Morten Hjorth-Jensen
2280 lines
200 KiB
HTML
2280 lines
200 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>Week 48: Gradient boosting and summary of course — 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>
|
||
<!--
|
||
this give us a css class that will be invisible only if js is disabled
|
||
-->
|
||
<noscript>
|
||
<style>
|
||
.pst-js-only { display: none !important; }
|
||
|
||
</style>
|
||
</noscript>
|
||
|
||
<!-- Loaded before other Sphinx assets -->
|
||
<link href="_static/styles/theme.css?digest=26a4bc78f4c0ddb94549" rel="stylesheet" />
|
||
<link href="_static/styles/pydata-sphinx-theme.css?digest=26a4bc78f4c0ddb94549" rel="stylesheet" />
|
||
|
||
<link rel="stylesheet" type="text/css" href="_static/pygments.css?v=fa44fd50" />
|
||
<link rel="stylesheet" type="text/css" href="_static/styles/sphinx-book-theme.css?v=a3416100" />
|
||
<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.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css?v=be8a1c11" />
|
||
<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" />
|
||
|
||
<!-- So that users can add custom icons -->
|
||
<script src="_static/scripts/fontawesome.js?digest=26a4bc78f4c0ddb94549"></script>
|
||
<!-- Pre-loaded scripts that we'll load fully later -->
|
||
<link rel="preload" as="script" href="_static/scripts/bootstrap.js?digest=26a4bc78f4c0ddb94549" />
|
||
<link rel="preload" as="script" href="_static/scripts/pydata-sphinx-theme.js?digest=26a4bc78f4c0ddb94549" />
|
||
|
||
<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 = 'week48';</script>
|
||
<link rel="index" title="Index" href="genindex.html" />
|
||
<link rel="search" title="Search" href="search.html" />
|
||
<link rel="next" title="Exercises week 48" href="exercisesweek48.html" />
|
||
<link rel="prev" title="Exercise week 47" href="exercisesweek47.html" />
|
||
<meta name="viewport" content="width=device-width, initial-scale=1"/>
|
||
<meta name="docsearch:language" content="en"/>
|
||
<meta name="docsearch:version" content="" />
|
||
</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>
|
||
|
||
|
||
<dialog id="pst-search-dialog">
|
||
|
||
<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"
|
||
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>
|
||
</dialog>
|
||
|
||
<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">
|
||
|
||
|
||
|
||
<dialog id="pst-primary-sidebar-modal"></dialog>
|
||
<div id="pst-primary-sidebar" 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"/>
|
||
<img src="_static/logo.png" class="logo__image only-dark pst-js-only" alt="Applied Data Analysis and Machine Learning - Home"/>
|
||
|
||
|
||
</a></div>
|
||
<div class="sidebar-primary-item">
|
||
|
||
<button class="btn search-button-field search-button__button pst-js-only" 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></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">Teaching schedule with links to material</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="nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="chapter6.html">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="current nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek34.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="exercisesweek35.html">Exercises week 35</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week35.html">Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Statistical interpretations</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek37.html">Exercises week 37</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week37.html">Week 37: Statistical interpretations and Resampling Methods</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek38.html">Exercises week 38</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week38.html">Week 38: Logistic Regression and Optimization</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek39.html">Exercises week 39</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week39.html">Week 39: Optimization and Gradient Methods</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week40.html">Week 40: Gradient descent methods (continued) and start Neural networks</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek41.html">Exercises week 41</a></li>
|
||
|
||
|
||
<li class="toctree-l1"><a class="reference internal" href="week41.html">Week 41 Neural networks and constructing a neural network code</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek42.html">Exercises week 42</a></li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="toctree-l1"><a class="reference internal" href="week42.html">Week 42 Constructing a Neural Network code with examples</a></li>
|
||
<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>
|
||
<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>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek43.html">Exercises week 43</a></li>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<li class="toctree-l1"><a class="reference internal" href="week44.html">Week 44, Convolutional Neural Networks (CNN)</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week45.html">Week 45, Convolutional Neural Networks (CCNs) and Recurrent Neural Networks (RNNs)</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="week46.html">Week 46: Decision Trees, Ensemble methods and Random Forests</a></li>
|
||
<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>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek47.html">Exercise week 47</a></li>
|
||
|
||
<li class="toctree-l1 current active"><a class="current reference internal" href="#">Week 48: Gradient boosting and summary of course</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="exercisesweek48.html">Exercises week 48</a></li>
|
||
|
||
</ul>
|
||
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Projects</span></p>
|
||
<ul class="nav bd-sidenav">
|
||
<li class="toctree-l1"><a class="reference internal" href="project1.html">Project 1 on Machine Learning, deadline October 7 (midnight), 2024</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="project2.html">Project 2 on Machine Learning, deadline November 4 (Midnight)</a></li>
|
||
<li class="toctree-l1"><a class="reference internal" href="project3.html">Project 3 on Machine Learning, deadline December 9 (midnight), 2024</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-launch-buttons">
|
||
<button class="btn dropdown-toggle" type="button" data-bs-toggle="dropdown" aria-expanded="false" aria-label="Launch interactive content">
|
||
<i class="fas fa-rocket"></i>
|
||
</button>
|
||
<ul class="dropdown-menu">
|
||
|
||
|
||
|
||
<li><a href="https://mybinder.org/v2/git/https%3A//compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/index.html/master?urlpath=tree/week48.ipynb" target="_blank"
|
||
class="btn btn-sm dropdown-item"
|
||
title="Launch on Binder"
|
||
data-bs-placement="left" data-bs-toggle="tooltip"
|
||
>
|
||
|
||
|
||
<span class="btn__icon-container">
|
||
|
||
<img alt="Binder logo" src="_static/images/logo_binder.svg">
|
||
</span>
|
||
<span class="btn__text-container">Binder</span>
|
||
</a>
|
||
</li>
|
||
|
||
</ul>
|
||
</div>
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<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/week48.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>
|
||
|
||
|
||
|
||
<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button pst-js-only" aria-label="Color mode" data-bs-title="Color mode" data-bs-placement="bottom" data-bs-toggle="tooltip">
|
||
<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light" title="Light"></i>
|
||
<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark" title="Dark"></i>
|
||
<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto" title="System Settings"></i>
|
||
</button>
|
||
|
||
|
||
<button class="btn btn-sm pst-navbar-icon search-button search-button__button pst-js-only" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
|
||
<i class="fa-solid fa-magnifying-glass fa-lg"></i>
|
||
</button>
|
||
<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>Week 48: Gradient boosting and summary of course</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="#overview-of-week-48">Overview of week 48</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lecture-monday-november-25">Lecture Monday, November 25</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lab-sessions">Lab sessions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forest-algorithm-reminder-from-last-week">Random Forest Algorithm, reminder from last week</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>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-an-adaboost-code-yourself">Making an ADAboost code yourself</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-basics-with-steepest-descent-functional-gradient-descent">Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-squared-error-again-steepest-descent">The Squared-Error again! Steepest Descent</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#steepest-descent-example">Steepest Descent Example</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-algorithm">Gradient Boosting, algorithm</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-examples-of-regression">Gradient Boosting, Examples of Regression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-classification-example">Gradient Boosting, Classification Example</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#xgboost-extreme-gradient-boosting">XGBoost: Extreme Gradient Boosting</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#xgboost-on-the-cancer-data">Xgboost on the Cancer Data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-making-our-own-code-for-a-regression-case">Gradient boosting, making our own code for a regression case</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary-of-course">Summary of course</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#what-me-worry-no-final-exam-in-this-course">What? Me worry? No final exam in this course!</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#topics-we-have-covered-this-year">Topics we have covered this year</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#statistical-analysis-and-optimization-of-data">Statistical analysis and optimization of data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#machine-learning">Machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-outcomes-and-overarching-aims-of-this-course">Learning outcomes and overarching aims of this course</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#perspective-on-machine-learning">Perspective on Machine Learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#machine-learning-research">Machine Learning Research</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#starting-your-machine-learning-project">Starting your Machine Learning Project</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#choose-a-model-and-algorithm">Choose a Model and Algorithm</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#preparing-your-data">Preparing Your Data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#which-activation-and-weights-to-choose-in-neural-networks">Which activation and weights to choose in neural networks</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#optimization-methods-and-hyperparameters">Optimization Methods and Hyperparameters</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling">Resampling</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-courses-on-data-science-and-machine-learning-at-uio">Other courses on Data science and Machine Learning at UiO</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#additional-courses-of-interest">Additional courses of interest</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#what-s-the-future-like">What’s the future like?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#types-of-machine-learning-a-repetition">Types of Machine Learning, a repetition</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-boltzmann-machines">Why Boltzmann machines?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#boltzmann-machines">Boltzmann Machines</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#some-similarities-and-differences-from-dnns">Some similarities and differences from DNNs</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#boltzmann-machines-bm">Boltzmann machines (BM)</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-standard-bm-setup">A standard BM setup</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-structure-of-the-rbm-network">The structure of the RBM network</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-network">The network</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#goals">Goals</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#joint-distribution">Joint distribution</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#network-elements-the-energy-function">Network Elements, the energy function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-different-types-of-rbms">Defining different types of RBMs</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-about-rbms">More about RBMs</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#autoencoders-overarching-view">Autoencoders: Overarching view</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bayesian-machine-learning">Bayesian Machine Learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#reinforcement-learning">Reinforcement Learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#transfer-learning">Transfer learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adversarial-learning">Adversarial learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dual-learning">Dual learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#distributed-machine-learning">Distributed machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#meta-learning">Meta learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-challenges-facing-machine-learning">The Challenges Facing Machine Learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#explainable-machine-learning">Explainable machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-machine-learning">Quantum machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-machine-learning-algorithms-based-on-linear-algebra">Quantum machine learning algorithms based on linear algebra</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-reinforcement-learning">Quantum reinforcement learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-deep-learning">Quantum deep learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#social-machine-learning">Social machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-last-words">The last words?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester">Best wishes to you all and thanks so much for your heroic efforts this semester</a></li>
|
||
</ul>
|
||
</nav>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
|
||
|
||
<div id="searchbox"></div>
|
||
<article class="bd-article">
|
||
|
||
<!-- HTML file automatically generated from DocOnce source (https://github.com/doconce/doconce/)
|
||
doconce format html week48.do.txt --no_mako -->
|
||
<!-- dom:TITLE: Week 48: Gradient boosting and summary of course --><section class="tex2jax_ignore mathjax_ignore" id="week-48-gradient-boosting-and-summary-of-course">
|
||
<h1>Week 48: Gradient boosting and summary of course<a class="headerlink" href="#week-48-gradient-boosting-and-summary-of-course" title="Link to this heading">#</a></h1>
|
||
<p><strong>Morten Hjorth-Jensen</strong>, Department of Physics and Center for Computing in Science Education, University of Oslo, Norway</p>
