An approach to roughness of fuzzy sets

In rough-set-based data analysis, the roughness of a set is traditionally used to express the degree of inexactness of the set arose due to incompleteness of available knowledge. Recently, an attempt of integration between the theories of fuzzy sets and rough sets has resulted in providing a roughness measure for fuzzy sets (Banerjee and Pal, 1996). 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. We introduce 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 proved. Furthermore, we also discuss some possible extensions of the so-called approximation quality, which may be used to describe the rough approximation quality of a fuzzy classification.