Morten Hjorth-Jensen
2974 lines
284 KiB
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Applied Data Analysis and Machine Learning
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">About the course</span></p>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Review of Statistics with Resampling Techniques and Linear Algebra</span></p>
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<li class="toctree-l1"><a class="reference internal" href="statistics.html">1. Elements of Probability Theory and Statistical Data Analysis</a></li>
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<li class="toctree-l1"><a class="reference internal" href="linalg.html">2. Linear Algebra, Handling of Arrays and more Python Features</a></li>
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</ul>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">From Regression to Support Vector Machines</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter1.html">3. Linear Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter2.html">4. Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter3.html">5. Resampling Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter4.html">6. Logistic Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapteroptimization.html">7. Optimization, the central part of any Machine Learning algortithm</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter5.html">8. Support Vector Machines, overarching aims</a></li>
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</ul>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Decision Trees, Ensemble Methods and Boosting</span></p>
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<ul class="nav bd-sidenav">
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<li class="toctree-l1"><a class="reference internal" href="chapter6.html">9. Decision trees, overarching aims</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter7.html">10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
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</ul>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Dimensionality Reduction</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter8.html">11. Basic ideas of the Principal Component Analysis (PCA)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="clustering.html">12. Clustering and Unsupervised Learning</a></li>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Deep Learning Methods</span></p>
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<li class="toctree-l1"><a class="reference internal" href="chapter9.html">13. Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter10.html">14. Building a Feed Forward Neural Network</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter11.html">15. Solving Differential Equations with Deep Learning</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter12.html">16. Convolutional Neural Networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="chapter13.html">17. Recurrent neural networks: Overarching view</a></li>
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</ul>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
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<ul class="current nav bd-sidenav">
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek34.html">Exercises week 34</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week34.html">Week 34: Introduction to the course, Logistics and Practicalities</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek35.html">Exercises week 35</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Statistical interpretations</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 37</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week37.html">Week 37: Statistical interpretations and Resampling Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek38.html">Exercises week 38</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week38.html">Week 38: Logistic Regression and Optimization</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek39.html">Exercises week 39</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week39.html">Week 39: Optimization and Gradient Methods</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week40.html">Week 40: Gradient descent methods (continued) and start Neural networks</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek41.html">Exercises week 41</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week41.html">Week 41 Neural networks and constructing a neural network code</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek42.html">Exercises week 42</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week42.html">Week 42 Constructing a Neural Network code with examples</a></li>
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<li class="toctree-l1"><a class="reference internal" href="additionweek42.html">Exercises Week 42: Logistic Regression and Optimization, reminders from week 38 and week 40</a></li>
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||
<li class="toctree-l1"><a class="reference internal" href="week43.html">Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek43.html">Exercises week 43</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week44.html">Week 44, Convolutional Neural Networks (CNN)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week45.html">Week 45, Convolutional Neural Networks (CCNs) and Recurrent Neural Networks (RNNs)</a></li>
|
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Week 46: Decision Trees, Ensemble methods and Random Forests</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week47.html">Week 47: From Decision Trees to Ensemble Methods, Random Forests and Boosting Methods</a></li>
|
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek47.html">Exercise week 47</a></li>
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<li class="toctree-l1"><a class="reference internal" href="week48.html">Week 48: Gradient boosting and summary of course</a></li>
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<li class="toctree-l1"><a class="reference internal" href="exercisesweek48.html">Exercises week 48</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project1.html">Project 1 on Machine Learning, deadline October 7 (midnight), 2024</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project2.html">Project 2 on Machine Learning, deadline November 4 (Midnight)</a></li>
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<li class="toctree-l1"><a class="reference internal" href="project3.html">Project 3 on Machine Learning, deadline December 9 (midnight), 2024</a></li>
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<div id="jb-print-docs-body" class="onlyprint">
|
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<h1>Week 46: Decision Trees, Ensemble methods and Random Forests</h1>
|
||
<!-- Table of contents -->
|
||
<div id="print-main-content">
|
||
<div id="jb-print-toc">
|
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|
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<div>
|
||
<h2> Contents </h2>
|
||
</div>
|
||
<nav aria-label="Page">
|
||
<ul class="visible nav section-nav flex-column">
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plan-for-week-46">Plan for week 46</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#decision-trees-overarching-aims">Decision trees, overarching aims</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basics-of-a-tree">Basics of a tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-typical-decision-tree-with-its-pertinent-jargon-classification-problem">A typical Decision Tree with its pertinent Jargon, Classification Problem</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#general-features">General Features</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-we-set-it-up">How do we set it up?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#decision-trees-and-regression">Decision trees and Regression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-a-tree-regression">Building a tree, regression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-top-down-approach-recursive-binary-splitting">A top-down approach, recursive binary splitting</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-a-tree">Making a tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pruning-the-tree">Pruning the tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cost-complexity-pruning">Cost complexity pruning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#schematic-regression-procedure">Schematic Regression Procedure</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-classification-tree">A Classification Tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#growing-a-classification-tree">Growing a classification tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#classification-tree-how-to-split-nodes">Classification tree, how to split nodes</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-tree-classification">Visualizing the Tree, Classification</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-tree-the-moons">Visualizing the Tree, The Moons</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-ways-of-visualizing-the-trees">Other ways of visualizing the trees</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#printing-out-as-text">Printing out as text</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#algorithms-for-setting-up-decision-trees">Algorithms for Setting up Decision Trees</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cart-algorithm-for-classification">The CART algorithm for Classification</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cart-algorithm-for-regression">The CART algorithm for Regression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-binary-splits">Why binary splits?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-a-tree-using-the-gini-index">Computing a Tree using the Gini Index</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-table">The Table</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-various-gini-indices">Computing the various Gini Indices</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-various-gini-indices-hours-slept">Computing the various Gini Indices, Hours slept</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-various-gini-indices-hours-studied">Computing the various Gini Indices, Hours studied</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-possible-code-using-scikit-learn">A possible code using Scikit-Learn</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#further-example-computing-the-gini-index">Further example: Computing the Gini index</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-python-code-to-read-in-data-and-perform-classification">Simple Python Code to read in Data and perform Classification</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-gini-factor">Computing the Gini Factor</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-trees">Regression trees</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#final-regressor-code">Final regressor code</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pros-and-cons-of-trees-pros">Pros and cons of trees, pros</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#disadvantages">Disadvantages</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods">Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#an-overview-of-ensemble-methods">An Overview of Ensemble Methods</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-voting">Why Voting?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tossing-coins">Tossing coins</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#standard-imports-first">Standard imports first</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-voting-example-head-or-tail">Simple Voting Example, head or tail</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-the-voting-classifier">Using the Voting Classifier</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#voting-and-bagging">Voting and Bagging</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bagging">Bagging</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-bagging">More bagging</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-your-own-bootstrap-changing-the-level-of-the-decision-tree">Making your own Bootstrap: Changing the Level of the Decision Tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forests">Random forests</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forest-algorithm">Random Forest Algorithm</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forests-compared-with-other-methods-on-the-cancer-data">Random Forests Compared with other Methods on the Cancer Data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#compare-bagging-on-trees-with-random-forests">Compare Bagging on Trees with Random Forests</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#boosting-a-bird-s-eye-view">Boosting, a Bird’s Eye View</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#what-is-boosting-additive-modelling-iterative-fitting">What is boosting? Additive Modelling/Iterative Fitting</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#iterative-fitting-regression-and-squared-error-cost-function">Iterative Fitting, Regression and Squared-error Cost Function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#squared-error-example-and-iterative-fitting">Squared-Error Example and Iterative Fitting</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#iterative-fitting-classification-and-adaboost">Iterative Fitting, Classification and AdaBoost</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adaptive-boosting-adaboost">Adaptive Boosting, AdaBoost</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-up-adaboost">Building up AdaBoost</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adaptive-boosting-adaboost-basic-algorithm">Adaptive boosting: AdaBoost, Basic Algorithm</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basic-steps-of-adaboost">Basic Steps of AdaBoost</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adaboost-examples">AdaBoost Examples</a></li>
|
||
</ul>
|
||
</nav>
|
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</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
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|
||
<div id="searchbox"></div>
|
||
<article class="bd-article">
|
||
|
||
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
|
||
doconce format html week46.do.txt --no_mako -->
|
||
<!-- dom:TITLE: Week 46: Decision Trees, Ensemble methods and Random Forests --><section class="tex2jax_ignore mathjax_ignore" id="week-46-decision-trees-ensemble-methods-and-random-forests">
|
||
<h1>Week 46: Decision Trees, Ensemble methods and Random Forests<a class="headerlink" href="#week-46-decision-trees-ensemble-methods-and-random-forests" title="Link to this heading">#</a></h1>
|
||
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</p>
|
||
<p>Date: <strong>Week 46, November 11-15</strong></p>
|
||
<section id="plan-for-week-46">
|
||
<h2>Plan for week 46<a class="headerlink" href="#plan-for-week-46" title="Link to this heading">#</a></h2>
|
||
<p><strong>Lab sessions on Tuesday and Wednesday.</strong></p>
|
||
<ol class="arabic simple">
|
||
<li><p>Work on and discussions of project 3</p></li>
|
||
</ol>
|
||
<p><strong>Material for the lecture on Monday November 11, 2024.</strong></p>
|
||
<p>Basics of decision trees, classification and regression algorithms and ensemble models</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Readings and Videos:</p></li>
|
||
</ol>
|
||
<p>a. Lecture notes at <a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/pub/week46/ipynb/week46.ipynb">CompPhysics/MachineLearning</a></p>
|
||
<!-- * [Video of lecture](https://youtu.be/PMswUwhYa7k) -->
|
||
<!-- * [Whiteboard notes](https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2023/NotesNov16.pdf) -->
|
||
<p>b. Video on Decision trees at <a class="reference external" href="https://www.youtube.com/watch?v=RmajweUFKvM&amp;ab_channel=Simplilearn">https://www.youtube.com/watch?v=RmajweUFKvM&ab_channel=Simplilearn</a></p>
|
||
<p>c. Decision Trees: Rashcka et al chapter 3 pages 86-98, and chapter 7 on Ensemble methods, Voting and Bagging and Gradient Boosting. See also lecture from STK-IN4300, lecture 7 at <a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_7.pdf">https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_7.pdf</a>.</p>
|
||
</section>
|
||
<section id="decision-trees-overarching-aims">
|
||
<h2>Decision trees, overarching aims<a class="headerlink" href="#decision-trees-overarching-aims" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
<p>Decision trees are supervised learning algorithms used for both,
|
||
classification and regression tasks.</p>
|
||
<p>The main idea of decision trees
|
||
is to find those descriptive features which contain the most
|
||
<strong>information</strong> 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.</p>
|
||
<p>The descriptive features which reproduce best the target/output features are normally said
|
||
to be the most informative ones. The process of finding the <strong>most
|
||
informative</strong> feature is done until we accomplish a stopping criteria
|
||
where we then finally end up in so called <strong>leaf nodes</strong>.</p>
|
||
</section>
|
||
<section id="basics-of-a-tree">
|
||
<h2>Basics of a tree<a class="headerlink" href="#basics-of-a-tree" title="Link to this heading">#</a></h2>
|
||
<p>A decision tree is typically divided into a <strong>root node</strong>, the <strong>interior nodes</strong>,
|
||
and the final <strong>leaf nodes</strong> or just <strong>leaves</strong>. These entities are then connected by so-called <strong>branches</strong>.</p>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="a-typical-decision-tree-with-its-pertinent-jargon-classification-problem">
|
||
<h2>A typical Decision Tree with its pertinent Jargon, Classification Problem<a class="headerlink" href="#a-typical-decision-tree-with-its-pertinent-jargon-classification-problem" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [DataFiles/cancer.png, width=600 frac=0.8] -->
|
||
<!-- begin figure -->
|
||
<p><img src="DataFiles/cancer.png" width="600"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure -->
|
||
<p>This tree was produced using the Wisconsin cancer data (discussed here as well, see code examples below) using <strong>Scikit-Learn</strong>’s decision tree classifier. Here we have used the so-called <strong>gini</strong> index (see below) to split the various branches.</p>
|
||
</section>
|
||
<section id="general-features">
|
||
<h2>General Features<a class="headerlink" href="#general-features" title="Link to this heading">#</a></h2>
|
||
<p>The overarching approach to decision trees is a top-down approach.</p>
|
||
<ul class="simple">
|
||
<li><p>A leaf provides the classification of a given instance.</p></li>
|
||
<li><p>A node specifies a test of some attribute of the instance.</p></li>
|
||
<li><p>A branch corresponds to a possible values of an attribute.</p></li>
|
||
<li><p>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.</p></li>
|
||
</ul>
|
||
<p>This process is then repeated for the subtree rooted at the new
|
||
node.</p>
|
||
</section>
|
||
<section id="how-do-we-set-it-up">
|
||
<h2>How do we set it up?<a class="headerlink" href="#how-do-we-set-it-up" title="Link to this heading">#</a></h2>
|
||
<p>In simplified terms, the process of training a decision tree and
|
||
predicting the target features of query instances is as follows:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature</p></li>
|
||
<li><p>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</p></li>
|
||
<li><p>Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the <em>predictions</em> we want to make for new query instances</p></li>
|
||
<li><p>Show query instances to the tree and run down the tree until we arrive at leaf nodes</p></li>
|
||
</ol>
|
||
<p>Then we are essentially done!</p>
|
||
</section>
|
||
<section id="decision-trees-and-regression">
|
||
<h2>Decision trees and Regression<a class="headerlink" href="#decision-trees-and-regression" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
|
||
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
|
||
|
||
<span class="n">steps</span><span class="o">=</span><span class="mi">250</span>
|
||
|
||
<span class="n">distance</span><span class="o">=</span><span class="mi">0</span>
|
||
<span class="n">x</span><span class="o">=</span><span class="mi">0</span>
|
||
<span class="n">distance_list</span><span class="o">=</span><span class="p">[]</span>
|
||
<span class="n">steps_list</span><span class="o">=</span><span class="p">[]</span>
|
||
