Validation of fuzzy and possibilistic clustering results

Unsupervised fuzzy clustering is an important tool for finding the meaningful patterns in data sets. In fuzzy clustering analyses, the performances of clustering algorithms are mostly compared using several internal fuzzy validity indices. However, since the well-known fuzzy indices have originally been proposed for working with membership degrees produced by the traditional Fuzzy c-means Clustering (FCM) algorithm, these indices cannot be used for possibilistic algorithms that produce typicality matrices instead of fuzzy membership matrices. Even more, the variants of FCM and PCM such as Possibilistic Fuzzy C-means (PFCM) and Fuzzy Possibilistic C-means (FPCM) simultaneously result with probabilistic and possibilistic membership degrees. Thus, some kind of validity indices are needed for working with both of these results. For this purpose, a few extended and generalized validity indices has been proposed in recent years. In this paper, the performances of these indices were examined for validating the clustering results from Unsupervised Possibilistic Fuzzy Clustering (UPFC), FCM and PCM algorithms. The findings showed that generalized versions of the fuzzy validity indices based on normalization of typicality degrees can be successfully used to validate the results from PCM, UPFC and the variants of FCM and PCM.