High
Strong
0
+
+
+
+
Simple Python Code to read in Data and perform Classification¶
+
+
+
# Common imports
+importnumpyasnp
+importpandasaspd
+importmatplotlib.pyplotasplt
+fromsklearn.treeimportDecisionTreeClassifier
+fromsklearn.model_selectionimporttrain_test_split
+fromsklearn.treeimportexport_graphviz
+fromsklearn.preprocessingimportStandardScaler,OneHotEncoder
+fromsklearn.composeimportColumnTransformer
+fromIPython.displayimportImage
+frompydotimportgraph_from_dot_data
+importos
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR="Results"
+FIGURE_ID="Results/FigureFiles"
+DATA_ID="DataFiles/"
+
+ifnotos.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+ifnotos.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+ifnotos.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+defimage_path(fig_id):
+ returnos.path.join(FIGURE_ID,fig_id)
+
+defdata_path(dat_id):
+ returnos.path.join(DATA_ID,dat_id)
+
+defsave_fig(fig_id):
+ plt.savefig(image_path(fig_id)+".png",format='png')
+
+infile=open(data_path("rideclass.csv"),'r')
+
+# Read the experimental data with Pandas
+fromIPython.displayimportdisplay
+ridedata=pd.read_csv(infile,names=('Outlook','Temperature','Humidity','Wind','Ride'))
+ridedata=pd.DataFrame(ridedata)
+
+# Features and targets
+X=ridedata.loc[:,ridedata.columns!='Ride'].values
+y=ridedata.loc[:,ridedata.columns=='Ride'].values
+
+# Create the encoder.
+encoder=OneHotEncoder(handle_unknown="ignore")
+# Assume for simplicity all features are categorical.
+encoder.fit(X)
+# Apply the encoder.
+X=encoder.transform(X)
+print(X)
+# Then do a Classification tree
+tree_clf=DecisionTreeClassifier(max_depth=2)
+tree_clf.fit(X,y)
+print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
+#transfer to a decision tree graph
+export_graphviz(
+ tree_clf,
+ out_file="DataFiles/ride.dot",
+ rounded=True,
+ filled=True
+)
+cmd='dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
+os.system(cmd)
+
The above functions (gini, entropy and misclassification error) are
+important components of the so-called CART algorithm. We will discuss
+this algorithm below after we have discussed the information gain
+algorithm ID3.
+
In the example here we have converted all our attributes into numerical values \(0,1,2\) etc.
+
+
+
# Split a dataset based on an attribute and an attribute value
+deftest_split(index,value,dataset):
+ left,right=list(),list()
+ forrowindataset:
+ ifrow[index]<value:
+ left.append(row)
+ else:
+ right.append(row)
+ returnleft,right
+
+# Calculate the Gini index for a split dataset
+defgini_index(groups,classes):
+ # count all samples at split point
+ n_instances=float(sum([len(group)forgroupingroups]))
+ # sum weighted Gini index for each group
+ gini=0.0
+ forgroupingroups:
+ size=float(len(group))
+ # avoid divide by zero
+ ifsize==0:
+ continue
+ score=0.0
+ # score the group based on the score for each class
+ forclass_valinclasses:
+ p=[row[-1]forrowingroup].count(class_val)/size
+ score+=p*p
+ # weight the group score by its relative size
+ gini+=(1.0-score)*(size/n_instances)
+ returngini
+
+# Select the best split point for a dataset
+defget_split(dataset):
+ class_values=list(set(row[-1]forrowindataset))
+ b_index,b_value,b_score,b_groups=999,999,999,None
+ forindexinrange(len(dataset[0])-1):
+ forrowindataset:
+ groups=test_split(index,row[index],dataset)
+ gini=gini_index(groups,class_values)
+ print('X%d < %.3f Gini=%.3f'%((index+1),row[index],gini))
+ ifgini<b_score:
+ b_index,b_value,b_score,b_groups=index,row[index],gini,groups
+ return{'index':b_index,'value':b_value,'groups':b_groups}
+
+dataset=[[0,0,0,0,0],
+ [0,0,0,1,1],
+ [1,0,0,0,1],
+ [2,1,0,0,1],
+ [2,2,1,0,1],
+ [2,2,1,1,0],
+ [1,2,1,1,1],
+ [0,1,0,0,0],
+ [0,2,1,0,1],
+ [2,1,1,0,1],
+ [0,1,1,1,1],
+ [1,1,0,1,1],
+ [1,0,1,0,1],
+ [2,1,0,1,0]]
+
+split=get_split(dataset)
+print('Split: [X%d < %.3f]'%((split['index']+1),split['value']))
+
White box, easy to interpret model. Some people believe that decision trees more closely mirror human decision-making than do the regression and classification approaches discussed earlier (think of support vector machines)
+
Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!
+
No feature normalization needed
+
Tree models can handle both continuous and categorical data (Classification and Regression Trees)
+
Can model nonlinear relationships
+
Can model interactions between the different descriptive features
+
Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)
Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches
+
If continuous features are used the tree may become quite large and hence less interpretable
+
Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented
+
Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests
+
Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones.
+
If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data
+
Features with many levels may be preferred over features with less levels since for them it is more easy to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain
+
+
However, by aggregating many decision trees, using methods like
+bagging, random forests, and boosting, the predictive performance of
+trees can be substantially improved.
+
+
+
Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods¶
+
As stated above and seen in many of the examples discussed here about
+a single decision tree, we often end up overfitting our training
+data. This normally means that we have a high variance. Can we reduce
+the variance of a statistical learning method?
+
This leads us to a set of different methods that can combine different
+machine learning algorithms or just use one of them to construct
+forests and jungles of trees, homogeneous ones or heterogenous
+ones. These methods are recognized by different names which we will
+try to explain here. These are
+
+
Voting classifiers
+
Bagging and Pasting
+
Random forests
+
Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)
The idea behind boosting, and voting as well can be phrased as follows:
+Can a group of people somehow arrive at highly
+reasoned decisions, despite the weak judgement of the individual
+members?
+
The aim is to create a good classifier by combining several weak classifiers.
+A weak classifier is a classifier which is able to produce results that are only slightly better than guessing at random.
+
The basic approach is to apply repeatedly (in boosting this is done in an iterative way) a weak classifier to modifications of the data.
+In voting we simply apply the law of large numbers while in boosting we give more weight to misclassified data in
+each iteration.
+
Decision trees play an important role as our weak classifier. They serve as the basic method.
The simplest case is a so-called voting ensemble. To illustrate this,
+think of yourself tossing coins with a biased outcome of 51 per cent
+for heads and 49% for tails. With only few tosses,
+you may not clearly see this distribution for heads and tails. However, after some
+thousands of tosses, there will be a clear majority of heads. With 2000 tosses
+you should see approximately 1020 heads and 980 tails.
+
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.
+
Another example would be to collect all polls before an
+election. Different polls may show different likelihoods for a
+candidate winning with say a majority of the popular vote. The majority vote
+would then consist in many polls indicating that this candidate will
+actually win.
+
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.
We can use the voting classifier on other data sets, here the exciting binary case of two distinct objects using the make moons functionality of Scikit-Learn.
The plain decision trees suffer from high
+variance. This means that if we split the training data into two parts
+at random, and fit a decision tree to both halves, the results that we
+get could be quite different. In contrast, a procedure with low
+variance will yield similar results if applied repeatedly to distinct
+data sets; linear regression tends to have low variance, if the ratio
+of \(n\) to \(p\) is moderately large.
+
Bootstrap aggregation, or just bagging, is a
+general-purpose procedure for reducing the variance of a statistical
+learning method.
Bagging typically results in improved accuracy
+over prediction using a single tree. Unfortunately, however, it can be
+difficult to interpret the resulting model. Recall that one of the
+advantages of decision trees is the attractive and easily interpreted
+diagram that results.
+
However, when we bag a large number of trees, it is no longer
+possible to represent the resulting statistical learning procedure
+using a single tree, and it is no longer clear which variables are
+most important to the procedure. Thus, bagging improves prediction
+accuracy at the expense of interpretability. Although the collection
+of bagged trees is much more difficult to interpret than a single
+tree, one can obtain an overall summary of the importance of each
+predictor using the MSE (for bagging regression trees) or the Gini
+index (for bagging classification trees). In the case of bagging
+regression trees, we can record the total amount that the MSE is
+decreased due to splits over a given predictor, averaged over all \(B\) possible
+trees. A large value indicates an important predictor. Similarly, in
+the context of bagging classification trees, we can add up the total
+amount that the Gini index is decreased by splits over a given
+predictor, averaged over all \(B\) trees.
+
+
+
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 \(n\)).
Random forests provide an improvement over bagged trees by way of a
+small tweak that decorrelates the trees.
+
As in bagging, we build a
+number of decision trees on bootstrapped training samples. But when
+building these decision trees, each time a split in a tree is
+considered, a random sample of \(m\) predictors is chosen as split
+candidates from the full set of \(p\) predictors. The split is allowed to
+use only one of those \(m\) predictors.
+
A fresh sample of \(m\) predictors is
+taken at each split, and typically we choose
+
+\[
+m\approx \sqrt{p}.
+\]
+
In building a random forest, at
+each split in the tree, the algorithm is not even allowed to consider
+a majority of the available predictors.
+
The reason for this is rather clever. Suppose that there is one very
+strong predictor in the data set, along with a number of other
+moderately strong predictors. Then in the collection of bagged
+variable importance random forest trees, most or all of the trees will
+use this strong predictor in the top split. Consequently, all of the
+bagged trees will look quite similar to each other. Hence the
+predictions from the bagged trees will be highly correlated.
+Unfortunately, averaging many highly correlated quantities does not
+lead to as large of a reduction in variance as averaging many
+uncorrelated quantities. In particular, this means that bagging will
+not lead to a substantial reduction in variance over a single tree in
+this setting.
The algorithm described here can be applied to both classification and regression problems.
+
We will grow of forest of say \(B\) trees.
+
+
For \(b=1:B\)
+
+
+
Draw a bootstrap sample from the training data organized in our \(\boldsymbol{X}\) matrix.
+
We grow then a random forest tree \(T_b\) based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached
+
+
+
we select \(m \le p\) variables at random from the \(p\) predictors/features
+
pick the best split point among the \(m\) features using for example the CART algorithm and create a new node
+
split the node into daughter nodes
+
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.
+
+
+
+
Random Forests Compared with other Methods on the Cancer Data¶
+
+
+
importmatplotlib.pyplotasplt
+importnumpyasnp
+fromsklearn.model_selectionimporttrain_test_split
+fromsklearn.datasetsimportload_breast_cancer
+fromsklearn.svmimportSVC
+fromsklearn.linear_modelimportLogisticRegression
+fromsklearn.treeimportDecisionTreeClassifier
+fromsklearn.ensembleimportBaggingClassifier
+
+# 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)
+#define methods
+# Logistic Regression
+logreg=LogisticRegression(solver='lbfgs')
+# Support vector machine
+svm=SVC(gamma='auto',C=100)
+# Decision Trees
+deep_tree_clf=DecisionTreeClassifier(max_depth=None)
+#Scale the data
+fromsklearn.preprocessingimportStandardScaler
+scaler=StandardScaler()
+scaler.fit(X_train)
+X_train_scaled=scaler.transform(X_train)
+X_test_scaled=scaler.transform(X_test)
+# Logistic Regression
+logreg.fit(X_train_scaled,y_train)
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+# Support Vector Machine
+svm.fit(X_train_scaled,y_train)
+print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+# Decision Trees
+deep_tree_clf.fit(X_train_scaled,y_train)
+print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
+
+
+fromsklearn.ensembleimportRandomForestClassifier
+fromsklearn.preprocessingimportLabelEncoder
+fromsklearn.model_selectionimportcross_validate
+# Data set not specificied
+#Instantiate the model with 500 trees and entropy as splitting criteria
+Random_Forest_model=RandomForestClassifier(n_estimators=500,criterion="entropy")
+Random_Forest_model.fit(X_train_scaled,y_train)
+#Cross validation
+accuracy=cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
+
+
+importscikitplotasskplt
+y_pred=Random_Forest_model.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test,y_pred,normalize=True)
+plt.show()
+y_probas=Random_Forest_model.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test,y_probas)
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test,y_probas)
+plt.show()
+
+
+
+
+
Recall that the cumulative gains curve shows the percentage of the
+overall number of cases in a given category gained by targeting a
+percentage of the total number of cases.
+
Similarly, the receiver operating characteristic curve, or ROC curve,
+displays the diagnostic ability of a binary classifier system as its
+discrimination threshold is varied. It plots the true positive rate against the false positive rate.
The basic idea is to combine weak classifiers in order to create a good
+classifier. With a weak classifier we often intend a classifier which
+produces results which are only slightly better than we would get by
+random guesses.
+
This is done by applying in an iterative way a weak (or a standard
+classifier like decision trees) to modify the data. In each iteration
+we emphasize those observations which are misclassified by weighting
+them with a factor.