<span class="k">while</span> <span class="n">x</span><span class="o"><</span><span class="n">steps</span><span class="p">:</span>
|
||
<span class="n">distance</span><span class="o">+=</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">distance_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">distance</span><span class="p">)</span>
|
||
<span class="n">x</span><span class="o">+=</span><span class="mi">1</span>
|
||
<span class="n">steps_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">steps_list</span><span class="p">,</span><span class="n">distance_list</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'green'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Random Walk Data"</span><span class="p">)</span>
|
||
|
||
<span class="n">steps_list</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">steps_list</span><span class="p">)</span>
|
||
<span class="n">distance_list</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">distance_list</span><span class="p">)</span>
|
||
|
||
<span class="n">X</span><span class="o">=</span><span class="n">steps_list</span><span class="p">[:,</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
|
||
|
||
<span class="c1">#Polynomial fits</span>
|
||
|
||
<span class="c1">#Degree 2</span>
|
||
<span class="n">poly_features</span><span class="o">=</span><span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">include_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<span class="n">X_poly</span><span class="o">=</span><span class="n">poly_features</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
|
||
<span class="n">lin_reg</span><span class="o">=</span><span class="n">LinearRegression</span><span class="p">()</span>
|
||
<span class="n">poly_fit</span><span class="o">=</span><span class="n">lin_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_poly</span><span class="p">,</span><span class="n">distance_list</span><span class="p">)</span>
|
||
<span class="n">b</span><span class="o">=</span><span class="n">lin_reg</span><span class="o">.</span><span class="n">coef_</span>
|
||
<span class="n">c</span><span class="o">=</span><span class="n">lin_reg</span><span class="o">.</span><span class="n">intercept_</span>
|
||
<span class="nb">print</span> <span class="p">(</span><span class="s2">"2nd degree coefficients:"</span><span class="p">)</span>
|
||
<span class="nb">print</span> <span class="p">(</span><span class="s2">"zero power: "</span><span class="p">,</span><span class="n">c</span><span class="p">)</span>
|
||
<span class="nb">print</span> <span class="p">(</span><span class="s2">"first power: "</span><span class="p">,</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||
<span class="nb">print</span> <span class="p">(</span><span class="s2">"second power: "</span><span class="p">,</span><span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||
|
||
<span class="n">z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">steps</span><span class="p">,</span> <span class="mf">.01</span><span class="p">)</span>
|
||
<span class="n">z_mod</span><span class="o">=</span><span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">z</span><span class="o">+</span><span class="n">c</span>
|
||
|
||
<span class="n">fit_mod</span><span class="o">=</span><span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">X</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">c</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z_mod</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"2nd Degree Fit"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Polynomial Regression"</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"Steps"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Distance"</span><span class="p">)</span>
|
||
|
||
<span class="c1">#Degree 10</span>
|
||
<span class="n">poly_features10</span><span class="o">=</span><span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">include_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<span class="n">X_poly10</span><span class="o">=</span><span class="n">poly_features10</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
|
||
<span class="n">poly_fit10</span><span class="o">=</span><span class="n">lin_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_poly10</span><span class="p">,</span><span class="n">distance_list</span><span class="p">)</span>
|
||
|
||
<span class="n">y_plot</span><span class="o">=</span><span class="n">poly_fit10</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_poly10</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y_plot</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'black'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"10th Degree Fit"</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
|
||
<span class="c1">#Decision Tree Regression</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
|
||
<span class="n">regr_1</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">regr_2</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
<span class="n">regr_3</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">11</span><span class="p">)</span>
|
||
<span class="n">regr_1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">distance_list</span><span class="p">)</span>
|
||
<span class="n">regr_2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">distance_list</span><span class="p">)</span>
|
||
<span class="n">regr_3</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">distance_list</span><span class="p">)</span>
|
||
|
||
<span class="n">X_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">steps</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">)[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
|
||
<span class="n">y_1</span> <span class="o">=</span> <span class="n">regr_1</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="n">y_2</span> <span class="o">=</span> <span class="n">regr_2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="n">y_3</span><span class="o">=</span><span class="n">regr_3</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Plot the results</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">distance_list</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mf">2.5</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"data"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_1</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"red"</span><span class="p">,</span>
|
||
<span class="n">label</span><span class="o">=</span><span class="s2">"max_depth=2"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"green"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"max_depth=5"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_3</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"m"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"max_depth=7"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"Data"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Darget"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Decision Tree Regression"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2nd degree coefficients:
|
||
zero power: -9.690066070734556
|
||
first power: 0.2579633987876525
|
||
second power: -0.0008519856133366335
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/8e8566a48272bbcd741d192ea8f10c53172ad3cba8f805ceb38c6695c7dd5c2d.png" src="_images/8e8566a48272bbcd741d192ea8f10c53172ad3cba8f805ceb38c6695c7dd5c2d.png" />
|
||
<img alt="_images/eb601133698cb5198be6ed45531c33666c9fa7198b8b8a8e59f3cf4e95ddc8eb.png" src="_images/eb601133698cb5198be6ed45531c33666c9fa7198b8b8a8e59f3cf4e95ddc8eb.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="building-a-tree-regression">
|
||
<h2>Building a tree, regression<a class="headerlink" href="#building-a-tree-regression" title="Link to this heading">#</a></h2>
|
||
<p>There are mainly two steps</p>
|
||
<ol class="arabic simple">
|
||
<li><p>We split the predictor space (the set of possible values <span class="math notranslate nohighlight">\(x_1,x_2,\dots, x_p\)</span>) into <span class="math notranslate nohighlight">\(J\)</span> distinct and non-non-overlapping regions, <span class="math notranslate nohighlight">\(R_1,R_2,\dots,R_J\)</span>.</p></li>
|
||
<li><p>For every observation that falls into the region <span class="math notranslate nohighlight">\(R_j\)</span> , we make the same prediction, which is simply the mean of the response values for the training observations in <span class="math notranslate nohighlight">\(R_j\)</span>.</p></li>
|
||
</ol>
|
||
<p>How do we construct the regions <span class="math notranslate nohighlight">\(R_1,\dots,R_J\)</span>? 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 <span class="math notranslate nohighlight">\(R_1,\dots,R_J\)</span> that minimize the
|
||
MSE, given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sum_{j=1}^J\sum_{i\in R_j}(y_i-\overline{y}_{R_j})^2,
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\overline{y}_{R_j}\)</span> is the mean response for the training observations
|
||
within box <span class="math notranslate nohighlight">\(j\)</span>.</p>
|
||
</section>
|
||
<section id="a-top-down-approach-recursive-binary-splitting">
|
||
<h2>A top-down approach, recursive binary splitting<a class="headerlink" href="#a-top-down-approach-recursive-binary-splitting" title="Link to this heading">#</a></h2>
|
||
<p>Unfortunately, it is computationally infeasible to consider every
|
||
possible partition of the feature space into <span class="math notranslate nohighlight">\(J\)</span> boxes. The common
|
||
strategy is to take a top-down approach</p>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="making-a-tree">
|
||
<h2>Making a tree<a class="headerlink" href="#making-a-tree" title="Link to this heading">#</a></h2>
|
||
<p>In order to implement the recursive binary splitting we start by selecting
|
||
the predictor <span class="math notranslate nohighlight">\(x_j\)</span> and a cutpoint <span class="math notranslate nohighlight">\(s\)</span> that splits the predictor space into two regions <span class="math notranslate nohighlight">\(R_1\)</span> and <span class="math notranslate nohighlight">\(R_2\)</span></p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\left\{X\vert x_j < s\right\},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\left\{X\vert x_j \geq s\right\},
|
||
\]</div>
|
||
<p>so that we obtain the lowest MSE, that is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sum_{i:x_i\in R_j}(y_i-\overline{y}_{R_1})^2+\sum_{i:x_i\in R_2}(y_i-\overline{y}_{R_2})^2,
|
||
\]</div>
|
||
<p>which we want to minimize by considering all predictors
|
||
<span class="math notranslate nohighlight">\(x_1,x_2,\dots,x_p\)</span>. We consider also all possible values of <span class="math notranslate nohighlight">\(s\)</span> 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.</p>
|
||
<p>For any <span class="math notranslate nohighlight">\(j\)</span> and <span class="math notranslate nohighlight">\(s\)</span>, we define the pair of half-planes where
|
||
<span class="math notranslate nohighlight">\(\overline{y}_{R_1}\)</span> is the mean response for the training
|
||
observations in <span class="math notranslate nohighlight">\(R_1(j,s)\)</span>, and <span class="math notranslate nohighlight">\(\overline{y}_{R_2}\)</span> is the mean
|
||
response for the training observations in <span class="math notranslate nohighlight">\(R_2(j,s)\)</span>.</p>
|
||
<p>Finding the values of <span class="math notranslate nohighlight">\(j\)</span> and <span class="math notranslate nohighlight">\(s\)</span> that minimize the above equation can be
|
||
done quite quickly, especially when the number of features <span class="math notranslate nohighlight">\(p\)</span> is not
|
||
too large.</p>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="pruning-the-tree">
|
||
<h2>Pruning the tree<a class="headerlink" href="#pruning-the-tree" title="Link to this heading">#</a></h2>
|
||
<p>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 <span class="math notranslate nohighlight">\(T_0\)</span> 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.</p>
|
||
<p>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 <span class="math notranslate nohighlight">\(\alpha\)</span>.</p>
|
||
<p>Read more at the following <a class="reference external" href="https://scikit-learn.org/stable/auto_examples/tree/plot_cost_complexity_pruning.html#sphx-glr-auto-examples-tree-plot-cost-complexity-pruning-py">Scikit-Learn link on pruning</a>.</p>
|
||
</section>
|
||
<section id="cost-complexity-pruning">
|
||
<h2>Cost complexity pruning<a class="headerlink" href="#cost-complexity-pruning" title="Link to this heading">#</a></h2>
|
||
<p>For each value of <span class="math notranslate nohighlight">\(\alpha\)</span> there corresponds a subtree <span class="math notranslate nohighlight">\(T \in T_0\)</span> such that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sum_{m=1}^{\overline{T}}\sum_{i:x_i\in R_m}(y_i-\overline{y}_{R_m})^2+\alpha\overline{T},
|
||
\]</div>
|
||
<p>is as small as possible. Here <span class="math notranslate nohighlight">\(\overline{T}\)</span> is
|
||
the number of terminal nodes of the tree <span class="math notranslate nohighlight">\(T\)</span> , <span class="math notranslate nohighlight">\(R_m\)</span> is the
|
||
rectangle (i.e. the subset of predictor space) corresponding to the <span class="math notranslate nohighlight">\(m\)</span>-th terminal node.</p>
|
||
<p>The tuning parameter <span class="math notranslate nohighlight">\(\alpha\)</span> controls a trade-off between the subtree’s
|
||
complexity and its fit to the training data. When <span class="math notranslate nohighlight">\(\alpha = 0\)</span>, then the
|
||
subtree <span class="math notranslate nohighlight">\(T\)</span> will simply equal <span class="math notranslate nohighlight">\(T_0\)</span>,
|
||
because then the above equation just measures the
|
||
training error.
|
||
However, as <span class="math notranslate nohighlight">\(\alpha\)</span> 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.</p>
|
||
<p>It turns out that as we increase <span class="math notranslate nohighlight">\(\alpha\)</span> 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 <span class="math notranslate nohighlight">\(\alpha\)</span> is
|
||
easy. We can select a value of <span class="math notranslate nohighlight">\(\alpha\)</span> using a validation set or using
|
||
cross-validation. We then return to the full data set and obtain the
|
||
subtree corresponding to <span class="math notranslate nohighlight">\(\alpha\)</span>.</p>
|
||
</section>
|
||
<section id="schematic-regression-procedure">
|
||
<h2>Schematic Regression Procedure<a class="headerlink" href="#schematic-regression-procedure" title="Link to this heading">#</a></h2>
|
||
<p><strong>Building a Regression Tree.</strong></p>
|
||
<ol class="arabic simple">
|
||
<li><p>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.</p></li>
|
||
<li><p>Apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of <span class="math notranslate nohighlight">\(\alpha\)</span>.</p></li>
|
||
<li><p>Use for example <span class="math notranslate nohighlight">\(K\)</span>-fold cross-validation to choose <span class="math notranslate nohighlight">\(\alpha\)</span>. Divide the training observations into <span class="math notranslate nohighlight">\(K\)</span> folds. For each <span class="math notranslate nohighlight">\(k=1,2,\dots,K\)</span> we:</p></li>
|
||
</ol>
|
||
<ul class="simple">
|
||
<li><p>repeat steps 1 and 2 on all but the <span class="math notranslate nohighlight">\(k\)</span>-th fold of the training data.</p></li>
|
||
<li><p>Then we valuate the mean squared prediction error on the data in the left-out <span class="math notranslate nohighlight">\(k\)</span>-th fold, as a function of <span class="math notranslate nohighlight">\(\alpha\)</span>.</p></li>
|
||
<li><p>Finally we average the results for each value of <span class="math notranslate nohighlight">\(\alpha\)</span>, and pick <span class="math notranslate nohighlight">\(\alpha\)</span> to minimize the average error.</p></li>
|
||
</ul>
|
||
<ol class="arabic simple" start="4">
|
||
<li><p>Return the subtree from Step 2 that corresponds to the chosen value of <span class="math notranslate nohighlight">\(\alpha\)</span>.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="a-classification-tree">
|
||
<h2>A Classification Tree<a class="headerlink" href="#a-classification-tree" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="growing-a-classification-tree">
|
||
<h2>Growing a classification tree<a class="headerlink" href="#growing-a-classification-tree" title="Link to this heading">#</a></h2>
|
||
<p>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 <strong>classification
|
||
error rate</strong>. 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.</p>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="classification-tree-how-to-split-nodes">
|
||
<h2>Classification tree, how to split nodes<a class="headerlink" href="#classification-tree-how-to-split-nodes" title="Link to this heading">#</a></h2>
|
||
<p>If our targets are the outcome of a classification process that takes
|
||
for example <span class="math notranslate nohighlight">\(k=1,2,\dots,K\)</span> values, the only thing we need to think of
|
||
is to set up the splitting criteria for each node.</p>
|
||
<p>We define a PDF <span class="math notranslate nohighlight">\(p_{mk}\)</span> that represents the number of observations of
|
||
a class <span class="math notranslate nohighlight">\(k\)</span> in a region <span class="math notranslate nohighlight">\(R_m\)</span> with <span class="math notranslate nohighlight">\(N_m\)</span> observations. We represent
|
||
this likelihood function in terms of the proportion <span class="math notranslate nohighlight">\(I(y_i=k)\)</span> of
|
||
observations of this class in the region <span class="math notranslate nohighlight">\(R_m\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i=k).
|
||
\]</div>
|
||
<p>We let <span class="math notranslate nohighlight">\(p_{mk}\)</span> represent the majority class of observations in region
|
||
<span class="math notranslate nohighlight">\(m\)</span>. The three most common ways of splitting a node are given by</p>
|
||
<ul class="simple">
|
||
<li><p>Misclassification error</p></li>
|
||
</ul>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i\ne k) = 1-p_{mk}.
|
||
\]</div>
|
||
<ul class="simple">
|
||
<li><p>Gini index <span class="math notranslate nohighlight">\(g\)</span></p></li>
|
||
</ul>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g = \sum_{k=1}^K p_{mk}(1-p_{mk}).
|
||
\]</div>
|
||
<ul class="simple">
|
||
<li><p>Information entropy or just entropy <span class="math notranslate nohighlight">\(s\)</span></p></li>
|
||
</ul>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
s = -\sum_{k=1}^K p_{mk}\log{p_{mk}}.
|
||
\]</div>
|
||
</section>
|
||
<section id="visualizing-the-tree-classification">
|
||
<h2>Visualizing the Tree, Classification<a class="headerlink" href="#visualizing-the-tree-classification" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">confusion_matrix</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
|
||
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Categorical</span><span class="o">.</span><span class="n">from_codes</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span> <span class="n">cancer</span><span class="o">.</span><span class="n">target_names</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">get_dummies</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
|
||
<span class="n">export_graphviz</span><span class="p">(</span>
|
||
<span class="n">tree_clf</span><span class="p">,</span>
|
||
<span class="n">out_file</span><span class="o">=</span><span class="s2">"DataFiles/cancer.dot"</span><span class="p">,</span>
|
||
<span class="n">feature_names</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">,</span>
|
||
<span class="n">class_names</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">target_names</span><span class="p">,</span>
|
||
<span class="n">rounded</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
|
||
<span class="n">filled</span><span class="o">=</span><span class="kc">True</span>
|
||
<span class="p">)</span>
|
||
<span class="n">cmd</span> <span class="o">=</span> <span class="s1">'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">cmd</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> mean radius mean texture mean perimeter mean area mean smoothness \
|
||
0 17.99 10.38 122.80 1001.0 0.11840
|
||
1 20.57 17.77 132.90 1326.0 0.08474
|
||
2 19.69 21.25 130.00 1203.0 0.10960
|
||
3 11.42 20.38 77.58 386.1 0.14250
|
||
4 20.29 14.34 135.10 1297.0 0.10030
|
||
.. ... ... ... ... ...