|
||
<p>Date: <strong>Nov 25, 2024</strong></p>
|
||
<p>Copyright 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license</p>
|
||
<section id="overview-of-week-48">
|
||
<h2>Overview of week 48<a class="headerlink" href="#overview-of-week-48" title="Link to this heading">#</a></h2>
|
||
</section>
|
||
<section id="lecture-monday-november-25">
|
||
<h2>Lecture Monday, November 25<a class="headerlink" href="#lecture-monday-november-25" title="Link to this heading">#</a></h2>
|
||
<p><strong>Plans for the lecture Monday 25 November, with video suggestions etc.</strong></p>
|
||
<ol class="arabic simple">
|
||
<li><p>Boosting and gradient boosting and ensemble models</p></li>
|
||
<li><p>Summary of course</p></li>
|
||
<li><p>Readings and Videos:</p></li>
|
||
</ol>
|
||
<p>a. These lecture notes at <a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/pub/week48/ipynb/week48.ipynb">CompPhysics/MachineLearning</a></p>
|
||
<p>b. See also lecture notes from week 47 at <a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/pub/week47/ipynb/week47.ipynb">CompPhysics/MachineLearning</a>. The lecture on Monday starts with a repetition on AdaBoost before we move over to gradient boosting with examples</p>
|
||
<p>c. Video of lecture at <a class="reference external" href="https://youtu.be/iTaRdAPQnDA">https://youtu.be/iTaRdAPQnDA</a></p>
|
||
<p>d. Whiteboard notes at <a class="github reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesNovember25.pdf">CompPhysics/MachineLearning</a></p>
|
||
<p>e. Video on Decision trees <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>f. Video on boosting methods <a class="reference external" href="https://www.youtube.com/watch?v=wPqtzj5VZus&amp;ab_channel=H2O.ai">https://www.youtube.com/watch?v=wPqtzj5VZus&ab_channel=H2O.ai</a></p>
|
||
<p>g. Video on AdaBoost <a class="reference external" href="https://www.youtube.com/watch?v=LsK-xG1cLYA">https://www.youtube.com/watch?v=LsK-xG1cLYA</a></p>
|
||
<p>h. Video on Gradient boost, part 1, parts 2-4 follow thereafter <a class="reference external" href="https://www.youtube.com/watch?v=3CC4N4z3GJc">https://www.youtube.com/watch?v=3CC4N4z3GJc</a></p>
|
||
<p>i. 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="lab-sessions">
|
||
<h2>Lab sessions<a class="headerlink" href="#lab-sessions" title="Link to this heading">#</a></h2>
|
||
<p><strong>Lab sessions on Tuesday and Wednesday.</strong></p>
|
||
<ul class="simple">
|
||
<li><p>Work and Discussion of project 3</p></li>
|
||
<li><p>Last weekly exercise</p></li>
|
||
<li><p>Lab sessions at usual times.</p></li>
|
||
<li><p>For the week of December 2-6, lab sessions start at 10am and end at 4pm, room FØ434, Tuesday and Wednesday</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="random-forest-algorithm-reminder-from-last-week">
|
||
<h2>Random Forest Algorithm, reminder from last week<a class="headerlink" href="#random-forest-algorithm-reminder-from-last-week" 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>
|
||
<ul class="simple">
|
||
<li><p>For <span class="math notranslate nohighlight">\(b=1:B\)</span></p></li>
|
||
</ul>
|
||
<p>a. Draw a bootstrap sample from the training data organized in our <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> matrix.</p>
|
||
<p>b. 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>
|
||
<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>
|
||
</ol>
|
||
<p>Finally we 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>
|
||
</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="o">%</span><span class="k">matplotlib</span> inline
|
||
|
||
<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: 0.92
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[1. 0.73333333 0.93333333 1. 1. 0.92857143
|
||
1. 0.92857143 0.92857143 0.92857143]
|
||
Test set accuracy with Random Forests and scaled data: 0.97
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/90075505602c3f17740e87e303ddff0ae0ff2ec0245679468ad8fb7cb2ba3b3a.png" src="_images/90075505602c3f17740e87e303ddff0ae0ff2ec0245679468ad8fb7cb2ba3b3a.png" />
|
||
<img alt="_images/a318a67ffc1f6a60c2418c7f612568dd9b06752eb57ba2c4643dcc6e7d1a01dc.png" src="_images/a318a67ffc1f6a60c2418c7f612568dd9b06752eb57ba2c4643dcc6e7d1a01dc.png" />
|
||
<img alt="_images/1d150add40cbaa91d293348b988a5fbd2e04c68c41f11a70ebf78ac5686e7e4a.png" src="_images/1d150add40cbaa91d293348b988a5fbd2e04c68c41f11a70ebf78ac5686e7e4a.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">\({\cal C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2\)</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">2</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">0.01</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="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</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</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/4b551662b71f36e771bedba88b8176e1bd7b16262de80f1445234d9175a77e42.png" src="_images/4b551662b71f36e771bedba88b8176e1bd7b16262de80f1445234d9175a77e42.png" />
|
||
<img alt="_images/42c69c0d24892802c490b4669010a3c7927840c483799216381719a88df1dcea.png" src="_images/42c69c0d24892802c490b4669010a3c7927840c483799216381719a88df1dcea.png" />
|
||
<img alt="_images/94ace5b7042b960e9356e09b25e6338703d1e93b5c6e886432ee07b71c47404b.png" src="_images/94ace5b7042b960e9356e09b25e6338703d1e93b5c6e886432ee07b71c47404b.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="making-an-adaboost-code-yourself">
|
||
<h2>Making an ADAboost code yourself<a class="headerlink" href="#making-an-adaboost-code-yourself" 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">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
<span class="k">class</span> <span class="nc">DecisionStump</span><span class="p">:</span>
|
||
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</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">weights</span><span class="p">):</span>
|
||
<span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">=</span> <span class="mi">0</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">threshold</span> <span class="o">=</span> <span class="kc">None</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">polarity</span> <span class="o">=</span> <span class="mi">1</span>
|
||
|
||
<span class="n">min_error</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">)</span>
|
||
|
||
<span class="k">for</span> <span class="n">feature</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
|
||
<span class="n">feature_values</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="n">feature</span><span class="p">])</span>
|
||
|
||
<span class="k">for</span> <span class="n">threshold</span> <span class="ow">in</span> <span class="n">feature_values</span><span class="p">:</span>
|
||
<span class="k">for</span> <span class="n">polarity</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
|
||
<span class="n">predictions</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
|
||
<span class="n">predictions</span><span class="p">[</span><span class="n">X</span><span class="p">[:,</span> <span class="n">feature</span><span class="p">]</span> <span class="o"><</span> <span class="n">threshold</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
|
||
<span class="n">predictions</span> <span class="o">*=</span> <span class="n">polarity</span>
|
||
|
||
<span class="n">error</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">weights</span><span class="p">[</span><span class="n">predictions</span> <span class="o">!=</span> <span class="n">y</span><span class="p">])</span>
|
||
|
||
<span class="k">if</span> <span class="n">error</span> <span class="o"><</span> <span class="n">min_error</span><span class="p">:</span>
|
||
<span class="n">min_error</span> <span class="o">=</span> <span class="n">error</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">((</span><span class="mi">1</span> <span class="o">-</span> <span class="n">error</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">error</span> <span class="o">+</span> <span class="mf">1e-10</span><span class="p">))</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">threshold</span> <span class="o">=</span> <span class="n">threshold</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">feature_index</span> <span class="o">=</span> <span class="n">feature</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">polarity</span> <span class="o">=</span> <span class="n">polarity</span>
|
||
|
||
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
|
||
<span class="n">m</span> <span class="o">=</span> <span class="n">X</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">predictions</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">m</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">polarity</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
|
||
<span class="n">predictions</span><span class="p">[</span><span class="n">X</span><span class="p">[:,</span> <span class="bp">self</span><span class="o">.</span><span class="n">feature_index</span><span class="p">]</span> <span class="o"><</span> <span class="bp">self</span><span class="o">.</span><span class="n">threshold</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">predictions</span><span class="p">[</span><span class="n">X</span><span class="p">[:,</span> <span class="bp">self</span><span class="o">.</span><span class="n">feature_index</span><span class="p">]</span> <span class="o">>=</span> <span class="bp">self</span><span class="o">.</span><span class="n">threshold</span><span class="p">]</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
|
||
<span class="k">return</span> <span class="n">predictions</span>
|
||
|
||
<span class="k">class</span> <span class="nc">AdaBoost</span><span class="p">:</span>
|
||
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</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">n_estimators</span><span class="p">):</span>
|
||
<span class="n">m</span> <span class="o">=</span> <span class="n">X</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="bp">self</span><span class="o">.</span><span class="n">alphas</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">models</span> <span class="o">=</span> <span class="p">[]</span>
|
||
|
||
<span class="n">weights</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">/</span> <span class="n">m</span>
|
||
|
||
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_estimators</span><span class="p">):</span>
|
||
<span class="n">stump</span> <span class="o">=</span> <span class="n">DecisionStump</span><span class="p">()</span>
|
||
<span class="n">stump</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">weights</span><span class="p">)</span>
|
||
<span class="n">predictions</span> <span class="o">=</span> <span class="n">stump</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
|
||
<span class="n">error</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">weights</span><span class="p">[</span><span class="n">predictions</span> <span class="o">!=</span> <span class="n">y</span><span class="p">])</span>
|
||
<span class="k">if</span> <span class="n">error</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="k">break</span>
|
||
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">models</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">stump</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">alphas</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">stump</span><span class="o">.</span><span class="n">alpha</span><span class="p">)</span>
|
||
|
||
<span class="n">weights</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">stump</span><span class="o">.</span><span class="n">alpha</span> <span class="o">*</span> <span class="n">y</span> <span class="o">*</span> <span class="n">predictions</span><span class="p">)</span>
|
||
<span class="n">weights</span> <span class="o">/=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">weights</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
|
||
<span class="n">final_predictions</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">X</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="k">for</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">model</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">alphas</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">models</span><span class="p">):</span>
|
||
<span class="n">final_predictions</span> <span class="o">+=</span> <span class="n">alpha</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</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">final_predictions</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Example dataset (X, y)</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</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="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">],</span> <span class="p">[</span><span class="mi">8</span><span class="p">],</span> <span class="p">[</span><span class="mi">9</span><span class="p">],</span> <span class="p">[</span><span class="mi">10</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">array</span><span class="p">([</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</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="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="mi">1</span><span class="p">])</span> <span class="c1"># Labels must be -1 or 1</span>
|
||
|
||
<span class="c1"># Train AdaBoost</span>
|
||
<span class="n">ada</span> <span class="o">=</span> <span class="n">AdaBoost</span><span class="p">()</span>
|
||
<span class="n">ada</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">n_estimators</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Predictions</span>
|
||
<span class="n">predictions</span> <span class="o">=</span> <span class="n">ada</span><span class="o">.</span><span class="n">predict</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="s2">"Predictions:"</span><span class="p">,</span> <span class="n">predictions</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>Predictions: [0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="gradient-boosting-basics-with-steepest-descent-functional-gradient-descent">
|
||
<h2>Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent<a class="headerlink" href="#gradient-boosting-basics-with-steepest-descent-functional-gradient-descent" title="Link to this heading">#</a></h2>
|
||
<p>Gradient boosting is again a similar technique to Adaptive boosting,
|
||
it combines so-called weak classifiers or regressors into a strong
|
||
method via a series of iterations.</p>
|
||
<p>In order to understand the method, let us illustrate its basics by
|
||
bringing back the essential steps in linear regression, where our cost
|
||
function was the least squares function.</p>
|
||
</section>
|
||
<section id="the-squared-error-again-steepest-descent">
|
||
<h2>The Squared-Error again! Steepest Descent<a class="headerlink" href="#the-squared-error-again-steepest-descent" title="Link to this heading">#</a></h2>
|
||
<p>We start again with our cost function <span class="math notranslate nohighlight">\({\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i))\)</span> where we want to minimize
|
||
This means that for every iteration, we need to optimize</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
||
\]</div>
|
||
<p>We define a real function <span class="math notranslate nohighlight">\(h_m(x)\)</span> that defines our final function <span class="math notranslate nohighlight">\(f_M(x)\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f_M(x) = \sum_{m=0}^M h_m(x).