+
+
+
What is boosting? Additive Modelling/Iterative Fitting¶
+
Boosting is a way of fitting an additive expansion in a set of
+elementary basis functions like for example some simple polynomials.
+Assume for example that we have a function
where \(\beta_m\) are the expansion parameters to be determined in a
+minimization process and \(b(x;\gamma_m)\) are some simple functions of
+the multivariable parameter \(x\) which is characterized by the
+parameters \(\gamma_m\).
+
As an example, consider the Sigmoid function we used in logistic
+regression. In that case, we can translate the function
+\(b(x;\gamma_m)\) into the Sigmoid function
+
+\[
+\sigma(t) = \frac{1}{1+\exp{(-t)}},
+\]
+
where \(t=\gamma_0+\gamma_1 x\) and the parameters \(\gamma_0\) and
+\(\gamma_1\) were determined by the Logistic Regression fitting
+algorithm.
+
As another example, consider the cost function we defined for linear regression
In this case the function \(f(x)\) was replaced by the design matrix
+\(\boldsymbol{X}\) and the unknown linear regression parameters \(\boldsymbol{\beta}\),
+that is \(\boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta}\). In linear regression we can
+simply invert a matrix and obtain the parameters \(\beta\) by
In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \(\beta_m\) and \(\gamma_m\).
+
+
+
Iterative Fitting, Regression and Squared-error Cost Function¶
+
The way we proceed is as follows (here we specialize to the squared-error cost function)
+
+
Establish a cost function, here \(C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2\) with \(f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)\).
+
Initialize with a guess \(f_0(x)\). It could be one or even zero or some random numbers.
+
For \(m=1:M\)
+
+
a. minimize \(\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2\) wrt \(\gamma\) and \(\beta\)
+
b. This gives the optimal values \(\beta_m\) and \(\gamma_m\)
+
c. Determine then the new values \(f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)\)
+
We could use any of the algorithms we have discussed till now. If we
+use trees, \(\gamma\) parameterizes the split variables and split points
+at the internal nodes, and the predictions at the terminal nodes.
We can then rewrite these equations as (defining \(\boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x})\) with \(\boldsymbol{e}\) being the unit vector)
which leads to \(\gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x})\). Inserting
+for \(\beta\) gives us an equation for \(\gamma\). This is a non-linear equation in the unknown \(\gamma\) and has to be solved numerically.
+
The solution to these two equations gives us in turn \(\beta_1\) and \(\gamma_1\) leading to the new expression for \(f_1(x)\) as
+\(f_1(x) = \beta_1(1+\gamma_1x)\). Doing this \(M\) times results in our final estimate for the function \(f\).
Let us consider a binary classification problem with two outcomes \(y_i \in \{-1,1\}\) and \(i=0,1,2,\dots,n-1\) as our set of
+observations. We define a classification function \(G(x)\) which produces a prediction taking one or the other of the two values
+\(\{-1,1\}\).
The iterative procedure starts with defining a weak classifier whose
+error rate is barely better than random guessing. The iterative
+procedure in boosting is to sequentially apply a weak
+classification algorithm to repeatedly modified versions of the data
+producing a sequence of weak classifiers \(G_m(x)\).
+
Here we will express our function \(f(x)\) in terms of \(G(x)\). That is
The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
+exponential cost/loss function defined as
We optimize \(\beta\) and \(G\) for each value of \(m=1:M\) as we did in the regression case.
+This is normally done in two steps. Let us however first rewrite the cost function as
The algorithm here is rather straightforward. Assume that our weak
+classifier is a decision tree and we consider a binary set of outputs
+with \(y_i \in \{-1,1\}\) and \(i=0,1,2,\dots,n-1\) as our set of
+observations. Our design matrix is given in terms of the
+feature/predictor vectors
+\(\boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}]\). Finally, we define also a
+classifier determined by our data via a function \(G(x)\). This function tells us how well we are able to classify our outputs/targets \(\boldsymbol{y}\).
+
We have already defined the misclassification error \(\mathrm{err}\) as
With the above definitions we are now ready to set up the algorithm for AdaBoost.
+The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
+
+
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 \(\sum_{i=0}^{n-1}w_i = 1\).
Then we start looping over all attempts at classifying, namely we start an iterative process for \(m=1:M\), where \(M\) is the final number of classifications. Our given classifier could for example be a plain decision tree.
+
+
a. Fit then a given classifier to the training set using the weights \(w_i\).
+
b. Compute then \(\mathrm{err}\) and figure out which events are classified properly and which are classified wrongly.
+
c. Define a quantity \(\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}\)
+
d. Set the new weights to \(w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}\).
+
+
Compute the new classifier \(G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)\).
+
+
For the iterations with \(m \le 2\) the weights are modified
+individually at each steps. The observations which were misclassified
+at iteration \(m-1\) have a weight which is larger than those which were
+classified properly. As this proceeds, the observations which were
+difficult to classifiy correctly are given a larger influence. Each
+new classification step \(m\) is then forced to concentrate on those
+observations that are missed in the previous iterations.
+
+
+
+
+
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/Project3.ipynb b/doc/LectureNotes/_build/jupyter_execute/Project3.ipynb
new file mode 100644
index 000000000..7fec35ceb
--- /dev/null
+++ b/doc/LectureNotes/_build/jupyter_execute/Project3.ipynb
@@ -0,0 +1,516 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "5831c36a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d774c8b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "# Project 3 on Machine Learning, deadline December 18 (midnight), 2023\n",
+ "**[Data Analysis and Machine Learning FYS-STK3155/FYS4155](http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html)**, Department of Physics, University of Oslo, Norway\n",
+ "\n",
+ "Date: **Nov 12, 2023**\n",
+ "\n",
+ "Copyright 1999-2023, [Data Analysis and Machine Learning FYS-STK3155/FYS4155](http://www.uio.no/studier/emner/matnat/fys/FYS3155/index-eng.html). Released under CC Attribution-NonCommercial 4.0 license"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5cbde03b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "# Paths for project 3"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5170403b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Defining the data sets to analyze yourself\n",
+ "\n",
+ "For project 3, you can propose own data sets that relate to your research interests or just use existing data sets from say\n",
+ "1. [Kaggle](https://www.kaggle.com/datasets) \n",
+ "\n",
+ "2. The [University of California at Irvine (UCI) with its machine learning repository](https://archive.ics.uci.edu/ml/index.php).\n",
+ "\n",
+ "3. Or other sources.\n",
+ "\n",
+ "The approach to the analysis of these new data sets should follow to a large extent what you did in projects 1 and 2. That is:\n",
+ "1. Whether you end up with a regression or a classification problem, you should employ at least two of the methods we have discussed among **linear regression (including Ridge and Lasso)**, **Logistic Regression**, **Neural Networks**, **Convolution Neural Networks**, **Recurrent Neural Networks**, and **Decision Trees, Random Forests, Bagging and Boosting**. You could for example explore all of the approaches from decision trees, via bagging and voting classifiers, to random forests, boosting and finally XGboost. If you wish to venture into **convolutional neural networks** or **recurrent neural networks**, or extensions of neural networkds, feel free to do so. You can also study unsupervised methods, although we in this course have mainly paid attention to supervised learning. \n",
+ "\n",
+ "For Boosting, feel also free to write your own codes.\n",
+ "\n",
+ "1. For project 3, you should feel free to use your own codes from projects 1 and 2, eventually write your own for Decision trees/random forests/bagging/boosting' or use the available functionality of **Scikit-Learn**, **Tensorflow**, etc. \n",
+ "\n",
+ "2. The estimates you used and tested in projects 1 and 2 should also be included, that is the $R2$-score, **MSE**, confusion matrix, accuracy score, information gain, ROC and Cumulative gains curves and other, cross-validation and/or bootstrap if these are relevant.\n",
+ "\n",
+ "3. Similarly, feel free to explore various activations functions in deep learning and various approachs to stochastic gradient descent approaches.\n",
+ "\n",
+ "4. If possible, you should link the data sets with exisiting research and analyses thereof. Scientific articles which have used Machine Learning algorithms to analyze the data are highly welcome. Perhaps you can improve previous analyses and even publish a new article? \n",
+ "\n",
+ "5. A critical assessment of the methods with ditto perspectives and recommendations is also something you need to include.\n",
+ "\n",
+ "All in all, the report should follow the same pattern as the two previous ones, with abstract, introduction, methods, code, results, conclusions etc..\n",
+ "\n",
+ "We propose also an alternative to the above. This is a project on using machine learning methods (neural networks mainly) to the solution of ordinary differential equations and partial differential equations, with a final twist on how to diagonalize a symmetric matrix with neural networks.\n",
+ "\n",
+ "This is a field with a large interest recently, spanning from studies of turbulence in fluid mechanics and meteorology to the solution of quantum mechanical systems. As reading background you can use the slides [from week 43](https://compphysics.github.io/MachineLearning/doc/pub/week43/html/week42.html) and/or the textbook by [Yadav et al](https://www.springer.com/gp/book/9789401798150)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e9fc8e8e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## The basic structure of your project\n",
+ "\n",
+ "Here follows a set up on how to structure your report and analyze the data you have opted for."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ce8b52a3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part a)\n",
+ "\n",
+ "The first part deals with structuring and reading the data, much along the same lines as done in projects 1 and 2. Explain how the data are produced and place them in a proper context."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "eb943f7a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part b)\n",
+ "\n",
+ "You need to include at least two central algorithms, or as an alternative explore methods from decisions tree to bagging, random forests and boosting. Explain the basics of the methods you have chosen to work with. This would be your theory part."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "17447dcc",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part c)\n",
+ "\n",
+ "Then describe your algorithm and its implementation and tests you have performed."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d9bea856",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part d)\n",
+ "\n",
+ "Then presents your results and findings, link with existing literature and more."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5b74282b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part e)\n",
+ "\n",
+ "Finally, here you should present a critical assessment of the methods you have studied and link your results with the existing literature."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "28f082d7",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Solving partial differential equations with neural networks\n",
+ "\n",
+ "For this variant of project 3, we will assume that you have some\n",
+ "background in the solution of partial differential equations using\n",
+ "finite difference schemes. We will study the solution of the diffusion\n",
+ "equation in one dimension using a standard explicit scheme and neural\n",
+ "networks to solve the same equations.\n",
+ "\n",
+ "For the explicit scheme, you can study for example chapter 10 of the lecture notes in [Computational Physics](https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Lectures/lectures2015.pdf) or alternative sources. For the solution of ordinary and partial differential equations using neural networks, the lectures by [Kristine Baluka Hein and included in the lectures of week 43](https://compphysics.github.io/MachineLearning/doc/pub/week42/html/week43.html) at this course are highly recommended.\n",
+ "\n",
+ "For the machine learning part you can use your own code from project 2 or the functionality of for example **Tensorflow/Keras**.."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5e855e7f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part a), setting up the problem\n",
+ "\n",
+ "The physical problem can be that of the temperature gradient in a rod of length $L=1$ at $x=0$ and $x=1$.\n",
+ "We are looking at a one-dimensional\n",
+ "problem"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bc4be75f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\frac{\\partial^2 u(x,t)}{\\partial x^2} =\\frac{\\partial u(x,t)}{\\partial t}, t> 0, x\\in [0,L]\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "caa1eca4",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "or"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "98a2bd5b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx} = u_t,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "980b949d",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "with initial conditions, i.e., the conditions at $t=0$,"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7401e9ec",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(x,0)= \\sin{(\\pi x)} \\hspace{0.5cm} 0 < x < L,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e953662e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "with $L=1$ the length of the $x$-region of interest. The \n",
+ "boundary conditions are"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "52122eb7",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(0,t)= 0 \\hspace{0.5cm} t \\ge 0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ab71a383",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "and"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4f886681",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u(L,t)= 0 \\hspace{0.5cm} t \\ge 0.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "83f8d12e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "The function $u(x,t)$ can be the temperature gradient of a rod.\n",
+ "As time increases, the velocity approaches a linear variation with $x$. \n",
+ "\n",
+ "We will limit ourselves to the so-called explicit forward Euler algorithm with discretized versions of time given by a forward formula and a centered difference in space resulting in"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4bcf494f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_t\\approx \\frac{u(x,t+\\Delta t)-u(x,t)}{\\Delta t}=\\frac{u(x_i,t_j+\\Delta t)-u(x_i,t_j)}{\\Delta t}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7f95982b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "and"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1ee81ac0",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx}\\approx \\frac{u(x+\\Delta x,t)-2u(x,t)+u(x-\\Delta x,t)}{\\Delta x^2},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bae7898a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "or"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5de23ea9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "u_{xx}\\approx \\frac{u(x_i+\\Delta x,t_j)-2u(x_i,t_j)+u(x_i-\\Delta x,t_j)}{\\Delta x^2}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4d5d56e3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "Write down the algorithm and the equations you need to implement.\n",