|
||
564 21.56 22.39 142.00 1479.0 0.11100
|
||
565 20.13 28.25 131.20 1261.0 0.09780
|
||
566 16.60 28.08 108.30 858.1 0.08455
|
||
567 20.60 29.33 140.10 1265.0 0.11780
|
||
568 7.76 24.54 47.92 181.0 0.05263
|
||
|
||
mean compactness mean concavity mean concave points mean symmetry \
|
||
0 0.27760 0.30010 0.14710 0.2419
|
||
1 0.07864 0.08690 0.07017 0.1812
|
||
2 0.15990 0.19740 0.12790 0.2069
|
||
3 0.28390 0.24140 0.10520 0.2597
|
||
4 0.13280 0.19800 0.10430 0.1809
|
||
.. ... ... ... ...
|
||
564 0.11590 0.24390 0.13890 0.1726
|
||
565 0.10340 0.14400 0.09791 0.1752
|
||
566 0.10230 0.09251 0.05302 0.1590
|
||
567 0.27700 0.35140 0.15200 0.2397
|
||
568 0.04362 0.00000 0.00000 0.1587
|
||
|
||
mean fractal dimension ... worst radius worst texture \
|
||
0 0.07871 ... 25.380 17.33
|
||
1 0.05667 ... 24.990 23.41
|
||
2 0.05999 ... 23.570 25.53
|
||
3 0.09744 ... 14.910 26.50
|
||
4 0.05883 ... 22.540 16.67
|
||
.. ... ... ... ...
|
||
564 0.05623 ... 25.450 26.40
|
||
565 0.05533 ... 23.690 38.25
|
||
566 0.05648 ... 18.980 34.12
|
||
567 0.07016 ... 25.740 39.42
|
||
568 0.05884 ... 9.456 30.37
|
||
|
||
worst perimeter worst area worst smoothness worst compactness \
|
||
0 184.60 2019.0 0.16220 0.66560
|
||
1 158.80 1956.0 0.12380 0.18660
|
||
2 152.50 1709.0 0.14440 0.42450
|
||
3 98.87 567.7 0.20980 0.86630
|
||
4 152.20 1575.0 0.13740 0.20500
|
||
.. ... ... ... ...
|
||
564 166.10 2027.0 0.14100 0.21130
|
||
565 155.00 1731.0 0.11660 0.19220
|
||
566 126.70 1124.0 0.11390 0.30940
|
||
567 184.60 1821.0 0.16500 0.86810
|
||
568 59.16 268.6 0.08996 0.06444
|
||
|
||
worst concavity worst concave points worst symmetry \
|
||
0 0.7119 0.2654 0.4601
|
||
1 0.2416 0.1860 0.2750
|
||
2 0.4504 0.2430 0.3613
|
||
3 0.6869 0.2575 0.6638
|
||
4 0.4000 0.1625 0.2364
|
||
.. ... ... ...
|
||
564 0.4107 0.2216 0.2060
|
||
565 0.3215 0.1628 0.2572
|
||
566 0.3403 0.1418 0.2218
|
||
567 0.9387 0.2650 0.4087
|
||
568 0.0000 0.0000 0.2871
|
||
|
||
worst fractal dimension
|
||
0 0.11890
|
||
1 0.08902
|
||
2 0.08758
|
||
3 0.17300
|
||
4 0.07678
|
||
.. ...
|
||
564 0.07115
|
||
565 0.06637
|
||
566 0.07820
|
||
567 0.12400
|
||
568 0.07039
|
||
|
||
[569 rows x 30 columns]
|
||
malignant benign
|
||
0 True False
|
||
1 True False
|
||
2 True False
|
||
3 True False
|
||
4 True False
|
||
.. ... ...
|
||
564 True False
|
||
565 True False
|
||
566 True False
|
||
567 True False
|
||
568 False True
|
||
|
||
[569 rows x 2 columns]
|
||
</pre></div>
|
||
</div>
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="visualizing-the-tree-the-moons">
|
||
<h2>Visualizing the Tree, The Moons<a class="headerlink" href="#visualizing-the-tree-the-moons" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_moons</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_moons</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">53</span><span class="p">)</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
|
||
<span class="n">export_graphviz</span><span class="p">(</span>
|
||
<span class="n">tree_clf</span><span class="p">,</span>
|
||
<span class="n">out_file</span><span class="o">=</span><span class="s2">"DataFiles/moons.dot"</span><span class="p">,</span>
|
||
<span class="n">rounded</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
|
||
<span class="n">filled</span><span class="o">=</span><span class="kc">True</span>
|
||
<span class="p">)</span>
|
||
<span class="n">cmd</span> <span class="o">=</span> <span class="s1">'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">cmd</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="other-ways-of-visualizing-the-trees">
|
||
<h2>Other ways of visualizing the trees<a class="headerlink" href="#other-ways-of-visualizing-the-trees" title="Link to this heading">#</a></h2>
|
||
<p><strong>Scikit-Learn</strong> has also another way to visualize the trees which is very useful, here with the Iris data.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">tree</span>
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">(</span><span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">DecisionTreeClassifier</span><span class="p">()</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="c1"># and then plot the tree</span>
|
||
<span class="n">tree</span><span class="o">.</span><span class="n">plot_tree</span><span class="p">(</span><span class="n">tree_clf</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[Text(0.5, 0.9166666666666666, 'x[2] <= 2.45\ngini = 0.667\nsamples = 150\nvalue = [50, 50, 50]'),
|
||
Text(0.4230769230769231, 0.75, 'gini = 0.0\nsamples = 50\nvalue = [50, 0, 0]'),
|
||
Text(0.5769230769230769, 0.75, 'x[3] <= 1.75\ngini = 0.5\nsamples = 100\nvalue = [0, 50, 50]'),
|
||
Text(0.3076923076923077, 0.5833333333333334, 'x[2] <= 4.95\ngini = 0.168\nsamples = 54\nvalue = [0, 49, 5]'),
|
||
Text(0.15384615384615385, 0.4166666666666667, 'x[3] <= 1.65\ngini = 0.041\nsamples = 48\nvalue = [0, 47, 1]'),
|
||
Text(0.07692307692307693, 0.25, 'gini = 0.0\nsamples = 47\nvalue = [0, 47, 0]'),
|
||
Text(0.23076923076923078, 0.25, 'gini = 0.0\nsamples = 1\nvalue = [0, 0, 1]'),
|
||
Text(0.46153846153846156, 0.4166666666666667, 'x[3] <= 1.55\ngini = 0.444\nsamples = 6\nvalue = [0, 2, 4]'),
|
||
Text(0.38461538461538464, 0.25, 'gini = 0.0\nsamples = 3\nvalue = [0, 0, 3]'),
|
||
Text(0.5384615384615384, 0.25, 'x[2] <= 5.45\ngini = 0.444\nsamples = 3\nvalue = [0, 2, 1]'),
|
||
Text(0.46153846153846156, 0.08333333333333333, 'gini = 0.0\nsamples = 2\nvalue = [0, 2, 0]'),
|
||
Text(0.6153846153846154, 0.08333333333333333, 'gini = 0.0\nsamples = 1\nvalue = [0, 0, 1]'),
|
||
Text(0.8461538461538461, 0.5833333333333334, 'x[2] <= 4.85\ngini = 0.043\nsamples = 46\nvalue = [0, 1, 45]'),
|
||
Text(0.7692307692307693, 0.4166666666666667, 'x[1] <= 3.1\ngini = 0.444\nsamples = 3\nvalue = [0, 1, 2]'),
|
||
Text(0.6923076923076923, 0.25, 'gini = 0.0\nsamples = 2\nvalue = [0, 0, 2]'),
|
||
Text(0.8461538461538461, 0.25, 'gini = 0.0\nsamples = 1\nvalue = [0, 1, 0]'),
|
||
Text(0.9230769230769231, 0.4166666666666667, 'gini = 0.0\nsamples = 43\nvalue = [0, 0, 43]')]
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/172361305e097693c6f402407343ec25829eaddb59c9d765e220d229a8125b56.png" src="_images/172361305e097693c6f402407343ec25829eaddb59c9d765e220d229a8125b56.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="printing-out-as-text">
|
||
<h2>Printing out as text<a class="headerlink" href="#printing-out-as-text" title="Link to this heading">#</a></h2>
|
||
<p>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:</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_text</span>
|
||
<span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</span>
|
||
<span class="n">decision_tree</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">decision_tree</span> <span class="o">=</span> <span class="n">decision_tree</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">)</span>
|
||
<span class="n">r</span> <span class="o">=</span> <span class="n">export_text</span><span class="p">(</span><span class="n">decision_tree</span><span class="p">,</span> <span class="n">feature_names</span><span class="o">=</span><span class="n">iris</span><span class="p">[</span><span class="s1">'feature_names'</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>|--- petal width (cm) <= 0.80
|
||
| |--- class: 0
|
||
|--- petal width (cm) > 0.80
|
||
| |--- petal width (cm) <= 1.75
|
||
| | |--- class: 1
|
||
| |--- petal width (cm) > 1.75
|
||
| | |--- class: 2
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="algorithms-for-setting-up-decision-trees">
|
||
<h2>Algorithms for Setting up Decision Trees<a class="headerlink" href="#algorithms-for-setting-up-decision-trees" title="Link to this heading">#</a></h2>
|
||
<p>Two algorithms stand out in the set up of decision trees:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>The CART (Classification And Regression Tree) algorithm for both classification and regression</p></li>
|
||
<li><p>The ID3 algorithm based on the computation of the information gain for classification</p></li>
|
||
</ol>
|
||
<p>We discuss both algorithms with applications here. The popular library
|
||
<strong>Scikit-Learn</strong> uses the CART algorithm. For classification problems
|
||
you can use either the <strong>gini</strong> index or the <strong>entropy</strong> to split a tree
|
||
in two branches.</p>
|
||
</section>
|
||
<section id="the-cart-algorithm-for-classification">
|
||
<h2>The CART algorithm for Classification<a class="headerlink" href="#the-cart-algorithm-for-classification" title="Link to this heading">#</a></h2>
|
||
<p>For classification, the CART algorithm splits the data set in two subsets using a single feature <span class="math notranslate nohighlight">\(k\)</span> and a threshold <span class="math notranslate nohighlight">\(t_k\)</span>.
|
||
This could be for example a threshold set by a number below a certain circumference of a malign tumor.</p>
|
||
<p>How do we find these two quantities?
|
||
We search for the pair <span class="math notranslate nohighlight">\((k,t_k)\)</span> that produces the purest subset using for example the <strong>gini</strong> factor <span class="math notranslate nohighlight">\(G\)</span>.
|
||
The cost function it tries to minimize is then</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}},
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(G_{\mathrm{left/right}}\)</span> measures the impurity of the left/right subset and <span class="math notranslate nohighlight">\(m_{\mathrm{left/right}}\)</span>
|
||
is the number of instances in the left/right subset</p>
|
||
<p>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
|
||
<span class="math notranslate nohighlight">\(max\_depth\)</span> hyperparameter), or if it cannot find a split that will reduce impurity. A few other
|
||
hyperparameters control additional stopping conditions such as the <span class="math notranslate nohighlight">\(min\_samples\_split\)</span>,
|
||
<span class="math notranslate nohighlight">\(min\_samples\_leaf\)</span>, <span class="math notranslate nohighlight">\(min\_weight\_fraction\_leaf\)</span>, and <span class="math notranslate nohighlight">\(max\_leaf\_nodes\)</span>.</p>
|
||
</section>
|
||
<section id="the-cart-algorithm-for-regression">
|
||
<h2>The CART algorithm for Regression<a class="headerlink" href="#the-cart-algorithm-for-regression" title="Link to this heading">#</a></h2>
|
||
<p>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 <strong>gini</strong> or <strong>entropy</strong> 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</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}.
