KarlHenrik
1534 lines
76 KiB
HTML
1534 lines
76 KiB
HTML
|
||
<!DOCTYPE html>
|
||
|
||
|
||
<html lang="en" data-content_root="./" >
|
||
|
||
<head>
|
||
<meta charset="utf-8" />
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
|
||
|
||
<title>9. Decision trees, overarching aims — Applied Data Analysis and Machine Learning</title>
|
||
|
||
|
||
|
||
<script data-cfasync="false">
|
||
document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
|
||
document.documentElement.dataset.theme = localStorage.getItem("theme") || "";
|
||
</script>
|
||
|
||
<!-- Loaded before other Sphinx assets -->
|
||
<link href="_static/styles/theme.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
|
||
<link href="_static/styles/bootstrap.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
|
||
<link href="_static/styles/pydata-sphinx-theme.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
|
||
|
||
|
||
<link href="_static/vendor/fontawesome/6.5.2/css/all.min.css?digest=dfe6caa3a7d634c4db9b" rel="stylesheet" />
|
||
<link rel="preload" as="font" type="font/woff2" crossorigin href="_static/vendor/fontawesome/6.5.2/webfonts/fa-solid-900.woff2" />
|
||
<link rel="preload" as="font" type="font/woff2" crossorigin href="_static/vendor/fontawesome/6.5.2/webfonts/fa-brands-400.woff2" />
|
||
<link rel="preload" as="font" type="font/woff2" crossorigin href="_static/vendor/fontawesome/6.5.2/webfonts/fa-regular-400.woff2" />
|
||
|
||
<link rel="stylesheet" type="text/css" href="_static/pygments.css?v=03e43079" />
|
||
<link rel="stylesheet" type="text/css" href="_static/styles/sphinx-book-theme.css?v=eba8b062" />
|
||
<link rel="stylesheet" type="text/css" href="_static/togglebutton.css?v=13237357" />
|
||
<link rel="stylesheet" type="text/css" href="_static/copybutton.css?v=76b2166b" />
|
||
<link rel="stylesheet" type="text/css" href="_static/mystnb.8ecb98da25f57f5357bf6f572d296f466b2cfe2517ffebfabe82451661e28f02.css" />
|
||
<link rel="stylesheet" type="text/css" href="_static/sphinx-thebe.css?v=4fa983c6" />
|
||
<link rel="stylesheet" type="text/css" href="_static/sphinx-design.min.css?v=95c83b7e" />
|
||
|
||
<!-- Pre-loaded scripts that we'll load fully later -->
|
||
<link rel="preload" as="script" href="_static/scripts/bootstrap.js?digest=dfe6caa3a7d634c4db9b" />
|
||
<link rel="preload" as="script" href="_static/scripts/pydata-sphinx-theme.js?digest=dfe6caa3a7d634c4db9b" />
|
||
<script src="_static/vendor/fontawesome/6.5.2/js/all.min.js?digest=dfe6caa3a7d634c4db9b"></script>
|
||
|
||
<script src="_static/documentation_options.js?v=9eb32ce0"></script>
|
||
<script src="_static/doctools.js?v=9a2dae69"></script>
|
||
<script src="_static/sphinx_highlight.js?v=dc90522c"></script>
|
||
<script src="_static/clipboard.min.js?v=a7894cd8"></script>
|
||
<script src="_static/copybutton.js?v=f281be69"></script>
|
||
<script src="_static/scripts/sphinx-book-theme.js?v=887ef09a"></script>
|
||
<script>let toggleHintShow = 'Click to show';</script>
|
||
<script>let toggleHintHide = 'Click to hide';</script>
|
||
<script>let toggleOpenOnPrint = 'true';</script>
|
||
<script src="_static/togglebutton.js?v=4a39c7ea"></script>
|
||
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown';</script>
|
||
<script src="_static/design-tabs.js?v=f930bc37"></script>
|
||
<script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"; const thebe_selector = ".thebe,.cell"; const thebe_selector_input = "pre"; const thebe_selector_output = ".output, .cell_output"</script>
|
||
<script async="async" src="_static/sphinx-thebe.js?v=c100c467"></script>
|
||
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown';</script>
|
||
<script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"; const thebe_selector = ".thebe,.cell"; const thebe_selector_input = "pre"; const thebe_selector_output = ".output, .cell_output"</script>
|
||
<script>window.MathJax = {"options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
|
||
<script defer="defer" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
|
||
<script>DOCUMENTATION_OPTIONS.pagename = 'chapter6';</script>
|
||
<link rel="index" title="Index" href="genindex.html" />
|
||
<link rel="search" title="Search" href="search.html" />
|
||
<link rel="next" title="10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods" href="chapter7.html" />
|
||
<link rel="prev" title="8. Support Vector Machines, overarching aims" href="chapter5.html" />
|
||
<meta name="viewport" content="width=device-width, initial-scale=1"/>
|
||
<meta name="docsearch:language" content="en"/>
|
||
</head>
|
||
|
||
|
||
<body data-bs-spy="scroll" data-bs-target=".bd-toc-nav" data-offset="180" data-bs-root-margin="0px 0px -60%" data-default-mode="">
|
||
|
||
|
||
|
||
<div id="pst-skip-link" class="skip-link d-print-none"><a href="#main-content">Skip to main content</a></div>
|
||
|
||
<div id="pst-scroll-pixel-helper"></div>
|
||
|
||
<button type="button" class="btn rounded-pill" id="pst-back-to-top">
|
||
<i class="fa-solid fa-arrow-up"></i>Back to top</button>
|
||
|
||
|
||
<input type="checkbox"
|
||
class="sidebar-toggle"
|
||
id="pst-primary-sidebar-checkbox"/>
|
||
<label class="overlay overlay-primary" for="pst-primary-sidebar-checkbox"></label>
|
||
|
||
<input type="checkbox"
|
||
class="sidebar-toggle"
|
||
id="pst-secondary-sidebar-checkbox"/>
|
||
<label class="overlay overlay-secondary" for="pst-secondary-sidebar-checkbox"></label>
|
||
|
||
<div class="search-button__wrapper">
|
||
<div class="search-button__overlay"></div>
|
||
<div class="search-button__search-container">
|
||
<form class="bd-search d-flex align-items-center"
|
||
action="search.html"
|
||
method="get">
|
||
<i class="fa-solid fa-magnifying-glass"></i>
|
||
<input type="search"
|
||
class="form-control"
|
||
name="q"
|
||
id="search-input"
|
||
placeholder="Search this book..."
