The Entropy Of Fuzzy Variables

Fuzzy variables are mathematical descriptions for fuzzy phenomena, and are de-fined as functions from a possibility space to the set of real numbers. Credibility theoryis an axiomatic fuzzy theory while entropy is an important index to measure the de-gree of uncertainty of a fuzzy variable. This dissertation, based on credibility theory,constructs the concept of entropy of fuzzy variables and discusses its properties.Fuzzy set theory, an extension of classical set theory, is used to deal with the so-called fuzzy sets whose members are ambiguous. The uncertainty of a fuzzy set istotally due to the lack of discrimination of its members. When the uncertainty of afuzzy variable is concerned, it depends on the fuzziness of its values and the amountof values that the fuzzy variable could take, which makes it different in measuring theuncertainty of a fuzzy variable from that of a fuzzy set, though fuzzy variable and fuzzyset share much in common. Combining credibility measure and Shannon function, anew concept of entropy is provided to measure the degree of uncertainty for discreteand continuous fuzzy variables. Following that, the theorem of maximum entropy andthe theorem of monotonicity are proved as well as the positivity of entropy. Then somenumerical examples are given in the sake of illustration and some special propertiesof the entropy of trapezoidal and triangular fuzzy variables are given. In addition, an-other new definition of fuzzy entropy, quadratic entropy, is proposed using credibilitymeasure and quadratic function. Moreover, the characteristics of entropy and quadraticentropy are discussed. Then the general definition of entropy for fuzzy variables isestablished. Accordingly, two ways of constructing general fuzzy entropy, multiplica-tive method and additive method, are proposed and the relations between them and thedefinitions of entropy and quadratic entropy are discussed.In conclusion, this dissertation brings forth the following new ideas in fuzzy the-ory: (a) the concepts of entropy and quadratic entropy of fuzzy variables are initiatedand their properties are investigated;(b) the constraints of fuzzy entropy are analyzedand general form of definition of fuzzy entropy is established;(c) two methods of con-structing general fuzzy entropy are provided.