|
||
\]</div>
|
||
<p>Here the MSE for a specific node is defined as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2,
|
||
\]</div>
|
||
<p>with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i,
|
||
\]</div>
|
||
<p>the mean value of all observations in a specific node.</p>
|
||
<p>Without any regularization, the regression task for decision trees,
|
||
just like for classification tasks, is prone to overfitting.</p>
|
||
</section>
|
||
<section id="why-binary-splits">
|
||
<h2>Why binary splits?<a class="headerlink" href="#why-binary-splits" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="computing-a-tree-using-the-gini-index">
|
||
<h2>Computing a Tree using the Gini Index<a class="headerlink" href="#computing-a-tree-using-the-gini-index" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
<p>We have three features/attributes</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Trend of average grades before present course, classified as either below or above the average grade of the whole class</p></li>
|
||
<li><p>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 <span class="math notranslate nohighlight">\(ECTS\)</span> 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.</p></li>
|
||
<li><p>The number of hours slept as high for more than <span class="math notranslate nohighlight">\(8\)</span> hours and below for less than 8 hours of sleep, classified again as either high or low</p></li>
|
||
<li><p>The final grade whether it is above or below average</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="the-table">
|
||
<h2>The Table<a class="headerlink" href="#the-table" title="Link to this heading">#</a></h2>
|
||
<table class="dotable" border="1">
|
||
<thead>
|
||
<tr><th align="center">Grade Trend</th> <th align="center">Hours slept</th> <th align="center">Hours Studied</th> <th align="center">Grade</th> </tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr><td align="center"> Above </td> <td align="center"> Low </td> <td align="center"> High </td> <td align="center"> Above </td> </tr>
|
||
<tr><td align="center"> Below </td> <td align="center"> High </td> <td align="center"> Low </td> <td align="center"> Below </td> </tr>
|
||
<tr><td align="center"> Above </td> <td align="center"> Low </td> <td align="center"> High </td> <td align="center"> Above </td> </tr>
|
||
<tr><td align="center"> Above </td> <td align="center"> High </td> <td align="center"> High </td> <td align="center"> Above </td> </tr>
|
||
<tr><td align="center"> Below </td> <td align="center"> Low </td> <td align="center"> High </td> <td align="center"> Below </td> </tr>
|
||
<tr><td align="center"> Above </td> <td align="center"> Low </td> <td align="center"> Low </td> <td align="center"> Below </td> </tr>
|
||
<tr><td align="center"> Below </td> <td align="center"> High </td> <td align="center"> High </td> <td align="center"> Below </td> </tr>
|
||
<tr><td align="center"> Below </td> <td align="center"> Low </td> <td align="center"> High </td> <td align="center"> Below </td> </tr>
|
||
<tr><td align="center"> Above </td> <td align="center"> Low </td> <td align="center"> Low </td> <td align="center"> Below </td> </tr>
|
||
<tr><td align="center"> Above </td> <td align="center"> High </td> <td align="center"> High </td> <td align="center"> Above </td> </tr>
|
||
</tbody>
|
||
</table></section>
|
||
<section id="computing-the-various-gini-indices">
|
||
<h2>Computing the various Gini Indices<a class="headerlink" href="#computing-the-various-gini-indices" title="Link to this heading">#</a></h2>
|
||
<p>In computations we will translate all classes into numbers. Being
|
||
these binary classes, they can easily be split into ones and zeros.</p>
|
||
<p><strong>Gini index for Average trend.</strong></p>
|
||
<p><a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf">See handwritten notes November 3</a></p>
|
||
</section>
|
||
<section id="computing-the-various-gini-indices-hours-slept">
|
||
<h2>Computing the various Gini Indices, Hours slept<a class="headerlink" href="#computing-the-various-gini-indices-hours-slept" title="Link to this heading">#</a></h2>
|
||
<p><strong>Gini index for hour slept.</strong></p>
|
||
<p><a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf">See handwritten notes November 3</a></p>
|
||
</section>
|
||
<section id="computing-the-various-gini-indices-hours-studied">
|
||
<h2>Computing the various Gini Indices, Hours studied<a class="headerlink" href="#computing-the-various-gini-indices-hours-studied" title="Link to this heading">#</a></h2>
|
||
<p><strong>Gini index for hour studied.</strong></p>
|
||
<p><a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2022/NotesNov32022.pdf">See handwritten notes November 3</a></p>
|
||
<p>For final tree, see the above handwritten notes</p>
|
||
</section>
|
||
<section id="a-possible-code-using-scikit-learn">
|
||
<h2>A possible code using Scikit-Learn<a class="headerlink" href="#a-possible-code-using-scikit-learn" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">OneHotEncoder</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.compose</span> <span class="kn">import</span> <span class="n">ColumnTransformer</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
|
||
<span class="c1"># Where to save the figures and data files</span>
|
||
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">"Results"</span>
|
||
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">"Results/FigureFiles"</span>
|
||
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">"DataFiles/"</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||
|
||
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">"grades.csv"</span><span class="p">),</span><span class="s1">'r'</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Read the experimental data with Pandas</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
|
||
<span class="n">grades</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">)</span>
|
||
<span class="n">grades</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">grades</span><span class="p">)</span>
|
||
<span class="n">display</span><span class="p">(</span><span class="n">grades</span><span class="p">)</span>
|
||
<span class="c1"># Features and targets</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">grades</span><span class="o">.</span><span class="n">loc</span><span class="p">[:,</span> <span class="n">grades</span><span class="o">.</span><span class="n">columns</span> <span class="o">!=</span> <span class="s1">'Grade'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">grades</span><span class="o">.</span><span class="n">loc</span><span class="p">[:,</span> <span class="n">grades</span><span class="o">.</span><span class="n">columns</span> <span class="o">==</span> <span class="s1">'Grade'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="c1"># Then do a Classification tree</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Train set accuracy with Decision Tree: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">tree_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">y</span><span class="p">)))</span>
|
||
<span class="c1">#transfer to a decision tree graph</span>
|
||
<span class="n">export_graphviz</span><span class="p">(</span>
|
||
<span class="n">tree_clf</span><span class="p">,</span>
|
||
<span class="n">out_file</span><span class="o">=</span><span class="s2">"DataFiles/grade.dot"</span><span class="p">,</span>
|
||
<span class="n">rounded</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
|
||
<span class="n">filled</span><span class="o">=</span><span class="kc">True</span>
|
||
<span class="p">)</span>
|
||
<span class="n">cmd</span> <span class="o">=</span> <span class="s1">'dot -Tpng DataFiles/grade.dot -o DataFiles/grades.png'</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">cmd</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_html"><div>
|
||
<style scoped>
|
||
.dataframe tbody tr th:only-of-type {
|
||
vertical-align: middle;
|
||
}
|
||
|
||
.dataframe tbody tr th {
|
||
vertical-align: top;
|
||
}
|
||
|
||
.dataframe thead th {
|
||
text-align: right;
|
||
}
|
||
</style>
|
||
<table border="1" class="dataframe">
|
||
<thead>
|
||
<tr style="text-align: right;">
|
||
<th></th>
|
||
<th>Grade Trend</th>
|
||
<th>Hours slept</th>
|
||
<th>Hours Studied</th>
|
||
<th>Grade</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<th>0</th>
|
||
<td>1</td>
|
||
<td>0</td>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr>
|
||
<th>1</th>
|
||
<td>0</td>
|
||
<td>1</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr>
|
||
<th>2</th>
|
||
<td>1</td>
|
||
<td>0</td>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr>
|
||
<th>3</th>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
</tr>
|
||
<tr>
|
||
<th>4</th>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>1</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr>
|
||
<th>5</th>
|
||
<td>1</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr>
|
||
<th>6</th>
|
||
<td>0</td>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr>
|
||
<th>7</th>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>1</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr>
|
||
<th>8</th>
|
||
<td>1</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr>
|
||
<th>9</th>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div></div><div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[1 0 1]
|
||
[0 1 0]
|
||
[1 0 1]
|
||
[1 1 1]
|
||
[0 0 1]
|
||
[1 0 0]
|
||
[0 1 1]
|
||
[0 0 1]
|
||
[1 0 0]
|
||
[1 1 1]]
|
||
Train set accuracy with Decision Tree: 1.00
|
||
</pre></div>
|
||
</div>
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="further-example-computing-the-gini-index">
|
||
<h2>Further example: Computing the Gini index<a class="headerlink" href="#further-example-computing-the-gini-index" title="Link to this heading">#</a></h2>
|
||
<p>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 <strong>outlook</strong>, <strong>temperature</strong>,
|
||
<strong>humidity</strong> and <strong>wind</strong>. 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.</p>
|
||
<p>The table here summarizes the various attributes and</p>
|
||
<table class="dotable" border="1">
|
||
<thead>
|
||
<tr><th align="center">Day</th> <th align="center">Outlook </th> <th align="center">Temperature</th> <th align="center">Humidity</th> <th align="center"> Wind </th> <th align="center">Ride</th> </tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr><td align="center"> 1 </td> <td align="center"> Sunny </td> <td align="center"> Hot </td> <td align="center"> High </td> <td align="center"> Weak </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 2 </td> <td align="center"> Sunny </td> <td align="center"> Hot </td> <td align="center"> High </td> <td align="center"> Strong </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 3 </td> <td align="center"> Overcast </td> <td align="center"> Hot </td> <td align="center"> High </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 4 </td> <td align="center"> Rain </td> <td align="center"> Mild </td> <td align="center"> High </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 5 </td> <td align="center"> Rain </td> <td align="center"> Cool </td> <td align="center"> Normal </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 6 </td> <td align="center"> Rain </td> <td align="center"> Cool </td> <td align="center"> Normal </td> <td align="center"> Strong </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 7 </td> <td align="center"> Overcast </td> <td align="center"> Cool </td> <td align="center"> Normal </td> <td align="center"> Strong </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 8 </td> <td align="center"> Sunny </td> <td align="center"> Mild </td> <td align="center"> High </td> <td align="center"> Weak </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 9 </td> <td align="center"> Sunny </td> <td align="center"> Cool </td> <td align="center"> Normal </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 10 </td> <td align="center"> Rain </td> <td align="center"> Mild </td> <td align="center"> Normal </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 11 </td> <td align="center"> Sunny </td> <td align="center"> Mild </td> <td align="center"> Normal </td> <td align="center"> Strong </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 12 </td> <td align="center"> Overcast </td> <td align="center"> Mild </td> <td align="center"> High </td> <td align="center"> Strong </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 13 </td> <td align="center"> Overcast </td> <td align="center"> Hot </td> <td align="center"> Normal </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 14 </td> <td align="center"> Rain </td> <td align="center"> Mild </td> <td align="center"> High </td> <td align="center"> Strong </td> <td align="center"> 0 </td> </tr>
|
||
</tbody>
|
||
</table></section>
|
||
<section id="simple-python-code-to-read-in-data-and-perform-classification">
|
||
<h2>Simple Python Code to read in Data and perform Classification<a class="headerlink" href="#simple-python-code-to-read-in-data-and-perform-classification" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">OneHotEncoder</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.compose</span> <span class="kn">import</span> <span class="n">ColumnTransformer</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
|
||
<span class="c1"># Where to save the figures and data files</span>
|
||
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">"Results"</span>
|
||
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">"Results/FigureFiles"</span>
|
||
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">"DataFiles/"</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||
|
||
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">"rideclass.csv"</span><span class="p">),</span><span class="s1">'r'</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Read the experimental data with Pandas</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
|
||
<span class="n">ridedata</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span><span class="n">names</span> <span class="o">=</span> <span class="p">(</span><span class="s1">'Outlook'</span><span class="p">,</span><span class="s1">'Temperature'</span><span class="p">,</span><span class="s1">'Humidity'</span><span class="p">,</span><span class="s1">'Wind'</span><span class="p">,</span><span class="s1">'Ride'</span><span class="p">))</span>
|
||
<span class="n">ridedata</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">ridedata</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Features and targets</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">ridedata</span><span class="o">.</span><span class="n">loc</span><span class="p">[:,</span> <span class="n">ridedata</span><span class="o">.</span><span class="n">columns</span> <span class="o">!=</span> <span class="s1">'Ride'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">ridedata</span><span class="o">.</span><span class="n">loc</span><span class="p">[:,</span> <span class="n">ridedata</span><span class="o">.</span><span class="n">columns</span> <span class="o">==</span> <span class="s1">'Ride'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
|
||
|
||
<span class="c1"># Create the encoder.</span>
|
||
<span class="n">encoder</span> <span class="o">=</span> <span class="n">OneHotEncoder</span><span class="p">(</span><span class="n">handle_unknown</span><span class="o">=</span><span class="s2">"ignore"</span><span class="p">)</span>
|
||
<span class="c1"># Assume for simplicity all features are categorical.</span>
|
||
<span class="n">encoder</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="c1"># Apply the encoder.</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">encoder</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="c1"># Then do a Classification tree</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Train set accuracy with Decision Tree: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">tree_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">y</span><span class="p">)))</span>
|
||
<span class="c1">#transfer to a decision tree graph</span>
|
||
<span class="n">export_graphviz</span><span class="p">(</span>
|
||
<span class="n">tree_clf</span><span class="p">,</span>
|
||
<span class="n">out_file</span><span class="o">=</span><span class="s2">"DataFiles/ride.dot"</span><span class="p">,</span>
|
||
<span class="n">rounded</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
|
||
<span class="n">filled</span><span class="o">=</span><span class="kc">True</span>
|
||
<span class="p">)</span>
|
||
<span class="n">cmd</span> <span class="o">=</span> <span class="s1">'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">cmd</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> (0, 0) 1.0
|
||
(0, 7) 1.0
|
||
(0, 9) 1.0
|
||
(0, 13) 1.0
|
||
(1, 3) 1.0
|
||
(1, 5) 1.0
|
||
(1, 8) 1.0
|
||
(1, 12) 1.0
|
||
(2, 3) 1.0
|
||
(2, 5) 1.0
|
||
(2, 8) 1.0
|
||
(2, 11) 1.0
|
||
(3, 1) 1.0
|
||
(3, 5) 1.0
|
||
(3, 8) 1.0
|
||
(3, 12) 1.0
|
||
(4, 2) 1.0
|
||
(4, 6) 1.0
|
||
(4, 8) 1.0
|
||
(4, 12) 1.0
|
||
(5, 2) 1.0
|
||
(5, 4) 1.0
|
||
(5, 10) 1.0
|
||
(5, 12) 1.0
|
||
(6, 2) 1.0
|
||
: :
|
||
(8, 12) 1.0
|
||
(9, 3) 1.0
|
||
(9, 4) 1.0
|
||
(9, 10) 1.0
|
||
(9, 12) 1.0
|
||
(10, 2) 1.0
|
||
(10, 6) 1.0
|
||
(10, 10) 1.0
|
||
(10, 12) 1.0
|
||
(11, 3) 1.0
|
||
(11, 6) 1.0
|
||
(11, 10) 1.0
|
||
(11, 11) 1.0
|
||
(12, 1) 1.0
|
||
(12, 6) 1.0
|
||
(12, 8) 1.0
|
||
(12, 11) 1.0
|
||
(13, 1) 1.0
|
||
(13, 5) 1.0
|
||
(13, 10) 1.0
|
||
(13, 12) 1.0
|
||
(14, 2) 1.0
|
||
(14, 6) 1.0
|
||
(14, 8) 1.0
|
||
(14, 11) 1.0
|
||
Train set accuracy with Decision Tree: 0.73
|
||
</pre></div>
|
||
</div>
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="computing-the-gini-factor">
|
||
<h2>Computing the Gini Factor<a class="headerlink" href="#computing-the-gini-factor" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
<p>In the example here we have converted all our attributes into numerical values <span class="math notranslate nohighlight">\(0,1,2\)</span> etc.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Split a dataset based on an attribute and an attribute value</span>
|
||
<span class="k">def</span> <span class="nf">test_split</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">value</span><span class="p">,</span> <span class="n">dataset</span><span class="p">):</span>
|
||
<span class="n">left</span><span class="p">,</span> <span class="n">right</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(),</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">dataset</span><span class="p">:</span>
|
||
<span class="k">if</span> <span class="n">row</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o"><</span> <span class="n">value</span><span class="p">:</span>
|
||
<span class="n">left</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">right</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span>
|
||
|
||
<span class="c1"># Calculate the Gini index for a split dataset</span>
|
||
<span class="k">def</span> <span class="nf">gini_index</span><span class="p">(</span><span class="n">groups</span><span class="p">,</span> <span class="n">classes</span><span class="p">):</span>
|
||
<span class="c1"># count all samples at split point</span>
|
||
<span class="n">n_instances</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="nb">sum</span><span class="p">([</span><span class="nb">len</span><span class="p">(</span><span class="n">group</span><span class="p">)</span> <span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">]))</span>
