Probabilistic rough sets characterized by fuzzy sets

Abstract

In this paper, fuzziness in probabilistic rough set is studied by fuzzy sets. we show that the variable precision approximation of a probabilistic rough set can be generalized from the vantage point of the cuts of a fuzzy set which is determined by the rough membership function. As a result, the fuzzy set can be used conveniently to describe the feature of rough set. Moreover we give a measure of fuzziness, fuzzy entropy, induced by roughness in a probabilistic rough set and make some characterizations of this measure. For three well-known entropy functions, we show that the finer the information granulation is, the less the fuzziness in a rough set. The superiority of fuzzy entropy to Pawlak’s accuracy measure is illustrated with examples. Finally, the fuzzy entropy of a rough classification is defined by the fuzzy entropy of corresponding rough sets, and show that one possible application of it is to measure the inconsistency in a decision table.