+ "Find also the analytical solution to the problem."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e52523ef",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part b)\n",
+ "\n",
+ "Implement the explicit scheme algorithm and perform tests of the solution \n",
+ "for $\\Delta x=1/10$, $\\Delta x=1/100$ using $\\Delta t$ as dictated by the stability limit of the explicit scheme. The stability criterion for the explicit scheme requires that $\\Delta t/\\Delta x^2 \\leq 1/2$. \n",
+ "\n",
+ "Study the solutions at two time points $t_1$ and $t_2$ where $u(x,t_1)$ is smooth but still significantly curved\n",
+ "and $u(x,t_2)$ is almost linear, close to the stationary state."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "678607d0",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part c) Neural networks\n",
+ "\n",
+ "Study now the lecture notes on solving ODEs and PDEs with neural\n",
+ "network and use either your own code from project 2 or the\n",
+ "functionality of tensorflow/keras to solve the same equation as in\n",
+ "part b). Discuss your results and compare them with the standard\n",
+ "explicit scheme. Include also the analytical solution and compare with\n",
+ "that."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "61d51d93",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "### Part d)\n",
+ "\n",
+ "Finally, present a critical assessment of the methods you have studied\n",
+ "and discuss the potential for the solving differential equations and\n",
+ "eigenvalue problems with machine learning methods."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d2163034",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Introduction to numerical projects\n",
+ "\n",
+ "Here follows a brief recipe and recommendation on how to write a report for each\n",
+ "project.\n",
+ "\n",
+ " * Give a short description of the nature of the problem and the eventual numerical methods you have used.\n",
+ "\n",
+ " * Describe the algorithm you have used and/or developed. Here you may find it convenient to use pseudocoding. In many cases you can describe the algorithm in the program itself.\n",
+ "\n",
+ " * Include the source code of your program. Comment your program properly.\n",
+ "\n",
+ " * If possible, try to find analytic solutions, or known limits in order to test your program when developing the code.\n",
+ "\n",
+ " * Include your results either in figure form or in a table. Remember to label your results. All tables and figures should have relevant captions and labels on the axes.\n",
+ "\n",
+ " * Try to evaluate the reliabilty and numerical stability/precision of your results. If possible, include a qualitative and/or quantitative discussion of the numerical stability, eventual loss of precision etc.\n",
+ "\n",
+ " * Try to give an interpretation of you results in your answers to the problems.\n",
+ "\n",
+ " * Critique: if possible include your comments and reflections about the exercise, whether you felt you learnt something, ideas for improvements and other thoughts you've made when solving the exercise. We wish to keep this course at the interactive level and your comments can help us improve it.\n",
+ "\n",
+ " * Try to establish a practice where you log your work at the computerlab. You may find such a logbook very handy at later stages in your work, especially when you don't properly remember what a previous test version of your program did. Here you could also record the time spent on solving the exercise, various algorithms you may have tested or other topics which you feel worthy of mentioning."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e31290fd",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Format for electronic delivery of report and programs\n",
+ "\n",
+ "The preferred format for the report is a PDF file. You can also use DOC or postscript formats or as an ipython notebook file. As programming language we prefer that you choose between C/C++, Fortran2008 or Python. The following prescription should be followed when preparing the report:\n",
+ "\n",
+ " * Use Canvas to hand in your projects, log in at with your normal UiO username and password.\n",
+ "\n",
+ " * Upload **only** the report file or the link to your GitHub/GitLab or similar typo of repos! For the source code file(s) you have developed please provide us with your link to your GitHub/GitLab or similar domain. The report file should include all of your discussions and a list of the codes you have developed. Do not include library files which are available at the course homepage, unless you have made specific changes to them.\n",
+ "\n",
+ " * In your GitHub/GitLab or similar repository, please include a folder which contains selected results. These can be in the form of output from your code for a selected set of runs and input parameters.\n",
+ "\n",
+ "Finally, \n",
+ "we encourage you to collaborate. Optimal working groups consist of \n",
+ "2-3 students. You can then hand in a common report."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "99e78b73",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Software and needed installations\n",
+ "\n",
+ "If you have Python installed (we recommend Python3) and you feel pretty familiar with installing different packages, \n",
+ "we recommend that you install the following Python packages via **pip** as\n",
+ "1. pip install numpy scipy matplotlib ipython scikit-learn tensorflow sympy pandas pillow\n",
+ "\n",
+ "For Python3, replace **pip** with **pip3**.\n",
+ "\n",
+ "See below for a discussion of **tensorflow** and **scikit-learn**. \n",
+ "\n",
+ "For OSX users we recommend also, after having installed Xcode, to install **brew**. Brew allows \n",
+ "for a seamless installation of additional software via for example\n",
+ "1. brew install python3\n",
+ "\n",
+ "For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution\n",
+ "you can use **pip** as well and simply install Python as \n",
+ "1. sudo apt-get install python3 (or python for python2.7)\n",
+ "\n",
+ "etc etc. \n",
+ "\n",
+ "If you don't want to install various Python packages with their dependencies separately, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely\n",
+ "1. [Anaconda](https://docs.anaconda.com/) Anaconda is an open source distribution of the Python and R programming languages for large-scale data processing, predictive analytics, and scientific computing, that aims to simplify package management and deployment. Package versions are managed by the package management system **conda**\n",
+ "\n",
+ "2. [Enthought canopy](https://www.enthought.com/product/canopy/) is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.\n",
+ "\n",
+ "Popular software packages written in Python for ML are\n",
+ "\n",
+ "* [Scikit-learn](http://scikit-learn.org/stable/), \n",
+ "\n",
+ "* [Tensorflow](https://www.tensorflow.org/),\n",
+ "\n",
+ "* [PyTorch](http://pytorch.org/) and \n",
+ "\n",
+ "* [Keras](https://keras.io/).\n",
+ "\n",
+ "These are all freely available at their respective GitHub sites. They \n",
+ "encompass communities of developers in the thousands or more. And the number\n",
+ "of code developers and contributors keeps increasing."
+ ]
+ }
+ ],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/Project3.txt b/doc/LectureNotes/_build/jupyter_execute/Project3.txt
new file mode 100644
index 000000000..e69de29bb
diff --git a/doc/LectureNotes/_build/jupyter_execute/week46.ipynb b/doc/LectureNotes/_build/jupyter_execute/week46.ipynb
new file mode 100644
index 000000000..29170b0e2
--- /dev/null
+++ b/doc/LectureNotes/_build/jupyter_execute/week46.ipynb
@@ -0,0 +1,3397 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "3ae913f7",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bde4c6f5",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "# Week 46: Decision Trees, Ensemble methods and Random Forests\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: **Week 46, November 13-17**"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9e846d96",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Plan for week 46\n",
+ "\n",
+ "**Active learning sessions on Tuesday and Wednesday.**\n",
+ "\n",
+ " * Work and Discussion of project 2\n",
+ "\n",
+ " * Discussion of project 3 as well\n",
+ "\n",
+ " \n",
+ "\n",
+ "**Material for the lecture on Thursday November 16, 2023.**\n",
+ "\n",
+ " * Thursday: Basics of decision trees, classification and regression algorithms and ensemble models \n",
+ "\n",
+ " * Readings and Videos:\n",
+ "\n",
+ " * These lecture notes\n",
+ "\n",
+ " * [Video of lecture to be added](https://youtu.be/)\n",
+ "\n",
+ " * [Video on Decision trees](https://www.youtube.com/watch?v=RmajweUFKvM&ab_channel=Simplilearn)\n",
+ "\n",
+ " * Decision Trees: 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b181bac8",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Decision trees, overarching aims\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",
+ "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**."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5c065b15",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0685a1f3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## A Sketch of a Tree, Regression problem\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9deb4609",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## A Sketch of a Tree, Classification problem\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "971fbe6d",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## A typical Decision Tree with its pertinent Jargon, Classification Problem\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "
Figure 1:
\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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3c3fee0c",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5d7a417e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## How do we set it up?\n",
+ "\n",
+ "In simplified terms, the process of training a decision tree and\n",
+ "predicting the target features of query instances is as follows:\n",
+ "\n",
+ "1. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature\n",
+ "\n",
+ "2. Train the decision tree model by continuously splitting the target feature along the values of the descriptive features using a measure of information gain during the training process\n",
+ "\n",
+ "3. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the *predictions* we want to make for new query instances\n",
+ "\n",
+ "4. Show query instances to the tree and run down the tree until we arrive at leaf nodes\n",
+ "\n",
+ "Then we are essentially done!"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "98739ae8",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Decision trees and Regression"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "2e75ab9e",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "2nd degree coefficients:\n",
+ "zero power: 4.888883015934703\n",
+ "first power: -0.10815559091341771\n",
+ "second power: 0.0005603761549585715\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "
"
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week46_11_2.png"
+ }
+ },
+ "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"
+ ]
+ },
+ "metadata": {
+ "filenames": {
+ "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/week46_43_1.png"
+ }
+ },
+ "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",
+ "id": "c1b460bb",
+ "metadata": {
+ "editable": true
+ },
+ "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": 5,
+ "id": "6f103d31",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "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",
+ "id": "573c8314",
+ "metadata": {
+ "editable": true
+ },
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "090434b9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "c9a39ce1",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "2d631ba5",
+ "metadata": {
+ "editable": true
+ },
+ "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 $max\\_leaf\\_nodes$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ffd719b4",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "90f1d88b",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "92376037",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "Here the MSE for a specific node is defined as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "24d1cf8d",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "a4216389",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "with"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ac393440",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\overline{y}_{\\mathrm{node}}=\\frac{1}{m_\\mathrm{node}}\\sum_{i\\in \\mathrm{node}}y_i,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d251764b",
+ "metadata": {
+ "editable": true
+ },
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1ab6171a",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Why binary splits?\n",
+ "\n",
+ "It is custom to split to a tree uising binary splits. The reason is\n",
+ "that multiway splits fragment the data too quickly, leaving\n",
+ "insufficient data at the next level down. Multiway splits can be\n",
+ "achieved by a series of binary split and this is normally preferred."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d0e81ec2",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Computing a Tree using the Gini Index\n",
+ "\n",
+ "Consider the following example with attributes/features and two\n",
+ "possible outcomes (classes) for each attribute. Assume we wish to find some\n",
+ "correlations between the average grade of a student as function of the\n",
+ "number of hours studied and hours slept. We want also to correlate the\n",
+ "grade in a given course with the general trend, whether the students\n",
+ "recently has gotten grades below average or above.\n",
+ "\n",
+ "We have three features/attributes\n",
+ "1. Trend of average grades before present course, classified as either below or above the average grade of the whole class \n",
+ "\n",
+ "2. The number of hours studies, classified again as either higher (more than 3 hours per day) or lower . Here we have used a standard for one $ECTS$ which is scaled to 25-30 hours of work for a semester which lasts 18 weeks, with 15 weeks of lectures and 3 weeks for exams, assuming a total of 30 ECTS per semester. \n",
+ "\n",
+ "3. The number of hours slept as high for more than $8$ hours and below for less than 8 hours of sleep, classified again as either high or low\n",
+ "\n",
+ "4. The final grade whether it is above or below average"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c2c41db9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## The Table\n",
+ "\n",
+ "
\n",
+ "\n",
+ "
Grade Trend
Hours slept
Hours Studied
Grade
\n",
+ "\n",
+ "\n",
+ "
Above
Low
High
Above
\n",
+ "
Below
High
Low
Below
\n",
+ "
Above
Low
High
Above
\n",
+ "
Above
High
High
Above
\n",
+ "
Below
Low
High
Below
\n",
+ "
Above
Low
Low
Below
\n",
+ "
Below
High
High
Below
\n",
+ "
Below
Low
High
Below
\n",
+ "
Above
Low
Low
Below
\n",
+ "
Above
High
High
Above
\n",
+ "\n",
+ "
"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8ba76cfd",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Computing the various Gini Indices\n",
+ "\n",
+ "In computations we will translate all classes into numbers. Being\n",
+ "these binary classes, they can easily be split into ones and zeros.\n",
+ "\n",
+ "**Gini index for Average trend.**\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a7a4d204",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Computing the various Gini Indices, Hours slept\n",
+ "\n",
+ "**Gini index for hour slept.**\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "51de5c69",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Computing the various Gini Indices, Hours studied\n",
+ "\n",
+ "**Gini index for hour studied.**\n",
+ "\n",
+ "[See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)\n",
+ "\n",
+ "For final tree, see the above handwritten notes"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9f4bf856",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## A possible code using Scikit-Learn"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "id": "020402b2",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [
+ {
+ "ename": "FileNotFoundError",
+ "evalue": "[Errno 2] No such file or directory: 'DataFiles/grades.csv'",