|
||
<span class="c1"># sum weighted Gini index for each group</span>
|
||
<span class="n">gini</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
<span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">:</span>
|
||
<span class="n">size</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">group</span><span class="p">))</span>
|
||
<span class="c1"># avoid divide by zero</span>
|
||
<span class="k">if</span> <span class="n">size</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="k">continue</span>
|
||
<span class="n">score</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
<span class="c1"># score the group based on the score for each class</span>
|
||
<span class="k">for</span> <span class="n">class_val</span> <span class="ow">in</span> <span class="n">classes</span><span class="p">:</span>
|
||
<span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="n">row</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">group</span><span class="p">]</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">class_val</span><span class="p">)</span> <span class="o">/</span> <span class="n">size</span>
|
||
<span class="n">score</span> <span class="o">+=</span> <span class="n">p</span> <span class="o">*</span> <span class="n">p</span>
|
||
<span class="c1"># weight the group score by its relative size</span>
|
||
<span class="n">gini</span> <span class="o">+=</span> <span class="p">(</span><span class="mf">1.0</span> <span class="o">-</span> <span class="n">score</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">size</span> <span class="o">/</span> <span class="n">n_instances</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">gini</span>
|
||
|
||
<span class="c1"># Select the best split point for a dataset</span>
|
||
<span class="k">def</span> <span class="nf">get_split</span><span class="p">(</span><span class="n">dataset</span><span class="p">):</span>
|
||
<span class="n">class_values</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">row</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">dataset</span><span class="p">))</span>
|
||
<span class="n">b_index</span><span class="p">,</span> <span class="n">b_value</span><span class="p">,</span> <span class="n">b_score</span><span class="p">,</span> <span class="n">b_groups</span> <span class="o">=</span> <span class="mi">999</span><span class="p">,</span> <span class="mi">999</span><span class="p">,</span> <span class="mi">999</span><span class="p">,</span> <span class="kc">None</span>
|
||
<span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">dataset</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">dataset</span><span class="p">:</span>
|
||
<span class="n">groups</span> <span class="o">=</span> <span class="n">test_split</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">row</span><span class="p">[</span><span class="n">index</span><span class="p">],</span> <span class="n">dataset</span><span class="p">)</span>
|
||
<span class="n">gini</span> <span class="o">=</span> <span class="n">gini_index</span><span class="p">(</span><span class="n">groups</span><span class="p">,</span> <span class="n">class_values</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'X</span><span class="si">%d</span><span class="s1"> < </span><span class="si">%.3f</span><span class="s1"> Gini=</span><span class="si">%.3f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">((</span><span class="n">index</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="n">row</span><span class="p">[</span><span class="n">index</span><span class="p">],</span> <span class="n">gini</span><span class="p">))</span>
|
||
<span class="k">if</span> <span class="n">gini</span> <span class="o"><</span> <span class="n">b_score</span><span class="p">:</span>
|
||
<span class="n">b_index</span><span class="p">,</span> <span class="n">b_value</span><span class="p">,</span> <span class="n">b_score</span><span class="p">,</span> <span class="n">b_groups</span> <span class="o">=</span> <span class="n">index</span><span class="p">,</span> <span class="n">row</span><span class="p">[</span><span class="n">index</span><span class="p">],</span> <span class="n">gini</span><span class="p">,</span> <span class="n">groups</span>
|
||
<span class="k">return</span> <span class="p">{</span><span class="s1">'index'</span><span class="p">:</span><span class="n">b_index</span><span class="p">,</span> <span class="s1">'value'</span><span class="p">:</span><span class="n">b_value</span><span class="p">,</span> <span class="s1">'groups'</span><span class="p">:</span><span class="n">b_groups</span><span class="p">}</span>
|
||
|
||
<span class="n">dataset</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]]</span>
|
||
|
||
<span class="n">split</span> <span class="o">=</span> <span class="n">get_split</span><span class="p">(</span><span class="n">dataset</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Split: [X</span><span class="si">%d</span><span class="s1"> < </span><span class="si">%.3f</span><span class="s1">]'</span> <span class="o">%</span> <span class="p">((</span><span class="n">split</span><span class="p">[</span><span class="s1">'index'</span><span class="p">]</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="n">split</span><span class="p">[</span><span class="s1">'value'</span><span class="p">]))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>X1 < 0.000 Gini=0.408
|
||
X1 < 0.000 Gini=0.408
|
||
X1 < 1.000 Gini=0.394
|
||
X1 < 2.000 Gini=0.394
|
||
X1 < 2.000 Gini=0.394
|
||
X1 < 2.000 Gini=0.394
|
||
X1 < 1.000 Gini=0.394
|
||
X1 < 0.000 Gini=0.408
|
||
X1 < 0.000 Gini=0.408
|
||
X1 < 2.000 Gini=0.394
|
||
X1 < 0.000 Gini=0.408
|
||
X1 < 1.000 Gini=0.394
|
||
X1 < 1.000 Gini=0.394
|
||
X1 < 2.000 Gini=0.394
|
||
X2 < 0.000 Gini=0.408
|
||
X2 < 0.000 Gini=0.408
|
||
X2 < 0.000 Gini=0.408
|
||
X2 < 1.000 Gini=0.407
|
||
X2 < 2.000 Gini=0.407
|
||
X2 < 2.000 Gini=0.407
|
||
X2 < 2.000 Gini=0.407
|
||
X2 < 1.000 Gini=0.407
|
||
X2 < 2.000 Gini=0.407
|
||
X2 < 1.000 Gini=0.407
|
||
X2 < 1.000 Gini=0.407
|
||
X2 < 1.000 Gini=0.407
|
||
X2 < 0.000 Gini=0.408
|
||
X2 < 1.000 Gini=0.407
|
||
X3 < 0.000 Gini=0.408
|
||
X3 < 0.000 Gini=0.408
|
||
X3 < 0.000 Gini=0.408
|
||
X3 < 0.000 Gini=0.408
|
||
X3 < 1.000 Gini=0.367
|
||
X3 < 1.000 Gini=0.367
|
||
X3 < 1.000 Gini=0.367
|
||
X3 < 0.000 Gini=0.408
|
||
X3 < 1.000 Gini=0.367
|
||
X3 < 1.000 Gini=0.367
|
||
X3 < 1.000 Gini=0.367
|
||
X3 < 0.000 Gini=0.408
|
||
X3 < 1.000 Gini=0.367
|
||
X3 < 0.000 Gini=0.408
|
||
X4 < 0.000 Gini=0.408
|
||
X4 < 1.000 Gini=0.405
|
||
X4 < 0.000 Gini=0.408
|
||
X4 < 0.000 Gini=0.408
|
||
X4 < 0.000 Gini=0.408
|
||
X4 < 1.000 Gini=0.405
|
||
X4 < 1.000 Gini=0.405
|
||
X4 < 0.000 Gini=0.408
|
||
X4 < 0.000 Gini=0.408
|
||
X4 < 0.000 Gini=0.408
|
||
X4 < 1.000 Gini=0.405
|
||
X4 < 1.000 Gini=0.405
|
||
X4 < 0.000 Gini=0.408
|
||
X4 < 1.000 Gini=0.405
|
||
Split: [X3 < 1.000]
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="regression-trees">
|
||
<h2>Regression trees<a class="headerlink" href="#regression-trees" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Quadratic training set + noise</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">m</span> <span class="o">=</span> <span class="mi">200</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span> <span class="o">*</span> <span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">10</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
|
||
|
||
<span class="n">tree_reg</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">tree_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_html"><style>#sk-container-id-1 {color: black;}#sk-container-id-1 pre{padding: 0;}#sk-container-id-1 div.sk-toggleable {background-color: white;}#sk-container-id-1 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-1 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-1 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-1 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-1 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-1 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-1 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-1 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-1 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-1 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-1 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-1 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-1 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-1 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-1 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-1 div.sk-item {position: relative;z-index: 1;}#sk-container-id-1 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-1 div.sk-item::before, #sk-container-id-1 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-1 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-1 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-1 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-1 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-1 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-1 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-1 div.sk-label-container {text-align: center;}#sk-container-id-1 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-1 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-1" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>DecisionTreeRegressor(max_depth=2, random_state=42)</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-1" type="checkbox" checked><label for="sk-estimator-id-1" class="sk-toggleable__label sk-toggleable__label-arrow">DecisionTreeRegressor</label><div class="sk-toggleable__content"><pre>DecisionTreeRegressor(max_depth=2, random_state=42)</pre></div></div></div></div></div></div></div>
|
||
</div>
|
||
</section>
|
||
<section id="final-regressor-code">
|
||
<h2>Final regressor code<a class="headerlink" href="#final-regressor-code" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
|
||
|
||
<span class="n">tree_reg1</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">tree_reg2</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
|
||
<span class="n">tree_reg1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="n">tree_reg2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">plot_regression_predictions</span><span class="p">(</span><span class="n">tree_reg</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">ylabel</span><span class="o">=</span><span class="s2">"$y$"</span><span class="p">):</span>
|
||
<span class="n">x1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">500</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">tree_reg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="n">axes</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$x_1$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="n">ylabel</span><span class="p">:</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="n">ylabel</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s2">"b."</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="s2">"r.-"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\hat</span><span class="si">{y}</span><span class="s2">$"</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
|
||
<span class="n">plot_regression_predictions</span><span class="p">(</span><span class="n">tree_reg1</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">split</span><span class="p">,</span> <span class="n">style</span> <span class="ow">in</span> <span class="p">((</span><span class="mf">0.1973</span><span class="p">,</span> <span class="s2">"k-"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.0917</span><span class="p">,</span> <span class="s2">"k--"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.7718</span><span class="p">,</span> <span class="s2">"k--"</span><span class="p">)):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">split</span><span class="p">,</span> <span class="n">split</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">style</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.21</span><span class="p">,</span> <span class="mf">0.65</span><span class="p">,</span> <span class="s2">"Depth=0"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="s2">"Depth=1"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.65</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="s2">"Depth=1"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper center"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"max_depth=2"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
|
||
<span class="n">plot_regression_predictions</span><span class="p">(</span><span class="n">tree_reg2</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">split</span><span class="p">,</span> <span class="n">style</span> <span class="ow">in</span> <span class="p">((</span><span class="mf">0.1973</span><span class="p">,</span> <span class="s2">"k-"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.0917</span><span class="p">,</span> <span class="s2">"k--"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.7718</span><span class="p">,</span> <span class="s2">"k--"</span><span class="p">)):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">split</span><span class="p">,</span> <span class="n">split</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">style</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">split</span> <span class="ow">in</span> <span class="p">(</span><span class="mf">0.0458</span><span class="p">,</span> <span class="mf">0.1298</span><span class="p">,</span> <span class="mf">0.2873</span><span class="p">,</span> <span class="mf">0.9040</span><span class="p">):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">split</span><span class="p">,</span> <span class="n">split</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="s2">"k:"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="s2">"Depth=2"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"max_depth=3"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<img alt="_images/b434099f6787c8e68e7df05b14ead06c81266c80052f890b8976683415965e55.png" src="_images/b434099f6787c8e68e7df05b14ead06c81266c80052f890b8976683415965e55.png" />
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">tree_reg1</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">tree_reg2</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">,</span> <span class="n">min_samples_leaf</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
|
||
<span class="n">tree_reg1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="n">tree_reg2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
|
||
<span class="n">x1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">500</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y_pred1</span> <span class="o">=</span> <span class="n">tree_reg1</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
|
||
<span class="n">y_pred2</span> <span class="o">=</span> <span class="n">tree_reg2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s2">"b."</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_pred1</span><span class="p">,</span> <span class="s2">"r.-"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\hat</span><span class="si">{y}</span><span class="s2">$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$x_1$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"$y$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper center"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"No restrictions"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s2">"b."</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_pred2</span><span class="p">,</span> <span class="s2">"r.-"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\hat</span><span class="si">{y}</span><span class="s2">$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$x_1$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"min_samples_leaf=</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">tree_reg2</span><span class="o">.</span><span class="n">min_samples_leaf</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<img alt="_images/7e2519ea00e4eb8c4d8f1fad48e4fb234d111eb4bdbd52208f7f47215db26976.png" src="_images/7e2519ea00e4eb8c4d8f1fad48e4fb234d111eb4bdbd52208f7f47215db26976.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="pros-and-cons-of-trees-pros">
|
||
<h2>Pros and cons of trees, pros<a class="headerlink" href="#pros-and-cons-of-trees-pros" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p>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)</p></li>
|
||
<li><p>Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!</p></li>
|
||
<li><p>No feature normalization needed</p></li>
|
||
<li><p>Tree models can handle both continuous and categorical data (Classification and Regression Trees)</p></li>
|
||
<li><p>Can model nonlinear relationships</p></li>
|
||
<li><p>Can model interactions between the different descriptive features</p></li>
|
||
<li><p>Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="disadvantages">
|
||
<h2>Disadvantages<a class="headerlink" href="#disadvantages" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p>Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches</p></li>
|
||
<li><p>If continuous features are used the tree may become quite large and hence less interpretable</p></li>
|
||
<li><p>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</p></li>
|
||
<li><p>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</p></li>
|
||
<li><p>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.</p></li>
|
||
<li><p>If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data</p></li>
|
||
<li><p>Features with many levels may be preferred over features with less levels since for them it is <em>more easy</em> 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</p></li>
|
||
</ul>
|
||
<p>However, by aggregating many decision trees, using methods like
|
||
bagging, random forests, and boosting, the predictive performance of
|
||
trees can be substantially improved.</p>
|
||
</section>
|
||
<section id="ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods">
|
||
<h2>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods<a class="headerlink" href="#ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods" title="Link to this heading">#</a></h2>
|
||
<p>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?</p>
|
||
<p>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</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Voting classifiers</p></li>
|
||
<li><p>Bagging and Pasting</p></li>
|
||
<li><p>Random forests</p></li>
|
||
<li><p>Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)</p></li>
|
||
</ol>
|
||
<p>We discuss these methods here.</p>
|
||
</section>
|
||
<section id="an-overview-of-ensemble-methods">
|
||
<h2>An Overview of Ensemble Methods<a class="headerlink" href="#an-overview-of-ensemble-methods" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [DataFiles/ensembleoverview.png, width=600 frac=0.8] -->
|
||
<!-- begin figure -->
|
||
<p><img src="DataFiles/ensembleoverview.png" width="600"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></section>
|
||
<section id="why-voting">
|
||
<h2>Why Voting?<a class="headerlink" href="#why-voting" title="Link to this heading">#</a></h2>
|
||
<p>The idea behind boosting, and voting as well can be phrased as follows:
|
||
<strong>Can a group of people somehow arrive at highly
|
||
reasoned decisions, despite the weak judgement of the individual
|
||
members?</strong></p>
|
||
<p>The aim is to create a good classifier by combining several weak classifiers.
