A novel fuzzy clustering algorithm based on a fuzzy scatter matrix with optimality tests

Most clustering algorithms are based on a within-cluster scatter matrix with a compactness measure. In this paper we propose a novel fuzzy clustering algorithm, called the fuzzy compactness and separation (FCS), based on a fuzzy scatter matrix in which the FCS algorithm is derived using compactness measure minimization and separation measure maximization. The compactness is measured using a fuzzy within-cluster variation. The separation is measured using a fuzzy between-cluster variation. The proposed FCS objective function is a modification of the FS validity index proposed by Fukuyama and Sugeno and also a generalization of the fuzzy c-means (FCM). The FCS algorithm assigns a crisp boundary (cluster kernel) for each cluster such that hard memberships and fuzzy memberships can co-exist in the clustering results. Thus, FCS can be seen as a clustering algorithm with a novel sense between the hard c-means and fuzzy c-means. The FCS optimality tests and parameter selection are also investigated. Some numerical examples are demonstrated to show its robust properties and effectiveness.