+ "output_type": "error",
+ "traceback": [
+ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+ "\u001b[0;31mFileNotFoundError\u001b[0m Traceback (most recent call last)",
+ "Input \u001b[0;32mIn [6]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 34\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21msave_fig\u001b[39m(fig_id):\n\u001b[1;32m 35\u001b[0m plt\u001b[38;5;241m.\u001b[39msavefig(image_path(fig_id) \u001b[38;5;241m+\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m.png\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;28mformat\u001b[39m\u001b[38;5;241m=\u001b[39m\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mpng\u001b[39m\u001b[38;5;124m'\u001b[39m)\n\u001b[0;32m---> 37\u001b[0m infile \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mopen\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43mdata_path\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mgrades.csv\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mr\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m)\u001b[49m\n\u001b[1;32m 39\u001b[0m \u001b[38;5;66;03m# Read the experimental data with Pandas\u001b[39;00m\n\u001b[1;32m 40\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01mIPython\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mdisplay\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m display\n",
+ "\u001b[0;31mFileNotFoundError\u001b[0m: [Errno 2] No such file or directory: 'DataFiles/grades.csv'"
+ ]
+ }
+ ],
+ "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(\"grades.csv\"),'r')\n",
+ "\n",
+ "# Read the experimental data with Pandas\n",
+ "from IPython.display import display\n",
+ "grades = pd.read_csv(infile)\n",
+ "grades = pd.DataFrame(grades)\n",
+ "display(grades)\n",
+ "# Features and targets\n",
+ "X = grades.loc[:, grades.columns != 'Grade'].values\n",
+ "y = grades.loc[:, grades.columns == 'Grade'].values\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/grade.dot\",\n",
+ " rounded=True,\n",
+ " filled=True\n",
+ ")\n",
+ "cmd = 'dot -Tpng DataFiles/grade.dot -o DataFiles/grades.png'\n",
+ "os.system(cmd)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "082c7f06",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Further example: Computing the Gini index\n",
+ "\n",
+ "The next 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",
+ "
Day
Outlook
Temperature
Humidity
Wind
Ride
\n",
+ "\n",
+ "\n",
+ "
1
Sunny
Hot
High
Weak
0
\n",
+ "
2
Sunny
Hot
High
Strong
1
\n",
+ "
3
Overcast
Hot
High
Weak
1
\n",
+ "
4
Rain
Mild
High
Weak
1
\n",
+ "
5
Rain
Cool
Normal
Weak
1
\n",
+ "
6
Rain
Cool
Normal
Strong
0
\n",
+ "
7
Overcast
Cool
Normal
Strong
1
\n",
+ "
8
Sunny
Mild
High
Weak
0
\n",
+ "
9
Sunny
Cool
Normal
Weak
1
\n",
+ "
10
Rain
Mild
Normal
Weak
1
\n",
+ "
11
Sunny
Mild
Normal
Strong
1
\n",
+ "
12
Overcast
Mild
High
Strong
1
\n",
+ "
13
Overcast
Hot
Normal
Weak
1
\n",
+ "
14
Rain
Mild
High
Strong
0
\n",
+ "\n",
+ "
"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b742d7cc",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Simple Python Code to read in Data and perform Classification"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "1f114a5a",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Common imports\n",
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "from sklearn.tree import DecisionTreeClassifier\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.tree import export_graphviz\n",
+ "from sklearn.preprocessing import StandardScaler, OneHotEncoder\n",
+ "from sklearn.compose import ColumnTransformer\n",
+ "from IPython.display import Image \n",
+ "from pydot import graph_from_dot_data\n",
+ "import os\n",
+ "\n",
+ "# Where to save the figures and data files\n",
+ "PROJECT_ROOT_DIR = \"Results\"\n",
+ "FIGURE_ID = \"Results/FigureFiles\"\n",
+ "DATA_ID = \"DataFiles/\"\n",
+ "\n",
+ "if not os.path.exists(PROJECT_ROOT_DIR):\n",
+ " os.mkdir(PROJECT_ROOT_DIR)\n",
+ "\n",
+ "if not os.path.exists(FIGURE_ID):\n",
+ " os.makedirs(FIGURE_ID)\n",
+ "\n",
+ "if not os.path.exists(DATA_ID):\n",
+ " os.makedirs(DATA_ID)\n",
+ "\n",
+ "def image_path(fig_id):\n",
+ " return os.path.join(FIGURE_ID, fig_id)\n",
+ "\n",
+ "def data_path(dat_id):\n",
+ " return os.path.join(DATA_ID, dat_id)\n",
+ "\n",
+ "def save_fig(fig_id):\n",
+ " plt.savefig(image_path(fig_id) + \".png\", format='png')\n",
+ "\n",
+ "infile = open(data_path(\"rideclass.csv\"),'r')\n",
+ "\n",
+ "# Read the experimental data with Pandas\n",
+ "from IPython.display import display\n",
+ "ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))\n",
+ "ridedata = pd.DataFrame(ridedata)\n",
+ "\n",
+ "# Features and targets\n",
+ "X = ridedata.loc[:, ridedata.columns != 'Ride'].values\n",
+ "y = ridedata.loc[:, ridedata.columns == 'Ride'].values\n",
+ "\n",
+ "# Create the encoder.\n",
+ "encoder = OneHotEncoder(handle_unknown=\"ignore\")\n",
+ "# Assume for simplicity all features are categorical.\n",
+ "encoder.fit(X) \n",
+ "# Apply the encoder.\n",
+ "X = encoder.transform(X)\n",
+ "print(X)\n",
+ "# Then do a Classification tree\n",
+ "tree_clf = DecisionTreeClassifier(max_depth=2)\n",
+ "tree_clf.fit(X, y)\n",
+ "print(\"Train set accuracy with Decision Tree: {:.2f}\".format(tree_clf.score(X,y)))\n",
+ "#transfer to a decision tree graph\n",
+ "export_graphviz(\n",
+ " tree_clf,\n",
+ " out_file=\"DataFiles/ride.dot\",\n",
+ " rounded=True,\n",
+ " filled=True\n",
+ ")\n",
+ "cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'\n",
+ "os.system(cmd)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3ae90db1",
+ "metadata": {
+ "editable": true
+ },
+ "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": 8,
+ "id": "22320738",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Split a dataset based on an attribute and an attribute value\n",
+ "def test_split(index, value, dataset):\n",
+ "\tleft, right = list(), list()\n",
+ "\tfor row in dataset:\n",
+ "\t\tif row[index] < value:\n",
+ "\t\t\tleft.append(row)\n",
+ "\t\telse:\n",
+ "\t\t\tright.append(row)\n",
+ "\treturn left, right\n",
+ " \n",
+ "# Calculate the Gini index for a split dataset\n",
+ "def gini_index(groups, classes):\n",
+ "\t# count all samples at split point\n",
+ "\tn_instances = float(sum([len(group) for group in groups]))\n",
+ "\t# sum weighted Gini index for each group\n",
+ "\tgini = 0.0\n",
+ "\tfor group in groups:\n",
+ "\t\tsize = float(len(group))\n",
+ "\t\t# avoid divide by zero\n",
+ "\t\tif size == 0:\n",
+ "\t\t\tcontinue\n",
+ "\t\tscore = 0.0\n",
+ "\t\t# score the group based on the score for each class\n",
+ "\t\tfor class_val in classes:\n",
+ "\t\t\tp = [row[-1] for row in group].count(class_val) / size\n",
+ "\t\t\tscore += p * p\n",
+ "\t\t# weight the group score by its relative size\n",
+ "\t\tgini += (1.0 - score) * (size / n_instances)\n",
+ "\treturn gini\n",
+ "\n",
+ "# Select the best split point for a dataset\n",
+ "def get_split(dataset):\n",
+ "\tclass_values = list(set(row[-1] for row in dataset))\n",
+ "\tb_index, b_value, b_score, b_groups = 999, 999, 999, None\n",
+ "\tfor index in range(len(dataset[0])-1):\n",
+ "\t\tfor row in dataset:\n",
+ "\t\t\tgroups = test_split(index, row[index], dataset)\n",
+ "\t\t\tgini = gini_index(groups, class_values)\n",
+ "\t\t\tprint('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))\n",
+ "\t\t\tif gini < b_score:\n",
+ "\t\t\t\tb_index, b_value, b_score, b_groups = index, row[index], gini, groups\n",
+ "\treturn {'index':b_index, 'value':b_value, 'groups':b_groups}\n",
+ " \n",
+ "dataset = [[0,0,0,0,0],\n",
+ " [0,0,0,1,1],\n",
+ " [1,0,0,0,1],\n",
+ " [2,1,0,0,1],\n",
+ " [2,2,1,0,1],\n",
+ " [2,2,1,1,0],\n",
+ " [1,2,1,1,1],\n",
+ " [0,1,0,0,0],\n",
+ " [0,2,1,0,1],\n",
+ " [2,1,1,0,1],\n",
+ " [0,1,1,1,1],\n",
+ " [1,1,0,1,1],\n",
+ " [1,0,1,0,1],\n",
+ " [2,1,0,1,0]]\n",
+ "\n",
+ "split = get_split(dataset)\n",
+ "print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ebd0ac9f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Regression trees"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "id": "bfdc5109",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Quadratic training set + noise\n",
+ "np.random.seed(42)\n",
+ "m = 200\n",
+ "X = np.random.rand(m, 1)\n",
+ "y = 4 * (X - 0.5) ** 2\n",
+ "y = y + np.random.randn(m, 1) / 10"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "id": "ab8bb0be",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.tree import DecisionTreeRegressor\n",
+ "\n",
+ "tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)\n",
+ "tree_reg.fit(X, y)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b102de57",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Final regressor code"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "id": "3aeec95d",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.tree import DecisionTreeRegressor\n",
+ "\n",
+ "tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)\n",
+ "tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)\n",
+ "tree_reg1.fit(X, y)\n",
+ "tree_reg2.fit(X, y)\n",
+ "\n",
+ "def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel=\"$y$\"):\n",
+ " x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)\n",
+ " y_pred = tree_reg.predict(x1)\n",
+ " plt.axis(axes)\n",
+ " plt.xlabel(\"$x_1$\", fontsize=18)\n",
+ " if ylabel:\n",
+ " plt.ylabel(ylabel, fontsize=18, rotation=0)\n",
+ " plt.plot(X, y, \"b.\")\n",
+ " plt.plot(x1, y_pred, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n",
+ "\n",
+ "plt.figure(figsize=(11, 4))\n",
+ "plt.subplot(121)\n",
+ "plot_regression_predictions(tree_reg1, X, y)\n",
+ "for split, style in ((0.1973, \"k-\"), (0.0917, \"k--\"), (0.7718, \"k--\")):\n",
+ " plt.plot([split, split], [-0.2, 1], style, linewidth=2)\n",
+ "plt.text(0.21, 0.65, \"Depth=0\", fontsize=15)\n",
+ "plt.text(0.01, 0.2, \"Depth=1\", fontsize=13)\n",
+ "plt.text(0.65, 0.8, \"Depth=1\", fontsize=13)\n",
+ "plt.legend(loc=\"upper center\", fontsize=18)\n",
+ "plt.title(\"max_depth=2\", fontsize=14)\n",
+ "\n",
+ "plt.subplot(122)\n",
+ "plot_regression_predictions(tree_reg2, X, y, ylabel=None)\n",
+ "for split, style in ((0.1973, \"k-\"), (0.0917, \"k--\"), (0.7718, \"k--\")):\n",
+ " plt.plot([split, split], [-0.2, 1], style, linewidth=2)\n",
+ "for split in (0.0458, 0.1298, 0.2873, 0.9040):\n",
+ " plt.plot([split, split], [-0.2, 1], \"k:\", linewidth=1)\n",
+ "plt.text(0.3, 0.5, \"Depth=2\", fontsize=13)\n",
+ "plt.title(\"max_depth=3\", fontsize=14)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "id": "638f8ac8",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "tree_reg1 = DecisionTreeRegressor(random_state=42)\n",
+ "tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)\n",
+ "tree_reg1.fit(X, y)\n",
+ "tree_reg2.fit(X, y)\n",
+ "\n",
+ "x1 = np.linspace(0, 1, 500).reshape(-1, 1)\n",
+ "y_pred1 = tree_reg1.predict(x1)\n",
+ "y_pred2 = tree_reg2.predict(x1)\n",
+ "\n",
+ "plt.figure(figsize=(11, 4))\n",
+ "\n",
+ "plt.subplot(121)\n",
+ "plt.plot(X, y, \"b.\")\n",
+ "plt.plot(x1, y_pred1, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n",
+ "plt.axis([0, 1, -0.2, 1.1])\n",
+ "plt.xlabel(\"$x_1$\", fontsize=18)\n",
+ "plt.ylabel(\"$y$\", fontsize=18, rotation=0)\n",
+ "plt.legend(loc=\"upper center\", fontsize=18)\n",
+ "plt.title(\"No restrictions\", fontsize=14)\n",
+ "\n",
+ "plt.subplot(122)\n",
+ "plt.plot(X, y, \"b.\")\n",
+ "plt.plot(x1, y_pred2, \"r.-\", linewidth=2, label=r\"$\\hat{y}$\")\n",
+ "plt.axis([0, 1, -0.2, 1.1])\n",
+ "plt.xlabel(\"$x_1$\", fontsize=18)\n",
+ "plt.title(\"min_samples_leaf={}\".format(tree_reg2.min_samples_leaf), fontsize=14)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "939c2f5c",
+ "metadata": {
+ "editable": true
+ },
+ "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)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "121bdc13",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f58b9924",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3369dc37",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## An Overview of Ensemble Methods\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "
Figure 1:
\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "247d8e1f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "418dee56",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Tossing coins\n",
+ "\n",
+ "The simplest case is a so-called voting ensemble. To illustrate this,\n",
+ "think of yourself tossing coins with a biased outcome of 51 per cent\n",
+ "for heads and 49% for tails. With only few tosses,\n",
+ "you may not clearly see this distribution for heads and tails. However, after some\n",
+ "thousands of tosses, there will be a clear majority of heads. With 2000 tosses\n",
+ "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\n",
+ "you do this ten thousand times, it is easy to see that there is a 97%\n",
+ "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,\n",
+ "clealry demostrating that after some tosses we see the [law of large](https://en.wikipedia.org/wiki/Law_of_large_numbers)\n",
+ "numbers kicking in."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "81c7b4f6",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Standard imports first"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "id": "c8635354",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "# Common imports\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",
+ "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')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6945cdc1",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Simple Voting Example, head or tail"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "id": "dbe486ef",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "\n",
+ "# Common imports\n",
+ "import numpy as np\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",
+ "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",
+ "id": "31814f98",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Using the Voting Classifier\n",
+ "\n",
+ "We can use the voting classifier on other data sets, here the exciting binary case of two distinct objects using the make moons functionality of **Scikit-Learn**."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "id": "20858a25",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.datasets import make_moons\n",
+ "\n",
+ "X, y = make_moons(n_samples=500, noise=0.30, random_state=42)\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)\n",
+ "\n",
+ "from sklearn.ensemble import RandomForestClassifier\n",
+ "from sklearn.ensemble import VotingClassifier\n",
+ "from sklearn.linear_model import LogisticRegression\n",
+ "from sklearn.svm import SVC\n",
+ "\n",
+ "log_clf = LogisticRegression(solver=\"liblinear\", random_state=42)\n",
+ "rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)\n",
+ "svm_clf = SVC(gamma=\"auto\", random_state=42)\n",
+ "\n",
+ "voting_clf = VotingClassifier(\n",
+ " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n",
+ " voting='hard')\n",
+ "\n",
+ "voting_clf.fit(X_train, y_train)\n",
+ "\n",
+ "from sklearn.metrics import accuracy_score\n",
+ "\n",
+ "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n",
+ " clf.fit(X_train, y_train)\n",
+ " y_pred = clf.predict(X_test)\n",
+ " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))\n",
+ "\n",
+ "log_clf = LogisticRegression(solver=\"liblinear\", random_state=42)\n",
+ "rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)\n",
+ "svm_clf = SVC(gamma=\"auto\", probability=True, random_state=42)\n",
+ "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",
+ "id": "69bde19e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Voting and Bagging"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "579fcaba",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.datasets import make_moons\n",
+ "\n",
+ "X, y = make_moons(n_samples=500, noise=0.30, random_state=42)\n",
+ "X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)\n",
+ "from sklearn.ensemble import RandomForestClassifier\n",
+ "from sklearn.ensemble import VotingClassifier\n",
+ "from sklearn.linear_model import LogisticRegression\n",
+ "from sklearn.svm import SVC\n",
+ "\n",
+ "log_clf = LogisticRegression(random_state=42)\n",
+ "rnd_clf = RandomForestClassifier(random_state=42)\n",
+ "svm_clf = SVC(random_state=42)\n",
+ "\n",
+ "voting_clf = VotingClassifier(\n",
+ " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n",
+ " voting='hard')\n",
