<!DOCTYPE html> <html> <head> <meta charset="utf-8" /> <meta name="author" content="Brandon Rozek"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <meta name="robots" content="noindex" /> <title>Brandon Rozek</title> <link rel="stylesheet" href="themes/bitsandpieces/styles/main.css" type="text/css" /> <link rel="stylesheet" href="themes/bitsandpieces/styles/highlightjs-github.css" type="text/css" /> </head> <body> <aside class="main-nav"> <nav> <ul> <li class="menuitem "> <a href="index.html%3Findex.html" data-shortcut=""> Home </a> </li> <li class="menuitem "> <a href="index.html%3Fcourses.html" data-shortcut=""> Courses </a> </li> <li class="menuitem "> <a href="index.html%3Flabaide.html" data-shortcut=""> Lab Aide </a> </li> <li class="menuitem "> <a href="index.html%3Fpresentations.html" data-shortcut=""> Presentations </a> </li> <li class="menuitem "> <a href="index.html%3Fresearch.html" data-shortcut=""> Research </a> </li> <li class="menuitem "> <a href="index.html%3Ftranscript.html" data-shortcut=""> Transcript </a> </li> </ul> </nav> </aside> <main class="main-content"> <article class="article"> <h1>K-Medoids</h1> <p>A medoid can be defined as the object of a cluster whose average dissimilarity to all the objects in the cluster is minimal.</p> <p>The K-medoids algorithm is related to k-means and the medoidshift algorithm. Both the k-means and k-medoids algorithms are partition and both attempt to minimize the distance between points in the cluster to it's center. In contrast to k-means, it chooses data points as centers and uses the Manhattan Norm to define the distance between data points instead of the Euclidean.</p> <p>This method is known to be more robust to noise and outliers compared to k-means since it minimizes the sum of pairwise dissimilarities instead of the sum of squared Euclidean distances.</p> <h2>Algorithms</h2> <p>There are several algorithms that have been created as an optimization to an exhaustive search. In this section, we'll discuss PAM and Voronoi iteration method.</p> <h3>Partitioning Around Medoids (PAM)</h3> <ol> <li>Select $k$ of the $n$ data points as medoids</li> <li>Associate each data point to the closes medoid</li> <li>While the cost of the configuration decreases: <ol> <li>For each medoid $m$, for each non-medoid data point $o$: <ol> <li>Swap $m$ and $o$, recompute the cost (sum of distances of points to their medoid)</li> <li>If the total cost of the configuration increased in the previous step, undo the swap</li> </ol></li> </ol></li> </ol> <h3>Voronoi Iteration Method</h3> <ol> <li>Select $k$ of the $n$ data points as medoids</li> <li>While the cost of the configuration decreases <ol> <li>In each cluster, make the point that minimizes the sum of distances within the cluster the medoid</li> <li>Reassign each point to the cluster defined by the closest medoid determined in the previous step.</li> </ol></li> </ol> <h3>Clustering Large Applications (CLARA</h3> <p>This is a variant of the PAM algorithm that relies on the sampling approach to handle large datasets. The cost of a particular cluster configuration is the mean cost of all the dissimilarities.</p> <h2>R Implementations</h2> <p>Both PAM and CLARA are defined in the <code>cluster</code> package in R.</p> <pre><code class="language-R">clara(x, k, metric = "euclidean", stand = FALSE, samples = 5, sampsize = min(n, 40 + 2 * k), trace = 0, medoids.x = TRUE, keep.data = medoids.x, rngR = FALSE)</code></pre> <pre><code class="language-R">pam(x, k, metric = "euclidean", stand = FALSE)</code></pre> </article> </main> <script src="themes/bitsandpieces/scripts/highlight.js"></script> <script src="themes/bitsandpieces/scripts/mousetrap.min.js"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ tex2jax: { inlineMath: [ ['$','$'], ["\\(","\\)"] ], processEscapes: true } }); </script> <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"> </script> <script> hljs.initHighlightingOnLoad(); document.querySelectorAll('.menuitem a').forEach(function(el) { if (el.getAttribute('data-shortcut').length > 0) { Mousetrap.bind(el.getAttribute('data-shortcut'), function() { location.assign(el.getAttribute('href')); }); } }); </script> </body> </html>