mirror of
https://github.com/Brandon-Rozek/website.git
synced 2024-12-23 08:42:49 +00:00
103 lines
4.4 KiB
HTML
103 lines
4.4 KiB
HTML
<!DOCTYPE html>
|
|
<html>
|
|
<head>
|
|
<meta charset="utf-8" />
|
|
<meta name="author" content="Fredrik Danielsson, http://lostkeys.se">
|
|
<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>Cluster Tendency</h1>
|
|
<p>This is the assessment of the suitability of clustering. Cluster Tendency determines whether the data has any inherent grouping structure.</p>
|
|
<p>This is a hard task since there are so many different definitions of clusters (portioning, hierarchical, density, graph, etc.) Even after fixing a cluster type, this is still hard in defining an appropriate null model for a data set.</p>
|
|
<p>One way we can go about measuring cluster tendency is to compare the data against random data. On average, random data should not contain clusters.</p>
|
|
<p>There are some clusterability assessment methods such as Spatial histogram, distance distribution and Hopkins statistic.</p>
|
|
<h2>Hopkins Statistic</h2>
|
|
<p>Let $X$ be the set of $n$ data points in $d$ dimensional space. Consider a random sample (without replacement) of $m << n$ data points. Also generate a set $Y$ of $m$ uniformly randomly distributed data points.</p>
|
|
<p>Now define two distance measures $u_i$ to be the distance of $y_i \in Y$ from its nearest neighbor in X and $w_i$ to be the distance of $x_i \in X$ from its nearest neighbor in X</p>
|
|
<p>We can then define Hopkins statistic as
|
|
$$
|
|
H = \frac{\sum_{i = 1}^m{u<em>i^d}}{\sum</em>{i = 1}^m{u<em>i^d} + \sum</em>{i =1}^m{w_i^d}}
|
|
$$</p>
|
|
<h3>Properties</h3>
|
|
<p>With this definition, uniform random data should tend to have values near 0.5, and clustered data should tend to have values nearer to 1.</p>
|
|
<h3>Drawbacks</h3>
|
|
<p>However, data containing a single Gaussian will also score close to one. As this statistic measures deviation from a uniform distribution. Making this statistic less useful in application as real data is usually not remotely uniform.</p>
|
|
<h2>Spatial Histogram Approach</h2>
|
|
<p>For this method, I'm not too sure how this works, but here are some key points I found.</p>
|
|
<p>Divide each dimension in equal width bins, and count how many points lie in each of the bins and obtain the empirical joint probability mass function.</p>
|
|
<p>Do the same for the randomly sampled data</p>
|
|
<p>Finally compute how much they differ using the Kullback-Leibler (KL) divergence value. If it differs greatly than we can say that the data is clusterable.</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>
|