lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
Files
31470981479f5fcb400b3d281d41e7d102543f77
FYS-STK4155/doc/LectureNotes/_build/html/clustering.html
T
Morten Hjorth-Jensen 3147098147 update
2024-11-17 15:55:37 +01:00

1000 lines
70 KiB
HTML
Raw Blame History

This file contains ambiguous Unicode characters
This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
<!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>12. Clustering and Unsupervised Learning &#8212; 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 = 'clustering';</script>
<link rel="index" title="Index" href="genindex.html" />
<link rel="search" title="Search" href="search.html" />
<link rel="next" title="13. Neural networks" href="chapter9.html" />
<link rel="prev" title="11. Basic ideas of the Principal Component Analysis (PCA)" href="chapter8.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="current 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 current active"><a class="current reference internal" href="#">12. Clustering and Unsupervised Learning</a></li>
</ul>
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Deep Learning Methods</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="chapter9.html">13. Neural networks</a></li>
<li class="toctree-l1"><a class="reference internal" href="chapter10.html">14. Building a Feed Forward Neural Network</a></li>
<li class="toctree-l1"><a class="reference internal" href="chapter11.html">15. Solving Differential Equations with Deep Learning</a></li>
<li class="toctree-l1"><a class="reference internal" href="chapter12.html">16. Convolutional Neural Networks</a></li>
<li class="toctree-l1"><a class="reference internal" href="chapter13.html">17. Recurrent neural networks: Overarching view</a></li>
</ul>
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Weekly material, notes and exercises</span></p>
<ul class="nav bd-sidenav">
<li class="toctree-l1"><a class="reference internal" href="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>
</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/clustering.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/clustering.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>Clustering and Unsupervised Learning</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="#codes-and-approaches">12.1. Codes and Approaches</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 clustering.do.txt --><section class="tex2jax_ignore mathjax_ignore" id="clustering-and-unsupervised-learning">
<h1><span class="section-number">12. </span>Clustering and Unsupervised Learning<a class="headerlink" href="#clustering-and-unsupervised-learning" title="Link to this heading">#</a></h1>
<p>In general terms cluster analysis, or clustering, is the task of grouping a
data-set into different distinct categories based on some measure of equality of
the data. This measure is often referred to as a <strong>metric</strong> or <strong>similarity
measure</strong> in the literature (note: sometimes we deal with a <strong>dissimilarity
measure</strong> instead). Usually, these metrics are formulated as some kind of
distance function between points in a high-dimensional space.</p>
<p>The simplest, and also the most
common is the <strong>Euclidean distance</strong>.</p>
<p>The simplest of all clustering algorithms is the <strong>k-means algorithm</strong>
, sometimes also referred to as <em>Lloyds algorithm</em>. It is the simplest and also
the most common. From its simplicity it obtains both strengths and weaknesses.
These will be discussed in more detail later. The <span class="math notranslate nohighlight">\(k\)</span>-means algorithm is a
<strong>centroid based</strong> clustering algorithm.</p>
<p>Assume, we are given <span class="math notranslate nohighlight">\(n\)</span> data points and we wish to split the data into <span class="math notranslate nohighlight">\(K &lt; n\)</span>
different categories, or clusters. We label each cluster by an integer</p>
<div class="math notranslate nohighlight">
\[
k\in\{1, \cdots, K \}.
\]</div>
<p>In the basic k-means algorithm each point is assigned to only
one cluster <span class="math notranslate nohighlight">\(k\)</span>, and these assignments are <em>non-injective</em> i.e. many-to-one. We
can think of these mappings as an encoder <span class="math notranslate nohighlight">\(k = C(i)\)</span>, which assigns the <span class="math notranslate nohighlight">\(i\)</span>-th
data-point <span class="math notranslate nohighlight">\(\bf x_i\)</span> to the <span class="math notranslate nohighlight">\(k\)</span>-th cluster.</p>
<p><span class="math notranslate nohighlight">\(k\)</span>-means algorithm in words:</p>
<ol class="arabic simple">
<li><p>We start with guesses / random initializations of our <span class="math notranslate nohighlight">\(k\)</span> cluster centers/centroids</p></li>
<li><p>For each centroid the points that are most similar are identified</p></li>
<li><p>Then we move / replace each centroid with a coordinate average of all the points that were assigned to that centroid.</p></li>
<li><p>Iterate 2-3 until the centroids no longer move (to some tolerance)</p></li>
</ol>
<p>We assume we have <span class="math notranslate nohighlight">\(n\)</span> data-points</p>
<!-- Equation labels as ordinary links -->
<div id="eq:kmeanspoints"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}\label{eq:kmeanspoints} \tag{1}
\boldsymbol{x_i} = \{x_{i, 1}, \cdots, x_{i, p}\}\in\mathbb{R}^p.