|
||
aria-label="Search this book..."
|
||
autocomplete="off"
|
||
autocorrect="off"
|
||
autocapitalize="off"
|
||
spellcheck="false"/>
|
||
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd>K</kbd></span>
|
||
</form></div>
|
||
</div>
|
||
|
||
<div class="pst-async-banner-revealer d-none">
|
||
<aside id="bd-header-version-warning" class="d-none d-print-none" aria-label="Version warning"></aside>
|
||
</div>
|
||
|
||
|
||
<header class="bd-header navbar navbar-expand-lg bd-navbar d-print-none">
|
||
</header>
|
||
|
||
|
||
<div class="bd-container">
|
||
<div class="bd-container__inner bd-page-width">
|
||
|
||
|
||
|
||
<div class="bd-sidebar-primary bd-sidebar">
|
||
|
||
|
||
|
||
<div class="sidebar-header-items sidebar-primary__section">
|
||
|
||
|
||
|
||
|
||
</div>
|
||
|
||
<div class="sidebar-primary-items__start sidebar-primary__section">
|
||
<div class="sidebar-primary-item">
|
||
|
||
|
||
|
||
|
||
|
||
<a class="navbar-brand logo" href="intro.html">
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<img src="_static/logo.png" class="logo__image only-light" alt="Applied Data Analysis and Machine Learning - Home"/>
|
||
<script>document.write(`<img src="_static/logo.png" class="logo__image only-dark" alt="Applied Data Analysis and Machine Learning - Home"/>`);</script>
|
||
|
||
|
||
</a></div>
|
||
<div class="sidebar-primary-item">
|
||
|
||
<script>
|
||
document.write(`
|
||
<button class="btn search-button-field search-button__button" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
|
||
<i class="fa-solid fa-magnifying-glass"></i>
|
||
<span class="search-button__default-text">Search</span>
|
||
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd class="kbd-shortcut__modifier">K</kbd></span>
|
||
</button>
|
||
`);
|
||
</script></div>
|
||
<div class="sidebar-primary-item"><nav class="bd-links bd-docs-nav" aria-label="Main">
|
||
<div class="bd-toc-item navbar-nav active">
|
||
|
||
<ul class="nav bd-sidenav bd-sidenav__home-link">
|
||
<li class="toctree-l1">
|
||
<a class="reference internal" href="intro.html">
|
||
Applied Data Analysis and Machine Learning
|
||
</a>
|
||
</li>
|
||
</ul>
|
||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">About the course</span></p>
|
||
<ul class="nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="schedule.html">Course setting</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="teachers.html">Teachers and Grading</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="textbooks.html">Textbooks</a></li>
|
||
|
||
</ul>
|
||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Review of Statistics with Resampling Techniques and Linear Algebra</span></p>
|
||
<ul class="nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="statistics.html">1. Elements of Probability Theory and Statistical Data Analysis</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="linalg.html">2. Linear Algebra, Handling of Arrays and more Python Features</a></li>
|
||
</ul>
|
||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">From Regression to Support Vector Machines</span></p>
|
||
<ul class="nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter1.html">3. Linear Regression</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter2.html">4. Ridge and Lasso Regression</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter3.html">5. Resampling Methods</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter4.html">6. Logistic Regression</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapteroptimization.html">7. Optimization, the central part of any Machine Learning algortithm</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter5.html">8. Support Vector Machines, overarching aims</a></li>
|
||
</ul>
|
||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Decision Trees, Ensemble Methods and Boosting</span></p>
|
||
<ul class="current nav bd-sidenav">
|
||
<li class="toctree-l1 current active"><a class="current reference internal" href="#">9. Decision trees, overarching aims</a></li>
|
||
<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>
|
||
</ul>
|
||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Dimensionality Reduction</span></p>
|
||
<ul class="nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter8.html">11. Basic ideas of the Principal Component Analysis (PCA)</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="clustering.html">12. Clustering and Unsupervised Learning</a></li>
|
||
</ul>
|
||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Deep Learning Methods</span></p>
|
||
<ul class="nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter9.html">13. Neural networks</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter10.html">14. Building a Feed Forward Neural Network</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter11.html">15. Solving Differential Equations with Deep Learning</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter12.html">16. Convolutional Neural Networks</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter13.html">17. Recurrent neural networks: Overarching view</a></li>
|
||
|
||
</ul>
|
||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
|
||
<ul class="nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="E1.html">Exercises week 34</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week34.html">Week 34: Introduction to the course, Logistics and Practicalities</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="E2.html">Exercises week 35</a></li>
|
||
</ul>
|
||
|
||
</div>
|
||
</nav></div>
|
||
</div>
|
||
|
||
|
||
<div class="sidebar-primary-items__end sidebar-primary__section">
