Research On Distance And Similarity Measures Between Uncertain Variables

There exists a great deal of uncertainties such as randomness, fuzziness and fuzzyrandomness in the fields of management sciences, computer sciences, sys-tern sciences, information sciences, and engineering applications, etc. Distances between fuzzy sets is a important topic for fuzzy set theory. Various distances between fuzzy sets were presented to reflect the difference level of fuzzy sets in the last decades. Similarity measures between fuzzy sets can be regarded as a dual concept of distance measures reflecting] nearness of fuzzy sets. Distance measures and similarity measures between uncertain variables have been widely studied and applied in a variety of areas such as pattern recognition, machine learning, decision making and market prediction etc.Concepts of distance and similarity of fuzzy sets given before are based on membership function on possibility measure, so they do not satisfy identification. In order to overcome this shortage, several definitions of distance measures and similarity measures between uncertain variables are put forward and correspond-ing proofs are given. Integral and differential of fuzzy processes based on new distances are discussed. The contents are described as follows:A new kind of distances between fuzzy variables, fuzzy random variables, random fuzzy variables and birandom variables are proposed and these distances fully satisfy the mathematical axioms of a distance metric. Furthermore, metric spaces of different uncertain variables are defined, the completeness of this space is proved and the properties of new distances are discussed. Finally, the distances between fuzzy vectors, fuzzy random vectors, random fuzzy vectors and birandom vectors are also given.The concept of continuity for fuzzy processes is proposed with our new dis-tances defined by the expected value operator and dealt with in the framework of credibility theory. Some properties of continuity for fuzzy processes are also proven. From this, the integral and differential of fuzzy processes are defined and their properties are discussed.The definition of the degree of similarity between fuzzy variables is intro-duced to reflect nearness of fuzzy variables] Four similarity measures between fuzzy variables are proposed and corresponding proofs are given. Similarity mea-sures between fuzzy vectors and their calculational methods are also presented. The proposed similarity measures are applied to pattern recognition.The definition of the degree of similarity between hybrid variables is in troduced based on chance theory. Several similarity measures between hybrid variables are proposed and corresponding proofs are given. Similarity measures between hybrid vectors are also discussed. Finally, a numerical example is given to illustrate the application of the proposed similarity measure.