Fuzzy Joint Entropy And Fuzzy Conditional Entropy Based On Credibility Distributions

Fuzzy entropy is a measure of uncertainty,many definitions of which have been presented in previous literature.Those definitions of entropy characterize the uncertainty resulting primarily from the linguistic vagueness,and the entropy vanishes when the fuzzy variable is an equipossible one.Credibility theory is a branch of mathematics for studying the behavior of fuzzy phenomena.The fuzzy entropy on the basis of credibility theory is defined from an information theoretical view.It meets the requirements of minimum,maximum and universality,and characterizes the uncertainty resulting from information deficiency which is caused by the impossibility to predict the specified value that a fuzzy variable takes.The purpose of this paper is to study the joint entropy for fuzzy vectors and the conditional entropy for a fuzzy variable given another.The paper is organized as follows.Chapter 1 introduces the basic concepts of fuzziness and fuzzy sets,and reviews the research progress of fuzzy measures and fuzzy entropy.Chapter 2 presents some concepts and theorems in credibility theory.Chapter 3 provides the definitions of joint entropy for both 2-dimensional discrete and continuous fuzzy vectors.The minimality,maximality and monotonicity of the joint entropy,as well as the expansibility and strict concavity of the joint entropy for 2-dimensional discrete fuzzy vectors,are investigated.It is also pointed in this chapter that the definitions and properties of the joint entropy for 2-dimensional fuzzy vectors can be expanded to the case of n-dimensional ones.In order to measure the uncertainty of a fuzzy variable given another,a definition of conditional entropy for discrete fuzzy variables is formulated in Chapter 4.The minimality and maximality of the conditional entropy are investigated.The sufficient and necessary condition for the maximum that the conditional entropy for independent simple variables takes is obtained.Chapter 5 gives a summary of this paper and shows the prospects in the further study of fuzzy entropy based on credibility distributions.