Relationships Between Fuzzy Probabilistic Approximation Spaces And Their Entropy Measurement

A fuzzy probability approximation space (FPA-space) is a approximation space (A-space) where three types of uncertainty (probability, fuzziness and roughness) are combined, which is obtained by putting probability distribution into a fuzzy approximation space (FA-space). This paper studies relationships between FPA-spaces and and their entropy measurement. Two types of fuzzy relation matrices are first defined by introducing the probability into a given fuzzy relation matrix in two ways, and on this basis, they are extended to two FA-spaces. Then, equality, dependence and independence between FPA-spaces are studied. Next, the distance between FPA-spaces is discussed. Moreover, the uncertainty for an FPA-space is measured by means of information entropy. Finally, the proposed information entropy is applied in the selection of classifier systems. Since fuzzy set theory, probability theory and rough set theory are aggregated together in an FPA-space, the obtained results of this paper may be helpful for dealing with practice problems with a sort of uncertainty.