Rough set models based on random fuzzy sets and belief function of fuzzy sets

Abstract

As knowledge blocks in databases are fuzzy and random patterns may emerge, we propose random fuzzy rough set models based on random fuzzy sets and fuzzy logic operators. We give some properties of random fuzzy rough set. We prove that the lower fuzzy measures and the upper fuzzy measures induced by lower approximation and upper approximation of fuzzy set are fuzzy belief measures and fuzzy plausibility measures, respectively, in particular, if a random fuzzy set degenerates a random fuzzy point set, then the lower fuzzy measures and upper fuzzy measures are fuzzy probability measures, and then the lower fuzzy measures and the upper fuzzy measures are exactly the infimum and supremum of the fuzzy probability measures induced by the selections of the random fuzzy set. We investigate the Gödel random fuzzy rough set model in detail, in which the implication operator is chosen as Gödel implication, and prove that a pair of fuzzy belief measure and fuzzy plausibility measure is induced essentially by the lower approximation and the upper approximation of a random fuzzy approximation space.