Uncertainty Measures For Set-valued Information Systems

Set-valued information systems are generalized models of single-valued information systems, it is an important model for dealing with incomplete and uncertain data. Uncertainty is critical focus problem in recent years. In this paper, the uncertainty measures in classical set-valued information systems and fuzzy set-valued information systems are studied, more details are as follows:As for classical set-valued information systems, in this paper, the uncertainty measures of set-valued information systems from the viewpoint of algebra and information are studied. First, a pair of approximation operations called a-lower and upper approximation are constructed and the properties are investigated. Then, based on the pair of approximation operations, accuracy and roughness in set-valued information systems are defined and the properties are investigated, accuracy and roughness can measure uncertainty for set-valued information systems are proved. Finally, pay attention to different sets of universe U with the same a-lower and upper approximation the powerset of U can be classified. Then, entropy and co-entropy are defined and the properties are investigated, entropy and co-entropy also can measure uncertainty for set-valued information systems are proved.As for set-valued information systems, in this paper, a fuzzy tolerance relation (a-fuzzy tolerance relation) for set-valued information systems is defined and the properties are inves-tigated. Two fuzzy partial-order relations on fuzzy tolerance granular structure are proposed, the relationship between them is derived. Fuzzy entropy measures and fuzzy granularity mea-sures of set-valued information systems are introduced based on the a-fuzzy tolerance relation, take advantage of two fuzzy partial-order relations on fuzzy tolerance granular structure, the properties are investigated. Finally, the relationship between fuzzy granularity, fuzzy granular-ity measures, fuzzy information entropy and fuzzy rough entropy are considered. These results are effective and suitable for measuring uncertainty for set-valued information systems with the a-fuzzy tolerance relation.