\end{equation}
\]</div>
<p>which we wish to group into <span class="math notranslate nohighlight">\(K &lt; n\)</span> clusters. For our dissimilarity measure we
use the <em>squared Euclidean distance</em></p>
<!-- Equation labels as ordinary links -->
<div id="eq:squaredeuclidean"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}\label{eq:squaredeuclidean} \tag{2}
d(\boldsymbol{x_i}, \boldsymbol{x_i'}) = \sum_{j=1}^p(x_{ij} - x_{i'j})^2
= ||\boldsymbol{x_i} - \boldsymbol{x_{i'}}||^2
\end{equation}
\]</div>
<p>We define the so called <em>within-cluster point scatter</em> which gives us a
measure of how close each data point assigned to the same cluster tends to be to
the all the others.</p>
<!-- Equation labels as ordinary links -->
<div id="eq:withincluster"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}\label{eq:withincluster} \tag{3}
W(C) = \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k}
\sum_{C(i')=k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}}) =
\sum_{k=1}^KN_k\sum_{C(i)=k}||\boldsymbol{x_i} - \boldsymbol{\overline{x_k}}||^2
\end{equation}
\]</div>
<p>where <span class="math notranslate nohighlight">\(\boldsymbol{\overline{x_k}}\)</span> is the mean vector associated with the <span class="math notranslate nohighlight">\(k\)</span>-th
cluster, and <span class="math notranslate nohighlight">\(N_k = \sum_{i=1}^nI(C(i) = k)\)</span>, where the <span class="math notranslate nohighlight">\(I()\)</span> notation is
similar to the Kronecker delta (<em>Commonly used in statistics, it just means that
when <span class="math notranslate nohighlight">\(i = k\)</span> we have the encoder <span class="math notranslate nohighlight">\(C(i)\)</span></em>). In other words, the within-cluster
scatter measures the compactness of each cluster with respect to the data points
assigned to each cluster. This is the quantity that the <span class="math notranslate nohighlight">\(k\)</span>-means algorithm aims
to minimize. We refer to this quantity <span class="math notranslate nohighlight">\(W(C)\)</span> as the within cluster scatter
because of its relation to the <em>total scatter</em>.</p>
<p>We have</p>
<!-- Equation labels as ordinary links -->
<div id="eq:totalscatter"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}\label{eq:totalscatter} \tag{4}
T = W(C) + B(C) = \frac{1}{2}\sum_{i=1}^n
\sum_{i'=1}^nd(\boldsymbol{x_i}, \boldsymbol{x_{i'}})
= \frac{1}{2}\sum_{k=1}^K\sum_{C(i)=k}
\Big(\sum_{C(i') = k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})
+ \sum_{C(i')\neq k}d(\boldsymbol{x_i}, \boldsymbol{x_{i'}})\Big).