+ "voting_clf.fit(X_train, y_train)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "id": "e19aed80",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.metrics import accuracy_score\n",
+ "\n",
+ "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n",
+ " clf.fit(X_train, y_train)\n",
+ " y_pred = clf.predict(X_test)\n",
+ " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "id": "5e7fff36",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "log_clf = LogisticRegression(random_state=42)\n",
+ "rnd_clf = RandomForestClassifier(random_state=42)\n",
+ "svm_clf = SVC(probability=True, random_state=42)\n",
+ "\n",
+ "voting_clf = VotingClassifier(\n",
+ " estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],\n",
+ " voting='soft')\n",
+ "voting_clf.fit(X_train, y_train)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "id": "12a367f0",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.metrics import accuracy_score\n",
+ "\n",
+ "for clf in (log_clf, rnd_clf, svm_clf, voting_clf):\n",
+ " clf.fit(X_train, y_train)\n",
+ " y_pred = clf.predict(X_test)\n",
+ " print(clf.__class__.__name__, accuracy_score(y_test, y_pred))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "70753458",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d37bbe68",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "498956cc",
+ "metadata": {
+ "editable": true
+ },
+ "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": 20,
+ "id": "452de3bb",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "\n",
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np\n",
+ "from sklearn.model_selection import train_test_split\n",
+ "from sklearn.pipeline import make_pipeline\n",
+ "from sklearn.utils import resample\n",
+ "from sklearn.tree import DecisionTreeRegressor\n",
+ "\n",
+ "n = 100\n",
+ "n_boostraps = 100\n",
+ "maxdepth = 8\n",
+ "\n",
+ "# Make data set.\n",
+ "x = np.linspace(-3, 3, n).reshape(-1, 1)\n",
+ "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)\n",
+ "error = np.zeros(maxdepth)\n",
+ "bias = np.zeros(maxdepth)\n",
+ "variance = np.zeros(maxdepth)\n",
+ "polydegree = np.zeros(maxdepth)\n",
+ "X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n",
+ "\n",
+ "from sklearn.preprocessing import StandardScaler\n",
+ "scaler = StandardScaler()\n",
+ "scaler.fit(X_train)\n",
+ "X_train_scaled = scaler.transform(X_train)\n",
+ "X_test_scaled = scaler.transform(X_test)\n",
+ "\n",
+ "# we produce a simple tree first as benchmark\n",
+ "simpletree = DecisionTreeRegressor(max_depth=3) \n",
+ "simpletree.fit(X_train_scaled, y_train)\n",
+ "simpleprediction = simpletree.predict(X_test_scaled)\n",
+ "for degree in range(1,maxdepth):\n",
+ " model = DecisionTreeRegressor(max_depth=degree) \n",
+ " y_pred = np.empty((y_test.shape[0], n_boostraps))\n",
+ " for i in range(n_boostraps):\n",
+ " x_, y_ = resample(X_train_scaled, y_train)\n",
+ " model.fit(x_, y_)\n",
+ " y_pred[:, i] = model.predict(X_test_scaled)#.ravel()\n",
+ "\n",
+ " polydegree[degree] = degree\n",
+ " error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )\n",
+ " bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )\n",
+ " variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )\n",
+ " print('Polynomial degree:', degree)\n",
+ " print('Error:', error[degree])\n",
+ " print('Bias^2:', bias[degree])\n",
+ " print('Var:', variance[degree])\n",
+ " print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))\n",
+ " \n",
+ "mse_simpletree= np.mean( np.mean((y_test - simpleprediction)**2))\n",
+ "print(\"Simple tree:\",mse_simpletree)\n",
+ "plt.xlim(1,maxdepth)\n",
+ "plt.plot(polydegree, error, label='MSE')\n",
+ "plt.plot(polydegree, bias, label='bias')\n",
+ "plt.plot(polydegree, variance, label='Variance')\n",
+ "plt.legend()\n",
+ "save_fig(\"baggingboot\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b7a39b9f",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "d39a6e13",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "m\\approx \\sqrt{p}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "42f8693b",
+ "metadata": {
+ "editable": true
+ },
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a25fd53e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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 from the training data organized in our $\\boldsymbol{X}$ matrix.\n",
+ "\n",
+ " * We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached\n",
+ "\n",
+ "1. we select $m \\le p$ variables at random from the $p$ predictors/features\n",
+ "\n",
+ "2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node\n",
+ "\n",
+ "3. split the node into daughter nodes\n",
+ "\n",
+ "4. Output then the ensemble of trees $\\{T_b\\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c72f80ca",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Random Forests Compared with other Methods on the Cancer Data"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "id": "9b20111b",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np\n",
+ "from sklearn.model_selection import train_test_split \n",
+ "from sklearn.datasets import load_breast_cancer\n",
+ "from sklearn.svm import SVC\n",
+ "from sklearn.linear_model import LogisticRegression\n",
+ "from sklearn.tree import DecisionTreeClassifier\n",
+ "from sklearn.ensemble import BaggingClassifier\n",
+ "\n",
+ "# Load the data\n",
+ "cancer = load_breast_cancer()\n",
+ "\n",
+ "X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)\n",
+ "print(X_train.shape)\n",
+ "print(X_test.shape)\n",
+ "#define methods\n",
+ "# Logistic Regression\n",
+ "logreg = LogisticRegression(solver='lbfgs')\n",
+ "# Support vector machine\n",
+ "svm = SVC(gamma='auto', C=100)\n",
+ "# Decision Trees\n",
+ "deep_tree_clf = DecisionTreeClassifier(max_depth=None)\n",
+ "#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",
+ "id": "a2a69156",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "Recall that the cumulative gains curve shows the percentage of the\n",
+ "overall number of cases in a given category *gained* by targeting a\n",
+ "percentage of the total number of cases.\n",
+ "\n",
+ "Similarly, the receiver operating characteristic curve, or ROC curve,\n",
+ "displays the diagnostic ability of a binary classifier system as its\n",
+ "discrimination threshold is varied. It plots the true positive rate against the false positive rate."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "935e5ca1",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Compare Bagging on Trees with Random Forests"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "id": "2b6604c5",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "bag_clf = BaggingClassifier(\n",
+ " DecisionTreeClassifier(splitter=\"random\", max_leaf_nodes=16, random_state=42),\n",
+ " n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 23,
+ "id": "b6412285",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "bag_clf.fit(X_train, y_train)\n",
+ "y_pred = bag_clf.predict(X_test)\n",
+ "from sklearn.ensemble import RandomForestClassifier\n",
+ "rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)\n",
+ "rnd_clf.fit(X_train, y_train)\n",
+ "y_pred_rf = rnd_clf.predict(X_test)\n",
+ "np.sum(y_pred == y_pred_rf) / len(y_pred)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c8adc4f4",
+ "metadata": {
+ "editable": true
+ },
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e0bd3a34",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "b93c398e",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7fc3ae93",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "e5c67b21",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\sigma(t) = \\frac{1}{1+\\exp{(-t)}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d573725",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "f8cf15d3",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "e734d8d4",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "0585b995",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\boldsymbol{\\beta}=\\left(\\boldsymbol{X}^T\\boldsymbol{X}\\right)^{-1}\\boldsymbol{X}^T\\boldsymbol{y}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b983e555",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\\beta_m$ and $\\gamma_m$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "203a7a1c",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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 $C(\\boldsymbol{y},\\boldsymbol{f}) = \\frac{1}{n} \\sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ 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",
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6d391d0c",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "3b98bb46",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "3d2a99d8",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "3af9f679",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "368371bc",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "and"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ebc23cc6",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "3285cd00",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "420c5c35",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\gamma \\boldsymbol{w}^T(\\boldsymbol{y}-\\beta\\gamma \\boldsymbol{w})=0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2a386bd3",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "which gives us $\\beta = \\boldsymbol{w}^T\\boldsymbol{y}/(\\boldsymbol{w}^T\\boldsymbol{w})$. Similarly we have"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cb75d383",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\beta\\gamma \\boldsymbol{x}^T(\\boldsymbol{y}-\\beta(1+\\gamma \\boldsymbol{x}))=0,\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d7aac8e6",
+ "metadata": {
+ "editable": true
+ },
+ "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 $f$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8211e34f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "0759bace",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\mathrm{\\overline{err}}=\\frac{1}{n} \\sum_{i=0}^{n-1} I(y_i\\ne G(x_i)).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6a08f8a3",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "ea71963f",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "f_M(x) = \\sum_{i=1}^M \\beta_m b(x;\\gamma_m),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "42c5df3b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "will be a function of"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "237180f9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "G_M(x) = \\mathrm{sign} \\sum_{i=1}^M \\alpha_m G_m(x).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2374907b",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Adaptive Boosting, AdaBoost\n",
+ "\n",
+ "In our iterative procedure we define thus"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "04048caf",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "f_m(x) = f_{m-1}(x)+\\beta_mG_m(x).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1a77afe0",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "981883bd",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "3d8d3830",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "fe3d598c",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "762ac6b8",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "where we have defined $w_i^m= \\exp{(-y_if_{m-1}(x_i))}$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "396805d8",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## Building up AdaBoost\n",
+ "\n",
+ "First, for any $\\beta > 0$, we optimize $G$ by setting"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "19ca93fe",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "5b78c347",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "742c335a",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "766b1fdd",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "which can be rewritten as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d9698851",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "8b9dc891",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "which leads to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "32e16984",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\beta_m = \\frac{1}{2}\\log{\\frac{1-\\mathrm{\\overline{err}}}{\\mathrm{\\overline{err}}}},\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "85898ea6",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "where we have redefined the error as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e9c70102",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "a00653d9",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "which leads to an update of"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e041fdeb",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "f_m(x) = f_{m-1}(x) +\\beta_m G_m(x).\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5f07c0b1",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "This leads to the new weights"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d13323ee",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "w_i^{m+1} = w_i^m \\exp{(-y_i\\beta_m G_m(x_i))}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c7237587",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "5eabf8df",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "$$\n",
+ "\\mathrm{err}=\\frac{1}{n}\\sum_{i=0}^{n-1}I(y_i\\ne G(x_i)),\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "309fe485",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "where the function $I()$ is one if we misclassify and zero if we classify correctly."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "344c2fc2",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## 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",
+ "id": "0af051ff",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "id": "ebf99e70",
+ "metadata": {
+ "editable": true
+ },
+ "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",
+ "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."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "38533606",
+ "metadata": {
+ "editable": true
+ },
+ "source": [
+ "## AdaBoost Examples\n",
+ "\n",
+ "Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "id": "58b8da05",
+ "metadata": {
+ "collapsed": false,
+ "editable": true
+ },
+ "outputs": [],
+ "source": [
+ "from sklearn.ensemble import AdaBoostClassifier\n",
+ "\n",
+ "ada_clf = AdaBoostClassifier(\n",
+ " DecisionTreeClassifier(max_depth=1), n_estimators=200,\n",
+ " algorithm=\"SAMME.R\", learning_rate=0.5, random_state=42)\n",
+ "ada_clf.fit(X_train, y_train)\n",
+ "\n",
+ "from sklearn.ensemble import AdaBoostClassifier\n",
+ "\n",
+ "ada_clf = AdaBoostClassifier(\n",
+ " DecisionTreeClassifier(max_depth=1), n_estimators=200,\n",
+ " algorithm=\"SAMME.R\", learning_rate=0.5, random_state=42)\n",
+ "ada_clf.fit(X_train_scaled, y_train)\n",
+ "y_pred = ada_clf.predict(X_test_scaled)\n",
+ "skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)\n",
+ "plt.show()\n",
+ "y_probas = ada_clf.predict_proba(X_test_scaled)\n",
+ "skplt.metrics.plot_roc(y_test, y_probas)\n",
+ "plt.show()\n",
+ "skplt.metrics.plot_cumulative_gain(y_test, y_probas)\n",
+ "plt.show()"
+ ]
+ }
+ ],
+ "metadata": {
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.9.10"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/doc/LectureNotes/_build/jupyter_execute/week46.py b/doc/LectureNotes/_build/jupyter_execute/week46.py
new file mode 100644
index 000000000..a917e985b
--- /dev/null
+++ b/doc/LectureNotes/_build/jupyter_execute/week46.py
@@ -0,0 +1,1863 @@
+#!/usr/bin/env python
+# coding: utf-8
+
+#
+#
+
+# # Week 46: Decision Trees, Ensemble methods and Random Forests
+# **Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
+#
+# Date: **Week 46, November 13-17**
+
+# ## Plan for week 46
+#
+# **Active learning sessions on Tuesday and Wednesday.**
+#
+# * Work and Discussion of project 2
+#
+# * Discussion of project 3 as well
+#
+#
+#
+# **Material for the lecture on Thursday November 16, 2023.**
+#
+# * Thursday: Basics of decision trees, classification and regression algorithms and ensemble models
+#
+# * Readings and Videos:
+#
+# * These lecture notes
+#
+# * [Video of lecture to be added](https://youtu.be/)
+#
+# * [Video on Decision trees](https://www.youtube.com/watch?v=RmajweUFKvM&ab_channel=Simplilearn)
+#
+# * Decision Trees: 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.
