Theoretical analysis of classic and capacity constrained fuzzy clustering

Abstract

In this paper we present a theoretical analysis on fuzzy centroid-based clustering methods. In addition to the formulation on the classical approaches, we consider constraints that may be useful in some practical applications, such as restrictions on the number of points in each group, and methods that deal with these constraints. We propose a more general formulation to the constrained clustering problem, where each point has an associated weight, and the sum of the weights of the points that compose each group is established a priori. For both classical and proposed approaches we discuss existence and uniqueness of solutions of the involved problems, providing mathematical foundations for the established formulas. Preliminary numerical experiments, performed by means of two-dimensional examples, are also presented.