\end{equation}
\]</div>
<p>This is a quantity that is conserved throughout the <span class="math notranslate nohighlight">\(k\)</span>-means algorithm. It can
be thought of as the total amount of information in the data, and it is composed
of the aforementioned within-cluster scatter and the <em>between-cluster scatter</em>
<span class="math notranslate nohighlight">\(B(C)\)</span>. In methods such as principle component analysis the total scatter is not
conserved.</p>
<p>Given a cluster mean <span class="math notranslate nohighlight">\(\boldsymbol{m_k}\)</span> we define the <strong>total cluster variance</strong></p>
<!-- Equation labels as ordinary links -->
<div id="eq:totalclustervariance"></div>
<div class="math notranslate nohighlight">
\[
\begin{equation}\label{eq:totalclustervariance} \tag{5}
\min_{C, \{\boldsymbol{m_k}\}_1^K}\sum_{k=1}^KN_k\sum||\boldsymbol{x_i} - \boldsymbol{m_k}||^2
\end{equation}
\]</div>
<p>Now we have all the pieces necessary to formally revisit the <span class="math notranslate nohighlight">\(k\)</span>-means algorithm.</p>
<p>The <span class="math notranslate nohighlight">\(k\)</span>-means clustering algorithm goes as follows</p>
<ol class="arabic simple">
<li><p>For a given cluster assignment <span class="math notranslate nohighlight">\(C\)</span>, and <span class="math notranslate nohighlight">\(k\)</span> cluster means <span class="math notranslate nohighlight">\(\left\{m_1, \cdots, m_k\right\}\)</span>. We minimize the total cluster variance with respect to the cluster means <span class="math notranslate nohighlight">\(\{m_k\}\)</span> yielding the means of the currently assigned clusters.</p></li>
<li><p>Given a current set of <span class="math notranslate nohighlight">\(k\)</span> means <span class="math notranslate nohighlight">\(\{m_k\}\)</span> the total cluster variance is minimized by assigning each observation to the closest (current) cluster mean. That is $<span class="math notranslate nohighlight">\(C(i) = \underset{1\leq k\leq K}{\mathrm{argmin}} ||\boldsymbol{x_i} - \boldsymbol{m_k}||^2\)</span>$</p></li>
<li><p>Steps 1 and 2 are repeated until the assignments do not change.</p></li>
</ol>
<section id="codes-and-approaches">
<h2><span class="section-number">12.1. </span>Codes and Approaches<a class="headerlink" href="#codes-and-approaches" title="Link to this heading">#</a></h2>
<ol class="arabic simple">
<li><p>Before we start we specify a number <span class="math notranslate nohighlight">\(k\)</span> which is the number of clusters we want to try to separate our data into.</p></li>
<li><p>We initially choose <span class="math notranslate nohighlight">\(k\)</span> random data points in our data as our initial centroids, <em>or means</em> (this is where the name comes from).</p></li>
<li><p>Assign each data point to their closest centroid, based on the squared Euclidean distance.</p></li>
<li><p>For each of the <span class="math notranslate nohighlight">\(k\)</span> cluster we update the centroid by calculating new mean values for all the data points in the cluster.</p></li>
<li><p>Iteratively minimize the within cluster scatter by performing steps (3, 4) until the new assignments stop changing (can be to some tolerance) or until a maximum number of iterations have passed.</p></li>
</ol>
<p>Let us now program the most basic version of the algorithm using nothing but
Python with numpy arrays. This code is kept intentionally simple to gradually
progress our understanding. There is no vectorization of any kind, and even most
helper functions are not utilized.</p>
<p>We need first a dataset to do our cluster analysis on. In our case
this is a plain <em>vanilla</em> data set using random numbers using a
Gaussian distribution.</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="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">time</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">tensorflow</span> <span class="k">as</span> <span class="nn">tf</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">image</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">KMeans</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2021</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Next we define functions, for ease of use later, to generate Gaussians and to
set up our toy data set.</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="k">def</span> <span class="nf">gaussian_points</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_points</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">mean_vector</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">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]),</span>
<span class="n">sample_variance</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Very simple custom function to generate gaussian distributed point clusters</span>
<span class="sd"> with variable dimension, number of points, means in each direction</span>
<span class="sd"> (must match dim) and sample variance.</span>
<span class="sd"> Inputs:</span>
<span class="sd"> dim (int)</span>
<span class="sd"> n_points (int)</span>
<span class="sd"> mean_vector (np.array) (where index 0 is x, index 1 is y etc.)</span>
<span class="sd"> sample_variance (float)</span>
<span class="sd"> Returns:</span>
<span class="sd"> data (np.array): with dimensions (dim x n_points)</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">mean_matrix</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">dim</span><span class="p">)</span> <span class="o">+</span> <span class="n">mean_vector</span>
<span class="n">covariance_matrix</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">dim</span><span class="p">)</span> <span class="o">*</span> <span class="n">sample_variance</span>
<span class="n">data</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">multivariate_normal</span><span class="p">(</span><span class="n">mean_matrix</span><span class="p">,</span> <span class="n">covariance_matrix</span><span class="p">,</span>
<span class="n">n_points</span><span class="p">)</span>
<span class="k">return</span> <span class="n">data</span>