+
+# ## Decision trees, overarching aims
+#
+# We start here with the most basic algorithm, the so-called decision
+# tree. With this basic algorithm we can in turn build more complex
+# networks, spanning from homogeneous and heterogenous forests (bagging,
+# random forests and more) to one of the most popular supervised
+# algorithms nowadays, the extreme gradient boosting, or just
+# XGBoost. But let us start with the simplest possible ingredient.
+#
+# Decision trees are supervised learning algorithms used for both,
+# classification and regression tasks.
+#
+# The main idea of decision trees
+# is to find those descriptive features which contain the most
+# **information** regarding the target feature and then split the dataset
+# along the values of these features such that the target feature values
+# for the resulting underlying datasets are as pure as possible.
+#
+# The descriptive features which reproduce best the target/output features are normally said
+# to be the most informative ones. The process of finding the **most
+# informative** feature is done until we accomplish a stopping criteria
+# where we then finally end up in so called **leaf nodes**.
+
+# ## Basics of a tree
+#
+# A decision tree is typically divided into a **root node**, the **interior nodes**,
+# and the final **leaf nodes** or just **leaves**. These entities are then connected by so-called **branches**.
+#
+# The leaf nodes
+# contain the predictions we will make for new query instances presented
+# to our trained model. This is possible since the model has
+# learned the underlying structure of the training data and hence can,
+# given some assumptions, make predictions about the target feature value
+# (class) of unseen query instances.
+
+# ## A Sketch of a Tree, Regression problem
+#
+# [See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)
+#
+#
+
+# ## A Sketch of a Tree, Classification problem
+#
+# [See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)
+#
+
+# ## A typical Decision Tree with its pertinent Jargon, Classification Problem
+#
+#
+#
+#
+#
Figure 1:
+#
+#
+# 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.
+
+# ## General Features
+#
+# The overarching approach to decision trees is a top-down approach.
+#
+# * A leaf provides the classification of a given instance.
+#
+# * A node specifies a test of some attribute of the instance.
+#
+# * A branch corresponds to a possible values of an attribute.
+#
+# * An instance is classified by starting at the root node of the tree, testing the attribute specified by this node, then moving down the tree branch corresponding to the value of the attribute in the given example.
+#
+# This process is then repeated for the subtree rooted at the new
+# node.
+
+# ## How do we set it up?
+#
+# In simplified terms, the process of training a decision tree and
+# predicting the target features of query instances is as follows:
+#
+# 1. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature
+#
+# 2. Train the decision tree model by continuously splitting the target feature along the values of the descriptive features using a measure of information gain during the training process
+#
+# 3. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the *predictions* we want to make for new query instances
+#
+# 4. Show query instances to the tree and run down the tree until we arrive at leaf nodes
+#
+# Then we are essentially done!
+
+# ## Decision trees and Regression
+
+# In[1]:
+
+
+get_ipython().run_line_magic('matplotlib', 'inline')
+
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.linear_model import LinearRegression
+
+steps=250
+
+distance=0
+x=0
+distance_list=[]
+steps_list=[]
+while x
+#
+#
Grade Trend
Hours slept
Hours Studied
Grade
+#
+#
+#
Above
Low
High
Above
+#
Below
High
Low
Below
+#
Above
Low
High
Above
+#
Above
High
High
Above
+#
Below
Low
High
Below
+#
Above
Low
Low
Below
+#
Below
High
High
Below
+#
Below
Low
High
Below
+#
Above
Low
Low
Below
+#
Above
High
High
Above
+#
+#
+
+# ## Computing the various Gini Indices
+#
+# In computations we will translate all classes into numbers. Being
+# these binary classes, they can easily be split into ones and zeros.
+#
+# **Gini index for Average trend.**
+#
+# [See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)
+
+# ## Computing the various Gini Indices, Hours slept
+#
+# **Gini index for hour slept.**
+#
+# [See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)
+
+# ## Computing the various Gini Indices, Hours studied
+#
+# **Gini index for hour studied.**
+#
+# [See handwritten notes November 3](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf)
+#
+# For final tree, see the above handwritten notes
+
+# ## A possible code using Scikit-Learn
+
+# In[6]:
+
+
+# Common imports
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.tree import DecisionTreeClassifier
+from sklearn.model_selection import train_test_split
+from sklearn.tree import export_graphviz
+from sklearn.preprocessing import StandardScaler, OneHotEncoder
+from sklearn.compose import ColumnTransformer
+from IPython.display import Image
+from pydot import graph_from_dot_data
+import os
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+ return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+ return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+ plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("grades.csv"),'r')
+
+# Read the experimental data with Pandas
+from IPython.display import display
+grades = pd.read_csv(infile)
+grades = pd.DataFrame(grades)
+display(grades)
+# Features and targets
+X = grades.loc[:, grades.columns != 'Grade'].values
+y = grades.loc[:, grades.columns == 'Grade'].values
+print(X)
+# Then do a Classification tree
+tree_clf = DecisionTreeClassifier(max_depth=2)
+tree_clf.fit(X, y)
+print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
+#transfer to a decision tree graph
+export_graphviz(
+ tree_clf,
+ out_file="DataFiles/grade.dot",
+ rounded=True,
+ filled=True
+)
+cmd = 'dot -Tpng DataFiles/grade.dot -o DataFiles/grades.png'
+os.system(cmd)
+
+
+# ## Further example: Computing the Gini index
+#
+# The next example we will look at is a classical one in many Machine
+# Learning applications. Based on various meteorological features, we
+# have several so-called attributes which decide whether we at the end
+# will do some outdoor activity like skiing, going for a bike ride etc
+# etc. The table here contains the feautures **outlook**, **temperature**,
+# **humidity** and **wind**. The target or output is whether we ride
+# (True=1) or whether we do something else that day (False=0). The
+# attributes for each feature are then sunny, overcast and rain for the
+# outlook, hot, cold and mild for temperature, high and normal for
+# humidity and weak and strong for wind.
+#
+# The table here summarizes the various attributes and
+#
+#
+#
Day
Outlook
Temperature
Humidity
Wind
Ride
+#
+#
+#
1
Sunny
Hot
High
Weak
0
+#
2
Sunny
Hot
High
Strong
1
+#
3
Overcast
Hot
High
Weak
1
+#
4
Rain
Mild
High
Weak
1
+#
5
Rain
Cool
Normal
Weak
1
+#
6
Rain
Cool
Normal
Strong
0
+#
7
Overcast
Cool
Normal
Strong
1
+#
8
Sunny
Mild
High
Weak
0
+#
9
Sunny
Cool
Normal
Weak
1
+#
10
Rain
Mild
Normal
Weak
1
+#
11
Sunny
Mild
Normal
Strong
1
+#
12
Overcast
Mild
High
Strong
1
+#
13
Overcast
Hot
Normal
Weak
1
+#
14
Rain
Mild
High
Strong
0
+#
+#
+
+# ## Simple Python Code to read in Data and perform Classification
+
+# In[7]:
+
+
+# Common imports
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.tree import DecisionTreeClassifier
+from sklearn.model_selection import train_test_split
+from sklearn.tree import export_graphviz
+from sklearn.preprocessing import StandardScaler, OneHotEncoder
+from sklearn.compose import ColumnTransformer
+from IPython.display import Image
+from pydot import graph_from_dot_data
+import os
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+ return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+ return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+ plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("rideclass.csv"),'r')
+
+# Read the experimental data with Pandas
+from IPython.display import display
+ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))
+ridedata = pd.DataFrame(ridedata)
+
+# Features and targets
+X = ridedata.loc[:, ridedata.columns != 'Ride'].values
+y = ridedata.loc[:, ridedata.columns == 'Ride'].values
+
+# Create the encoder.
+encoder = OneHotEncoder(handle_unknown="ignore")
+# Assume for simplicity all features are categorical.
+encoder.fit(X)
+# Apply the encoder.
+X = encoder.transform(X)
+print(X)
+# Then do a Classification tree
+tree_clf = DecisionTreeClassifier(max_depth=2)
+tree_clf.fit(X, y)
+print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
+#transfer to a decision tree graph
+export_graphviz(
+ tree_clf,
+ out_file="DataFiles/ride.dot",
+ rounded=True,
+ filled=True
+)
+cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
+os.system(cmd)
+
+
+# ## Computing the Gini Factor
+#
+# The above functions (gini, entropy and misclassification error) are
+# important components of the so-called CART algorithm. We will discuss
+# this algorithm below after we have discussed the information gain
+# algorithm ID3.
+#
+# In the example here we have converted all our attributes into numerical values $0,1,2$ etc.
