On cluster validity for the fuzzy c-means model

Many functionals have been proposed for validation of partitions of object data produced by the fuzzy c-means (FCM) clustering algorithm. We examine the role a subtle but important parameter-the weighting exponent m of the FCM model-plays in determining the validity of FCM partitions. The functionals considered are the partition coefficient and entropy indexes of Bezdek, the Xie-Beni, and extended Xie-Beni indexes, and the FukuyamaSugeno index. Limit analysis indicates, and numerical experiments confirm, that the FukuyamaSugeno index is sensitive to both high and low values of m and may be unreliable because of this. Of the indexes tested, the Xie-Beni index provided the best response over a wide range of choices for the number of clusters, (%lo), and for m from 1.01-7. Finally, our calculations suggest that the best choice for m is probably in the interval [U, 2.51, whose mean and midpoint, m = 2, have often been the preferred choice for many users of FCM.