|
||
<strong>A weak classifier is a classifier which is able to produce results that are only slightly better than guessing at random.</strong></p>
|
||
<p>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.</p>
|
||
<p>Decision trees play an important role as our weak classifier. They serve as the basic method.</p>
|
||
</section>
|
||
<section id="tossing-coins">
|
||
<h2>Tossing coins<a class="headerlink" href="#tossing-coins" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
<p>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.</p>
|
||
<p>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.</p>
|
||
<p>The example here shows how we can implement the coin tossing case,
|
||
clealry demostrating that after some tosses we see the <a class="reference external" href="https://en.wikipedia.org/wiki/Law_of_large_numbers">law of large</a>
|
||
numbers kicking in.</p>
|
||
</section>
|
||
<section id="standard-imports-first">
|
||
<h2>Standard imports first<a class="headerlink" href="#standard-imports-first" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">OneHotEncoder</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.compose</span> <span class="kn">import</span> <span class="n">ColumnTransformer</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
|
||
<span class="c1"># Where to save the figures and data files</span>
|
||
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">"Results"</span>
|
||
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">"Results/FigureFiles"</span>
|
||
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">"DataFiles/"</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="simple-voting-example-head-or-tail">
|
||
<h2>Simple Voting Example, head or tail<a class="headerlink" href="#simple-voting-example-head-or-tail" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib.colors</span> <span class="kn">import</span> <span class="n">ListedColormap</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'axes.labelsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">14</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'xtick.labelsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">12</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'ytick.labelsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">12</span>
|
||
|
||
<span class="n">heads_proba</span> <span class="o">=</span> <span class="mf">0.51</span>
|
||
<span class="n">coin_tosses</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">10000</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="o"><</span> <span class="n">heads_proba</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">int32</span><span class="p">)</span>
|
||
<span class="n">cumulative_heads_ratio</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">coin_tosses</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10001</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span><span class="mf">3.5</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">cumulative_heads_ratio</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.51</span><span class="p">,</span> <span class="mf">0.51</span><span class="p">],</span> <span class="s2">"k--"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"51%"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">],</span> <span class="s2">"k-"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"50%"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"Number of coin tosses"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Heads ratio"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"lower right"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10000</span><span class="p">,</span> <span class="mf">0.42</span><span class="p">,</span> <span class="mf">0.58</span><span class="p">])</span>
|
||
<span class="n">save_fig</span><span class="p">(</span><span class="s2">"votingsimple"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<img alt="_images/4487bd34c409fa3530783c8c994facb4e72f15c9cf61774524b892ac4c45590f.png" src="_images/4487bd34c409fa3530783c8c994facb4e72f15c9cf61774524b892ac4c45590f.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="using-the-voting-classifier">
|
||
<h2>Using the Voting Classifier<a class="headerlink" href="#using-the-voting-classifier" title="Link to this heading">#</a></h2>
|
||
<p>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 <strong>Scikit-Learn</strong>.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_moons</span>
|
||
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_moons</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mf">0.30</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">RandomForestClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">VotingClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">SVC</span>
|
||
|
||
<span class="n">log_clf</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s2">"liblinear"</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">rnd_clf</span> <span class="o">=</span> <span class="n">RandomForestClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">svm_clf</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
|
||
<span class="n">voting_clf</span> <span class="o">=</span> <span class="n">VotingClassifier</span><span class="p">(</span>
|
||
<span class="n">estimators</span><span class="o">=</span><span class="p">[(</span><span class="s1">'lr'</span><span class="p">,</span> <span class="n">log_clf</span><span class="p">),</span> <span class="p">(</span><span class="s1">'rf'</span><span class="p">,</span> <span class="n">rnd_clf</span><span class="p">),</span> <span class="p">(</span><span class="s1">'svc'</span><span class="p">,</span> <span class="n">svm_clf</span><span class="p">)],</span>
|
||
<span class="n">voting</span><span class="o">=</span><span class="s1">'hard'</span><span class="p">)</span>
|
||
|
||
<span class="n">voting_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
|
||
|
||
<span class="k">for</span> <span class="n">clf</span> <span class="ow">in</span> <span class="p">(</span><span class="n">log_clf</span><span class="p">,</span> <span class="n">rnd_clf</span><span class="p">,</span> <span class="n">svm_clf</span><span class="p">,</span> <span class="n">voting_clf</span><span class="p">):</span>
|
||
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span>
|
||
|
||
<span class="n">log_clf</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s2">"liblinear"</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">rnd_clf</span> <span class="o">=</span> <span class="n">RandomForestClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">svm_clf</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="s2">"auto"</span><span class="p">,</span> <span class="n">probability</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">voting_clf</span> <span class="o">=</span> <span class="n">VotingClassifier</span><span class="p">(</span>
|
||
<span class="n">estimators</span><span class="o">=</span><span class="p">[(</span><span class="s1">'lr'</span><span class="p">,</span> <span class="n">log_clf</span><span class="p">),</span> <span class="p">(</span><span class="s1">'rf'</span><span class="p">,</span> <span class="n">rnd_clf</span><span class="p">),</span> <span class="p">(</span><span class="s1">'svc'</span><span class="p">,</span> <span class="n">svm_clf</span><span class="p">)],</span>
|
||
<span class="n">voting</span><span class="o">=</span><span class="s1">'soft'</span><span class="p">)</span>
|
||
<span class="n">voting_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
|
||
|
||
<span class="k">for</span> <span class="n">clf</span> <span class="ow">in</span> <span class="p">(</span><span class="n">log_clf</span><span class="p">,</span> <span class="n">rnd_clf</span><span class="p">,</span> <span class="n">svm_clf</span><span class="p">,</span> <span class="n">voting_clf</span><span class="p">):</span>
|
||
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>LogisticRegression 0.864
|
||
RandomForestClassifier 0.872
|
||
SVC 0.888
|
||
VotingClassifier 0.896
|
||
LogisticRegression 0.864
|
||
RandomForestClassifier 0.872
|
||
SVC 0.888
|
||
VotingClassifier 0.912
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="voting-and-bagging">
|
||
<h2>Voting and Bagging<a class="headerlink" href="#voting-and-bagging" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_moons</span>
|
||
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_moons</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mf">0.30</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">RandomForestClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">VotingClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">SVC</span>
|
||
|
||
<span class="n">log_clf</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">rnd_clf</span> <span class="o">=</span> <span class="n">RandomForestClassifier</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">svm_clf</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
|
||
<span class="n">voting_clf</span> <span class="o">=</span> <span class="n">VotingClassifier</span><span class="p">(</span>
|
||
<span class="n">estimators</span><span class="o">=</span><span class="p">[(</span><span class="s1">'lr'</span><span class="p">,</span> <span class="n">log_clf</span><span class="p">),</span> <span class="p">(</span><span class="s1">'rf'</span><span class="p">,</span> <span class="n">rnd_clf</span><span class="p">),</span> <span class="p">(</span><span class="s1">'svc'</span><span class="p">,</span> <span class="n">svm_clf</span><span class="p">)],</span>
|
||
<span class="n">voting</span><span class="o">=</span><span class="s1">'hard'</span><span class="p">)</span>
|
||
<span class="n">voting_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_html"><style>#sk-container-id-2 {color: black;}#sk-container-id-2 pre{padding: 0;}#sk-container-id-2 div.sk-toggleable {background-color: white;}#sk-container-id-2 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-2 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-2 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-2 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-2 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-2 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-2 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-2 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-2 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-2 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-2 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-2 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-2 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-2 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-2 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-2 div.sk-item {position: relative;z-index: 1;}#sk-container-id-2 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-2 div.sk-item::before, #sk-container-id-2 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-2 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-2 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-2 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-2 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-2 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-2 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-2 div.sk-label-container {text-align: center;}#sk-container-id-2 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-2 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-2" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>VotingClassifier(estimators=[('lr', LogisticRegression(random_state=42)),
|
||
('rf', RandomForestClassifier(random_state=42)),
|
||
('svc', SVC(random_state=42))])</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-2" type="checkbox" ><label for="sk-estimator-id-2" class="sk-toggleable__label sk-toggleable__label-arrow">VotingClassifier</label><div class="sk-toggleable__content"><pre>VotingClassifier(estimators=[('lr', LogisticRegression(random_state=42)),
|
||
('rf', RandomForestClassifier(random_state=42)),
|
||
('svc', SVC(random_state=42))])</pre></div></div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><label>lr</label></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-3" type="checkbox" ><label for="sk-estimator-id-3" class="sk-toggleable__label sk-toggleable__label-arrow">LogisticRegression</label><div class="sk-toggleable__content"><pre>LogisticRegression(random_state=42)</pre></div></div></div></div></div></div><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><label>rf</label></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-4" type="checkbox" ><label for="sk-estimator-id-4" class="sk-toggleable__label sk-toggleable__label-arrow">RandomForestClassifier</label><div class="sk-toggleable__content"><pre>RandomForestClassifier(random_state=42)</pre></div></div></div></div></div></div><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><label>svc</label></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-5" type="checkbox" ><label for="sk-estimator-id-5" class="sk-toggleable__label sk-toggleable__label-arrow">SVC</label><div class="sk-toggleable__content"><pre>SVC(random_state=42)</pre></div></div></div></div></div></div></div></div></div></div></div></div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
|
||
|
||
<span class="k">for</span> <span class="n">clf</span> <span class="ow">in</span> <span class="p">(</span><span class="n">log_clf</span><span class="p">,</span> <span class="n">rnd_clf</span><span class="p">,</span> <span class="n">svm_clf</span><span class="p">,</span> <span class="n">voting_clf</span><span class="p">):</span>
|
||
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>LogisticRegression 0.864
|
||
RandomForestClassifier 0.896
|
||
SVC 0.896
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>VotingClassifier 0.912
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">log_clf</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">rnd_clf</span> <span class="o">=</span> <span class="n">RandomForestClassifier</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">svm_clf</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">probability</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
|
||
<span class="n">voting_clf</span> <span class="o">=</span> <span class="n">VotingClassifier</span><span class="p">(</span>
|
||
<span class="n">estimators</span><span class="o">=</span><span class="p">[(</span><span class="s1">'lr'</span><span class="p">,</span> <span class="n">log_clf</span><span class="p">),</span> <span class="p">(</span><span class="s1">'rf'</span><span class="p">,</span> <span class="n">rnd_clf</span><span class="p">),</span> <span class="p">(</span><span class="s1">'svc'</span><span class="p">,</span> <span class="n">svm_clf</span><span class="p">)],</span>
|
||
<span class="n">voting</span><span class="o">=</span><span class="s1">'soft'</span><span class="p">)</span>
|
||
<span class="n">voting_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_html"><style>#sk-container-id-3 {color: black;}#sk-container-id-3 pre{padding: 0;}#sk-container-id-3 div.sk-toggleable {background-color: white;}#sk-container-id-3 label.sk-toggleable__label {cursor: pointer;display: block;width: 100%;margin-bottom: 0;padding: 0.3em;box-sizing: border-box;text-align: center;}#sk-container-id-3 label.sk-toggleable__label-arrow:before {content: "▸";float: left;margin-right: 0.25em;color: #696969;}#sk-container-id-3 label.sk-toggleable__label-arrow:hover:before {color: black;}#sk-container-id-3 div.sk-estimator:hover label.sk-toggleable__label-arrow:before {color: black;}#sk-container-id-3 div.sk-toggleable__content {max-height: 0;max-width: 0;overflow: hidden;text-align: left;background-color: #f0f8ff;}#sk-container-id-3 div.sk-toggleable__content pre {margin: 0.2em;color: black;border-radius: 0.25em;background-color: #f0f8ff;}#sk-container-id-3 input.sk-toggleable__control:checked~div.sk-toggleable__content {max-height: 200px;max-width: 100%;overflow: auto;}#sk-container-id-3 input.sk-toggleable__control:checked~label.sk-toggleable__label-arrow:before {content: "▾";}#sk-container-id-3 div.sk-estimator input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-3 div.sk-label input.sk-toggleable__control:checked~label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-3 input.sk-hidden--visually {border: 0;clip: rect(1px 1px 1px 1px);clip: rect(1px, 1px, 1px, 1px);height: 1px;margin: -1px;overflow: hidden;padding: 0;position: absolute;width: 1px;}#sk-container-id-3 div.sk-estimator {font-family: monospace;background-color: #f0f8ff;border: 1px dotted black;border-radius: 0.25em;box-sizing: border-box;margin-bottom: 0.5em;}#sk-container-id-3 div.sk-estimator:hover {background-color: #d4ebff;}#sk-container-id-3 div.sk-parallel-item::after {content: "";width: 100%;border-bottom: 1px solid gray;flex-grow: 1;}#sk-container-id-3 div.sk-label:hover label.sk-toggleable__label {background-color: #d4ebff;}#sk-container-id-3 div.sk-serial::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: 0;}#sk-container-id-3 div.sk-serial {display: flex;flex-direction: column;align-items: center;background-color: white;padding-right: 0.2em;padding-left: 0.2em;position: relative;}#sk-container-id-3 div.sk-item {position: relative;z-index: 1;}#sk-container-id-3 div.sk-parallel {display: flex;align-items: stretch;justify-content: center;background-color: white;position: relative;}#sk-container-id-3 div.sk-item::before, #sk-container-id-3 div.sk-parallel-item::before {content: "";position: absolute;border-left: 1px solid gray;box-sizing: border-box;top: 0;bottom: 0;left: 50%;z-index: -1;}#sk-container-id-3 div.sk-parallel-item {display: flex;flex-direction: column;z-index: 1;position: relative;background-color: white;}#sk-container-id-3 div.sk-parallel-item:first-child::after {align-self: flex-end;width: 50%;}#sk-container-id-3 div.sk-parallel-item:last-child::after {align-self: flex-start;width: 50%;}#sk-container-id-3 div.sk-parallel-item:only-child::after {width: 0;}#sk-container-id-3 div.sk-dashed-wrapped {border: 1px dashed gray;margin: 0 0.4em 0.5em 0.4em;box-sizing: border-box;padding-bottom: 0.4em;background-color: white;}#sk-container-id-3 div.sk-label label {font-family: monospace;font-weight: bold;display: inline-block;line-height: 1.2em;}#sk-container-id-3 div.sk-label-container {text-align: center;}#sk-container-id-3 div.sk-container {/* jupyter's `normalize.less` sets `[hidden] { display: none; }` but bootstrap.min.css set `[hidden] { display: none !important; }` so we also need the `!important` here to be able to override the default hidden behavior on the sphinx rendered scikit-learn.org. See: https://github.com/scikit-learn/scikit-learn/issues/21755 */display: inline-block !important;position: relative;}#sk-container-id-3 div.sk-text-repr-fallback {display: none;}</style><div id="sk-container-id-3" class="sk-top-container"><div class="sk-text-repr-fallback"><pre>VotingClassifier(estimators=[('lr', LogisticRegression(random_state=42)),
|
||
('rf', RandomForestClassifier(random_state=42)),
|
||
('svc', SVC(probability=True, random_state=42))],
|
||
voting='soft')</pre><b>In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook. <br />On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.</b></div><div class="sk-container" hidden><div class="sk-item sk-dashed-wrapped"><div class="sk-label-container"><div class="sk-label sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-6" type="checkbox" ><label for="sk-estimator-id-6" class="sk-toggleable__label sk-toggleable__label-arrow">VotingClassifier</label><div class="sk-toggleable__content"><pre>VotingClassifier(estimators=[('lr', LogisticRegression(random_state=42)),
|
||
('rf', RandomForestClassifier(random_state=42)),
|
||
('svc', SVC(probability=True, random_state=42))],
|
||
voting='soft')</pre></div></div></div><div class="sk-parallel"><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><label>lr</label></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-7" type="checkbox" ><label for="sk-estimator-id-7" class="sk-toggleable__label sk-toggleable__label-arrow">LogisticRegression</label><div class="sk-toggleable__content"><pre>LogisticRegression(random_state=42)</pre></div></div></div></div></div></div><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><label>rf</label></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-8" type="checkbox" ><label for="sk-estimator-id-8" class="sk-toggleable__label sk-toggleable__label-arrow">RandomForestClassifier</label><div class="sk-toggleable__content"><pre>RandomForestClassifier(random_state=42)</pre></div></div></div></div></div></div><div class="sk-parallel-item"><div class="sk-item"><div class="sk-label-container"><div class="sk-label sk-toggleable"><label>svc</label></div></div><div class="sk-serial"><div class="sk-item"><div class="sk-estimator sk-toggleable"><input class="sk-toggleable__control sk-hidden--visually" id="sk-estimator-id-9" type="checkbox" ><label for="sk-estimator-id-9" class="sk-toggleable__label sk-toggleable__label-arrow">SVC</label><div class="sk-toggleable__content"><pre>SVC(probability=True, random_state=42)</pre></div></div></div></div></div></div></div></div></div></div></div></div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">accuracy_score</span>
|
||
|
||
<span class="k">for</span> <span class="n">clf</span> <span class="ow">in</span> <span class="p">(</span><span class="n">log_clf</span><span class="p">,</span> <span class="n">rnd_clf</span><span class="p">,</span> <span class="n">svm_clf</span><span class="p">,</span> <span class="n">voting_clf</span><span class="p">):</span>
|
||
<span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">clf</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">accuracy_score</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>LogisticRegression 0.864
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>RandomForestClassifier 0.896
|
||
SVC 0.896
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>VotingClassifier 0.92
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="bagging">
|
||
