Validity index for crisp and fuzzy clusters

In this article, a cluster validity index and its fuzzification is described, which can provide a measure of goodness of clustering on different partitions of a data set. The maximum value of this index, called the PBM-index, across the hierarchy provides the best partitioning. The index is defined as a product of three factors, maximization of which ensures the formation of a small number of compact clusters with large separation between at least two clusters. We have used both the k-means and the expectation maximization algorithms as underlying crisp clustering techniques. For fuzzy clustering, we have utilized the well-known fuzzy c-means algorithm. Results demonstrating the superiority of the PBM-index in appropriately determining the number of clusters, as compared to three other well-known measures, the Davies–Bouldin index, Dunn's index and the Xie–Beni index, are provided for several artificial and real-life data sets.