Uncertainty Measurement Of Fuzzy Relations

Relationship is an important concept to describe data,which is widely used in the field of artificial intelligence.Entropy is an important method to measure the uncertainty of relation,which is paid more and more attention by scholars and widely used.In practical application,most of the data are fuzzy relations rather than clear equivalent relations,so it is necessary to study the entropy of fuzzy relations.At present,there are some researches on fuzzy equivalence relation,clear equivalence relation,fuzzy similarity relation and uncertainty measurement of other relations.However,the study on the uncertainty measurement of fuzzy relation especially fuzzy binary relation is not enough.Therefore,it is necessary to study the entropy of fuzzy relation and then extend the entropy to fuzzy information system.In this paper,a new definition of entropy based on Shannon entropy is given from the perspective of relation,and a new definition of entropy derived from entropy is also presented to calculate the information of fuzzy and indistinguishable relation.Based on the new definition,some related definitions and properties of Shannon entropy are extended in fuzzy binary relation.The definitions of joint entropy,conditional entropy and mutual information correlation are given,and their related properties are discussed.Furthermore,the related definitions are extended to fuzzy information systems.The related definitions of joint entropy,conditional entropy and mutual information in fuzzy systems are given,and their related properties are discussed.At the same time,the attribute reduction based on entropy is defined.Finally,the concepts of relative entropy,relative joint entropy and relative conditional entropy are defined,and the related properties are discussed.