Comparing soft clusters and partitions

Previously, we presented a method for comparing soft partitions (i.e. crisp, probabilistic, fuzzy and possibilistic) to a known crisp reference partition. Many of the classical indices that have been used with outputs of crisp clustering algorithms were generalized so that they are applicable for candidate partitions of any type. In particular, focus was placed on generalizations of the Rand index. In this article, we extend our prior work by (1) investigating the behavior of the soft Rand for comparing non-crisp, specifically possibilistic, partitions and (2) we demonstrate how the possibilistic Rand and visual assessment of (cluster) tendency (VAT) algorithm can be used to discover the number of actual clusters and coincident clusters for outputs from the possibilistic c-means (PCM) algorithm.