This is a followup to this question. I am currently trying to implement the C-Index in order to find a near-optimal number of clusters from a hierarchy of clusters. I do this by calculating the C-Index for every step of the (agglomerative) hierarchical clustering. The problem is that the C-Index is minimal (0 to be exact) for very degenerated clusterings. Consider this:

$c = \frac{S-S_{min}}{S_{max}-S_{min}}$

In this case $S$ is the sum of all distances between pairs of observations in the same cluster over all clusters. Let $n$ be the number of these pairs. $S_{min}$ and $S_{max}$ are the sums of $n$ lowest/highest distances across all pairs of observations. In the first step of the hierarchical clustering, the two closest observations (minimal distance) are merged into a cluster. Let $d$ be the distance between these observations. Now there is one pair of observations in the same cluster, so $n=1$ (all other clusters are singletons). Consequently $S=d$. The problem is that $S_{min}$ also equals $d$, because $d$ is the smallest distance (that is why the observations where merged first). So for this case, the C-Index is always 0. It stays 0 as long as only singleton clusters are merged. This means the optimal clustering according the C-Index would always consist of a bunch of clusters containing two observations, and the rest singletons. Does this mean that the C-Index is not applicable to hierarchical clustering? Am I doing something wrong? I have searched a lot, but could not find any suitable explanation. Can someone refer me to some resource that is freely available on the internet? Or, if not, at least a book I may try to get at my universities library?

Thanks in advance!

  • $\begingroup$ Your observation is correct, but it is all fine with C-index. C-index is 0 when the observed clustering solution does not differ from the theoretically "ideal" best one under the given (observed) number of within-cluster distances. Consider a dataset which all consists of tight pairs of objects, and the pairs are quite far apart. Hierarchical clustering under virtually any linkage method will first - on initial steps - "collect" the objects into these pairs. And all that time C-index will remain 0. Later, the clustering will come to merge between the apart pairs: C-index will abrubtly worsen. $\endgroup$
    – ttnphns
    May 8 '18 at 9:49
  • $\begingroup$ Algorithm to compute C-index is shown here stats.stackexchange.com/q/343878/3277. $\endgroup$
    – ttnphns
    May 8 '18 at 9:50
  • $\begingroup$ P.S. Don't forget that C-Index is the lower (closer to 0) is the better! $\endgroup$
    – ttnphns
    May 8 '18 at 10:39

This may be one of the cases where there's more art than science to clustering. I would suggest that you let your clustering algorithm run for a short time before letting the C-Index calculations kick in. "Short time" may be after processing a few pairs, just when it starts to exceed 0, or some other heuristic. (After all you don't expect to stop at 1 or 2 clusters, otherwise a different separation algorithm may have been deployed.)

For a book recommendation, I can suggest:

You can scan/search the available contents on google books to see if it might meet your needs. It's worked as a reference for me in the past.

  • $\begingroup$ Oops, you're using agglomerative methods, so the "1 or 2 clusters" part doesn't make sense -- the "inverse" applies, you don't want n-1 or n-2 singletons, etc, i.e. letting clustering work for a bit before applying validity criteria shouldn't be problematic. $\endgroup$
    – ars
    Sep 14 '10 at 1:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.