<!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-Medians</h1> <p>This is a variation of k-means clustering where instead of calculating the mean for each cluster to determine its centroid we are going to calculate the median instead.</p> <p>This has the effect of minimizing error over all the clusters with respect to the Manhattan norm as opposed to the Euclidean squared norm which is minimized in K-means</p> <h3>Algorithm</h3> <p>Given an initial set of $k$ medians, the algorithm proceeds by alternating between two steps.</p> <p><strong>Assignment step</strong>: Assign each observation to the cluster whose median has the leas Manhattan distance.</p> <ul> <li>Intuitively this is finding the nearest median</li> </ul> <p><strong>Update Step</strong>: Calculate the new medians to be the centroids of the observations in the new clusters</p> <p>The algorithm is known to have converged when assignments no longer change. There is no guarantee that the optimum is found using this algorithm. </p> <p>The result depends on the initial clusters. It is common to run this multiple times with different starting conditions.</p> </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>