I am using Silhouette cluster validation for each repetition (for a specific K) of k-means, k-modes and k-medoids.
All the definitions of Silhouette I see calculate the distance of each point to others points within the same cluster, then compare it to the distance of the same point with points in other clusters, in turn.
From wikipedia:
$ a(i) = $ average distance between point $i$ and other points in the same cluster
$ b(i) = $ average distance between point $i$ and all points in the closest cluster
$ s(i) = \frac{b(i) - a(i)}{\max\{a(i),b(i)\}} $
I would expect similar results by, instead, comparing each point with its own centroid, and then the centroids of other clusters, in turn.
Is this a good approach, or do I risk losing information? It's much more efficient computationally (with my dataset, it's a concern).
a
andb
in it. So,a
could be distance to the centroid of the home cluster andb
be the distance to the closest foreign centroid. I've implemented this and some other versions for SPSS (see Clustering criterions on my web page). One should keep in mind, thought, that different versions of Silhouette criterion cannot be compared to each other numerically. $\endgroup$