|
||
\]</div>
|
||
<p>In the steepest decent approach we approximate <span class="math notranslate nohighlight">\(h_m(x) = -\rho_m g_m(x)\)</span>, where <span class="math notranslate nohighlight">\(\rho_m\)</span> is a scalar and <span class="math notranslate nohighlight">\(g_m(x)\)</span> the gradient defined as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.
|
||
\]</div>
|
||
<p>With the new gradient we can update <span class="math notranslate nohighlight">\(f_m(x) = f_{m-1}(x) -\rho_m g_m(x)\)</span>. Using the above squared-error function we see that
|
||
the gradient is <span class="math notranslate nohighlight">\(g_m(x_i) = -2(y_i-f(x_i))\)</span>.</p>
|
||
<p>Choosing <span class="math notranslate nohighlight">\(f_0(x)=0\)</span> we obtain <span class="math notranslate nohighlight">\(g_m(x) = -2y_i\)</span> and inserting this into the minimization problem for the cost function we have</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.
|
||
\]</div>
|
||
</section>
|
||
<section id="steepest-descent-example">
|
||
<h2>Steepest Descent Example<a class="headerlink" href="#steepest-descent-example" title="Link to this heading">#</a></h2>
|
||
<p>Optimizing with respect to <span class="math notranslate nohighlight">\(\rho\)</span> we obtain (taking the derivative) that <span class="math notranslate nohighlight">\(\rho_1 = -1/2\)</span>. We have then that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i.
|
||
\]</div>
|
||
<p>We can then proceed and compute</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,
|
||
\]</div>
|
||
<p>and find a new value for <span class="math notranslate nohighlight">\(\rho_2=-1/2\)</span> and continue till we have reached <span class="math notranslate nohighlight">\(m=M\)</span>. We can modify the steepest descent method, or steepest boosting, by introducing what is called <strong>gradient boosting</strong>.</p>
|
||
</section>
|
||
<section id="gradient-boosting-algorithm">
|
||
<h2>Gradient Boosting, algorithm<a class="headerlink" href="#gradient-boosting-algorithm" title="Link to this heading">#</a></h2>
|
||
<p>Steepest descent is however not much used, since it only optimizes <span class="math notranslate nohighlight">\(f\)</span> at a fixed set of <span class="math notranslate nohighlight">\(n\)</span> points,
|
||
so we do not learn a function that can generalize. However, we can modify the algorithm by
|
||
fitting a weak learner to approximate the negative gradient signal.</p>
|
||
<p>Suppose we have a cost function <span class="math notranslate nohighlight">\(C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i))\)</span> where <span class="math notranslate nohighlight">\(y_i\)</span> is our target and <span class="math notranslate nohighlight">\(f(x_i)\)</span> the function which is meant to model <span class="math notranslate nohighlight">\(y_i\)</span>. The above cost function could be our standard squared-error function</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
||
\]</div>
|
||
<p>The way we proceed in an iterative fashion is to</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Initialize our estimate <span class="math notranslate nohighlight">\(f_0(x)\)</span>.</p></li>
|
||
<li><p>For <span class="math notranslate nohighlight">\(m=1:M\)</span>, we</p></li>
|
||
</ol>
|
||
<p>a. compute the negative gradient vector <span class="math notranslate nohighlight">\(\boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x)\)</span> at <span class="math notranslate nohighlight">\(f(x) = f_{m-1}(x)\)</span>;</p>
|
||
<p>b. fit the so-called base-learner to the negative gradient <span class="math notranslate nohighlight">\(h_m(u_m,x)\)</span>;</p>
|
||
<p>c. update the estimate <span class="math notranslate nohighlight">\(f_m(x) = f_{m-1}(x)+h_m(u_m,x)\)</span>;</p>
|
||
<ol class="arabic simple" start="4">
|
||
<li><p>The final estimate is then <span class="math notranslate nohighlight">\(f_M(x) = \sum_{m=1}^M h_m(u_m,x)\)</span>.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="gradient-boosting-examples-of-regression">
|
||
<h2>Gradient Boosting, Examples of Regression<a class="headerlink" href="#gradient-boosting-examples-of-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="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.ensemble</span> <span class="kn">import</span> <span class="n">GradientBoostingRegressor</span>
|
||
<span class="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</span>
|
||
|
||
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
|
||
<span class="n">maxdegree</span> <span class="o">=</span> <span class="mi">6</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">maxdegree</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">maxdegree</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">maxdegree</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">maxdegree</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="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">maxdegree</span><span class="p">):</span>
|
||
<span class="n">model</span> <span class="o">=</span> <span class="n">GradientBoostingRegressor</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">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">1.0</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_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">model</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">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="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="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="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Max depth:'</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">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">maxdegree</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">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">'Error'</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">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>Max depth: 1
|
||
Error: 0.4010613825254484
|
||
Bias^2: 0.2079593417034804
|
||
Var: 0.19310204082196794
|
||
0.4010613825254484 >= 0.2079593417034804 + 0.19310204082196794 = 0.4010613825254483
|
||
Max depth: 2
|
||
Error: 0.4250776117755916
|
||
Bias^2: 0.2080984218270197
|
||
Var: 0.21697918994857185
|
||
0.4250776117755916 >= 0.2080984218270197 + 0.21697918994857185 = 0.42507761177559156
|
||
Max depth: 3
|
||
Error: 0.4250796355306808
|
||
Bias^2: 0.2080985447081304
|
||
Var: 0.21698109082255032
|
||
0.4250796355306808 >= 0.2080985447081304 + 0.21698109082255032 = 0.42507963553068073
|
||
Max depth: 4
|
||
Error: 0.4250796355306808
|
||
Bias^2: 0.2080985447081304
|
||
Var: 0.21698109082255038
|
||
0.4250796355306808 >= 0.2080985447081304 + 0.21698109082255038 = 0.4250796355306808
|
||
Max depth: 5
|
||
Error: 0.42507963553068073
|
||
Bias^2: 0.2080985447081304
|
||
Var: 0.21698109082255032
|
||
0.42507963553068073 >= 0.2080985447081304 + 0.21698109082255032 = 0.42507963553068073
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:424: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
|
||
y = column_or_1d(y, warn=True)
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:424: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
|
||
y = column_or_1d(y, warn=True)
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:424: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
|
||
y = column_or_1d(y, warn=True)
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:424: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
|
||
y = column_or_1d(y, warn=True)
|
||
/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/ensemble/_gb.py:424: DataConversionWarning: A column-vector y was passed when a 1d array was expected. Please change the shape of y to (n_samples, ), for example using ravel().
|
||
y = column_or_1d(y, warn=True)
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/801d16873af4f485d9c398ac11811a7d9c4f25dd40c995cf7565bd81af8071c5.png" src="_images/801d16873af4f485d9c398ac11811a7d9c4f25dd40c995cf7565bd81af8071c5.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="gradient-boosting-classification-example">
|
||
<h2>Gradient Boosting, Classification Example<a class="headerlink" href="#gradient-boosting-classification-example" 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">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">GradientBoostingClassifier</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"># 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">#now 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="n">gd_clf</span> <span class="o">=</span> <span class="n">GradientBoostingClassifier</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">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">1.0</span><span class="p">)</span>
|
||
<span class="n">gd_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="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">gd_clf</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 Gradient boosting 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">gd_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">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">gd_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">gd_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">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(426, 30)
|
||
(143, 30)
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.93333333 0.93333333 0.86666667 1. 1. 0.92857143
|
||
1. 0.92857143 0.85714286 0.92857143]
|
||
Test set accuracy with Gradient boosting and scaled data: 0.97
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/90075505602c3f17740e87e303ddff0ae0ff2ec0245679468ad8fb7cb2ba3b3a.png" src="_images/90075505602c3f17740e87e303ddff0ae0ff2ec0245679468ad8fb7cb2ba3b3a.png" />
|
||
<img alt="_images/de0019c6a0f8206c4c2b0f29eedfececf3b0ad5a07a733217a9ed5b8e4cae808.png" src="_images/de0019c6a0f8206c4c2b0f29eedfececf3b0ad5a07a733217a9ed5b8e4cae808.png" />
|
||
<img alt="_images/45972a93ed8e1f6ed66fe9c322a65b549b39dd2c80e16d5081151b2bc713b669.png" src="_images/45972a93ed8e1f6ed66fe9c322a65b549b39dd2c80e16d5081151b2bc713b669.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="xgboost-extreme-gradient-boosting">
|
||
<h2>XGBoost: Extreme Gradient Boosting<a class="headerlink" href="#xgboost-extreme-gradient-boosting" title="Link to this heading">#</a></h2>
|
||
<p><a class="reference external" href="https://github.com/dmlc/xgboost">XGBoost</a> or Extreme Gradient
|
||
Boosting, is an optimized distributed gradient boosting library
|
||
designed to be highly efficient, flexible and portable. It implements
|
||
machine learning algorithms under the Gradient Boosting
|
||
framework. XGBoost provides a parallel tree boosting that solve many
|
||
data science problems in a fast and accurate way. See the <a class="reference external" href="https://arxiv.org/abs/1603.02754">article by Chen and Guestrin</a>.</p>
|
||
<p>The authors design and build a highly scalable end-to-end tree
|
||
boosting system. It has a theoretically justified weighted quantile
|
||
sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.</p>
|
||
<p>It is now the algorithm which wins essentially all ML competitions!!!</p>
|
||
</section>
|
||
<section id="xgboost-on-the-cancer-data">
|
||
<h2>Xgboost on the Cancer Data<a class="headerlink" href="#xgboost-on-the-cancer-data" title="Link to this heading">#</a></h2>
|
||
<p>As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.</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.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</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="kn">import</span> <span class="nn">scikitplot</span> <span class="k">as</span> <span class="nn">skplt</span>
|
||
<span class="kn">import</span> <span class="nn">xgboost</span> <span class="k">as</span> <span class="nn">xgb</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">#now 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="n">xg_clf</span> <span class="o">=</span> <span class="n">xgb</span><span class="o">.</span><span class="n">XGBClassifier</span><span class="p">()</span>
|
||
<span class="n">xg_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_test</span> <span class="o">=</span> <span class="n">xg_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="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Gradient Boosting 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">xg_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">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">xg_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">xg_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>
|
||
|
||
|
||
<span class="n">xgb</span><span class="o">.</span><span class="n">plot_tree</span><span class="p">(</span><span class="n">xg_clf</span><span class="p">,</span><span class="n">num_trees</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">rcParams</span><span class="p">[</span><span class="s1">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">50</span><span class="p">,</span> <span class="mi">10</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">xgb</span><span class="o">.</span><span class="n">plot_importance</span><span class="p">(</span><span class="n">xg_clf</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'figure.figsize'</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</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 with Gradient Boosting and scaled data: 1.00
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/bae16b30a452ef5c69ddef05f0de3bbba23676920f8affa40a5a964be4504dd8.png" src="_images/bae16b30a452ef5c69ddef05f0de3bbba23676920f8affa40a5a964be4504dd8.png" />
|
||
<img alt="_images/429975495ea036cdaa180dafa8c2225aba5cd58c9c9a67fe5998acd764007772.png" src="_images/429975495ea036cdaa180dafa8c2225aba5cd58c9c9a67fe5998acd764007772.png" />
|
||
<img alt="_images/1baae63cbdd15ce7abeb765e3d3adddddca04d8f36b9292d03ad703844b287d4.png" src="_images/1baae63cbdd15ce7abeb765e3d3adddddca04d8f36b9292d03ad703844b287d4.png" />
|
||
<img alt="_images/3a39cedd7df8be523efd25c1ba51760790f33378dd9e7a9d1eff178bc4b629e1.png" src="_images/3a39cedd7df8be523efd25c1ba51760790f33378dd9e7a9d1eff178bc4b629e1.png" />
|
||
<img alt="_images/a9306da9994b47ad9b17c6ed3244dbd079e10a74b85153831b07c9570b8677c3.png" src="_images/a9306da9994b47ad9b17c6ed3244dbd079e10a74b85153831b07c9570b8677c3.png" />
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="gradient-boosting-making-our-own-code-for-a-regression-case">
|
||
<h2>Gradient boosting, making our own code for a regression case<a class="headerlink" href="#gradient-boosting-making-our-own-code-for-a-regression-case" 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">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="k">class</span> <span class="nc">DecisionTreeRegressor</span><span class="p">:</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</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="bp">self</span><span class="o">.</span><span class="n">max_depth</span> <span class="o">=</span> <span class="n">max_depth</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">tree</span> <span class="o">=</span> <span class="kc">None</span>
|
||
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</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="bp">self</span><span class="o">.</span><span class="n">tree</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_grow_tree</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">_grow_tree</span><span class="p">(</span><span class="bp">self</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">depth</span><span class="o">=</span><span class="mi">0</span><span class="p">):</span>
|
||
<span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span>