<h2>Bagging<a class="headerlink" href="#bagging" title="Link to this heading">#</a></h2>
|
||
<p>The <strong>plain</strong> 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 <span class="math notranslate nohighlight">\(n\)</span> to <span class="math notranslate nohighlight">\(p\)</span> is moderately large.</p>
|
||
<p><strong>Bootstrap aggregation</strong>, or just <strong>bagging</strong>, is a
|
||
general-purpose procedure for reducing the variance of a statistical
|
||
learning method.</p>
|
||
</section>
|
||
<section id="more-bagging">
|
||
<h2>More bagging<a class="headerlink" href="#more-bagging" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
<p>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 <span class="math notranslate nohighlight">\(B\)</span> 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 <span class="math notranslate nohighlight">\(B\)</span> trees.</p>
|
||
</section>
|
||
<section id="making-your-own-bootstrap-changing-the-level-of-the-decision-tree">
|
||
<h2>Making your own Bootstrap: Changing the Level of the Decision Tree<a class="headerlink" href="#making-your-own-bootstrap-changing-the-level-of-the-decision-tree" title="Link to this heading">#</a></h2>
|
||
<p>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 <span class="math notranslate nohighlight">\(n\)</span>).</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.pipeline</span> <span class="kn">import</span> <span class="n">make_pipeline</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">resample</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">n_boostraps</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">maxdepth</span> <span class="o">=</span> <span class="mi">8</span>
|
||
|
||
<span class="c1"># Make data set.</span>
|
||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="n">error</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdepth</span><span class="p">)</span>
|
||
<span class="n">bias</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdepth</span><span class="p">)</span>
|
||
<span class="n">variance</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdepth</span><span class="p">)</span>
|
||
<span class="n">polydegree</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">maxdepth</span><span class="p">)</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
|
||
<span class="c1"># we produce a simple tree first as benchmark</span>
|
||
<span class="n">simpletree</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
|
||
<span class="n">simpletree</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">simpleprediction</span> <span class="o">=</span> <span class="n">simpletree</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">degree</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">maxdepth</span><span class="p">):</span>
|
||
<span class="n">model</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="n">y_test</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">n_boostraps</span><span class="p">))</span>
|
||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_boostraps</span><span class="p">):</span>
|
||
<span class="n">x_</span><span class="p">,</span> <span class="n">y_</span> <span class="o">=</span> <span class="n">resample</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">x_</span><span class="p">,</span> <span class="n">y_</span><span class="p">)</span>
|
||
<span class="n">y_pred</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span><span class="c1">#.ravel()</span>
|
||
|
||
<span class="n">polydegree</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">degree</span>
|
||
<span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">y_pred</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="p">)</span>
|
||
<span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="p">(</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span> <span class="p">)</span>
|
||
<span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_pred</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">keepdims</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span> <span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Polynomial degree:'</span><span class="p">,</span> <span class="n">degree</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Error:'</span><span class="p">,</span> <span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Bias^2:'</span><span class="p">,</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Var:'</span><span class="p">,</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'</span><span class="si">{}</span><span class="s1"> >= </span><span class="si">{}</span><span class="s1"> + </span><span class="si">{}</span><span class="s1"> = </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">error</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">],</span> <span class="n">bias</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span><span class="o">+</span><span class="n">variance</span><span class="p">[</span><span class="n">degree</span><span class="p">]))</span>
|
||
|
||
<span class="n">mse_simpletree</span><span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">y_test</span> <span class="o">-</span> <span class="n">simpleprediction</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Simple tree:"</span><span class="p">,</span><span class="n">mse_simpletree</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">maxdepth</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">error</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'MSE'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">bias</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'bias'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">polydegree</span><span class="p">,</span> <span class="n">variance</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'Variance'</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<span class="n">save_fig</span><span class="p">(</span><span class="s2">"baggingboot"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 1
|
||
Error: 0.06380941468319971
|
||
Bias^2: 0.05160313473529168
|
||
Var: 0.01220627994790804
|
||
0.06380941468319971 >= 0.05160313473529168 + 0.01220627994790804 = 0.06380941468319971
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 2
|
||
Error: 0.043464037468677004
|
||
Bias^2: 0.02659851591375224
|
||
Var: 0.01686552155492476
|
||
0.043464037468677004 >= 0.02659851591375224 + 0.01686552155492476 = 0.043464037468677
|
||
Polynomial degree: 3
|
||
Error: 0.020716391693769383
|
||
Bias^2: 0.01159033914386312
|
||
Var: 0.00912605254990626
|
||
0.020716391693769383 >= 0.01159033914386312 + 0.00912605254990626 = 0.02071639169376938
|
||
Polynomial degree: 4
|
||
Error: 0.02063627410934057
|
||
Bias^2: 0.0117496656370668
|
||
Var: 0.008886608472273775
|
||
0.02063627410934057 >= 0.0117496656370668 + 0.008886608472273775 = 0.020636274109340574
|
||
Polynomial degree: 5
|
||
Error: 0.02087627881701288
|
||
Bias^2: 0.01349183949256158
|
||
Var: 0.007384439324451296
|
||
0.02087627881701288 >= 0.01349183949256158 + 0.007384439324451296 = 0.020876278817012876
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 6
|
||
Error: 0.02069601123831537
|
||
Bias^2: 0.013918526350129823
|
||
Var: 0.0067774848881855445
|
||
0.02069601123831537 >= 0.013918526350129823 + 0.0067774848881855445 = 0.020696011238315368
|
||
Polynomial degree: 7
|
||
Error: 0.022964339924731444
|
||
Bias^2: 0.01550381208433455
|
||
Var: 0.007460527840396904
|
||
0.022964339924731444 >= 0.01550381208433455 + 0.007460527840396904 = 0.022964339924731455
|
||
Simple tree: 0.5148389267750961
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/952ed963a54362216823127e5133c43d0231100f817ed80c14c38a6cac43de84.png" src="_images/952ed963a54362216823127e5133c43d0231100f817ed80c14c38a6cac43de84.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="random-forests">
|
||
<h2>Random forests<a class="headerlink" href="#random-forests" title="Link to this heading">#</a></h2>
|
||
<p>Random forests provide an improvement over bagged trees by way of a
|
||
small tweak that decorrelates the trees.</p>
|
||
<p>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 <span class="math notranslate nohighlight">\(m\)</span> predictors is chosen as split
|
||
candidates from the full set of <span class="math notranslate nohighlight">\(p\)</span> predictors. The split is allowed to
|
||
use only one of those <span class="math notranslate nohighlight">\(m\)</span> predictors.</p>
|
||
<p>A fresh sample of <span class="math notranslate nohighlight">\(m\)</span> predictors is
|
||
taken at each split, and typically we choose</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
m\approx \sqrt{p}.
|
||
\]</div>
|
||
<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.</p>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="random-forest-algorithm">
|
||
<h2>Random Forest Algorithm<a class="headerlink" href="#random-forest-algorithm" title="Link to this heading">#</a></h2>
|
||
<p>The algorithm described here can be applied to both classification and regression problems.</p>
|
||
<p>We will grow of forest of say <span class="math notranslate nohighlight">\(B\)</span> trees.</p>
|
||
<ol class="arabic simple">
|
||
<li><p>For <span class="math notranslate nohighlight">\(b=1:B\)</span></p></li>
|
||
</ol>
|
||
<ul class="simple">
|
||
<li><p>Draw a bootstrap sample from the training data organized in our <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> matrix.</p></li>
|
||
<li><p>We grow then a random forest tree <span class="math notranslate nohighlight">\(T_b\)</span> based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached</p></li>
|
||
</ul>
|
||
<ol class="arabic simple">
|
||
<li><p>we select <span class="math notranslate nohighlight">\(m \le p\)</span> variables at random from the <span class="math notranslate nohighlight">\(p\)</span> predictors/features</p></li>
|
||
<li><p>pick the best split point among the <span class="math notranslate nohighlight">\(m\)</span> features using for example the CART algorithm and create a new node</p></li>
|
||
<li><p>split the node into daughter nodes</p></li>
|
||
<li><p>Output then the ensemble of trees <span class="math notranslate nohighlight">\(\{T_b\}_1^{B}\)</span> and make predictions for either a regression type of problem or a classification type of problem.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="random-forests-compared-with-other-methods-on-the-cancer-data">
|
||
<h2>Random Forests Compared with other Methods on the Cancer Data<a class="headerlink" href="#random-forests-compared-with-other-methods-on-the-cancer-data" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">SVC</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">BaggingClassifier</span>
|
||
|
||
<span class="c1"># Load the data</span>
|
||
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="c1">#define methods</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s1">'lbfgs'</span><span class="p">)</span>
|
||
<span class="c1"># Support vector machine</span>
|
||
<span class="n">svm</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="s1">'auto'</span><span class="p">,</span> <span class="n">C</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
|
||
<span class="c1"># Decision Trees</span>
|
||
<span class="n">deep_tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
|
||
<span class="c1">#Scale the data</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy Logistic Regression with scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1"># Support Vector Machine</span>
|
||
<span class="n">svm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy SVM with scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1"># Decision Trees</span>
|
||
<span class="n">deep_tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Decision Trees and scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">deep_tree_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">RandomForestClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">LabelEncoder</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">cross_validate</span>
|
||
<span class="c1"># Data set not specificied</span>
|
||
<span class="c1">#Instantiate the model with 500 trees and entropy as splitting criteria</span>
|
||
<span class="n">Random_Forest_model</span> <span class="o">=</span> <span class="n">RandomForestClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span><span class="n">criterion</span><span class="o">=</span><span class="s2">"entropy"</span><span class="p">)</span>
|
||
<span class="n">Random_Forest_model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="c1">#Cross validation</span>
|
||
<span class="n">accuracy</span> <span class="o">=</span> <span class="n">cross_validate</span><span class="p">(</span><span class="n">Random_Forest_model</span><span class="p">,</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">,</span><span class="n">cv</span><span class="o">=</span><span class="mi">10</span><span class="p">)[</span><span class="s1">'test_score'</span><span class="p">]</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">accuracy</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Random Forests and scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">Random_Forest_model</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
|
||
|
||
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">Random_Forest_model</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_confusion_matrix</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<span class="n">y_probas</span> <span class="o">=</span> <span class="n">Random_Forest_model</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_roc</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_cumulative_gain</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(426, 30)
|
||
(143, 30)
|
||
Test set accuracy Logistic Regression with scaled data: 0.96
|
||
Test set accuracy SVM with scaled data: 0.96
|
||
Test set accuracy with Decision Trees and scaled data: 0.87
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.93333333 0.73333333 0.93333333 1. 1. 0.92857143
|
||
1. 0.92857143 0.92857143 0.92857143]
|
||
Test set accuracy with Random Forests and scaled data: 0.98
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/d0ba7e7eb7be3f1dd0167b214330cb8a120c81e1e5c41e618498569e1aa1811c.png" src="_images/d0ba7e7eb7be3f1dd0167b214330cb8a120c81e1e5c41e618498569e1aa1811c.png" />
|
||
<img alt="_images/c89eee05eda5c57a33a2e04b4e0c14bd10b376394187ac1139a6ccce72e994f6.png" src="_images/c89eee05eda5c57a33a2e04b4e0c14bd10b376394187ac1139a6ccce72e994f6.png" />
|
||
<img alt="_images/9eb742fd240fc7f76c7356289f9cfe1f38dd45a48f502525f273d527cd6629e7.png" src="_images/9eb742fd240fc7f76c7356289f9cfe1f38dd45a48f502525f273d527cd6629e7.png" />
|
||
</div>
|
||
</div>
|
||
<p>Recall that the cumulative gains curve shows the percentage of the
|
||
overall number of cases in a given category <em>gained</em> by targeting a
|
||
percentage of the total number of cases.</p>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="compare-bagging-on-trees-with-random-forests">
|
||
<h2>Compare Bagging on Trees with Random Forests<a class="headerlink" href="#compare-bagging-on-trees-with-random-forests" title="Link to this heading">#</a></h2>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">bag_clf</span> <span class="o">=</span> <span class="n">BaggingClassifier</span><span class="p">(</span>
|
||
<span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">splitter</span><span class="o">=</span><span class="s2">"random"</span><span class="p">,</span> <span class="n">max_leaf_nodes</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">),</span>
|
||
<span class="n">n_estimators</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">max_samples</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">bootstrap</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">n_jobs</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">bag_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">bag_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">RandomForestClassifier</span>
|
||
<span class="n">rnd_clf</span> <span class="o">=</span> <span class="n">RandomForestClassifier</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">500</span><span class="p">,</span> <span class="n">max_leaf_nodes</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="n">n_jobs</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">rnd_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">y_pred_rf</span> <span class="o">=</span> <span class="n">rnd_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">y_pred</span> <span class="o">==</span> <span class="n">y_pred_rf</span><span class="p">)</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">y_pred</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.9790209790209791
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="boosting-a-bird-s-eye-view">
|
||
<h2>Boosting, a Bird’s Eye View<a class="headerlink" href="#boosting-a-bird-s-eye-view" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
<p>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.</p>
|
||
</section>
|
||
<section id="what-is-boosting-additive-modelling-iterative-fitting">
|
||
<h2>What is boosting? Additive Modelling/Iterative Fitting<a class="headerlink" href="#what-is-boosting-additive-modelling-iterative-fitting" title="Link to this heading">#</a></h2>
|
||
<p>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</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\beta_m\)</span> are the expansion parameters to be determined in a
|
||
minimization process and <span class="math notranslate nohighlight">\(b(x;\gamma_m)\)</span> are some simple functions of
|
||
the multivariable parameter <span class="math notranslate nohighlight">\(x\)</span> which is characterized by the
|
||
parameters <span class="math notranslate nohighlight">\(\gamma_m\)</span>.</p>
|
||
<p>As an example, consider the Sigmoid function we used in logistic
|
||
regression. In that case, we can translate the function
|
||
<span class="math notranslate nohighlight">\(b(x;\gamma_m)\)</span> into the Sigmoid function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sigma(t) = \frac{1}{1+\exp{(-t)}},
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(t=\gamma_0+\gamma_1 x\)</span> and the parameters <span class="math notranslate nohighlight">\(\gamma_0\)</span> and
|
||
<span class="math notranslate nohighlight">\(\gamma_1\)</span> were determined by the Logistic Regression fitting
|
||
algorithm.</p>
|
||
<p>As another example, consider the cost function we defined for linear regression</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
||
\]</div>
|
||
<p>In this case the function <span class="math notranslate nohighlight">\(f(x)\)</span> was replaced by the design matrix
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> and the unknown linear regression parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>,
|
||
that is <span class="math notranslate nohighlight">\(\boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta}\)</span>. In linear regression we can
|
||
simply invert a matrix and obtain the parameters <span class="math notranslate nohighlight">\(\beta\)</span> by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
|
||
\]</div>
|
||
<p>In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters <span class="math notranslate nohighlight">\(\beta_m\)</span> and <span class="math notranslate nohighlight">\(\gamma_m\)</span>.</p>
|
||
</section>
|
||
<section id="iterative-fitting-regression-and-squared-error-cost-function">
|
||
<h2>Iterative Fitting, Regression and Squared-error Cost Function<a class="headerlink" href="#iterative-fitting-regression-and-squared-error-cost-function" title="Link to this heading">#</a></h2>
|
||
<p>The way we proceed is as follows (here we specialize to the squared-error cost function)</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Establish a cost function, here <span class="math notranslate nohighlight">\(C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2\)</span> with <span class="math notranslate nohighlight">\(f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)\)</span>.</p></li>
|
||
<li><p>Initialize with a guess <span class="math notranslate nohighlight">\(f_0(x)\)</span>. It could be one or even zero or some random numbers.</p></li>
|
||
<li><p>For <span class="math notranslate nohighlight">\(m=1:M\)</span></p></li>
|
||
</ol>
|
||
<p>a. minimize <span class="math notranslate nohighlight">\(\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2\)</span> wrt <span class="math notranslate nohighlight">\(\gamma\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span></p>
|
||
<p>b. This gives the optimal values <span class="math notranslate nohighlight">\(\beta_m\)</span> and <span class="math notranslate nohighlight">\(\gamma_m\)</span></p>
|
||
<p>c. Determine then the new values <span class="math notranslate nohighlight">\(f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)\)</span></p>
|
||
<p>We could use any of the algorithms we have discussed till now. If we
|
||
use trees, <span class="math notranslate nohighlight">\(\gamma\)</span> parameterizes the split variables and split points
|
||
at the internal nodes, and the predictions at the terminal nodes.</p>
|
||
</section>
|
||
<section id="squared-error-example-and-iterative-fitting">
|
||
<h2>Squared-Error Example and Iterative Fitting<a class="headerlink" href="#squared-error-example-and-iterative-fitting" title="Link to this heading">#</a></h2>
|
||
<p>To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.</p>
|
||
<p>For simplicity we assume also that our functions <span class="math notranslate nohighlight">\(b(x;\gamma)=1+\gamma x\)</span>.</p>
|
||
<p>This means that for every iteration <span class="math notranslate nohighlight">\(m\)</span>, we need to optimize</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
|
||
\]</div>
|
||
<p>We start our iteration by simply setting <span class="math notranslate nohighlight">\(f_0(x)=0\)</span>.
