Figure 17.11 shows the first three steps of a centroid clustering. The first two iterations form the clusters with centroid and with centroid because the pairs and have the highest centroid similarities. In the third iteration, the highest centroid similarity is between and producing the cluster with centroid .
Like GAAC, centroid clustering is not best-merge persistent and therefore (Exercise 17.10 ).
In contrast to the other three HAC algorithms, centroid clustering is not monotonic. So-called inversions can occur: Similarity can increase during clustering as in the example in Figure 17.12 , where we define similarity as negative distance. In the first merge, the similarity of and is . In the second merge, the similarity of the centroid of and (the circle) and is . This is an example of an inversion: similarity increases in this sequence of two clustering steps. In a monotonic HAC algorithm, similarity is monotonically decreasing from iteration to iteration.
Increasing similarity in a series of HAC clustering steps contradicts the fundamental assumption that small clusters are more coherent than large clusters. An inversion in a dendrogram shows up as a horizontal merge line that is lower than the previous merge line. All merge lines in and 17.5 are higher than their predecessors because single-link and complete-link clustering are monotonic clustering algorithms.
Despite its non-monotonicity, centroid clustering is often used because its similarity measure - the similarity of two centroids - is conceptually simpler than the average of all pairwise similarities in GAAC. Figure 17.11 is all one needs to understand centroid clustering. There is no equally simple graph that would explain how GAAC works.