|
||
</div>
|
||
|
||
<div id="rtd-footer-container"></div>
|
||
|
||
|
||
</div>
|
||
|
||
<main id="main-content" class="bd-main" role="main">
|
||
|
||
|
||
|
||
<div class="sbt-scroll-pixel-helper"></div>
|
||
|
||
<div class="bd-content">
|
||
<div class="bd-article-container">
|
||
|
||
<div class="bd-header-article d-print-none">
|
||
<div class="header-article-items header-article__inner">
|
||
|
||
<div class="header-article-items__start">
|
||
|
||
<div class="header-article-item"><button class="sidebar-toggle primary-toggle btn btn-sm" title="Toggle primary sidebar" data-bs-placement="bottom" data-bs-toggle="tooltip">
|
||
<span class="fa-solid fa-bars"></span>
|
||
</button></div>
|
||
|
||
</div>
|
||
|
||
|
||
<div class="header-article-items__end">
|
||
|
||
<div class="header-article-item">
|
||
|
||
<div class="article-header-buttons">
|
||
|
||
|
||
|
||
|
||
|
||
<div class="dropdown dropdown-download-buttons">
|
||
<button class="btn dropdown-toggle" type="button" data-bs-toggle="dropdown" aria-expanded="false" aria-label="Download this page">
|
||
<i class="fas fa-download"></i>
|
||
</button>
|
||
<ul class="dropdown-menu">
|
||
|
||
|
||
|
||
<li><a href="_sources/chapter6.ipynb" target="_blank"
|
||
class="btn btn-sm btn-download-source-button dropdown-item"
|
||
title="Download source file"
|
||
data-bs-placement="left" data-bs-toggle="tooltip"
|
||
>
|
||
|
||
|
||
<span class="btn__icon-container">
|
||
<i class="fas fa-file"></i>
|
||
</span>
|
||
<span class="btn__text-container">.ipynb</span>
|
||
</a>
|
||
</li>
|
||
|
||
|
||
|
||
|
||
<li>
|
||
<button onclick="window.print()"
|
||
class="btn btn-sm btn-download-pdf-button dropdown-item"
|
||
title="Print to PDF"
|
||
data-bs-placement="left" data-bs-toggle="tooltip"
|
||
>
|
||
|
||
|
||
<span class="btn__icon-container">
|
||
<i class="fas fa-file-pdf"></i>
|
||
</span>
|
||
<span class="btn__text-container">.pdf</span>
|
||
</button>
|
||
</li>
|
||
|
||
</ul>
|
||
</div>
|
||
|
||
|
||
|
||
|
||
<button onclick="toggleFullScreen()"
|
||
class="btn btn-sm btn-fullscreen-button"
|
||
title="Fullscreen mode"
|
||
data-bs-placement="bottom" data-bs-toggle="tooltip"
|
||
>
|
||
|
||
|
||
<span class="btn__icon-container">
|
||
<i class="fas fa-expand"></i>
|
||
</span>
|
||
|
||
</button>
|
||
|
||
|
||
|
||
<script>
|
||
document.write(`
|
||
<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button" title="light/dark" aria-label="light/dark" data-bs-placement="bottom" data-bs-toggle="tooltip">
|
||
<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light"></i>
|
||
<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark"></i>
|
||
<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto"></i>
|
||
</button>
|
||
`);
|
||
</script>
|
||
|
||
|
||
<script>
|
||
document.write(`
|
||
<button class="btn btn-sm pst-navbar-icon search-button search-button__button" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
|
||
<i class="fa-solid fa-magnifying-glass fa-lg"></i>
|
||
</button>
|
||
`);
|
||
</script>
|
||
<button class="sidebar-toggle secondary-toggle btn btn-sm" title="Toggle secondary sidebar" data-bs-placement="bottom" data-bs-toggle="tooltip">
|
||
<span class="fa-solid fa-list"></span>
|
||
</button>
|
||
</div></div>
|
||
|
||
</div>
|
||
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="jb-print-docs-body" class="onlyprint">
|
||
<h1>Decision trees, overarching aims</h1>
|
||
<!-- Table of contents -->
|
||
<div id="print-main-content">
|
||
<div id="jb-print-toc">
|
||
|
||
<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="#basics-of-a-tree">9.1. Basics of a tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#general-features">9.2. General Features</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-a-tree-regression">9.3. Building a tree, regression</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#making-a-tree">9.3.1. Making a tree</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#schematic-regression-procedure">9.3.2. Schematic Regression Procedure</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-classification-tree">9.4. A Classification Tree</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-tree-classification">9.4.1. Visualizing the Tree, Classification</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#other-ways-of-visualizing-the-trees">9.4.2. Other ways of visualizing the trees</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#algorithms-for-setting-up-decision-trees">9.5. Algorithms for Setting up Decision Trees</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cart-algorithm-for-classification">9.5.1. The CART algorithm for Classification</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cart-algorithm-for-regression">9.5.2. The CART algorithm for Regression</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-gini-index">9.5.3. Computing the Gini index</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-python-code-to-read-in-data-and-perform-classification">9.5.4. Simple Python Code to read in Data and perform Classification</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#entropy-and-the-id3-algorithm">9.6. Entropy and the ID3 algorithm</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cancer-data-again-now-with-decision-trees-and-other-methods">9.6.1. Cancer Data again now with Decision Trees and other Methods</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#another-example-the-moons-again">9.6.2. Another example, the moons again</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pros-and-cons-of-trees-pros">9.7. Pros and cons of trees, pros</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#disadvantages">9.7.1. Disadvantages</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="searchbox"></div>