|
||
<span class="k">if</span> <span class="n">depth</span> <span class="o"><</span> <span class="bp">self</span><span class="o">.</span><span class="n">max_depth</span><span class="p">:</span>
|
||
<span class="n">best_feature</span><span class="p">,</span> <span class="n">best_threshold</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_best_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="k">if</span> <span class="n">best_feature</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
|
||
<span class="n">left_indices</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,</span> <span class="n">best_feature</span><span class="p">]</span> <span class="o"><</span> <span class="n">best_threshold</span>
|
||
<span class="n">right_indices</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,</span> <span class="n">best_feature</span><span class="p">]</span> <span class="o">>=</span> <span class="n">best_threshold</span>
|
||
<span class="n">left_child</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_grow_tree</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">left_indices</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">left_indices</span><span class="p">],</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">right_child</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_grow_tree</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">right_indices</span><span class="p">],</span> <span class="n">y</span><span class="p">[</span><span class="n">right_indices</span><span class="p">],</span> <span class="n">depth</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="p">(</span><span class="n">best_feature</span><span class="p">,</span> <span class="n">best_threshold</span><span class="p">,</span> <span class="n">left_child</span><span class="p">,</span> <span class="n">right_child</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||
<span class="k">def</span> <span class="nf">_best_split</span><span class="p">(</span><span class="bp">self</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">best_mse</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">)</span>
|
||
<span class="n">best_feature</span><span class="p">,</span> <span class="n">best_threshold</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span>
|
||
<span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">shape</span>
|
||
|
||
<span class="k">for</span> <span class="n">feature</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_features</span><span class="p">):</span>
|
||
<span class="n">thresholds</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="n">feature</span><span class="p">])</span>
|
||
<span class="k">for</span> <span class="n">threshold</span> <span class="ow">in</span> <span class="n">thresholds</span><span class="p">:</span>
|
||
<span class="n">left_indices</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,</span> <span class="n">feature</span><span class="p">]</span> <span class="o"><</span> <span class="n">threshold</span>
|
||
<span class="n">right_indices</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,</span> <span class="n">feature</span><span class="p">]</span> <span class="o">>=</span> <span class="n">threshold</span>
|
||
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="n">left_indices</span><span class="p">])</span> <span class="o">></span> <span class="mi">0</span> <span class="ow">and</span> <span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="n">right_indices</span><span class="p">])</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="n">left_mse</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</span><span class="p">[</span><span class="n">left_indices</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">y</span><span class="p">[</span><span class="n">left_indices</span><span class="p">]))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">right_mse</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</span><span class="p">[</span><span class="n">right_indices</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">y</span><span class="p">[</span><span class="n">right_indices</span><span class="p">]))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">mse</span> <span class="o">=</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="n">left_indices</span><span class="p">])</span> <span class="o">*</span> <span class="n">left_mse</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="n">right_indices</span><span class="p">])</span> <span class="o">*</span> <span class="n">right_mse</span><span class="p">)</span> <span class="o">/</span> <span class="n">n_samples</span>
|
||
|
||
<span class="k">if</span> <span class="n">mse</span> <span class="o"><</span> <span class="n">best_mse</span><span class="p">:</span>
|
||
<span class="n">best_mse</span> <span class="o">=</span> <span class="n">mse</span>
|
||
<span class="n">best_feature</span> <span class="o">=</span> <span class="n">feature</span>
|
||
<span class="n">best_threshold</span> <span class="o">=</span> <span class="n">threshold</span>
|
||
<span class="k">return</span> <span class="n">best_feature</span><span class="p">,</span> <span class="n">best_threshold</span>
|
||
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="bp">self</span><span class="o">.</span><span class="n">_predict_sample</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">tree</span><span class="p">)</span> <span class="k">for</span> <span class="n">sample</span> <span class="ow">in</span> <span class="n">X</span><span class="p">])</span>
|
||
<span class="k">def</span> <span class="nf">_predict_sample</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">sample</span><span class="p">,</span> <span class="n">node</span><span class="p">):</span>
|
||
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">node</span><span class="p">,</span> <span class="nb">tuple</span><span class="p">):</span>
|
||
<span class="n">feature</span><span class="p">,</span> <span class="n">threshold</span><span class="p">,</span> <span class="n">left_child</span><span class="p">,</span> <span class="n">right_child</span> <span class="o">=</span> <span class="n">node</span>
|
||
<span class="k">if</span> <span class="n">sample</span><span class="p">[</span><span class="n">feature</span><span class="p">]</span> <span class="o"><</span> <span class="n">threshold</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_predict_sample</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="n">left_child</span><span class="p">)</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_predict_sample</span><span class="p">(</span><span class="n">sample</span><span class="p">,</span> <span class="n">right_child</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">node</span>
|
||
<span class="k">class</span> <span class="nc">GradientBoostingRegressor</span><span class="p">:</span>
|
||
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.1</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="bp">self</span><span class="o">.</span><span class="n">n_estimators</span> <span class="o">=</span> <span class="n">n_estimators</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">learning_rate</span> <span class="o">=</span> <span class="n">learning_rate</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">max_depth</span> <span class="o">=</span> <span class="n">max_depth</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">models</span> <span class="o">=</span> <span class="p">[]</span>
|
||
<span class="k">def</span> <span class="nf">fit</span><span class="p">(</span><span class="bp">self</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="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_estimators</span><span class="p">):</span>
|
||
<span class="n">residuals</span> <span class="o">=</span> <span class="n">y</span> <span class="o">-</span> <span class="n">y_pred</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="bp">self</span><span class="o">.</span><span class="n">max_depth</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">residuals</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">learning_rate</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</span><span class="p">)</span>
|
||
<span class="bp">self</span><span class="o">.</span><span class="n">models</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">model</span><span class="p">)</span>
|
||
<span class="k">def</span> <span class="nf">predict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</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">zeros</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="mi">0</span><span class="p">])</span>
|
||
<span class="k">for</span> <span class="n">model</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">models</span><span class="p">:</span>
|
||
<span class="n">y_pred</span> <span class="o">+=</span> <span class="bp">self</span><span class="o">.</span><span class="n">learning_rate</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</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">y_pred</span>
|
||
<span class="c1"># Example usage</span>
|
||
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"__main__"</span><span class="p">:</span>
|
||
<span class="c1"># Sample data</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</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="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</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">array</span><span class="p">([</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">1.7</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">,</span> <span class="mf">3.7</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">])</span>
|
||
<span class="n">model</span> <span class="o">=</span> <span class="n">GradientBoostingRegressor</span><span class="p">(</span><span class="n">n_estimators</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">learning_rate</span><span class="o">=</span><span class="mf">0.1</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">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">predictions</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</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Predictions:"</span><span class="p">,</span> <span class="n">predictions</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>Predictions: [1.49999399 1.69995637 3.49991627 3.6998795 4.99984484]
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</section>
|
||
<section id="summary-of-course">
|
||
<h2>Summary of course<a class="headerlink" href="#summary-of-course" title="Link to this heading">#</a></h2>
|
||
</section>
|
||
<section id="what-me-worry-no-final-exam-in-this-course">
|
||
<h2>What? Me worry? No final exam in this course!<a class="headerlink" href="#what-me-worry-no-final-exam-in-this-course" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [figures/exam1.jpeg, width=500 frac=0.6] -->
|
||
<!-- begin figure -->
|
||
<p><img src="figures/exam1.jpeg" width="500"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></section>
|
||
<section id="topics-we-have-covered-this-year">
|
||
<h2>Topics we have covered this year<a class="headerlink" href="#topics-we-have-covered-this-year" title="Link to this heading">#</a></h2>
|
||
<p>The course has two central parts</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Statistical analysis and optimization of data</p></li>
|
||
<li><p>Machine learning</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="statistical-analysis-and-optimization-of-data">
|
||
<h2>Statistical analysis and optimization of data<a class="headerlink" href="#statistical-analysis-and-optimization-of-data" title="Link to this heading">#</a></h2>
|
||
<p>The following topics have been discussed:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Basic concepts, expectation values, variance, covariance, correlation functions and errors;</p></li>
|
||
<li><p>Simpler models, binomial distribution, the Poisson distribution, simple and multivariate normal distributions;</p></li>
|
||
<li><p>Central elements from linear algebra, matrix inversion and SVD</p></li>
|
||
<li><p>Gradient methods for data optimization</p></li>
|
||
<li><p>Estimation of errors using cross-validation, bootstrapping and jackknife methods;</p></li>
|
||
<li><p>Practical optimization using Singular-value decomposition and least squares for parameterizing data.</p></li>
|
||
<li><p>Not discussed: Principal Component Analysis to reduce the number of features.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="machine-learning">
|
||
<h2>Machine learning<a class="headerlink" href="#machine-learning" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p>Linear methods for regression and classification:</p></li>
|
||
</ul>
|
||
<p>a. Ordinary Least Squares</p>
|
||
<p>b. Ridge regression</p>
|
||
<p>c. Lasso regression</p>
|
||
<p>d. Logistic regression</p>
|
||
<ul class="simple">
|
||
<li><p>Neural networks and deep learning:</p></li>
|
||
</ul>
|
||
<p>a. Feed Forward Neural Networks</p>
|
||
<p>b. Convolutional Neural Networks</p>
|
||
<p>c. Recurrent Neural Networks</p>
|
||
<ul class="simple">
|
||
<li><p>Decisions trees and ensemble methods:</p></li>
|
||
</ul>
|
||
<p>a. Decision trees</p>
|
||
<p>b. Bagging and voting</p>
|
||
<p>c. Random forests</p>
|
||
<p>d. Boosting and gradient boosting</p>
|
||
<ul class="simple">
|
||
<li><p>Not discussed this year: Support vector machines</p></li>
|
||
</ul>
|
||
<p>a. Binary classification and multiclass classification</p>
|
||
<p>b. Kernel methods</p>
|
||
<p>c. Regression</p>
|
||
</section>
|
||
<section id="learning-outcomes-and-overarching-aims-of-this-course">
|
||
<h2>Learning outcomes and overarching aims of this course<a class="headerlink" href="#learning-outcomes-and-overarching-aims-of-this-course" title="Link to this heading">#</a></h2>
|
||
<p>The course introduces a variety of central algorithms and methods
|
||
essential for studies of data analysis and machine learning. The
|
||
course is project based and through the various projects, normally
|
||
three, you will be exposed to fundamental research problems
|
||
in these fields, with the aim to reproduce state of the art scientific
|
||
results. The students will learn to develop and structure large codes
|
||
for studying these systems, get acquainted with computing facilities
|
||
and learn to handle large scientific projects. A good scientific and
|
||
ethical conduct is emphasized throughout the course.</p>
|
||
<ul class="simple">
|
||
<li><p>Understand linear methods for regression and classification;</p></li>
|
||
<li><p>Learn about neural network;</p></li>
|
||
<li><p>Learn about bagging, boosting and trees</p></li>
|
||
</ul>
|
||
<!-- * Support vector machines -->
|
||
<ul class="simple">
|
||
<li><p>Learn about basic data analysis;</p></li>
|
||
<li><p>Be capable of extending the acquired knowledge to other systems and cases;</p></li>
|
||
<li><p>Have an understanding of central algorithms used in data analysis and machine learning;</p></li>
|
||
<li><p>Work on numerical projects to illustrate the theory. The projects play a central role.</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="perspective-on-machine-learning">
|
||
<h2>Perspective on Machine Learning<a class="headerlink" href="#perspective-on-machine-learning" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p>Rapidly emerging application area</p></li>
|
||
<li><p>Experiment AND theory are evolving in many many fields.</p></li>
|
||
<li><p>Requires education/retraining for more widespread adoption</p></li>
|
||
<li><p>A lot of “word-of-mouth” development methods</p></li>
|
||
</ol>
|
||
<p>Huge amounts of data sets require automation, classical analysis tools often inadequate.