<span class="k">def</span> <span class="nf">generate_simple_clustering_dataset</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_points</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">plotting</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="n">return_data</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Toy model to illustrate k-means clustering</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">data1</span> <span class="o">=</span> <span class="n">gaussian_points</span><span class="p">(</span><span class="n">mean_vector</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">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">]))</span>
<span class="n">data2</span> <span class="o">=</span> <span class="n">gaussian_points</span><span class="p">()</span>
<span class="n">data3</span> <span class="o">=</span> <span class="n">gaussian_points</span><span class="p">(</span><span class="n">mean_vector</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="mf">4.5</span><span class="p">]))</span>
<span class="n">data4</span> <span class="o">=</span> <span class="n">gaussian_points</span><span class="p">(</span><span class="n">mean_vector</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">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">data1</span><span class="p">,</span> <span class="n">data2</span><span class="p">,</span> <span class="n">data3</span><span class="p">,</span> <span class="n">data4</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="k">if</span> <span class="n">plotting</span><span class="p">:</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">data</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">data</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">&#39;Toy Model Dataset&#39;</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="k">if</span> <span class="n">return_data</span><span class="p">:</span>
<span class="k">return</span> <span class="n">data</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">generate_simple_clustering_dataset</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/386392c9fed6728256cbe115938f9a087bb6a8f2a530f613d32298404dd706c4.png" src="_images/386392c9fed6728256cbe115938f9a087bb6a8f2a530f613d32298404dd706c4.png" />
</div>
</div>
<p>With the above dataset we start
implementing the <span class="math notranslate nohighlight">\(k\)</span>-means algorithm.</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="n">n_samples</span><span class="p">,</span> <span class="n">dimensions</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span>
<span class="n">n_clusters</span> <span class="o">=</span> <span class="mi">4</span>
<span class="c1"># we randomly initialize our centroids</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">2021</span><span class="p">)</span>
<span class="n">centroids</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_clusters</span><span class="p">,</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span><span class="p">),</span> <span class="p">:]</span>
<span class="n">distances</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">n_samples</span><span class="p">,</span> <span class="n">n_clusters</span><span class="p">))</span>
<span class="c1"># first we need to calculate the distance to each centroid from our data</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="n">dist</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">dimensions</span><span class="p">):</span>
<span class="n">dist</span> <span class="o">+=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">]</span> <span class="o">-</span> <span class="n">centroids</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">d</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
<span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">dist</span>
<span class="c1"># we initialize an array to keep track of to which cluster each point belongs</span>
<span class="c1"># the way we set it up here the index tracks which point and the value which</span>
<span class="c1"># cluster the point belongs to</span>
<span class="n">cluster_labels</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">n_samples</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="s1">&#39;int&#39;</span><span class="p">)</span>
<span class="c1"># next we loop through our samples and for every point assign it to the cluster</span>
<span class="c1"># to which it has the smallest distance to</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="c1"># tracking variables (all of this is basically just an argmin)</span>
<span class="n">smallest</span> <span class="o">=</span> <span class="mf">1e10</span>
<span class="n">smallest_row_index</span> <span class="o">=</span> <span class="mf">1e10</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="k">if</span> <span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">smallest</span><span class="p">:</span>
<span class="n">smallest</span> <span class="o">=</span> <span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
<span class="n">smallest_row_index</span> <span class="o">=</span> <span class="n">k</span>
<span class="n">cluster_labels</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">smallest_row_index</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">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">()</span>
<span class="n">unique_cluster_labels</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">cluster_labels</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">unique_cluster_labels</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">cluster_labels</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="n">data</span><span class="p">[</span><span class="n">cluster_labels</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="n">label</span> <span class="o">=</span> <span class="n">i</span><span class="p">,</span>
<span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.2</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">centroids</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">centroids</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;First Grouping of Points to Centroids&quot;</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/08a3e33657ef89b511498459dfb46eb29d76afce76a3d347fcb1177b92a2ff9b.png" src="_images/08a3e33657ef89b511498459dfb46eb29d76afce76a3d347fcb1177b92a2ff9b.png" />
</div>
</div>
<p>So what do we have so far? We have ‘picked’ <span class="math notranslate nohighlight">\(k\)</span> centroids at random from our
data points. There are other ways of more intelligently choosing their
initializations, however for our purposes randomly is fine. Then we have
initialized an array ‘distances’ which holds the information of the distance,
