<p>Recall in the previous section that we spoke about Monothetic and Polythetic methods. Monothetic methods only looks at a single variable at a time while Polythetic looks at multiple variables simultaneously. In this section, we will speak more about polythetic divisive methods.</p>
<h2>Polythetic Divisive Methods</h2>
<p>Polythetic methods operate via a distance matrix.</p>
<p>This procedure avoids considering all possible splits by </p>
<ol>
<li>Finding the object that is furthest away from the others within a group and using that as a seed for a splinter group.</li>
<li>Each object is then considered for entry to that separate splinter group: any that are closer to the splinter group than the main group is moved to the splinter one. </li>
<li>The step is then repeated.</li>
</ol>
<p>This process has been developed into a program named <code>DIANA</code> (DIvisive ANAlysis Clustering) which is implemented in <code>R</code>.</p>
<h3>Similarities to Politics</h3>
<p>This somewhat resembles a way a political party might split due to inner conflicts.</p>
<p>Firstly, the most discontented member leaves the party and starts a new one, and then some others follow him until a kind of equilibrium is attained.</p>
<h2>Methods for Large Data Sets</h2>
<p>There are two common hierarchical methods used for large data sets <code>BIRCH</code> and <code>CURE</code>. Both of these algorithms employ a pre-clustering phase in where dense regions are summarized, the summaries being then clustered using a hierarchical method based on centroids.</p>
<h3>CURE</h3>
<ol>
<li><code>CURE</code> starts with a random sample of points and represents clusters by a smaller number of points that capture the shape of the cluster</li>
<li>Which are then shrunk towards the centroid as to dampen the effect of the outliers</li>
<li>Hierarchical clustering then operates on the representative points</li>
</ol>
<p><code>CURE</code> has been shown to be able to cope with arbitrary-shaped clusters and in that respect may be superior to <code>BIRCH</code>, although it does require judgment as to the number of clusters and also a parameter which favors either more or less compact clusters.</p>
<h2>Revisiting Topics: Cluster Dissimilarity</h2>
<p>In order to decide where clusters should be combined (for agglomerative), or where a cluster should be split (for divisive), a measure of dissimilarity between sets of observations is required.</p>
<p>In most methods of hierarchical clustering this is achieved by a use of an appropriate </p>
<ul>
<li>Metric (a measure of distance between pairs of observations)</li>
<li>Linkage Criterion (which specifies the dissimilarities of sets as functions of pairwise distances observations in the sets)</li>
</ul>
<h2>Advantages of Hierarchical Clustering</h2>
<ul>
<li>Any valid measure of distance measure can be used</li>
<li>In most cases, the observations themselves are not required, just hte matrix of distances
<ul>
<li>This can have the advantage of only having to store a distance matrix in memory as opposed to a n-dimensional matrix.</li>
