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    <p>Well there are two practical solutions to the the problem of intelligent selection of the number of centroids (k) in common use.</p> <p>The first is to <strong>PCA</strong> your data, and the output from PCA--which is the principal components (eigenvectors) and their cumulate contribution to the variation observed in the data--obviously suggests an optimal number of centroids. (E.g., if 95% of the variability in your data is explained by the first three principal components, then k=3 is a wise choice for k-means.)</p> <p>The second commonly used practical solution to intelligently estimate k is is a revised implementation of the k-means algorithm, called <strong>k-means++</strong>. In essence, k-means++ just differs from the original k-means by the additional of a pre-processing step. During this step, the number and initial position of the centroids and estimated. </p> <p>The algorithm that k-means++ relies on to do this is straightforward to understand and to implement in code. A good source for both is a 2007 <a href="http://lingpipe-blog.com/2009/03/23/arthur-vassilvitskii-2007-kmeans-the-advantages-of-careful-seeding/" rel="nofollow">Post</a> in the <em>LingPipe Blog</em>, which offers an excellent explanation of k-means++ as well as includes a citation to the original paper that first introduced this technique.</p> <p>Aside from providing an optimal choice for k, k-means++ is apparently superior to the original k-means in both performance (roughly 1/2 processing time compared with k-means in one published comparison) and accuracy (three orders of magnitude improvement in error in the same comparison study).</p>
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