<em>or dissimilarity</em>, of every point to of our centroids. Finally, we have
initialized an array ‘cluster_labels’ which according to our distances array
holds the information of to which centroid every point is assigned. This was the
first pass of our algorithm. Essentially, all we need to do now is repeat the
distance and assignment steps above until we have reached a desired convergence
or a maximum amount of iterations.</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="n">max_iterations</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">tolerance</span> <span class="o">=</span> <span class="mf">1e-8</span>
<span class="k">for</span> <span class="n">iteration</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_iterations</span><span class="p">):</span>
<span class="n">prev_centroids</span> <span class="o">=</span> <span class="n">centroids</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="c1"># this array will be used to update our centroid positions</span>
<span class="n">vector_mean</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">dimensions</span><span class="p">)</span>
<span class="n">mean_divisor</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="k">if</span> <span class="n">cluster_labels</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">==</span> <span class="n">k</span><span class="p">:</span>
<span class="n">vector_mean</span> <span class="o">+=</span> <span class="n">data</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="p">:]</span>
<span class="n">mean_divisor</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="c1"># update according to the k means</span>
<span class="n">centroids</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">vector_mean</span> <span class="o">/</span> <span class="n">mean_divisor</span>
<span class="c1"># we find the dissimilarity</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="n">dist</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">dimensions</span><span class="p">):</span>
<span class="n">dist</span> <span class="o">+=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">]</span> <span class="o">-</span> <span class="n">centroids</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">d</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
<span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">dist</span>
<span class="c1"># assign each point</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="n">smallest</span> <span class="o">=</span> <span class="mf">1e10</span>
<span class="n">smallest_row_index</span> <span class="o">=</span> <span class="mf">1e10</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="k">if</span> <span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">smallest</span><span class="p">:</span>
<span class="n">smallest</span> <span class="o">=</span> <span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
<span class="n">smallest_row_index</span> <span class="o">=</span> <span class="n">k</span>
<span class="n">cluster_labels</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">smallest_row_index</span>
<span class="c1"># convergence criteria</span>
<span class="n">centroid_difference</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">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">centroids</span> <span class="o">-</span> <span class="n">prev_centroids</span><span class="p">))</span>
<span class="k">if</span> <span class="n">centroid_difference</span> <span class="o">&lt;</span> <span class="n">tolerance</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Converged at iteration </span><span class="si">{</span><span class="n">iteration</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
<span class="k">break</span>
<span class="k">elif</span> <span class="n">iteration</span> <span class="o">==</span> <span class="n">max_iterations</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Did not converge in </span><span class="si">{</span><span class="n">max_iterations</span><span class="si">}</span><span class="s1"> iterations&#39;</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>Converged at iteration 5
</pre></div>
</div>
</div>
</div>
<p>We now have a simple , un-optimized <span class="math notranslate nohighlight">\(k\)</span>-means
clustering implementation. Lets plot the final result</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="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">()</span>
<span class="n">unique_cluster_labels</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">cluster_labels</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">unique_cluster_labels</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">cluster_labels</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="n">data</span><span class="p">[</span><span class="n">cluster_labels</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="n">label</span> <span class="o">=</span> <span class="n">i</span><span class="p">,</span>
<span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.2</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">centroids</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">centroids</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Final Result of K-means Clustering&quot;</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/cd302a02ed7196309f1e57024d9834c2540eab56bf7818d2267779cea5221451.png" src="_images/cd302a02ed7196309f1e57024d9834c2540eab56bf7818d2267779cea5221451.png" />
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">naive_kmeans</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">n_clusters</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span> <span class="n">max_iterations</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">tolerance</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">):</span>
<span class="n">start_time</span> <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
<span class="n">n_samples</span><span class="p">,</span> <span class="n">dimensions</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">shape</span>
<span class="n">n_clusters</span> <span class="o">=</span> <span class="mi">4</span>
<span class="c1">#np.random.seed(2021)</span>
<span class="n">centroids</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_clusters</span><span class="p">,</span> <span class="n">replace</span><span class="o">=</span><span class="kc">False</span><span class="p">),</span> <span class="p">:]</span>
