Optimal Number of Classes in Fuzzy Partitions

This paper proposes a cluster validation procedure allowing to obtain the optimal number of clusters on a set of fuzzy partitions. Such a procedure is established considering fuzzy classification systems endowed with a dissimilarity function that, in turn, generates a dissimilarity matrix. Establishing a dissimilarity matrix for the case of a crisp partition, we propose an optimization problem comparing the characteristic polynomials of the fuzzy partition and crisp partition. Based on the above, we propose a definition for the optimal number of fuzzy classes in a fuzzy partition. Our approach is illustrated through an example on image analysis by the fuzzy c-means algorithm.