|
||
<article class="bd-article">
|
||
|
||
<section class="tex2jax_ignore mathjax_ignore" id="decision-trees-overarching-aims">
|
||
<h1><span class="section-number">9. </span>Decision trees, overarching aims<a class="headerlink" href="#decision-trees-overarching-aims" title="Link to this heading">#</a></h1>
|
||
<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 id="basics-of-a-tree">
|
||
<h2><span class="section-number">9.1. </span>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="general-features">
|
||
<h2><span class="section-number">9.2. </span>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>
|
||
<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>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>%matplotlib inline
|
||
|
||
import numpy as np
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.preprocessing import PolynomialFeatures
|
||
from sklearn.linear_model import LinearRegression
|
||
|
||
steps=250
|
||
|
||
distance=0
|
||
x=0
|
||
distance_list=[]
|
||
steps_list=[]
|
||
while x<steps:
|
||
distance+=np.random.randint(-1,2)
|
||
distance_list.append(distance)
|
||
x+=1
|
||
steps_list.append(x)
|
||
plt.plot(steps_list,distance_list, color='green', label="Random Walk Data")
|
||
|
||
steps_list=np.asarray(steps_list)
|
||
distance_list=np.asarray(distance_list)
|
||
|
||
X=steps_list[:,np.newaxis]
|
||
|
||
#Polynomial fits
|
||
|
||
#Degree 2
|
||
poly_features=PolynomialFeatures(degree=2, include_bias=False)
|
||
X_poly=poly_features.fit_transform(X)
|
||
|
||
lin_reg=LinearRegression()
|
||
poly_fit=lin_reg.fit(X_poly,distance_list)
|
||
b=lin_reg.coef_
|
||
c=lin_reg.intercept_
|
||
print ("2nd degree coefficients:")
|
||
print ("zero power: ",c)
|
||
print ("first power: ", b[0])
|
||
print ("second power: ",b[1])
|
||
|
||
z = np.arange(0, steps, .01)
|
||
z_mod=b[1]*z**2+b[0]*z+c
|
||
|
||
fit_mod=b[1]*X**2+b[0]*X+c
|
||
plt.plot(z, z_mod, color='r', label="2nd Degree Fit")
|
||
plt.title("Polynomial Regression")
|
||
|
||
plt.xlabel("Steps")
|
||
plt.ylabel("Distance")
|
||
|
||
#Degree 10
|
||
poly_features10=PolynomialFeatures(degree=10, include_bias=False)
|
||
X_poly10=poly_features10.fit_transform(X)
|
||
|
||
poly_fit10=lin_reg.fit(X_poly10,distance_list)
|
||
|
||
y_plot=poly_fit10.predict(X_poly10)
|
||
plt.plot(X, y_plot, color='black', label="10th Degree Fit")
|
||
|
||
plt.legend()
|
||
plt.show()
|
||
|
||
|
||
#Decision Tree Regression
|
||
from sklearn.tree import DecisionTreeRegressor
|
||
regr_1=DecisionTreeRegressor(max_depth=2)
|
||
regr_2=DecisionTreeRegressor(max_depth=5)
|
||
regr_3=DecisionTreeRegressor(max_depth=7)
|
||
regr_1.fit(X, distance_list)
|
||
regr_2.fit(X, distance_list)
|
||
regr_3.fit(X, distance_list)
|
||
|
||
X_test = np.arange(0.0, steps, 0.01)[:, np.newaxis]
|
||
y_1 = regr_1.predict(X_test)
|
||
y_2 = regr_2.predict(X_test)
|
||
y_3=regr_3.predict(X_test)
|
||
|
||
# Plot the results
|
||
plt.figure()
|
||
plt.scatter(X, distance_list, s=2.5, c="black", label="data")
|
||
plt.plot(X_test, y_1, color="red",
|
||
label="max_depth=2", linewidth=2)
|
||
plt.plot(X_test, y_2, color="green", label="max_depth=5", linewidth=2)
|
||
plt.plot(X_test, y_3, color="m", label="max_depth=7", linewidth=2)
|
||
|
||
plt.xlabel("Data")
|
||
plt.ylabel("Darget")
|
||
plt.title("Decision Tree Regression")
|
||
plt.legend()
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="building-a-tree-regression">
|
||
<h2><span class="section-number">9.3. </span>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>
|
||
<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 id="making-a-tree">
|
||
<h3><span class="section-number">9.3.1. </span>Making a tree<a class="headerlink" href="#making-a-tree" title="Link to this heading">#</a></h3>
|
||
<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>
|
||
<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>
|
||
<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">
|
||
<h3><span class="section-number">9.3.2. </span>Schematic Regression Procedure<a class="headerlink" href="#schematic-regression-procedure" title="Link to this heading">#</a></h3>
|
||
<p>Building a Regression Tree</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>
|
||
<p>!eblock</p>
|
||
</section>
|
||
</section>
|
||
<section id="a-classification-tree">
|
||
<h2><span class="section-number">9.4. </span>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>
|
||
<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>
|
||
<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 id="visualizing-the-tree-classification">
|
||
<h3><span class="section-number">9.4.1. </span>Visualizing the Tree, Classification<a class="headerlink" href="#visualizing-the-tree-classification" title="Link to this heading">#</a></h3>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import os
|
||
from sklearn.datasets import load_breast_cancer
|
||
from sklearn.tree import DecisionTreeClassifier
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.metrics import confusion_matrix
|
||
from sklearn.tree import export_graphviz
|
||
|
||
from IPython.display import Image
|
||
from pydot import graph_from_dot_data
|
||
import pandas as pd
|
||
import numpy as np
|
||
|
||
|
||
cancer = load_breast_cancer()
|
||
X = pd.DataFrame(cancer.data, columns=cancer.feature_names)
|
||
print(X)
|
||
y = pd.Categorical.from_codes(cancer.target, cancer.target_names)
|
||
y = pd.get_dummies(y)
|
||
print(y)