|
||
Taking the derivatives with respect to <span class="math notranslate nohighlight">\(\beta\)</span> and <span class="math notranslate nohighlight">\(\gamma\)</span> we obtain</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
|
||
\]</div>
|
||
<p>We can then rewrite these equations as (defining <span class="math notranslate nohighlight">\(\boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x})\)</span> with <span class="math notranslate nohighlight">\(\boldsymbol{e}\)</span> being the unit vector)</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0,
|
||
\]</div>
|
||
<p>which gives us <span class="math notranslate nohighlight">\(\beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w})\)</span>. Similarly we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,
|
||
\]</div>
|
||
<p>which leads to <span class="math notranslate nohighlight">\(\gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x})\)</span>. Inserting
|
||
for <span class="math notranslate nohighlight">\(\beta\)</span> gives us an equation for <span class="math notranslate nohighlight">\(\gamma\)</span>. This is a non-linear equation in the unknown <span class="math notranslate nohighlight">\(\gamma\)</span> and has to be solved numerically.</p>
|
||
<p>The solution to these two equations gives us in turn <span class="math notranslate nohighlight">\(\beta_1\)</span> and <span class="math notranslate nohighlight">\(\gamma_1\)</span> leading to the new expression for <span class="math notranslate nohighlight">\(f_1(x)\)</span> as
|
||
<span class="math notranslate nohighlight">\(f_1(x) = \beta_1(1+\gamma_1x)\)</span>. Doing this <span class="math notranslate nohighlight">\(M\)</span> times results in our final estimate for the function <span class="math notranslate nohighlight">\(f\)</span>.</p>
|
||
</section>
|
||
<section id="iterative-fitting-classification-and-adaboost">
|
||
<h2>Iterative Fitting, Classification and AdaBoost<a class="headerlink" href="#iterative-fitting-classification-and-adaboost" title="Link to this heading">#</a></h2>
|
||
<p>Let us consider a binary classification problem with two outcomes <span class="math notranslate nohighlight">\(y_i \in \{-1,1\}\)</span> and <span class="math notranslate nohighlight">\(i=0,1,2,\dots,n-1\)</span> as our set of
|
||
observations. We define a classification function <span class="math notranslate nohighlight">\(G(x)\)</span> which produces a prediction taking one or the other of the two values
|
||
<span class="math notranslate nohighlight">\(\{-1,1\}\)</span>.</p>
|
||
<p>The error rate of the training sample is then</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
|
||
\]</div>
|
||
<p>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 <span class="math notranslate nohighlight">\(G_m(x)\)</span>.</p>
|
||
<p>Here we will express our function <span class="math notranslate nohighlight">\(f(x)\)</span> in terms of <span class="math notranslate nohighlight">\(G(x)\)</span>. That is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
|
||
\]</div>
|
||
<p>will be a function of</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
|
||
\]</div>
|
||
</section>
|
||
<section id="adaptive-boosting-adaboost">
|
||
<h2>Adaptive Boosting, AdaBoost<a class="headerlink" href="#adaptive-boosting-adaboost" title="Link to this heading">#</a></h2>
|
||
<p>In our iterative procedure we define thus</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
|
||
\]</div>
|
||
<p>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</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
|
||
\]</div>
|
||
<p>We optimize <span class="math notranslate nohighlight">\(\beta\)</span> and <span class="math notranslate nohighlight">\(G\)</span> for each value of <span class="math notranslate nohighlight">\(m=1:M\)</span> as we did in the regression case.
|
||
This is normally done in two steps. Let us however first rewrite the cost function as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},
|
||
\]</div>
|
||
<p>where we have defined <span class="math notranslate nohighlight">\(w_i^m= \exp{(-y_if_{m-1}(x_i))}\)</span>.</p>
|
||
</section>
|
||
<section id="building-up-adaboost">
|
||
<h2>Building up AdaBoost<a class="headerlink" href="#building-up-adaboost" title="Link to this heading">#</a></h2>
|
||
<p>First, for any <span class="math notranslate nohighlight">\(\beta > 0\)</span>, we optimize <span class="math notranslate nohighlight">\(G\)</span> by setting</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),
|
||
\]</div>
|
||
<p>which is the classifier that minimizes the weighted error rate in predicting <span class="math notranslate nohighlight">\(y\)</span>.</p>
|
||
<p>We can do this by rewriting</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,
|
||
\]</div>
|
||
<p>which can be rewritten as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,
|
||
\]</div>
|
||
<p>which leads to</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},
|
||
\]</div>
|
||
<p>where we have redefined the error as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},
|
||
\]</div>
|
||
<p>which leads to an update of</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f_m(x) = f_{m-1}(x) +\beta_m G_m(x).
|
||
\]</div>
|
||
<p>This leads to the new weights</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
|
||
\]</div>
|
||
</section>
|
||
<section id="adaptive-boosting-adaboost-basic-algorithm">
|
||
<h2>Adaptive boosting: AdaBoost, Basic Algorithm<a class="headerlink" href="#adaptive-boosting-adaboost-basic-algorithm" title="Link to this heading">#</a></h2>
|
||
<p>The algorithm here is rather straightforward. Assume that our weak
|
||
classifier is a decision tree and we consider a binary set of outputs
|
||
with <span class="math notranslate nohighlight">\(y_i \in \{-1,1\}\)</span> and <span class="math notranslate nohighlight">\(i=0,1,2,\dots,n-1\)</span> as our set of
|
||
observations. Our design matrix is given in terms of the
|
||
feature/predictor vectors
|
||
<span class="math notranslate nohighlight">\(\boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}]\)</span>. Finally, we define also a
|
||
classifier determined by our data via a function <span class="math notranslate nohighlight">\(G(x)\)</span>. This function tells us how well we are able to classify our outputs/targets <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span>.</p>
|
||
<p>We have already defined the misclassification error <span class="math notranslate nohighlight">\(\mathrm{err}\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),
|
||
\]</div>
|
||
<p>where the function <span class="math notranslate nohighlight">\(I()\)</span> is one if we misclassify and zero if we classify correctly.</p>
|
||
</section>
|
||
<section id="basic-steps-of-adaboost">
|
||
<h2>Basic Steps of AdaBoost<a class="headerlink" href="#basic-steps-of-adaboost" title="Link to this heading">#</a></h2>
|
||
<p>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.</p>
|
||
<ol class="arabic simple">
|
||
<li><p>We start by initializing all weights to <span class="math notranslate nohighlight">\(w_i = 1/n\)</span>, with <span class="math notranslate nohighlight">\(i=0,1,2,\dots n-1\)</span>. It is easy to see that we must have <span class="math notranslate nohighlight">\(\sum_{i=0}^{n-1}w_i = 1\)</span>.</p></li>
|
||
<li><p>We rewrite the misclassification error as</p></li>
|
||
</ol>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
|
||
\]</div>
|
||
<ol class="arabic simple">
|
||
<li><p>Then we start looping over all attempts at classifying, namely we start an iterative process for <span class="math notranslate nohighlight">\(m=1:M\)</span>, where <span class="math notranslate nohighlight">\(M\)</span> is the final number of classifications. Our given classifier could for example be a plain decision tree.</p></li>
|
||
</ol>
|
||
<p>a. Fit then a given classifier to the training set using the weights <span class="math notranslate nohighlight">\(w_i\)</span>.</p>
|
||
<p>b. Compute then <span class="math notranslate nohighlight">\(\mathrm{err}\)</span> and figure out which events are classified properly and which are classified wrongly.</p>
|
||
<p>c. Define a quantity <span class="math notranslate nohighlight">\(\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}\)</span></p>
|
||
<p>d. Set the new weights to <span class="math notranslate nohighlight">\(w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}\)</span>.</p>
|
||
<ol class="arabic simple" start="5">
|
||
<li><p>Compute the new classifier <span class="math notranslate nohighlight">\(G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)\)</span>.</p></li>
|
||
</ol>
|
||
<p>For the iterations with <span class="math notranslate nohighlight">\(m \le 2\)</span> the weights are modified
|
||
individually at each steps. The observations which were misclassified
|
||
at iteration <span class="math notranslate nohighlight">\(m-1\)</span> 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 <span class="math notranslate nohighlight">\(m\)</span> is then forced to concentrate on those
|
||
observations that are missed in the previous iterations.</p>
|
||
</section>
|
||
<section id="adaboost-examples">
|
||
<h2>AdaBoost Examples<a class="headerlink" href="#adaboost-examples" title="Link to this heading">#</a></h2>
|
||
<p>Using <strong>Scikit-Learn</strong> it is easy to apply the adaptive boosting algorithm, as done here.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">AdaBoostClassifier</span>
|
||
|
||
<span class="n">ada_clf</span> <span class="o">=</span> <span class="n">AdaBoostClassifier</span><span class="p">(</span>
|
||
<span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">n_estimators</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span>
|
||
<span class="n">algorithm</span><span class="o">=</span><span class="s2">"SAMME.R"</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">ada_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">AdaBoostClassifier</span>
|
||
|
||
<span class="n">ada_clf</span> <span class="o">=</span> <span class="n">AdaBoostClassifier</span><span class="p">(</span>
|
||
<span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">n_estimators</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span>
|
||
<span class="n">algorithm</span><span class="o">=</span><span class="s2">"SAMME.R"</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">ada_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">ada_clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_confusion_matrix</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">normalize</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<span class="n">y_probas</span> <span class="o">=</span> <span class="n">ada_clf</span><span class="o">.</span><span class="n">predict_proba</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">)</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_roc</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
<span class="n">skplt</span><span class="o">.</span><span class="n">metrics</span><span class="o">.</span><span class="n">plot_cumulative_gain</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">y_probas</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<img alt="_images/08ce20b8f2341c5e6ead650fbba3602c707e3ca168faad434e34c469f37aa5de.png" src="_images/08ce20b8f2341c5e6ead650fbba3602c707e3ca168faad434e34c469f37aa5de.png" />
|
||
<img alt="_images/1bb5f2c222dcf4a7a2066d4ed1f977dcb80f0628c532295b2468161bee6d0553.png" src="_images/1bb5f2c222dcf4a7a2066d4ed1f977dcb80f0628c532295b2468161bee6d0553.png" />
|
||
<img alt="_images/3116048d0a45b8a3babe592f2b2d3c74a850e5c08ac9dbceb36b74aca2db8423.png" src="_images/3116048d0a45b8a3babe592f2b2d3c74a850e5c08ac9dbceb36b74aca2db8423.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
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<i class="fa-solid fa-list"></i> Contents
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<ul class="visible nav section-nav flex-column">
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#plan-for-week-46">Plan for week 46</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#decision-trees-overarching-aims">Decision trees, overarching aims</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basics-of-a-tree">Basics of a tree</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-typical-decision-tree-with-its-pertinent-jargon-classification-problem">A typical Decision Tree with its pertinent Jargon, Classification Problem</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#general-features">General Features</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#how-do-we-set-it-up">How do we set it up?</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#decision-trees-and-regression">Decision trees and Regression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-a-tree-regression">Building a tree, regression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-top-down-approach-recursive-binary-splitting">A top-down approach, recursive binary splitting</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-a-tree">Making a tree</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pruning-the-tree">Pruning the tree</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cost-complexity-pruning">Cost complexity pruning</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#schematic-regression-procedure">Schematic Regression Procedure</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-classification-tree">A Classification Tree</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#growing-a-classification-tree">Growing a classification tree</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#classification-tree-how-to-split-nodes">Classification tree, how to split nodes</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-tree-classification">Visualizing the Tree, Classification</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-tree-the-moons">Visualizing the Tree, The Moons</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-ways-of-visualizing-the-trees">Other ways of visualizing the trees</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#printing-out-as-text">Printing out as text</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#algorithms-for-setting-up-decision-trees">Algorithms for Setting up Decision Trees</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cart-algorithm-for-classification">The CART algorithm for Classification</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cart-algorithm-for-regression">The CART algorithm for Regression</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-binary-splits">Why binary splits?</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-a-tree-using-the-gini-index">Computing a Tree using the Gini Index</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-table">The Table</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-various-gini-indices">Computing the various Gini Indices</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-various-gini-indices-hours-slept">Computing the various Gini Indices, Hours slept</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-various-gini-indices-hours-studied">Computing the various Gini Indices, Hours studied</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-possible-code-using-scikit-learn">A possible code using Scikit-Learn</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#further-example-computing-the-gini-index">Further example: Computing the Gini index</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-python-code-to-read-in-data-and-perform-classification">Simple Python Code to read in Data and perform Classification</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-gini-factor">Computing the Gini Factor</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#regression-trees">Regression trees</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#final-regressor-code">Final regressor code</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pros-and-cons-of-trees-pros">Pros and cons of trees, pros</a></li>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#disadvantages">Disadvantages</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ensemble-methods-from-a-single-tree-to-many-trees-and-extreme-boosting-meet-the-jungle-of-methods">Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#an-overview-of-ensemble-methods">An Overview of Ensemble Methods</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-voting">Why Voting?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tossing-coins">Tossing coins</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#standard-imports-first">Standard imports first</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-voting-example-head-or-tail">Simple Voting Example, head or tail</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#using-the-voting-classifier">Using the Voting Classifier</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#voting-and-bagging">Voting and Bagging</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bagging">Bagging</a></li>
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||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-bagging">More bagging</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-your-own-bootstrap-changing-the-level-of-the-decision-tree">Making your own Bootstrap: Changing the Level of the Decision Tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forests">Random forests</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forest-algorithm">Random Forest Algorithm</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forests-compared-with-other-methods-on-the-cancer-data">Random Forests Compared with other Methods on the Cancer Data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#compare-bagging-on-trees-with-random-forests">Compare Bagging on Trees with Random Forests</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#boosting-a-bird-s-eye-view">Boosting, a Bird’s Eye View</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#what-is-boosting-additive-modelling-iterative-fitting">What is boosting? Additive Modelling/Iterative Fitting</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#iterative-fitting-regression-and-squared-error-cost-function">Iterative Fitting, Regression and Squared-error Cost Function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#squared-error-example-and-iterative-fitting">Squared-Error Example and Iterative Fitting</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#iterative-fitting-classification-and-adaboost">Iterative Fitting, Classification and AdaBoost</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adaptive-boosting-adaboost">Adaptive Boosting, AdaBoost</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-up-adaboost">Building up AdaBoost</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adaptive-boosting-adaboost-basic-algorithm">Adaptive boosting: AdaBoost, Basic Algorithm</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basic-steps-of-adaboost">Basic Steps of AdaBoost</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adaboost-examples">AdaBoost Examples</a></li>
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