+
+# In[8]:
+
+
+# Split a dataset based on an attribute and an attribute value
+def test_split(index, value, dataset):
+ left, right = list(), list()
+ for row in dataset:
+ if row[index] < value:
+ left.append(row)
+ else:
+ right.append(row)
+ return left, right
+
+# Calculate the Gini index for a split dataset
+def gini_index(groups, classes):
+ # count all samples at split point
+ n_instances = float(sum([len(group) for group in groups]))
+ # sum weighted Gini index for each group
+ gini = 0.0
+ for group in groups:
+ size = float(len(group))
+ # avoid divide by zero
+ if size == 0:
+ continue
+ score = 0.0
+ # score the group based on the score for each class
+ for class_val in classes:
+ p = [row[-1] for row in group].count(class_val) / size
+ score += p * p
+ # weight the group score by its relative size
+ gini += (1.0 - score) * (size / n_instances)
+ return gini
+
+# Select the best split point for a dataset
+def get_split(dataset):
+ class_values = list(set(row[-1] for row in dataset))
+ b_index, b_value, b_score, b_groups = 999, 999, 999, None
+ for index in range(len(dataset[0])-1):
+ for row in dataset:
+ groups = test_split(index, row[index], dataset)
+ gini = gini_index(groups, class_values)
+ print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
+ if gini < b_score:
+ b_index, b_value, b_score, b_groups = index, row[index], gini, groups
+ return {'index':b_index, 'value':b_value, 'groups':b_groups}
+
+dataset = [[0,0,0,0,0],
+ [0,0,0,1,1],
+ [1,0,0,0,1],
+ [2,1,0,0,1],
+ [2,2,1,0,1],
+ [2,2,1,1,0],
+ [1,2,1,1,1],
+ [0,1,0,0,0],
+ [0,2,1,0,1],
+ [2,1,1,0,1],
+ [0,1,1,1,1],
+ [1,1,0,1,1],
+ [1,0,1,0,1],
+ [2,1,0,1,0]]
+
+split = get_split(dataset)
+print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))
+
+
+# ## Regression trees
+
+# In[9]:
+
+
+# Quadratic training set + noise
+np.random.seed(42)
+m = 200
+X = np.random.rand(m, 1)
+y = 4 * (X - 0.5) ** 2
+y = y + np.random.randn(m, 1) / 10
+
+
+# In[10]:
+
+
+from sklearn.tree import DecisionTreeRegressor
+
+tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)
+tree_reg.fit(X, y)
+
+
+# ## Final regressor code
+
+# In[11]:
+
+
+from sklearn.tree import DecisionTreeRegressor
+
+tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)
+tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)
+tree_reg1.fit(X, y)
+tree_reg2.fit(X, y)
+
+def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel="$y$"):
+ x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)
+ y_pred = tree_reg.predict(x1)
+ plt.axis(axes)
+ plt.xlabel("$x_1$", fontsize=18)
+ if ylabel:
+ plt.ylabel(ylabel, fontsize=18, rotation=0)
+ plt.plot(X, y, "b.")
+ plt.plot(x1, y_pred, "r.-", linewidth=2, label=r"$\hat{y}$")
+
+plt.figure(figsize=(11, 4))
+plt.subplot(121)
+plot_regression_predictions(tree_reg1, X, y)
+for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
+ plt.plot([split, split], [-0.2, 1], style, linewidth=2)
+plt.text(0.21, 0.65, "Depth=0", fontsize=15)
+plt.text(0.01, 0.2, "Depth=1", fontsize=13)
+plt.text(0.65, 0.8, "Depth=1", fontsize=13)
+plt.legend(loc="upper center", fontsize=18)
+plt.title("max_depth=2", fontsize=14)
+
+plt.subplot(122)
+plot_regression_predictions(tree_reg2, X, y, ylabel=None)
+for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
+ plt.plot([split, split], [-0.2, 1], style, linewidth=2)
+for split in (0.0458, 0.1298, 0.2873, 0.9040):
+ plt.plot([split, split], [-0.2, 1], "k:", linewidth=1)
+plt.text(0.3, 0.5, "Depth=2", fontsize=13)
+plt.title("max_depth=3", fontsize=14)
+
+plt.show()
+
+
+# In[12]:
+
+
+tree_reg1 = DecisionTreeRegressor(random_state=42)
+tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)
+tree_reg1.fit(X, y)
+tree_reg2.fit(X, y)
+
+x1 = np.linspace(0, 1, 500).reshape(-1, 1)
+y_pred1 = tree_reg1.predict(x1)
+y_pred2 = tree_reg2.predict(x1)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.plot(X, y, "b.")
+plt.plot(x1, y_pred1, "r.-", linewidth=2, label=r"$\hat{y}$")
+plt.axis([0, 1, -0.2, 1.1])
+plt.xlabel("$x_1$", fontsize=18)
+plt.ylabel("$y$", fontsize=18, rotation=0)
+plt.legend(loc="upper center", fontsize=18)
+plt.title("No restrictions", fontsize=14)
+
+plt.subplot(122)
+plt.plot(X, y, "b.")
+plt.plot(x1, y_pred2, "r.-", linewidth=2, label=r"$\hat{y}$")
+plt.axis([0, 1, -0.2, 1.1])
+plt.xlabel("$x_1$", fontsize=18)
+plt.title("min_samples_leaf={}".format(tree_reg2.min_samples_leaf), fontsize=14)
+
+plt.show()
+
+
+# ## Pros and cons of trees, pros
+#
+# * White box, easy to interpret model. Some people believe that decision trees more closely mirror human decision-making than do the regression and classification approaches discussed earlier (think of support vector machines)
+#
+# * Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!
+#
+# * No feature normalization needed
+#
+# * Tree models can handle both continuous and categorical data (Classification and Regression Trees)
+#
+# * Can model nonlinear relationships
+#
+# * Can model interactions between the different descriptive features
+#
+# * Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)
+
+# ## Disadvantages
+#
+# * Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches
+#
+# * If continuous features are used the tree may become quite large and hence less interpretable
+#
+# * Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented
+#
+# * Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests
+#
+# * Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones.
+#
+# * If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data
+#
+# * Features with many levels may be preferred over features with less levels since for them it is *more easy* to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain
+#
+# However, by aggregating many decision trees, using methods like
+# bagging, random forests, and boosting, the predictive performance of
+# trees can be substantially improved.
+
+# ## Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
+#
+# As stated above and seen in many of the examples discussed here about
+# a single decision tree, we often end up overfitting our training
+# data. This normally means that we have a high variance. Can we reduce
+# the variance of a statistical learning method?
+#
+# This leads us to a set of different methods that can combine different
+# machine learning algorithms or just use one of them to construct
+# forests and jungles of trees, homogeneous ones or heterogenous
+# ones. These methods are recognized by different names which we will
+# try to explain here. These are
+#
+# 1. Voting classifiers
+#
+# 2. Bagging and Pasting
+#
+# 3. Random forests
+#
+# 4. Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)
+#
+# We discuss these methods here.
+
+# ## An Overview of Ensemble Methods
+#
+#
+#
+#
+#
Figure 1:
+#
+
+# ## Why Voting?
+#
+# The idea behind boosting, and voting as well can be phrased as follows:
+# **Can a group of people somehow arrive at highly
+# reasoned decisions, despite the weak judgement of the individual
+# members?**
+#
+# The aim is to create a good classifier by combining several weak classifiers.
+# **A weak classifier is a classifier which is able to produce results that are only slightly better than guessing at random.**
+#
+# The basic approach is to apply repeatedly (in boosting this is done in an iterative way) a weak classifier to modifications of the data.
+# In voting we simply apply the law of large numbers while in boosting we give more weight to misclassified data in
+# each iteration.
+#
+# Decision trees play an important role as our weak classifier. They serve as the basic method.
+
+# ## Tossing coins
+#
+# The simplest case is a so-called voting ensemble. To illustrate this,
+# think of yourself tossing coins with a biased outcome of 51 per cent
+# for heads and 49% for tails. With only few tosses,
+# you may not clearly see this distribution for heads and tails. However, after some
+# thousands of tosses, there will be a clear majority of heads. With 2000 tosses
+# you should see approximately 1020 heads and 980 tails.
+#
+# 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.
+#
+# Another example would be to collect all polls before an
+# election. Different polls may show different likelihoods for a
+# candidate winning with say a majority of the popular vote. The majority vote
+# would then consist in many polls indicating that this candidate will
+# actually win.
+#
+# The example here shows how we can implement the coin tossing case,
+# clealry demostrating that after some tosses we see the [law of large](https://en.wikipedia.org/wiki/Law_of_large_numbers)
+# numbers kicking in.
+
+# ## Standard imports first
+
+# In[13]:
+
+
+# Common imports
+from IPython.display import Image
+from pydot import graph_from_dot_data
+import pandas as pd
+import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.tree import DecisionTreeClassifier
+from sklearn.model_selection import train_test_split
+from sklearn.tree import export_graphviz
+from sklearn.preprocessing import StandardScaler, OneHotEncoder
+from sklearn.compose import ColumnTransformer
+from IPython.display import Image
+from pydot import graph_from_dot_data
+import os
+
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
+
+if not os.path.exists(PROJECT_ROOT_DIR):
+ os.mkdir(PROJECT_ROOT_DIR)
+
+if not os.path.exists(FIGURE_ID):
+ os.makedirs(FIGURE_ID)
+
+if not os.path.exists(DATA_ID):
+ os.makedirs(DATA_ID)
+
+def image_path(fig_id):
+ return os.path.join(FIGURE_ID, fig_id)
+
+def data_path(dat_id):
+ return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+ plt.savefig(image_path(fig_id) + ".png", format='png')
+
+
+# ## Simple Voting Example, head or tail
+
+# In[14]:
+
+
+
+# Common imports
+import numpy as np
+import matplotlib
+import matplotlib.pyplot as plt
+from matplotlib.colors import ListedColormap
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
+
+heads_proba = 0.51
+coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
+cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)
+plt.figure(figsize=(8,3.5))
+plt.plot(cumulative_heads_ratio)
+plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%")
+plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%")
+plt.xlabel("Number of coin tosses")
+plt.ylabel("Heads ratio")
+plt.legend(loc="lower right")
+plt.axis([0, 10000, 0.42, 0.58])
+save_fig("votingsimple")
+plt.show()
+
+
+# ## Using the Voting Classifier
+#
+# We can use the voting classifier on other data sets, here the exciting binary case of two distinct objects using the make moons functionality of **Scikit-Learn**.
+
+# In[15]:
+
+
+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))
+
+
+# ## Voting and Bagging
+
+# In[16]:
+
+
+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)
+
+
+# In[17]:
+
+
+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))
+
+
+# In[18]:
+
+
+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)
+
+
+# In[19]:
+
+
+from sklearn.metrics import accuracy_score
+
+for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
+ clf.fit(X_train, y_train)
+ y_pred = clf.predict(X_test)
+ print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
+
+
+# ## Bagging
+#
+# The **plain** decision trees suffer from high
+# variance. This means that if we split the training data into two parts
+# at random, and fit a decision tree to both halves, the results that we
+# get could be quite different. In contrast, a procedure with low
+# variance will yield similar results if applied repeatedly to distinct
+# data sets; linear regression tends to have low variance, if the ratio
+# of $n$ to $p$ is moderately large.
+#
+# **Bootstrap aggregation**, or just **bagging**, is a
+# general-purpose procedure for reducing the variance of a statistical
+# learning method.
+
+# ## More bagging
+#
+# Bagging typically results in improved accuracy
+# over prediction using a single tree. Unfortunately, however, it can be
+# difficult to interpret the resulting model. Recall that one of the
+# advantages of decision trees is the attractive and easily interpreted
+# diagram that results.
+#
+# However, when we bag a large number of trees, it is no longer
+# possible to represent the resulting statistical learning procedure
+# using a single tree, and it is no longer clear which variables are
+# most important to the procedure. Thus, bagging improves prediction
+# accuracy at the expense of interpretability. Although the collection
+# of bagged trees is much more difficult to interpret than a single
+# tree, one can obtain an overall summary of the importance of each
+# predictor using the MSE (for bagging regression trees) or the Gini
+# index (for bagging classification trees). In the case of bagging
+# regression trees, we can record the total amount that the MSE is
+# decreased due to splits over a given predictor, averaged over all $B$ possible
+# trees. A large value indicates an important predictor. Similarly, in
+# the context of bagging classification trees, we can add up the total
+# amount that the Gini index is decreased by splits over a given
+# predictor, averaged over all $B$ trees.
+
+# ## 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 $n$).
+
+# In[20]:
+
+
+
+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("Simple tree:",mse_simpletree)
+plt.xlim(1,maxdepth)
+plt.plot(polydegree, error, label='MSE')
+plt.plot(polydegree, bias, label='bias')
+plt.plot(polydegree, variance, label='Variance')
+plt.legend()
+save_fig("baggingboot")
+plt.show()
+
+
+# ## Random forests
+#
+# Random forests provide an improvement over bagged trees by way of a
+# small tweak that decorrelates the trees.
+#
+# As in bagging, we build a
+# number of decision trees on bootstrapped training samples. But when
+# building these decision trees, each time a split in a tree is
+# considered, a random sample of $m$ predictors is chosen as split
+# candidates from the full set of $p$ predictors. The split is allowed to
+# use only one of those $m$ predictors.
+#
+# A fresh sample of $m$ predictors is
+# taken at each split, and typically we choose
+
+# $$
+# m\approx \sqrt{p}.
+# $$
+
+# In building a random forest, at
+# each split in the tree, the algorithm is not even allowed to consider
+# a majority of the available predictors.