|
||
High energy physics hit this wall in the 90’s.
|
||
In 2009 single top quark production was determined via <a class="reference external" href="https://arxiv.org/pdf/0903.0850.pdf">Boosted decision trees, Bayesian
|
||
Neural Networks, etc.</a></p>
|
||
</section>
|
||
<section id="machine-learning-research">
|
||
<h2>Machine Learning Research<a class="headerlink" href="#machine-learning-research" title="Link to this heading">#</a></h2>
|
||
<p>Where to find recent results:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Conference proceedings, arXiv and blog posts!</p></li>
|
||
<li><p><strong>NIPS</strong>: <a class="reference external" href="https://papers.nips.cc">Neural Information Processing Systems</a></p></li>
|
||
<li><p><strong>ICLR</strong>: <a class="reference external" href="https://openreview.net/group?id=ICLR.cc/2018/Conference#accepted-oral-papers">International Conference on Learning Representations</a></p></li>
|
||
<li><p><strong>ICML</strong>: International Conference on Machine Learning</p></li>
|
||
<li><p><a class="reference external" href="http://www.jmlr.org/papers/v19/">Journal of Machine Learning Research</a></p></li>
|
||
<li><p><a class="reference external" href="https://arxiv.org/list/cs.LG/recent">Follow ML on ArXiv</a></p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="starting-your-machine-learning-project">
|
||
<h2>Starting your Machine Learning Project<a class="headerlink" href="#starting-your-machine-learning-project" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p>Identify problem type: classification, regression</p></li>
|
||
<li><p>Consider your data carefully</p></li>
|
||
<li><p>Choose a simple model that fits 1 and 2</p></li>
|
||
<li><p>Consider your data carefully again! Think of data representation more carefully.</p></li>
|
||
<li><p>Based on your results, feedback loop to earliest possible point</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="choose-a-model-and-algorithm">
|
||
<h2>Choose a Model and Algorithm<a class="headerlink" href="#choose-a-model-and-algorithm" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p>Supervised?</p></li>
|
||
<li><p>Start with the simplest model that fits your problem</p></li>
|
||
<li><p>Start with minimal processing of data</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="preparing-your-data">
|
||
<h2>Preparing Your Data<a class="headerlink" href="#preparing-your-data" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p>Shuffle your data</p></li>
|
||
<li><p>Mean center your data</p>
|
||
<ul>
|
||
<li><p>Why?</p></li>
|
||
</ul>
|
||
</li>
|
||
<li><p>Normalize the variance</p>
|
||
<ul>
|
||
<li><p>Why?</p></li>
|
||
</ul>
|
||
</li>
|
||
<li><p><strong>Whitening</strong></p>
|
||
<ul>
|
||
<li><p>Decorrelates data</p></li>
|
||
<li><p>Can be hit or miss</p></li>
|
||
</ul>
|
||
</li>
|
||
<li><p>When to do train/test split?</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="which-activation-and-weights-to-choose-in-neural-networks">
|
||
<h2>Which activation and weights to choose in neural networks<a class="headerlink" href="#which-activation-and-weights-to-choose-in-neural-networks" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p>RELU? ELU? GELU? etc</p></li>
|
||
<li><p>Sigmoid or Tanh?</p></li>
|
||
<li><p>Set all weights to 0? Terrible idea</p></li>
|
||
<li><p>Set all weights to random values? Small random values</p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="optimization-methods-and-hyperparameters">
|
||
<h2>Optimization Methods and Hyperparameters<a class="headerlink" href="#optimization-methods-and-hyperparameters" title="Link to this heading">#</a></h2>
|
||
<ul class="simple">
|
||
<li><p>Stochastic gradient descent</p></li>
|
||
<li><p>Stochastic gradient descent + momentum</p></li>
|
||
<li><p>State-of-the-art approaches:</p></li>
|
||
</ul>
|
||
<p>a. RMSProp</p>
|
||
<p>b. Adam</p>
|
||
<p>c. and more</p>
|
||
<p>Which regularization and hyperparameters? <span class="math notranslate nohighlight">\(L_1\)</span> or <span class="math notranslate nohighlight">\(L_2\)</span>, soft
|
||
classifiers, depths of trees and many other. Need to explore a large
|
||
set of hyperparameters and regularization methods.</p>
|
||
</section>
|
||
<section id="resampling">
|
||
<h2>Resampling<a class="headerlink" href="#resampling" title="Link to this heading">#</a></h2>
|
||
<p>When do we resample?</p>
|
||
<ol class="arabic simple">
|
||
<li><p><a class="reference external" href="https://www.cambridge.org/core/books/bootstrap-methods-and-their-application/ED2FD043579F27952363566DC09CBD6A">Bootstrap</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.youtube.com/watch?v=fSytzGwwBVw&amp;ab_channel=StatQuestwithJoshStarmer">Cross-validation</a></p></li>
|
||
<li><p>Jackknife and many other</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="other-courses-on-data-science-and-machine-learning-at-uio">
|
||
<h2>Other courses on Data science and Machine Learning at UiO<a class="headerlink" href="#other-courses-on-data-science-and-machine-learning-at-uio" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/fys/FYS5429/index-eng.html">FYS5429 – Advanced machine learning and data analysis for the physical sciences</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN3050/index-eng.html">IN3050/IN4050 Introduction to Artificial Intelligence and Machine Learning</a>. Introductory course in machine learning and AI</p></li>
|
||
<li><p><a class="reference external" href="http://www.uio.no/studier/emner/matnat/math/STK-INF3000/index-eng.html">STK-INF3000/4000 Selected Topics in Data Science</a>. The course provides insight into selected contemporary relevant topics within Data Science.</p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN4080/index.html">IN4080 Natural Language Processing</a>. Probabilistic and machine learning techniques applied to natural language processing.</p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/STK-IN4300/index-eng.html">STK-IN4300 – Statistical learning methods in Data Science</a>. An advanced introduction to statistical and machine learning. For students with a good mathematics and statistics background.</p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN-STK5000/index-eng.html">IN-STK5000 Responsible Data Science</a>. Methods for adaptive collection and processing of data based on machine learning techniques.</p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN4310/index.html">IN4310 – Machine Learning for Image Analysis</a>. An introduction to deep learning with particular emphasis on applications within Image analysis, but useful for other application areas too.</p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN5310/index.html">IN5310 – Advanced Deep Learning for Image Analysis</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/ifi/IN5490/index.html">IN5490 – Advanced Topics in Artificial Intelligence for Intelligent Systems</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/its/TEK5040/">TEK5040 – Deep learning for autonomous systems</a>. The course addresses advanced algorithms and architectures for deep learning with neural networks. The course provides an introduction to how deep-learning techniques can be used in the construction of key parts of advanced autonomous systems that exist in physical environments and cyber environments.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="additional-courses-of-interest">
|
||
<h2>Additional courses of interest<a class="headerlink" href="#additional-courses-of-interest" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/STK4051/index-eng.html">STK4051 Computational Statistics</a></p></li>
|
||
<li><p><a class="reference external" href="https://www.uio.no/studier/emner/matnat/math/STK4021/index-eng.html">STK4021 Applied Bayesian Analysis and Numerical Methods</a></p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="what-s-the-future-like">
|
||
<h2>What’s the future like?<a class="headerlink" href="#what-s-the-future-like" title="Link to this heading">#</a></h2>
|
||
<p>Based on multi-layer nonlinear neural networks, deep learning can
|
||
learn directly from raw data, automatically extract and abstract
|
||
features from layer to layer, and then achieve the goal of regression,
|
||
classification, or ranking. Deep learning has made breakthroughs in
|
||
computer vision, speech processing and natural language, and reached
|
||
or even surpassed human level. The success of deep learning is mainly
|
||
due to the three factors: big data, big model, and big computing.</p>
|
||
<p>In the past few decades, many different architectures of deep neural
|
||
networks have been proposed, such as</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Convolutional neural networks, which are mostly used in image and video data processing, and have also been applied to sequential data such as text processing;</p></li>
|
||
<li><p>Recurrent neural networks, which can process sequential data of variable length and have been widely used in natural language understanding and speech processing;</p></li>
|
||
<li><p>Encoder-decoder framework, which is mostly used for image or sequence generation, such as machine translation, text summarization, and image captioning.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="types-of-machine-learning-a-repetition">
|
||
<h2>Types of Machine Learning, a repetition<a class="headerlink" href="#types-of-machine-learning-a-repetition" title="Link to this heading">#</a></h2>
|
||
<p>The approaches to machine learning are many, but are often split into two main categories.
|
||
In <em>supervised learning</em> we know the answer to a problem,
|
||
and let the computer deduce the logic behind it. On the other hand, <em>unsupervised learning</em>
|
||
is a method for finding patterns and relationship in data sets without any prior knowledge of the system.
|
||
Some authours also operate with a third category, namely <em>reinforcement learning</em>. This is a paradigm
|
||
of learning inspired by behavioural psychology, where learning is achieved by trial-and-error,
|
||
solely from rewards and punishment.</p>
|
||
<p>Another way to categorize machine learning tasks is to consider the desired output of a system.