<span class="n">distances</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">n_samples</span><span class="p">,</span> <span class="n">n_clusters</span><span class="p">))</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="n">dist</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">dimensions</span><span class="p">):</span>
<span class="n">dist</span> <span class="o">+=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">]</span> <span class="o">-</span> <span class="n">centroids</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">d</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
<span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">dist</span>
<span class="n">cluster_labels</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">n_samples</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="s1">&#39;int&#39;</span><span class="p">)</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="n">smallest</span> <span class="o">=</span> <span class="mf">1e10</span>
<span class="n">smallest_row_index</span> <span class="o">=</span> <span class="mf">1e10</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="k">if</span> <span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">smallest</span><span class="p">:</span>
<span class="n">smallest</span> <span class="o">=</span> <span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
<span class="n">smallest_row_index</span> <span class="o">=</span> <span class="n">k</span>
<span class="n">cluster_labels</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">smallest_row_index</span>
<span class="k">for</span> <span class="n">iteration</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">max_iterations</span><span class="p">):</span>
<span class="n">prev_centroids</span> <span class="o">=</span> <span class="n">centroids</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="n">vector_mean</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">dimensions</span><span class="p">)</span>
<span class="n">mean_divisor</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="k">if</span> <span class="n">cluster_labels</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">==</span> <span class="n">k</span><span class="p">:</span>
<span class="n">vector_mean</span> <span class="o">+=</span> <span class="n">data</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="p">:]</span>
<span class="n">mean_divisor</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">centroids</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">vector_mean</span> <span class="o">/</span> <span class="n">mean_divisor</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="n">dist</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">dimensions</span><span class="p">):</span>
<span class="n">dist</span> <span class="o">+=</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">d</span><span class="p">]</span> <span class="o">-</span> <span class="n">centroids</span><span class="p">[</span><span class="n">k</span><span class="p">,</span> <span class="n">d</span><span class="p">])</span><span class="o">**</span><span class="mi">2</span>
<span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="n">dist</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_samples</span><span class="p">):</span>
<span class="n">smallest</span> <span class="o">=</span> <span class="mf">1e10</span>
<span class="n">smallest_row_index</span> <span class="o">=</span> <span class="mf">1e10</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
<span class="k">if</span> <span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span> <span class="o">&lt;</span> <span class="n">smallest</span><span class="p">:</span>
<span class="n">smallest</span> <span class="o">=</span> <span class="n">distances</span><span class="p">[</span><span class="n">n</span><span class="p">,</span> <span class="n">k</span><span class="p">]</span>
<span class="n">smallest_row_index</span> <span class="o">=</span> <span class="n">k</span>
<span class="n">cluster_labels</span><span class="p">[</span><span class="n">n</span><span class="p">]</span> <span class="o">=</span> <span class="n">smallest_row_index</span>
<span class="n">centroid_difference</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">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">centroids</span> <span class="o">-</span> <span class="n">prev_centroids</span><span class="p">))</span>
<span class="k">if</span> <span class="n">centroid_difference</span> <span class="o">&lt;</span> <span class="n">tolerance</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Converged at iteration </span><span class="si">{</span><span class="n">iteration</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Runtime: </span><span class="si">{</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">start_time</span><span class="si">}</span><span class="s1"> seconds&#39;</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cluster_labels</span><span class="p">,</span> <span class="n">centroids</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Did not converge in </span><span class="si">{</span><span class="n">max_iterations</span><span class="si">}</span><span class="s1"> iterations&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Runtime: </span><span class="si">{</span><span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">start_time</span><span class="si">}</span><span class="s1"> seconds&#39;</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cluster_labels</span><span class="p">,</span> <span class="n">centroids</span>
</pre></div>
</div>
</div>
</div>
</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="chapter8.html"
title="previous page">
<i class="fa-solid fa-angle-left"></i>
<div class="prev-next-info">
<p class="prev-next-subtitle">previous</p>
<p class="prev-next-title"><span class="section-number">11. </span>Basic ideas of the Principal Component Analysis (PCA)</p>
</div>
</a>
<a class="right-next"
href="chapter9.html"
title="next page">
<div class="prev-next-info">
<p class="prev-next-subtitle">next</p>
<p class="prev-next-title"><span class="section-number">13. </span>Neural networks</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="#codes-and-approaches">12.1. Codes and Approaches</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>
Reference in New Issue View Git Blame Copy Permalink