|
||
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)
|
||
tree_clf = DecisionTreeClassifier(max_depth=5)
|
||
tree_clf.fit(X_train, y_train)
|
||
|
||
export_graphviz(
|
||
tree_clf,
|
||
out_file="DataFiles/cancer.dot",
|
||
feature_names=cancer.feature_names,
|
||
class_names=cancer.target_names,
|
||
rounded=True,
|
||
filled=True
|
||
)
|
||
cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
|
||
os.system(cmd)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Common imports
|
||
import numpy as np
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.tree import DecisionTreeClassifier
|
||
from sklearn.datasets import make_moons
|
||
from sklearn.tree import export_graphviz
|
||
from pydot import graph_from_dot_data
|
||
import pandas as pd
|
||
import os
|
||
|
||
np.random.seed(42)
|
||
X, y = make_moons(n_samples=100, noise=0.25, random_state=53)
|
||
X_train, X_test, y_train, y_test = train_test_split(X,y,random_state=0)
|
||
tree_clf = DecisionTreeClassifier(max_depth=5)
|
||
tree_clf.fit(X_train, y_train)
|
||
|
||
export_graphviz(
|
||
tree_clf,
|
||
out_file="DataFiles/moons.dot",
|
||
rounded=True,
|
||
filled=True
|
||
)
|
||
cmd = 'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'
|
||
os.system(cmd)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="other-ways-of-visualizing-the-trees">
|
||
<h3><span class="section-number">9.4.2. </span>Other ways of visualizing the trees<a class="headerlink" href="#other-ways-of-visualizing-the-trees" title="Link to this heading">#</a></h3>
|
||
<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-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import load_iris
|
||
from sklearn import tree
|
||
X, y = load_iris(return_X_y=True)
|
||
tree_clf = tree.DecisionTreeClassifier()
|
||
tree_clf = tree_clf.fit(X, y)
|
||
# and then plot the tree
|
||
tree.plot_tree(tree_clf)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<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-none notranslate"><div class="highlight"><pre><span></span>from sklearn.datasets import load_iris
|
||
from sklearn.tree import DecisionTreeClassifier
|
||
from sklearn.tree import export_text
|
||
iris = load_iris()
|
||
decision_tree = DecisionTreeClassifier(random_state=0, max_depth=2)
|
||
decision_tree = decision_tree.fit(iris.data, iris.target)
|
||
r = export_text(decision_tree, feature_names=iris['feature_names'])
|
||
print(r)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
<section id="algorithms-for-setting-up-decision-trees">
|
||
<h2><span class="section-number">9.5. </span>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 id="the-cart-algorithm-for-classification">
|
||
<h3><span class="section-number">9.5.1. </span>The CART algorithm for Classification<a class="headerlink" href="#the-cart-algorithm-for-classification" title="Link to this heading">#</a></h3>
|
||
<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">
|
||
<h3><span class="section-number">9.5.2. </span>The CART algorithm for Regression<a class="headerlink" href="#the-cart-algorithm-for-regression" title="Link to this heading">#</a></h3>
|
||
<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="computing-the-gini-index">
|
||
<h3><span class="section-number">9.5.3. </span>Computing the Gini index<a class="headerlink" href="#computing-the-gini-index" title="Link to this heading">#</a></h3>
|
||
<p>The example we will look at is a classical one in many Machine
|
||
Learning applications. Based on various meteorological features, we
|
||
have several so-called attributes which decide whether we at the end
|
||
will do some outdoor activity like skiing, going for a bike ride etc
|
||
etc. The table here contains the feautures <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 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">
|
||
<h3><span class="section-number">9.5.4. </span>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></h3>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Common imports
|
||
import numpy as np
|
||
import pandas as pd
|
||
import matplotlib.pyplot as plt
|
||
from sklearn.tree import DecisionTreeClassifier
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.tree import export_graphviz
|
||
from sklearn.preprocessing import StandardScaler, OneHotEncoder
|
||
from sklearn.compose import ColumnTransformer
|
||
from IPython.display import Image
|
||
from pydot import graph_from_dot_data
|
||
import os
|
||
|
||
# Where to save the figures and data files
|
||
PROJECT_ROOT_DIR = "Results"
|
||
FIGURE_ID = "Results/FigureFiles"
|
||
DATA_ID = "DataFiles/"
|
||
|
||
if not os.path.exists(PROJECT_ROOT_DIR):
|
||
os.mkdir(PROJECT_ROOT_DIR)
|
||
|
||
if not os.path.exists(FIGURE_ID):
|
||
os.makedirs(FIGURE_ID)
|
||
|
||
if not os.path.exists(DATA_ID):
|
||
os.makedirs(DATA_ID)
|
||
|
||
def image_path(fig_id):
|
||
return os.path.join(FIGURE_ID, fig_id)
|
||
|
||
def data_path(dat_id):
|
||
return os.path.join(DATA_ID, dat_id)
|
||
|
||
def save_fig(fig_id):
|
||
plt.savefig(image_path(fig_id) + ".png", format='png')
|
||
|
||
infile = open(data_path("rideclass.csv"),'r')
|
||
|
||
# Read the experimental data with Pandas
|
||
from IPython.display import display
|
||
ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))
|
||
ridedata = pd.DataFrame(ridedata)
|
||
|
||
# Features and targets
|
||
X = ridedata.loc[:, ridedata.columns != 'Ride'].values
|
||
y = ridedata.loc[:, ridedata.columns == 'Ride'].values
|
||
|
||
# Create the encoder.
|
||
encoder = OneHotEncoder(handle_unknown="ignore")
|
||
# Assume for simplicity all features are categorical.
|
||
encoder.fit(X)
