A roughness measure for fuzzy sets

Recently, an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets [M. Banerjee, S.K. Pal, Roughness of a fuzzy set, Information Sciences 93 (1996) 235-246]. Essentially, Banerjee and Pal's roughness measure depends on parameters that are designed as thresholds of definiteness and possibility in membership of the objects to a fuzzy set. In this paper we first remark that this measure of roughness has several undesirable properties, and then propose a parameter-free roughness measure for fuzzy sets based on the notion of the mass assignment of a fuzzy set. Several interesting properties of this new measure are examined. Furthermore, we also discuss how the proposed approach is used to describe the rough approximation quality of a fuzzy classification.