|
||
Some of the most common tasks are:</p>
|
||
<ul class="simple">
|
||
<li><p>Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.</p></li>
|
||
<li><p>Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.</p></li>
|
||
<li><p>Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.</p></li>
|
||
<li><p>Other unsupervised learning algortihms like <strong>Boltzmann machines</strong></p></li>
|
||
</ul>
|
||
</section>
|
||
<section id="why-boltzmann-machines">
|
||
<h2>Why Boltzmann machines?<a class="headerlink" href="#why-boltzmann-machines" title="Link to this heading">#</a></h2>
|
||
<p>What is known as restricted Boltzmann Machines (RMB) have received a lot of attention lately.
|
||
One of the major reasons is that they can be stacked layer-wise to build deep neural networks that capture complicated statistics.</p>
|
||
<p>The original RBMs had just one visible layer and a hidden layer, but recently so-called Gaussian-binary RBMs have gained quite some popularity in imaging since they are capable of modeling continuous data that are common to natural images.</p>
|
||
<p>Furthermore, they have been used to solve complicated <a class="reference external" href="https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045002">quantum mechanical many-particle problems or classical statistical physics problems like the Ising and Potts classes of models</a>.</p>
|
||
</section>
|
||
<section id="boltzmann-machines">
|
||
<h2>Boltzmann Machines<a class="headerlink" href="#boltzmann-machines" title="Link to this heading">#</a></h2>
|
||
<p>Why use a generative model rather than the more well known discriminative deep neural networks (DNN)?</p>
|
||
<ul class="simple">
|
||
<li><p>Discriminitave methods have several limitations: They are mainly supervised learning methods, thus requiring labeled data. And there are tasks they cannot accomplish, like drawing new examples from an unknown probability distribution.</p></li>
|
||
<li><p>A generative model can learn to represent and sample from a probability distribution. The core idea is to learn a parametric model of the probability distribution from which the training data was drawn. As an example</p></li>
|
||
</ul>
|
||
<p>a. A model for images could learn to draw new examples of cats and dogs, given a training dataset of images of cats and dogs.</p>
|
||
<p>b. Generate a sample of an ordered or disordered phase, having been given samples of such phases.</p>
|
||
<p>c. Model the trial function for <a class="reference external" href="https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.91.045002">Monte Carlo calculations</a>.</p>
|
||
</section>
|
||
<section id="some-similarities-and-differences-from-dnns">
|
||
<h2>Some similarities and differences from DNNs<a class="headerlink" href="#some-similarities-and-differences-from-dnns" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p>Both use gradient-descent based learning procedures for minimizing cost functions</p></li>
|
||
<li><p>Energy based models don’t use backpropagation and automatic differentiation for computing gradients, instead turning to Markov Chain Monte Carlo methods.</p></li>
|
||
<li><p>DNNs often have several hidden layers. A restricted Boltzmann machine has only one hidden layer, however several RBMs can be stacked to make up Deep Belief Networks, of which they constitute the building blocks.</p></li>
|
||
</ol>
|
||
<p>History: The RBM was developed by amongst others <a class="reference external" href="https://en.wikipedia.org/wiki/Geoffrey_Hinton">Geoffrey Hinton</a>, called by some the “Godfather of Deep Learning”, working with the University of Toronto and Google.</p>
|
||
</section>
|
||
<section id="boltzmann-machines-bm">
|
||
<h2>Boltzmann machines (BM)<a class="headerlink" href="#boltzmann-machines-bm" title="Link to this heading">#</a></h2>
|
||
<p>A BM is what we would call an undirected probabilistic graphical model
|
||
with stochastic continuous or discrete units.</p>
|
||
<p>It is interpreted as a stochastic recurrent neural network where the
|
||
state of each unit(neurons/nodes) depends on the units it is connected
|
||
to. The weights in the network represent thus the strength of the
|
||
interaction between various units/nodes.</p>
|
||
<p>It turns into a Hopfield network if we choose deterministic rather
|
||
than stochastic units. In contrast to a Hopfield network, a BM is a
|
||
so-called generative model. It allows us to generate new samples from
|
||
the learned distribution.</p>
|
||
</section>
|
||
<section id="a-standard-bm-setup">
|
||
<h2>A standard BM setup<a class="headerlink" href="#a-standard-bm-setup" title="Link to this heading">#</a></h2>
|
||
<p>A standard BM network is divided into a set of observable and visible units <span class="math notranslate nohighlight">\(\hat{x}\)</span> and a set of unknown hidden units/nodes <span class="math notranslate nohighlight">\(\hat{h}\)</span>.</p>
|
||
<p>Additionally there can be bias nodes for the hidden and visible layers. These biases are normally set to <span class="math notranslate nohighlight">\(1\)</span>.</p>
|
||
<p>BMs are stackable, meaning they cwe can train a BM which serves as input to another BM. We can construct deep networks for learning complex PDFs. The layers can be trained one after another, a feature which makes them popular in deep learning</p>
|
||
<p>However, they are often hard to train. This leads to the introduction of so-called restricted BMs, or RBMS.
|
||
Here we take away all lateral connections between nodes in the visible layer as well as connections between nodes in the hidden layer. The network is illustrated in the figure below.</p>
|
||
</section>
|
||
<section id="the-structure-of-the-rbm-network">
|
||
<h2>The structure of the RBM network<a class="headerlink" href="#the-structure-of-the-rbm-network" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [figures/RBM.png, width=800 frac=1.0] -->
|
||
<!-- begin figure -->
|
||
<p><img src="figures/RBM.png" width="800"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></section>
|
||
<section id="the-network">
|
||
<h2>The network<a class="headerlink" href="#the-network" title="Link to this heading">#</a></h2>
|
||
<p><strong>The network layers</strong>:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>A function <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> that represents the visible layer, a vector of <span class="math notranslate nohighlight">\(M\)</span> elements (nodes). This layer represents both what the RBM might be given as training input, and what we want it to be able to reconstruct. This might for example be given by the pixels of an image or coefficients representing speech, or the coordinates of a quantum mechanical state function.</p></li>
|
||
<li><p>The function <span class="math notranslate nohighlight">\(\mathbf{h}\)</span> represents the hidden, or latent, layer. A vector of <span class="math notranslate nohighlight">\(N\)</span> elements (nodes). Also called “feature detectors”.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="goals">
|
||
<h2>Goals<a class="headerlink" href="#goals" title="Link to this heading">#</a></h2>
|
||
<p>The goal of the hidden layer is to increase the model’s expressive
|
||
power. We encode complex interactions between visible variables by
|
||
introducing additional, hidden variables that interact with visible
|
||
degrees of freedom in a simple manner, yet still reproduce the complex
|
||
correlations between visible degrees in the data once marginalized
|
||
over (integrated out).</p>
|
||
<p><strong>The network parameters, to be optimized/learned</strong>:</p>
|
||
<ol class="arabic simple">
|
||
<li><p><span class="math notranslate nohighlight">\(\mathbf{a}\)</span> represents the visible bias, a vector of same length as <span class="math notranslate nohighlight">\(\mathbf{x}\)</span>.</p></li>
|
||
<li><p><span class="math notranslate nohighlight">\(\mathbf{b}\)</span> represents the hidden bias, a vector of same lenght as <span class="math notranslate nohighlight">\(\mathbf{h}\)</span>.</p></li>
|
||
<li><p><span class="math notranslate nohighlight">\(W\)</span> represents the interaction weights, a matrix of size <span class="math notranslate nohighlight">\(M\times N\)</span>.</p></li>
|
||
</ol>
|
||
</section>
|
||
<section id="joint-distribution">
|
||
<h2>Joint distribution<a class="headerlink" href="#joint-distribution" title="Link to this heading">#</a></h2>
|
||
<p>The restricted Boltzmann machine is described by a Boltzmann distribution</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto1"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
P_{rbm}(\mathbf{x},\mathbf{h}) = \frac{1}{Z} e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})},
|
||
\label{_auto1} \tag{1}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(Z\)</span> is the normalization constant or partition function, defined as</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto2"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
Z = \int \int e^{-\frac{1}{T_0}E(\mathbf{x},\mathbf{h})} d\mathbf{x} d\mathbf{h}.
|
||
\label{_auto2} \tag{2}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>It is common to ignore <span class="math notranslate nohighlight">\(T_0\)</span> by setting it to one.</p>
|
||
</section>
|
||
<section id="network-elements-the-energy-function">
|
||
<h2>Network Elements, the energy function<a class="headerlink" href="#network-elements-the-energy-function" title="Link to this heading">#</a></h2>
|
||
<p>The function <span class="math notranslate nohighlight">\(E(\mathbf{x},\mathbf{h})\)</span> gives the <strong>energy</strong> of a
|
||
configuration (pair of vectors) <span class="math notranslate nohighlight">\((\mathbf{x}, \mathbf{h})\)</span>. The lower
|
||
the energy of a configuration, the higher the probability of it. This
|
||
function also depends on the parameters <span class="math notranslate nohighlight">\(\mathbf{a}\)</span>, <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> and
|
||
<span class="math notranslate nohighlight">\(W\)</span>. Thus, when we adjust them during the learning procedure, we are
|
||
adjusting the energy function to best fit our problem.</p>
|
||
<p>An expression for the energy function is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
E(\hat{x},\hat{h}) = -\sum_{ia}^{NA}b_i^a \alpha_i^a(x_i)-\sum_{jd}^{MD}c_j^d \beta_j^d(h_j)-\sum_{ijad}^{NAMD}b_i^a \alpha_i^a(x_i)c_j^d \beta_j^d(h_j)w_{ij}^{ad}.
|
||
\]</div>
|
||
<p>Here <span class="math notranslate nohighlight">\(\beta_j^d(h_j)\)</span> and <span class="math notranslate nohighlight">\(\alpha_i^a(x_j)\)</span> are so-called transfer functions that map a given input value to a desired feature value. The labels <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(d\)</span> denote that there can be multiple transfer functions per variable. The first sum depends only on the visible units. The second on the hidden ones. <strong>Note</strong> that there is no connection between nodes in a layer.</p>
|
||
<p>The quantities <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(c\)</span> can be interpreted as the visible and hidden biases, respectively.</p>
|
||
<p>The connection between the nodes in the two layers is given by the weights <span class="math notranslate nohighlight">\(w_{ij}\)</span>.</p>
|
||
</section>
|
||
<section id="defining-different-types-of-rbms">
|
||
<h2>Defining different types of RBMs<a class="headerlink" href="#defining-different-types-of-rbms" title="Link to this heading">#</a></h2>
|
||
<p>There are different variants of RBMs, and the differences lie in the types of visible and hidden units we choose as well as in the implementation of the energy function <span class="math notranslate nohighlight">\(E(\mathbf{x},\mathbf{h})\)</span>.</p>
|
||
<p><strong>Binary-Binary RBM:</strong></p>
|
||
<p>RBMs were first developed using binary units in both the visible and hidden layer. The corresponding energy function is defined as follows:</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto3"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
E(\mathbf{x}, \mathbf{h}) = - \sum_i^M x_i a_i- \sum_j^N b_j h_j - \sum_{i,j}^{M,N} x_i w_{ij} h_j,
|
||
\label{_auto3} \tag{3}
|
||
\end{equation}
|
||
\]</div>
|
||
<p>where the binary values taken on by the nodes are most commonly 0 and 1.</p>
|
||
<p><strong>Gaussian-Binary RBM:</strong></p>
|
||
<p>Another varient is the RBM where the visible units are Gaussian while the hidden units remain binary:</p>
|
||
<!-- Equation labels as ordinary links -->
|
||
<div id="_auto4"></div>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\begin{equation}
|
||
E(\mathbf{x}, \mathbf{h}) = \sum_i^M \frac{(x_i - a_i)^2}{2\sigma_i^2} - \sum_j^N b_j h_j - \sum_{i,j}^{M,N} \frac{x_i w_{ij} h_j}{\sigma_i^2}.