|
||
# Apply the encoder.
|
||
X = encoder.transform(X)
|
||
print(X)
|
||
# Then do a Classification tree
|
||
tree_clf = DecisionTreeClassifier(max_depth=2)
|
||
tree_clf.fit(X, y)
|
||
print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
|
||
#transfer to a decision tree graph
|
||
export_graphviz(
|
||
tree_clf,
|
||
out_file="DataFiles/ride.dot",
|
||
rounded=True,
|
||
filled=True
|
||
)
|
||
cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
|
||
os.system(cmd)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<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-none notranslate"><div class="highlight"><pre><span></span># Split a dataset based on an attribute and an attribute value
|
||
def test_split(index, value, dataset):
|
||
left, right = list(), list()
|
||
for row in dataset:
|
||
if row[index] < value:
|
||
left.append(row)
|
||
else:
|
||
right.append(row)
|
||
return left, right
|
||
|
||
# Calculate the Gini index for a split dataset
|
||
def gini_index(groups, classes):
|
||
# count all samples at split point
|
||
n_instances = float(sum([len(group) for group in groups]))
|
||
# sum weighted Gini index for each group
|
||
gini = 0.0
|
||
for group in groups:
|
||
size = float(len(group))
|
||
# avoid divide by zero
|
||
if size == 0:
|
||
continue
|
||
score = 0.0
|
||
# score the group based on the score for each class
|
||
for class_val in classes:
|
||
p = [row[-1] for row in group].count(class_val) / size
|
||
score += p * p
|
||
# weight the group score by its relative size
|
||
gini += (1.0 - score) * (size / n_instances)
|
||
return gini
|
||
|
||
# Select the best split point for a dataset
|
||
def get_split(dataset):
|
||
class_values = list(set(row[-1] for row in dataset))
|
||
b_index, b_value, b_score, b_groups = 999, 999, 999, None
|
||
for index in range(len(dataset[0])-1):
|
||
for row in dataset:
|
||
groups = test_split(index, row[index], dataset)
|
||
gini = gini_index(groups, class_values)
|
||
print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
|
||
if gini < b_score:
|
||
b_index, b_value, b_score, b_groups = index, row[index], gini, groups
|
||
return {'index':b_index, 'value':b_value, 'groups':b_groups}
|
||
|
||
dataset = [[0,0,0,0,0],
|
||
[0,0,0,1,1],
|
||
[1,0,0,0,1],
|
||
[2,1,0,0,1],
|
||
[2,2,1,0,1],
|
||
[2,2,1,1,0],
|
||
[1,2,1,1,1],
|
||
[0,1,0,0,0],
|
||
[0,2,1,0,1],
|
||
[2,1,1,0,1],
|
||
[0,1,1,1,1],
|
||
[1,1,0,1,1],
|
||
[1,0,1,0,1],
|
||
[2,1,0,1,0]]
|
||
|
||
split = get_split(dataset)
|
||
print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
<section id="entropy-and-the-id3-algorithm">
|
||
<h2><span class="section-number">9.6. </span>Entropy and the ID3 algorithm<a class="headerlink" href="#entropy-and-the-id3-algorithm" title="Link to this heading">#</a></h2>
|
||
<p>The ID3 algorithm learns decision trees by constructing
|
||
them in a top down way, beginning with the question <strong>which attribute should be tested at the root of the tree</strong>?</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.</p></li>
|
||
<li><p>The best attribute is selected and used as the test at the root node of the tree.</p></li>
|
||
<li><p>A descendant of the root node is then created for each possible value of this attribute.</p></li>
|
||
<li><p>Training examples are sorted to the appropriate descendant node.</p></li>
|
||
<li><p>The entire process is then repeated using the training examples associated with each descendant node to select the best attribute to test at that point in the tree.</p></li>
|
||
<li><p>This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.</p></li>
|
||
</ol>
|
||
<p>The ID3 algorithm selects which attribute to test at each node in the
|
||
tree.</p>
|
||
<p>We would like to select the attribute that is most useful for classifying
|
||
examples.</p>
|
||
<p>What is a good quantitative measure of the worth of an attribute?</p>
|
||
<p>Information gain measures how well a given attribute separates the
|
||
training examples according to their target classification.</p>
|
||
<p>The ID3 algorithm uses this information gain measure to select among the candidate
|
||
attributes at each step while growing the tree.</p>
|
||
<section id="cancer-data-again-now-with-decision-trees-and-other-methods">
|
||
<h3><span class="section-number">9.6.1. </span>Cancer Data again now with Decision Trees and other Methods<a class="headerlink" href="#cancer-data-again-now-with-decision-trees-and-other-methods" title="Link to this heading">#</a></h3>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import matplotlib.pyplot as plt
|
||
import numpy as np
|
||
from sklearn.model_selection import train_test_split
|
||
from sklearn.datasets import load_breast_cancer
|
||
from sklearn.svm import SVC
|
||
from sklearn.linear_model import LogisticRegression
|
||
from sklearn.tree import DecisionTreeClassifier
|
||
|
||
# Load the data
|
||
cancer = load_breast_cancer()
|
||
|
||
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
|
||
print(X_train.shape)
|
||
print(X_test.shape)
|
||
# Logistic Regression
|
||
logreg = LogisticRegression(solver='lbfgs')
|
||
logreg.fit(X_train, y_train)
|
||
print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
|
||
# Support vector machine
|
||
svm = SVC(gamma='auto', C=100)
|
||
svm.fit(X_train, y_train)
|
||
print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
|
||
# Decision Trees
|
||
deep_tree_clf = DecisionTreeClassifier(max_depth=None)
|
||
deep_tree_clf.fit(X_train, y_train)
|
||
print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
|
||
#now scale the data
|
||
from sklearn.preprocessing import StandardScaler
|
||
scaler = StandardScaler()
|
||
scaler.fit(X_train)
|
||
X_train_scaled = scaler.transform(X_train)
|
||
X_test_scaled = scaler.transform(X_test)
|
||
# Logistic Regression
|
||
logreg.fit(X_train_scaled, y_train)
|
||
print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
|
||
# Support Vector Machine
|
||
svm.fit(X_train_scaled, y_train)
|
||
print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
|
||
