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Measures of uncertainty for a fuzzy probabilistic information system

Fuzzy probabilistic information system (FPIS), a combination of some fuzzy relations on the same universe which satisfies probability distribution, can be regarded as an information system (IS) with fuzzy relations under the probability environment. This means that three types of uncertainty: roughness fuzziness and probability are integrated into an IS. This paper aims to measure the uncertainty of a FPIS. Firstly, information granulation in a FPIS is proposed. Then, information entropy and rough entropy in a FPIS are proposed. Moreover, information amount in a FPIS is also investigated. Finally, the feasibility of the proposed measures are verified by a numerical example, and the statistical analysis is carried out. Those obtained results may be useful for understanding the essence of uncertainty for a FPIS.