|
||
\label{_auto4} \tag{4}
|
||
\end{equation}
|
||
\]</div>
|
||
</section>
|
||
<section id="more-about-rbms">
|
||
<h2>More about RBMs<a class="headerlink" href="#more-about-rbms" title="Link to this heading">#</a></h2>
|
||
<ol class="arabic simple">
|
||
<li><p>Useful when we model continuous data (i.e., we wish <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> to be continuous)</p></li>
|
||
<li><p>Requires a smaller learning rate, since there’s no upper bound to the value a component might take in the reconstruction</p></li>
|
||
</ol>
|
||
<p>Other types of units include:</p>
|
||
<ol class="arabic simple">
|
||
<li><p>Softmax and multinomial units</p></li>
|
||
<li><p>Gaussian visible and hidden units</p></li>
|
||
<li><p>Binomial units</p></li>
|
||
<li><p>Rectified linear units</p></li>
|
||
</ol>
|
||
<p>To read more, see <a class="reference external" href="https://github.com/CompPhysics/ComputationalPhysics2/blob/gh-pages/doc/pub/notebook2/ipynb/notebook2.ipynb">Lectures on Boltzmann machines in Physics</a>.</p>
|
||
</section>
|
||
<section id="autoencoders-overarching-view">
|
||
<h2>Autoencoders: Overarching view<a class="headerlink" href="#autoencoders-overarching-view" title="Link to this heading">#</a></h2>
|
||
<p>Autoencoders are artificial neural networks capable of learning
|
||
efficient representations of the input data (these representations are called codings) without
|
||
any supervision (i.e., the training set is unlabeled). These codings
|
||
typically have a much lower dimensionality than the input data, making
|
||
autoencoders useful for dimensionality reduction.</p>
|
||
<p>More importantly, autoencoders act as powerful feature detectors, and
|
||
they can be used for unsupervised pretraining of deep neural networks.</p>
|
||
<p>Lastly, they are capable of randomly generating new data that looks
|
||
very similar to the training data; this is called a generative
|
||
model. For example, you could train an autoencoder on pictures of
|
||
faces, and it would then be able to generate new faces. Surprisingly,
|
||
autoencoders work by simply learning to copy their inputs to their
|
||
outputs. This may sound like a trivial task, but we will see that
|
||
constraining the network in various ways can make it rather
|
||
difficult. For example, you can limit the size of the internal
|
||
representation, or you can add noise to the inputs and train the
|
||
network to recover the original inputs. These constraints prevent the
|
||
autoencoder from trivially copying the inputs directly to the outputs,
|
||
which forces it to learn efficient ways of representing the data. In
|
||
short, the codings are byproducts of the autoencoder’s attempt to
|
||
learn the identity function under some constraints.</p>
|
||
<p><a class="reference external" href="https://www.coursera.org/lecture/building-deep-learning-models-with-tensorflow/autoencoders-1U4L3">Video on autoencoders</a></p>
|
||
<p>See also A. Geron’s textbook, chapter 15.</p>
|
||
</section>
|
||
<section id="bayesian-machine-learning">
|
||
<h2>Bayesian Machine Learning<a class="headerlink" href="#bayesian-machine-learning" title="Link to this heading">#</a></h2>
|
||
<p>This is an important topic if we aim at extracting a probability
|
||
distribution. This gives us also a confidence interval and error
|
||
estimates.</p>
|
||
<p>Bayesian machine learning allows us to encode our prior beliefs about
|
||
what those models should look like, independent of what the data tells
|
||
us. This is especially useful when we don’t have a ton of data to
|
||
confidently learn our model.</p>
|
||
<p><a class="reference external" href="https://www.youtube.com/watch?v=E1qhGw8QxqY&amp;ab_channel=AndrewGordonWilson">Video on Bayesian deep learning</a></p>
|
||
<p>See also the <a class="reference external" href="https://github.com/CompPhysics/MachineLearning/blob/master/doc/Articles/lec03.pdf">slides here</a>.</p>
|
||
</section>
|
||
<section id="reinforcement-learning">
|
||
<h2>Reinforcement Learning<a class="headerlink" href="#reinforcement-learning" title="Link to this heading">#</a></h2>
|
||
<p>Reinforcement Learning (RL) is one of the most exciting fields of
|
||
Machine Learning today, and also one of the oldest. It has been around
|
||
since the 1950s, producing many interesting applications over the
|
||
years.</p>
|
||
<p>It studies
|
||
how agents take actions based on trial and error, so as to maximize
|
||
some notion of cumulative reward in a dynamic system or
|
||
environment. Due to its generality, the problem has also been studied
|
||
in many other disciplines, such as game theory, control theory,
|
||
operations research, information theory, multi-agent systems, swarm
|
||
intelligence, statistics, and genetic algorithms.</p>
|
||
<p>In March 2016, AlphaGo, a computer program that plays the board game
|
||
Go, beat Lee Sedol in a five-game match. This was the first time a
|
||
computer Go program had beaten a 9-dan (highest rank) professional
|
||
without handicaps. AlphaGo is based on deep convolutional neural
|
||
networks and reinforcement learning. AlphaGo’s victory was a major
|
||
milestone in artificial intelligence and it has also made
|
||
reinforcement learning a hot research area in the field of machine
|
||
learning.</p>
|
||
<p><a class="reference external" href="https://www.youtube.com/watch?v=FgzM3zpZ55o&amp;ab_channel=stanfordonline">Lecture on Reinforcement Learning</a>.</p>
|
||
<p>See also A. Geron’s textbook, chapter 16.</p>
|
||
</section>
|
||
<section id="transfer-learning">
|
||
<h2>Transfer learning<a class="headerlink" href="#transfer-learning" title="Link to this heading">#</a></h2>
|
||
<p>The goal of transfer learning is to transfer the model or knowledge
|
||
obtained from a source task to the target task, in order to resolve
|
||
the issues of insufficient training data in the target task. The
|
||
rationality of doing so lies in that usually the source and target
|
||
tasks have inter-correlations, and therefore either the features,
|
||
samples, or models in the source task might provide useful information
|
||
for us to better solve the target task. Transfer learning is a hot
|
||
research topic in recent years, with many problems still waiting to be studied.</p>
|
||
<p><a class="reference external" href="https://www.ias.edu/video/machinelearning/2020/0331-SamoryKpotufe">Lecture on transfer learning</a>.</p>
|
||
</section>
|
||
<section id="adversarial-learning">
|
||
<h2>Adversarial learning<a class="headerlink" href="#adversarial-learning" title="Link to this heading">#</a></h2>
|
||
<p>The conventional deep generative model has a potential problem: the
|
||
model tends to generate extreme instances to maximize the
|
||
probabilistic likelihood, which will hurt its performance. Adversarial
|
||
learning utilizes the adversarial behaviors (e.g., generating
|
||
adversarial instances or training an adversarial model) to enhance the
|
||
robustness of the model and improve the quality of the generated
|
||
data. In recent years, one of the most promising unsupervised learning
|
||
technologies, generative adversarial networks (GAN), has already been
|
||
successfully applied to image, speech, and text.</p>
|
||
<p><a class="reference external" href="https://www.youtube.com/watch?v=CIfsB_EYsVI&amp;ab_channel=StanfordUniversitySchoolofEngineering">Lecture on adversial learning</a>.</p>
|
||
</section>
|
||
<section id="dual-learning">
|
||
<h2>Dual learning<a class="headerlink" href="#dual-learning" title="Link to this heading">#</a></h2>
|
||
<p>Dual learning is a new learning paradigm, the basic idea of which is
|
||
to use the primal-dual structure between machine learning tasks to
|
||
obtain effective feedback/regularization, and guide and strengthen the
|
||
learning process, thus reducing the requirement of large-scale labeled
|
||
data for deep learning. The idea of dual learning has been applied to
|
||
many problems in machine learning, including machine translation,
|
||
image style conversion, question answering and generation, image
|
||
classification and generation, text classification and generation,
|
||
image-to-text, and text-to-image.</p>
|
||
</section>
|
||
<section id="distributed-machine-learning">
|
||
<h2>Distributed machine learning<a class="headerlink" href="#distributed-machine-learning" title="Link to this heading">#</a></h2>
|
||
<p>Distributed computation will speed up machine learning algorithms,
|
||
significantly improve their efficiency, and thus enlarge their
|
||
application. When distributed meets machine learning, more than just
|
||
implementing the machine learning algorithms in parallel is required.</p>
|
||
</section>
|
||
<section id="meta-learning">
|
||
<h2>Meta learning<a class="headerlink" href="#meta-learning" title="Link to this heading">#</a></h2>
|
||
<p>Meta learning is an emerging research direction in machine
|
||
learning. Roughly speaking, meta learning concerns learning how to
|
||
learn, and focuses on the understanding and adaptation of the learning
|
||
itself, instead of just completing a specific learning task. That is,
|
||
a meta learner needs to be able to evaluate its own learning methods
|
||
and adjust its own learning methods according to specific learning
|
||
tasks.</p>
|
||
</section>
|
||
<section id="the-challenges-facing-machine-learning">
|
||
<h2>The Challenges Facing Machine Learning<a class="headerlink" href="#the-challenges-facing-machine-learning" title="Link to this heading">#</a></h2>
|
||
<p>While there has been much progress in machine learning, there are also challenges.</p>
|
||
<p>For example, the mainstream machine learning technologies are
|
||
black-box approaches, making us concerned about their potential
|
||
risks. To tackle this challenge, we may want to make machine learning
|
||
more explainable and controllable. As another example, the
|
||
computational complexity of machine learning algorithms is usually
|
||
very high and we may want to invent lightweight algorithms or
|
||
implementations. Furthermore, in many domains such as physics,
|
||
chemistry, biology, and social sciences, people usually seek elegantly
|
||
simple equations (e.g., the Schrödinger equation) to uncover the
|
||
underlying laws behind various phenomena. In the field of machine
|
||
learning, can we reveal simple laws instead of designing more complex
|
||
models for data fitting? Although there are many challenges, we are
|
||
still very optimistic about the future of machine learning. As we look
|
||
forward to the future, here are what we think the research hotspots in
|
||
the next ten years will be.</p>
|
||
<p>See the article on <a class="reference external" href="https://www.frontiersin.org/articles/10.3389/frai.2020.00025/full">Discovery of Physics From Data: Universal Laws and Discrepancies</a></p>
|
||
</section>
|
||
<section id="explainable-machine-learning">
|
||
<h2>Explainable machine learning<a class="headerlink" href="#explainable-machine-learning" title="Link to this heading">#</a></h2>
|
||
<p>Machine learning, especially deep learning, evolves rapidly. The
|
||
ability gap between machine and human on many complex cognitive tasks
|
||
becomes narrower and narrower. However, we are still in the very early
|
||
stage in terms of explaining why those effective models work and how
|
||
they work.</p>
|
||
<p><strong>What is missing: the gap between correlation and causation</strong>. Standard Machine Learning is based on what e have called a frequentist approach.</p>
|
||
<p>Most
|
||
machine learning techniques, especially the statistical ones, depend
|
||
highly on correlations in data sets to make predictions and analyses. In
|
||
contrast, rational humans tend to reply on clear and trustworthy
|
||
causality relations obtained via logical reasoning on real and clear
|
||
facts. It is one of the core goals of explainable machine learning to
|
||
transition from solving problems by data correlation to solving
|
||
problems by logical reasoning.</p>
|
||
<p><strong>Bayesian Machine Learning is one of the exciting research directions in this field</strong>.</p>
|
||
</section>
|
||
<section id="quantum-machine-learning">
|
||
<h2>Quantum machine learning<a class="headerlink" href="#quantum-machine-learning" title="Link to this heading">#</a></h2>
|
||
<p>Quantum machine learning is an emerging interdisciplinary research
|
||
area at the intersection of quantum computing and machine learning.</p>
|
||
<p>Quantum computers use effects such as quantum coherence and quantum
|
||
entanglement to process information, which is fundamentally different
|
||
from classical computers. Quantum algorithms have surpassed the best
|
||
classical algorithms in several problems (e.g., searching for an
|
||
unsorted database, inverting a sparse matrix), which we call quantum
|
||
acceleration.</p>
|
||
<p>When quantum computing meets machine learning, it can be a mutually
|
||
beneficial and reinforcing process, as it allows us to take advantage
|
||
of quantum computing to improve the performance of classical machine
|
||
learning algorithms. In addition, we can also use the machine learning
|
||