# Decision Trees
|
||
deep_tree_clf.fit(X_train_scaled, y_train)
|
||
print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="another-example-the-moons-again">
|
||
<h3><span class="section-number">9.6.2. </span>Another example, the moons again<a class="headerlink" href="#another-example-the-moons-again" title="Link to this heading">#</a></h3>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from __future__ import division, print_function, unicode_literals
|
||
|
||
# Common imports
|
||
import numpy as np
|
||
import os
|
||
|
||
# to make this notebook's output stable across runs
|
||
np.random.seed(42)
|
||
|
||
# To plot pretty figures
|
||
import matplotlib
|
||
import matplotlib.pyplot as plt
|
||
from matplotlib.colors import ListedColormap
|
||
plt.rcParams['axes.labelsize'] = 14
|
||
plt.rcParams['xtick.labelsize'] = 12
|
||
plt.rcParams['ytick.labelsize'] = 12
|
||
|
||
|
||
from sklearn.svm import SVC
|
||
from sklearn import datasets
|
||
from sklearn.tree import DecisionTreeClassifier
|
||
from sklearn.datasets import make_moons
|
||
from sklearn.tree import export_graphviz
|
||
|
||
Xm, ym = make_moons(n_samples=100, noise=0.25, random_state=53)
|
||
|
||
deep_tree_clf1 = DecisionTreeClassifier(random_state=42)
|
||
deep_tree_clf2 = DecisionTreeClassifier(min_samples_leaf=4, random_state=42)
|
||
deep_tree_clf1.fit(Xm, ym)
|
||
deep_tree_clf2.fit(Xm, ym)
|
||
|
||
|
||
def plot_decision_boundary(clf, X, y, axes=[0, 7.5, 0, 3], iris=True, legend=False, plot_training=True):
|
||
x1s = np.linspace(axes[0], axes[1], 100)
|
||
x2s = np.linspace(axes[2], axes[3], 100)
|
||
x1, x2 = np.meshgrid(x1s, x2s)
|
||
X_new = np.c_[x1.ravel(), x2.ravel()]
|
||
y_pred = clf.predict(X_new).reshape(x1.shape)
|
||
custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
|
||
plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
|
||
if not iris:
|
||
custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
|
||
plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
|
||
if plot_training:
|
||
plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", label="Iris-Setosa")
|
||
plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", label="Iris-Versicolor")
|
||
plt.plot(X[:, 0][y==2], X[:, 1][y==2], "g^", label="Iris-Virginica")
|
||
plt.axis(axes)
|
||
if iris:
|
||
plt.xlabel("Petal length", fontsize=14)
|
||
plt.ylabel("Petal width", fontsize=14)
|
||
else:
|
||
plt.xlabel(r"$x_1$", fontsize=18)
|
||
plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
|
||
if legend:
|
||
plt.legend(loc="lower right", fontsize=14)
|
||
plt.figure(figsize=(11, 4))
|
||
plt.subplot(121)
|
||
plot_decision_boundary(deep_tree_clf1, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
|
||
plt.title("No restrictions", fontsize=16)
|
||
plt.subplot(122)
|
||
plot_decision_boundary(deep_tree_clf2, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
|
||
plt.title("min_samples_leaf = {}".format(deep_tree_clf2.min_samples_leaf), fontsize=14)
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>np.random.seed(6)
|
||
Xs = np.random.rand(100, 2) - 0.5
|
||
ys = (Xs[:, 0] > 0).astype(np.float32) * 2
|
||
|
||
angle = np.pi/4
|
||
rotation_matrix = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])
|
||
Xsr = Xs.dot(rotation_matrix)
|
||
|
||
tree_clf_s = DecisionTreeClassifier(random_state=42)
|
||
tree_clf_s.fit(Xs, ys)
|
||
tree_clf_sr = DecisionTreeClassifier(random_state=42)
|
||
tree_clf_sr.fit(Xsr, ys)
|
||
|
||
plt.figure(figsize=(11, 4))
|
||
plt.subplot(121)
|
||
plot_decision_boundary(tree_clf_s, Xs, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
|
||
plt.subplot(122)
|
||
plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
|
||
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Quadratic training set + noise
|
||
np.random.seed(42)
|
||
m = 200
|
||
X = np.random.rand(m, 1)
|
||
y = 4 * (X - 0.5) ** 2
|
||
y = y + np.random.randn(m, 1) / 10
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.tree import DecisionTreeRegressor
|
||
|
||
tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)
|
||
tree_reg.fit(X, y)
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.tree import DecisionTreeRegressor
|
||
|
||
tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)
|
||
tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)
|
||
tree_reg1.fit(X, y)
|
||
tree_reg2.fit(X, y)
|
||
|
||
def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel="$y$"):
|
||
x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)
|
||
y_pred = tree_reg.predict(x1)
|
||
plt.axis(axes)
|
||
plt.xlabel("$x_1$", fontsize=18)
|
||
if ylabel:
|
||
plt.ylabel(ylabel, fontsize=18, rotation=0)
|
||
plt.plot(X, y, "b.")
|
||
plt.plot(x1, y_pred, "r.-", linewidth=2, label=r"$\hat{y}$")
|
||
|
||
plt.figure(figsize=(11, 4))
|
||
plt.subplot(121)
|
||
plot_regression_predictions(tree_reg1, X, y)
|
||
for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
|
||
plt.plot([split, split], [-0.2, 1], style, linewidth=2)
|
||
plt.text(0.21, 0.65, "Depth=0", fontsize=15)
|
||
plt.text(0.01, 0.2, "Depth=1", fontsize=13)
|
||
plt.text(0.65, 0.8, "Depth=1", fontsize=13)
|
||
plt.legend(loc="upper center", fontsize=18)
|
||
plt.title("max_depth=2", fontsize=14)
|
||
|
||
plt.subplot(122)
|
||
plot_regression_predictions(tree_reg2, X, y, ylabel=None)
|
||
for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
|
||
plt.plot([split, split], [-0.2, 1], style, linewidth=2)
|
||
for split in (0.0458, 0.1298, 0.2873, 0.9040):
|
||
plt.plot([split, split], [-0.2, 1], "k:", linewidth=1)
|
||
plt.text(0.3, 0.5, "Depth=2", fontsize=13)
|
||
plt.title("max_depth=3", fontsize=14)
|
||
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>tree_reg1 = DecisionTreeRegressor(random_state=42)
|
||
tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)
|
||
tree_reg1.fit(X, y)
|
||
tree_reg2.fit(X, y)
|
||
|
||
x1 = np.linspace(0, 1, 500).reshape(-1, 1)
|
||
y_pred1 = tree_reg1.predict(x1)
|
||
y_pred2 = tree_reg2.predict(x1)
|
||
|
||
plt.figure(figsize=(11, 4))
|
||
|
||
plt.subplot(121)
|
||
plt.plot(X, y, "b.")