+#
+# The reason for this is rather clever. Suppose that there is one very
+# strong predictor in the data set, along with a number of other
+# moderately strong predictors. Then in the collection of bagged
+# variable importance random forest trees, most or all of the trees will
+# use this strong predictor in the top split. Consequently, all of the
+# bagged trees will look quite similar to each other. Hence the
+# predictions from the bagged trees will be highly correlated.
+# Unfortunately, averaging many highly correlated quantities does not
+# lead to as large of a reduction in variance as averaging many
+# uncorrelated quantities. In particular, this means that bagging will
+# not lead to a substantial reduction in variance over a single tree in
+# this setting.
+
+# ## Random Forest Algorithm
+# The algorithm described here can be applied to both classification and regression problems.
+#
+# We will grow of forest of say $B$ trees.
+# 1. For $b=1:B$
+#
+# * Draw a bootstrap sample from the training data organized in our $\boldsymbol{X}$ matrix.
+#
+# * We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached
+#
+# 1. we select $m \le p$ variables at random from the $p$ predictors/features
+#
+# 2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node
+#
+# 3. split the node into daughter nodes
+#
+# 4. Output then the ensemble of trees $\{T_b\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem.
+
+# ## Random Forests Compared with other Methods on the Cancer Data
+
+# In[21]:
+
+
+import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import train_test_split
+from sklearn.datasets import load_breast_cancer
+from sklearn.svm import SVC
+from sklearn.linear_model import LogisticRegression
+from sklearn.tree import DecisionTreeClassifier
+from sklearn.ensemble import BaggingClassifier
+
+# Load the data
+cancer = load_breast_cancer()
+
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+#define methods
+# Logistic Regression
+logreg = LogisticRegression(solver='lbfgs')
+# Support vector machine
+svm = SVC(gamma='auto', C=100)
+# Decision Trees
+deep_tree_clf = DecisionTreeClassifier(max_depth=None)
+#Scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Logistic Regression
+logreg.fit(X_train_scaled, y_train)
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+# Support Vector Machine
+svm.fit(X_train_scaled, y_train)
+print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+# Decision Trees
+deep_tree_clf.fit(X_train_scaled, y_train)
+print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
+
+
+from sklearn.ensemble import RandomForestClassifier
+from sklearn.preprocessing import LabelEncoder
+from sklearn.model_selection import cross_validate
+# Data set not specificied
+#Instantiate the model with 500 trees and entropy as splitting criteria
+Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy")
+Random_Forest_model.fit(X_train_scaled, y_train)
+#Cross validation
+accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
+print(accuracy)
+print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
+
+
+import scikitplot as skplt
+y_pred = Random_Forest_model.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+plt.show()
+y_probas = Random_Forest_model.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test, y_probas)
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test, y_probas)
+plt.show()
+
+
+# Recall that the cumulative gains curve shows the percentage of the
+# overall number of cases in a given category *gained* by targeting a
+# percentage of the total number of cases.
+#
+# Similarly, the receiver operating characteristic curve, or ROC curve,
+# displays the diagnostic ability of a binary classifier system as its
+# discrimination threshold is varied. It plots the true positive rate against the false positive rate.
+
+# ## Compare Bagging on Trees with Random Forests
+
+# In[22]:
+
+
+bag_clf = BaggingClassifier(
+ DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
+ n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
+
+
+# In[23]:
+
+
+bag_clf.fit(X_train, y_train)
+y_pred = bag_clf.predict(X_test)
+from sklearn.ensemble import RandomForestClassifier
+rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
+rnd_clf.fit(X_train, y_train)
+y_pred_rf = rnd_clf.predict(X_test)
+np.sum(y_pred == y_pred_rf) / len(y_pred)
+
+
+# ## Boosting, a Bird's Eye View
+#
+# The basic idea is to combine weak classifiers in order to create a good
+# classifier. With a weak classifier we often intend a classifier which
+# produces results which are only slightly better than we would get by
+# random guesses.
+#
+# This is done by applying in an iterative way a weak (or a standard
+# classifier like decision trees) to modify the data. In each iteration
+# we emphasize those observations which are misclassified by weighting
+# them with a factor.
+
+# ## What is boosting? Additive Modelling/Iterative Fitting
+#
+# Boosting is a way of fitting an additive expansion in a set of
+# elementary basis functions like for example some simple polynomials.
+# Assume for example that we have a function
+
+# $$
+# f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
+# $$
+
+# where $\beta_m$ are the expansion parameters to be determined in a
+# minimization process and $b(x;\gamma_m)$ are some simple functions of
+# the multivariable parameter $x$ which is characterized by the
+# parameters $\gamma_m$.
+#
+# As an example, consider the Sigmoid function we used in logistic
+# regression. In that case, we can translate the function
+# $b(x;\gamma_m)$ into the Sigmoid function
+
+# $$
+# \sigma(t) = \frac{1}{1+\exp{(-t)}},
+# $$
+
+# where $t=\gamma_0+\gamma_1 x$ and the parameters $\gamma_0$ and
+# $\gamma_1$ were determined by the Logistic Regression fitting
+# algorithm.
+#
+# As another example, consider the cost function we defined for linear regression
+
+# $$
+# C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
+# $$
+
+# In this case the function $f(x)$ was replaced by the design matrix
+# $\boldsymbol{X}$ and the unknown linear regression parameters $\boldsymbol{\beta}$,
+# that is $\boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta}$. In linear regression we can
+# simply invert a matrix and obtain the parameters $\beta$ by
+
+# $$
+# \boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
+# $$
+
+# In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\beta_m$ and $\gamma_m$.
+
+# ## Iterative Fitting, Regression and Squared-error Cost Function
+#
+# The way we proceed is as follows (here we specialize to the squared-error cost function)
+#
+# 1. Establish a cost function, here $C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ with $f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)$.
+#
+# 2. Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers.
+#
+# 3. For $m=1:M$
+#
+# a. minimize $\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2$ wrt $\gamma$ and $\beta$
+#
+# b. This gives the optimal values $\beta_m$ and $\gamma_m$
+#
+# c. Determine then the new values $f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)$
+#
+# We could use any of the algorithms we have discussed till now. If we
+# use trees, $\gamma$ parameterizes the split variables and split points
+# at the internal nodes, and the predictions at the terminal nodes.
+
+# ## Squared-Error Example and Iterative Fitting
+#
+# To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
+#
+# For simplicity we assume also that our functions $b(x;\gamma)=1+\gamma x$.
+#
+# This means that for every iteration $m$, we need to optimize
+
+# $$
+# (\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
+# $$
+
+# We start our iteration by simply setting $f_0(x)=0$.
+# Taking the derivatives with respect to $\beta$ and $\gamma$ we obtain
+
+# $$
+# \frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
+# $$
+
+# and
+
+# $$
+# \frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
+# $$
+
+# We can then rewrite these equations as (defining $\boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x})$ with $\boldsymbol{e}$ being the unit vector)
+
+# $$
+# \gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0,
+# $$
+
+# which gives us $\beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w})$. Similarly we have
+
+# $$
+# \beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,
+# $$
+
+# which leads to $\gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x})$. Inserting
+# for $\beta$ gives us an equation for $\gamma$. This is a non-linear equation in the unknown $\gamma$ and has to be solved numerically.
+#
+# The solution to these two equations gives us in turn $\beta_1$ and $\gamma_1$ leading to the new expression for $f_1(x)$ as
+# $f_1(x) = \beta_1(1+\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$.
+
+# ## Iterative Fitting, Classification and AdaBoost
+#
+# Let us consider a binary classification problem with two outcomes $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of
+# observations. We define a classification function $G(x)$ which produces a prediction taking one or the other of the two values
+# $\{-1,1\}$.
+#
+# The error rate of the training sample is then
+
+# $$
+# \mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
+# $$
+
+# The iterative procedure starts with defining a weak classifier whose
+# error rate is barely better than random guessing. The iterative
+# procedure in boosting is to sequentially apply a weak
+# classification algorithm to repeatedly modified versions of the data
+# producing a sequence of weak classifiers $G_m(x)$.
+#
+# Here we will express our function $f(x)$ in terms of $G(x)$. That is
+
+# $$
+# f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
+# $$
+
+# will be a function of
+
+# $$
+# G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
+# $$
+
+# ## Adaptive Boosting, AdaBoost
+#
+# In our iterative procedure we define thus
+
+# $$
+# f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
+# $$
+
+# The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
+# exponential cost/loss function defined as
+
+# $$
+# C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
+# $$
+
+# We optimize $\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case.
+# This is normally done in two steps. Let us however first rewrite the cost function as
+
+# $$
+# C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},
+# $$
+
+# where we have defined $w_i^m= \exp{(-y_if_{m-1}(x_i))}$.
+
+# ## Building up AdaBoost
+#
+# First, for any $\beta > 0$, we optimize $G$ by setting
+
+# $$
+# G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),
+# $$
+
+# which is the classifier that minimizes the weighted error rate in predicting $y$.
+#
+# We can do this by rewriting
+
+# $$
+# \exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,
+# $$
+
+# which can be rewritten as
+
+# $$
+# (\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,
+# $$
+
+# which leads to
+
+# $$
+# \beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},
+# $$
+
+# where we have redefined the error as
+
+# $$
+# \mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},
+# $$
+
+# which leads to an update of
+
+# $$
+# f_m(x) = f_{m-1}(x) +\beta_m G_m(x).
+# $$
+
+# This leads to the new weights
+
+# $$
+# w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
+# $$
+
+# ## Adaptive boosting: AdaBoost, Basic Algorithm
+#
+# The algorithm here is rather straightforward. Assume that our weak
+# classifier is a decision tree and we consider a binary set of outputs
+# with $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of
+# observations. Our design matrix is given in terms of the
+# feature/predictor vectors
+# $\boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}]$. Finally, we define also a
+# classifier determined by our data via a function $G(x)$. This function tells us how well we are able to classify our outputs/targets $\boldsymbol{y}$.
+#
+# We have already defined the misclassification error $\mathrm{err}$ as
+
+# $$
+# \mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),
+# $$
+
+# where the function $I()$ is one if we misclassify and zero if we classify correctly.
+
+# ## Basic Steps of AdaBoost
+#
+# With the above definitions we are now ready to set up the algorithm for AdaBoost.
+# The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
+# 1. We start by initializing all weights to $w_i = 1/n$, with $i=0,1,2,\dots n-1$. It is easy to see that we must have $\sum_{i=0}^{n-1}w_i = 1$.
+#
+# 2. We rewrite the misclassification error as
+
+# $$
+# \mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
+# $$
+
+# 1. Then we start looping over all attempts at classifying, namely we start an iterative process for $m=1:M$, where $M$ is the final number of classifications. Our given classifier could for example be a plain decision tree.
+#
+# a. Fit then a given classifier to the training set using the weights $w_i$.
+#
+# b. Compute then $\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly.
+#
+# c. Define a quantity $\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}$
+#
+# d. Set the new weights to $w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}$.
+#
+# 5. Compute the new classifier $G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)$.
+#
+# For the iterations with $m \le 2$ the weights are modified
+# individually at each steps. The observations which were misclassified
+# at iteration $m-1$ have a weight which is larger than those which were
+# classified properly. As this proceeds, the observations which were
+# difficult to classifiy correctly are given a larger influence. Each
+# new classification step $m$ is then forced to concentrate on those
+# observations that are missed in the previous iterations.
+
+# ## AdaBoost Examples
+#
+# Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here.
+
+# In[24]:
+
+
+from sklearn.ensemble import AdaBoostClassifier
+
+ada_clf = AdaBoostClassifier(
+ DecisionTreeClassifier(max_depth=1), n_estimators=200,
+ algorithm="SAMME.R", learning_rate=0.5, random_state=42)
+ada_clf.fit(X_train, y_train)
+
+from sklearn.ensemble import AdaBoostClassifier
+
+ada_clf = AdaBoostClassifier(
+ DecisionTreeClassifier(max_depth=1), n_estimators=200,
+ algorithm="SAMME.R", learning_rate=0.5, random_state=42)
+ada_clf.fit(X_train_scaled, y_train)
+y_pred = ada_clf.predict(X_test_scaled)
+skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
+plt.show()
+y_probas = ada_clf.predict_proba(X_test_scaled)
+skplt.metrics.plot_roc(y_test, y_probas)
+plt.show()
+skplt.metrics.plot_cumulative_gain(y_test, y_probas)
+plt.show()
+
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diff --git a/doc/LectureNotes/_toc.yml b/doc/LectureNotes/_toc.yml
index 5dd39d924..5f660faf5 100644
--- a/doc/LectureNotes/_toc.yml
+++ b/doc/LectureNotes/_toc.yml
@@ -62,8 +62,10 @@ parts:
- file: week43.ipynb
- file: week44.ipynb
- file: week45.ipynb
+ - file: week46.ipynb
- caption: Projects
numbered: false
chapters:
- file: project1.ipynb
- file: project2.ipynb
+ - file: project3.ipynb
+
+
+ ![](\"DataFiles/cancer.png\")
![](\"DataFiles/ensembleoverview.png\")
+
+
+