algorithms (on classic computers) to analyze and improve quantum
|
||
computing systems.</p>
|
||
<p><a class="reference external" href="https://www.youtube.com/watch?v=Xh9pUu3-WxM&amp;ab_channel=InstituteforPure%26AppliedMathematics%28IPAM%29">Lecture on Quantum ML</a>.</p>
|
||
<p><a class="reference external" href="https://physics.aps.org/articles/v13/179?utm_campaign=weekly&amp;utm_medium=email&amp;utm_source=emailalert">Read interview with Maria Schuld on her work on Quantum Machine Learning</a>. See also <a class="reference external" href="https://www.springer.com/gp/book/9783319964232">her recent textbook</a>.</p>
|
||
</section>
|
||
<section id="quantum-machine-learning-algorithms-based-on-linear-algebra">
|
||
<h2>Quantum machine learning algorithms based on linear algebra<a class="headerlink" href="#quantum-machine-learning-algorithms-based-on-linear-algebra" title="Link to this heading">#</a></h2>
|
||
<p>Many quantum machine learning algorithms are based on variants of
|
||
quantum algorithms for solving linear equations, which can efficiently
|
||
solve N-variable linear equations with complexity of O(log2 N) under
|
||
certain conditions. The quantum matrix inversion algorithm can
|
||
accelerate many machine learning methods, such as least square linear
|
||
regression, least square version of support vector machine, Gaussian
|
||
process, and more. The training of these algorithms can be simplified
|
||
to solve linear equations. The key bottleneck of this type of quantum
|
||
machine learning algorithms is data input—that is, how to initialize
|
||
the quantum system with the entire data set. Although efficient
|
||
data-input algorithms exist for certain situations, how to efficiently
|
||
input data into a quantum system is as yet unknown for most cases.</p>
|
||
</section>
|
||
<section id="quantum-reinforcement-learning">
|
||
<h2>Quantum reinforcement learning<a class="headerlink" href="#quantum-reinforcement-learning" title="Link to this heading">#</a></h2>
|
||
<p>In quantum reinforcement learning, a quantum agent interacts with the
|
||
classical environment to obtain rewards from the environment, so as to
|
||
adjust and improve its behavioral strategies. In some cases, it
|
||
achieves quantum acceleration by the quantum processing capabilities
|
||
of the agent or the possibility of exploring the environment through
|
||
quantum superposition. Such algorithms have been proposed in
|
||
superconducting circuits and systems of trapped ions.</p>
|
||
</section>
|
||
<section id="quantum-deep-learning">
|
||
<h2>Quantum deep learning<a class="headerlink" href="#quantum-deep-learning" title="Link to this heading">#</a></h2>
|
||
<p>Dedicated quantum information processors, such as quantum annealers
|
||
and programmable photonic circuits, are well suited for building deep
|
||
quantum networks. The simplest deep quantum network is the Boltzmann
|
||
machine. The classical Boltzmann machine consists of bits with tunable
|
||
interactions and is trained by adjusting the interaction of these bits
|
||
so that the distribution of its expression conforms to the statistics
|
||
of the data. To quantize the Boltzmann machine, the neural network can
|
||
simply be represented as a set of interacting quantum spins that
|
||
correspond to an adjustable Ising model. Then, by initializing the
|
||
input neurons in the Boltzmann machine to a fixed state and allowing
|
||
the system to heat up, we can read out the output qubits to get the
|
||
result.</p>
|
||
</section>
|
||
<section id="social-machine-learning">
|
||
<h2>Social machine learning<a class="headerlink" href="#social-machine-learning" title="Link to this heading">#</a></h2>
|
||
<p>Machine learning aims to imitate how humans
|
||
learn. While we have developed successful machine learning algorithms,
|
||
until now we have ignored one important fact: humans are social. Each
|
||
of us is one part of the total society and it is difficult for us to
|
||
live, learn, and improve ourselves, alone and isolated. Therefore, we
|
||
should design machines with social properties. Can we let machines
|
||
evolve by imitating human society so as to achieve more effective,
|
||
intelligent, interpretable “social machine learning”?</p>
|
||
<p>And much more.</p>
|
||
</section>
|
||
<section id="the-last-words">
|
||
<h2>The last words?<a class="headerlink" href="#the-last-words" title="Link to this heading">#</a></h2>
|
||
<p>Early computer scientist Alan Kay said, <strong>The best way to predict the
|
||
future is to create it</strong>. Therefore, all machine learning
|
||
practitioners, whether scholars or engineers, professors or students,
|
||
need to work together to advance these important research
|
||
topics. Together, we will not just predict the future, but create it.</p>
|
||
</section>
|
||
<section id="best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester">
|
||
<h2>Best wishes to you all and thanks so much for your heroic efforts this semester<a class="headerlink" href="#best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester" title="Link to this heading">#</a></h2>
|
||
<!-- dom:FIGURE: [figures/Nebbdyr2.png, width=500 frac=0.6] -->
|
||
<!-- begin figure -->
|
||
<p><img src="figures/Nebbdyr2.png" width="500"><p style="font-size: 0.9em"><i>Figure 1: </i></p></p>
|
||
<!-- end figure --></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="exercisesweek47.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">Exercise week 47</p>
|
||
</div>
|
||
</a>
|
||
<a class="right-next"
|
||
href="exercisesweek48.html"
|
||
title="next page">
|
||
<div class="prev-next-info">
|
||
<p class="prev-next-subtitle">next</p>
|
||
<p class="prev-next-title">Exercises week 48</p>
|
||
</div>
|
||
<i class="fa-solid fa-angle-right"></i>
|
||
</a>
|
||
</div>
|
||
</footer>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
<dialog id="pst-secondary-sidebar-modal"></dialog>
|
||
<div id="pst-secondary-sidebar" 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="#overview-of-week-48">Overview of week 48</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lecture-monday-november-25">Lecture Monday, November 25</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#lab-sessions">Lab sessions</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#random-forest-algorithm-reminder-from-last-week">Random Forest Algorithm, reminder from last week</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>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-an-adaboost-code-yourself">Making an ADAboost code yourself</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-basics-with-steepest-descent-functional-gradient-descent">Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-squared-error-again-steepest-descent">The Squared-Error again! Steepest Descent</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#steepest-descent-example">Steepest Descent Example</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-algorithm">Gradient Boosting, algorithm</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-examples-of-regression">Gradient Boosting, Examples of Regression</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-classification-example">Gradient Boosting, Classification Example</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#xgboost-extreme-gradient-boosting">XGBoost: Extreme Gradient Boosting</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#xgboost-on-the-cancer-data">Xgboost on the Cancer Data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#gradient-boosting-making-our-own-code-for-a-regression-case">Gradient boosting, making our own code for a regression case</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary-of-course">Summary of course</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#what-me-worry-no-final-exam-in-this-course">What? Me worry? No final exam in this course!</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#topics-we-have-covered-this-year">Topics we have covered this year</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#statistical-analysis-and-optimization-of-data">Statistical analysis and optimization of data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#machine-learning">Machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-outcomes-and-overarching-aims-of-this-course">Learning outcomes and overarching aims of this course</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#perspective-on-machine-learning">Perspective on Machine Learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#machine-learning-research">Machine Learning Research</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#starting-your-machine-learning-project">Starting your Machine Learning Project</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#choose-a-model-and-algorithm">Choose a Model and Algorithm</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#preparing-your-data">Preparing Your Data</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#which-activation-and-weights-to-choose-in-neural-networks">Which activation and weights to choose in neural networks</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#optimization-methods-and-hyperparameters">Optimization Methods and Hyperparameters</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#resampling">Resampling</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-courses-on-data-science-and-machine-learning-at-uio">Other courses on Data science and Machine Learning at UiO</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#additional-courses-of-interest">Additional courses of interest</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#what-s-the-future-like">What’s the future like?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#types-of-machine-learning-a-repetition">Types of Machine Learning, a repetition</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#why-boltzmann-machines">Why Boltzmann machines?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#boltzmann-machines">Boltzmann Machines</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#some-similarities-and-differences-from-dnns">Some similarities and differences from DNNs</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#boltzmann-machines-bm">Boltzmann machines (BM)</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-standard-bm-setup">A standard BM setup</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-structure-of-the-rbm-network">The structure of the RBM network</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-network">The network</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#goals">Goals</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#joint-distribution">Joint distribution</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#network-elements-the-energy-function">Network Elements, the energy function</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-different-types-of-rbms">Defining different types of RBMs</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#more-about-rbms">More about RBMs</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#autoencoders-overarching-view">Autoencoders: Overarching view</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bayesian-machine-learning">Bayesian Machine Learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#reinforcement-learning">Reinforcement Learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#transfer-learning">Transfer learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#adversarial-learning">Adversarial learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dual-learning">Dual learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#distributed-machine-learning">Distributed machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#meta-learning">Meta learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-challenges-facing-machine-learning">The Challenges Facing Machine Learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#explainable-machine-learning">Explainable machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-machine-learning">Quantum machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-machine-learning-algorithms-based-on-linear-algebra">Quantum machine learning algorithms based on linear algebra</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-reinforcement-learning">Quantum reinforcement learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#quantum-deep-learning">Quantum deep learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#social-machine-learning">Social machine learning</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-last-words">The last words?</a></li>
|
||
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#best-wishes-to-you-all-and-thanks-so-much-for-your-heroic-efforts-this-semester">Best wishes to you all and thanks so much for your heroic efforts this semester</a></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 defer src="_static/scripts/bootstrap.js?digest=26a4bc78f4c0ddb94549"></script>
|
||
<script defer src="_static/scripts/pydata-sphinx-theme.js?digest=26a4bc78f4c0ddb94549"></script>
|
||
|
||
<footer class="bd-footer">
|
||
</footer>
|
||
</body>
|
||
</html> |