|
||
plt.plot(x1, y_pred1, "r.-", linewidth=2, label=r"$\hat{y}$")
|
||
plt.axis([0, 1, -0.2, 1.1])
|
||
plt.xlabel("$x_1$", fontsize=18)
|
||
plt.ylabel("$y$", fontsize=18, rotation=0)
|
||
plt.legend(loc="upper center", fontsize=18)
|
||
plt.title("No restrictions", fontsize=14)
|
||
|
||
plt.subplot(122)
|
||
plt.plot(X, y, "b.")
|
||
plt.plot(x1, y_pred2, "r.-", linewidth=2, label=r"$\hat{y}$")
|
||
plt.axis([0, 1, -0.2, 1.1])
|
||
plt.xlabel("$x_1$", fontsize=18)
|
||
plt.title("min_samples_leaf={}".format(tree_reg2.min_samples_leaf), fontsize=14)
|
||
|
||
plt.show()
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
</section>
|
||
<section id="pros-and-cons-of-trees-pros">
|
||
<h2><span class="section-number">9.7. </span>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 id="disadvantages">
|
||
<h3><span class="section-number">9.7.1. </span>Disadvantages<a class="headerlink" href="#disadvantages" title="Link to this heading">#</a></h3>
|
||
<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>
|
||
</section>
|
||
|
||
<script type="text/x-thebe-config">
|
||
{
|
||
requestKernel: true,
|
||
binderOptions: {
|
||
repo: "binder-examples/jupyter-stacks-datascience",
|
||
ref: "master",
|
||
},
|
||
codeMirrorConfig: {
|
||
theme: "abcdef",
|
||
mode: "python"
|
||
},
|
||
kernelOptions: {
|
||
name: "python3",
|
||
path: "./."
|
||
},
|
||
predefinedOutput: true
|
||
}
|
||
</script>
|
||
<script>kernelName = 'python3'</script>
|
||
|
||
</article>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<footer class="prev-next-footer d-print-none">
|
||
|
||
<div class="prev-next-area">
|
||
<a class="left-prev"
|
||
href="chapter5.html"
|
||
title="previous page">
|
||
<i class="fa-solid fa-angle-left"></i>
|
||
<div class="prev-next-info">
|
||
<p class="prev-next-subtitle">previous</p>
|
||
<p class="prev-next-title"><span class="section-number">8. </span>Support Vector Machines, overarching aims</p>
|
||
</div>
|
||
</a>
|
||
<a class="right-next"
|
||
href="chapter7.html"
|
||
title="next page">
|
||
<div class="prev-next-info">
|
||
<p class="prev-next-subtitle">next</p>
|
||
<p class="prev-next-title"><span class="section-number">10. </span>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</p>
|
||
</div>
|
||
<i class="fa-solid fa-angle-right"></i>
|
||
</a>
|
||
</div>
|
||
</footer>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
<div class="bd-sidebar-secondary bd-toc"><div class="sidebar-secondary-items sidebar-secondary__inner">
|
||
|
||
|
||
<div class="sidebar-secondary-item">
|
||
<div class="page-toc tocsection onthispage">
|
||
<i class="fa-solid fa-list"></i> Contents
|
||
</div>
|
||
<nav class="bd-toc-nav page-toc">
|
||
<ul class="visible nav section-nav flex-column">
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basics-of-a-tree">9.1. Basics of a tree</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#general-features">9.2. General Features</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#building-a-tree-regression">9.3. Building a tree, regression</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#making-a-tree">9.3.1. Making a tree</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#schematic-regression-procedure">9.3.2. Schematic Regression Procedure</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-classification-tree">9.4. A Classification Tree</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-tree-classification">9.4.1. Visualizing the Tree, Classification</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#other-ways-of-visualizing-the-trees">9.4.2. Other ways of visualizing the trees</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#algorithms-for-setting-up-decision-trees">9.5. Algorithms for Setting up Decision Trees</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cart-algorithm-for-classification">9.5.1. The CART algorithm for Classification</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-cart-algorithm-for-regression">9.5.2. The CART algorithm for Regression</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-gini-index">9.5.3. Computing the Gini index</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#simple-python-code-to-read-in-data-and-perform-classification">9.5.4. Simple Python Code to read in Data and perform Classification</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#entropy-and-the-id3-algorithm">9.6. Entropy and the ID3 algorithm</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cancer-data-again-now-with-decision-trees-and-other-methods">9.6.1. Cancer Data again now with Decision Trees and other Methods</a></li>
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#another-example-the-moons-again">9.6.2. Another example, the moons again</a></li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pros-and-cons-of-trees-pros">9.7. Pros and cons of trees, pros</a><ul class="nav section-nav flex-column">
|
||
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#disadvantages">9.7.1. Disadvantages</a></li>
|
||
</ul>
|
||
</li>
|
||
</ul>
|
||
</nav></div>
|
||
|
||
</div></div>
|
||
|
||
|
||
</div>
|
||
<footer class="bd-footer-content">
|
||
|
||
<div class="bd-footer-content__inner container">
|
||
|
||
<div class="footer-item">
|
||
|
||
<p class="component-author">
|
||
By Morten Hjorth-Jensen
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="footer-item">
|
||
|
||
|
||
<p class="copyright">
|
||
|
||
© Copyright 2023.
|
||
<br/>
|
||
|
||
</p>
|
||
|
||
</div>
|
||
|
||
<div class="footer-item">
|
||
|
||
</div>
|
||
|
||
<div class="footer-item">
|
||
|
||
</div>
|
||
|
||
</div>
|
||
</footer>
|
||
|
||
|
||
</main>
|
||
</div>
|
||
</div>
|
||
|
||
<!-- Scripts loaded after <body> so the DOM is not blocked -->
|
||
<script src="_static/scripts/bootstrap.js?digest=dfe6caa3a7d634c4db9b"></script>
|
||
<script src="_static/scripts/pydata-sphinx-theme.js?digest=dfe6caa3a7d634c4db9b"></script>
|
||
|
||
<footer class="bd-footer">
|
||
</footer>
|
||
</body>
|
||
</html> |