The fuzzy c+2-means: solving the ambiguity rejection in clustering

摘要

In this paper we deal with the clustering problem whose goal consists of computing a partition of a family of patterns into disjoint classes. The method that we propose is formulated as a constrained minimization problem, whose solution depends on a fuzzy objective function in which reject options are introduced. Two types of rejection have been included: the ambiguity rejection which concerns patterns lying near the class boundaries and the distance rejection dealing with patterns that are far away from all the classes. To compute these rejections, we propose an extension of the fuzzy c-means (FcM) algorithm of Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, 1981. This algorithm is called the fuzzy c+2-means (Fc+2M). These measures allow to manage uncertainty due both to imprecise and incomplete definition of the classes. The advantages of our method are (1) the degree of membership to the reject classes for a pattern xk are learned in the iterative clustering problem; (2) it is not necessary to compute other characteristics to determine the reject and ambiguity degrees; (3) the partial ambiguity rejections introduce a discounting process between the classical FcM membership functions in order to avoid the memberships to be spread across the classes; (4) the membership functions are more immune to noise and correspond more closely to the notion of compatibility. Preliminary computational experiences on the developed algorithm are encouraging and compared favorably with results from other methods as FcM, FPcM and F(c+1)M (fuzzy c+1-means: clustering with solely distance rejection) algorithms on the same data sets. The differences in the performance can be attributed to the fact that ambiguous patterns are less accounted in for the computing of the centers.