Similarity Measures On Fuzzy Reasoning

Fuzzy set theory is a mathematical tool to deal with uncertain problems. It has broad applications in the field of control, decision-making, reasoning and so on. This dissertation is devoted to some related research to similarity measures of fuzzy sets. We present some approaches to constructing similarity measures of fuzzy sets. We examine relationships among similarity measures, divergence measures, subsethood measures and fuzzy entropies of fuzzy sets. We also examine how similarity measures of fuzzy sets are used in fuzzy reasoning.The first part of the dissertation is the part of theory. We present some approaches to constructing similarity measures of fuzzy sets. There exist many similarity measures of fuzzy sets in the literature. However, there exists no uniform computational formula for similarity measures of fuzzy sets and it is therefore hard to make comparisons between different similarity measures of fuzzy sets. Thus it is significance to find a uniform computational formula for similarity measures of fuzzy sets. The following results are obtained in this part.In Chapter 3 we propose different methods for the construction of fuzzy equivalencies from biresiduations, automorphisms, fuzzy negations, t-norms and t-conorms. We give two manners of generating similarity measures of fuzzy sets from fuzzy equivalencies. One is by aggregating fuzzy equivalencies. The other is by reconstructing two formulae of fuzzy equivalencies. They are general in the sense that by using different fuzzy equivalencies one gets the distance and the set-theoretic, as well as the implications based similarity measures designed in the literature. We discuss and compare different properties of the two kinds of similarity measures.In Chapter 4 we propose a definition of similarity measure of Interval-valued fuzzy sets (IVFSs, for short). In the proposed definition, the reflexivity, commutativity and transitivity properties are demanded. Then we present three approaches to constructing similarity measures of IVFSs. Based on the proposed approaches, three computation formulae for calculating the similarity degree between IVFSs are obtained. Two of them are constructed by similarity measures of fuzzy sets. The last one is obtained by combining two similarity measures of IVFSs. It is shown that most of similarity measures given in the literature can be constructed by our proposed formulae. Finally, we discuss properties of similarity measures of IVFSs as a whole.Since dissimilarity function plays an important role in constructing divergence measures, our initial aim in Chapter 5 is to present some ways of generating dissimilarity functions. Then we propose two approaches to constructing divergence measure in the sense of Liu and that in the sense of Montes. The construction is based on the use of dissimilarity functions and fuzzy equivalencies. We consider the form of divergence measure not only in the case where the universal set is finite but also in the case where the universal set is a measure space. We examine the properties of divergence measures as a whole. Meanwhile, we propose several propositions to show that similarity measures, subsethood measures and fuzzy entropies can be transformed by each other based on their axiomatic definitions. Some new formulae to calculate similarity measures, subsethood measures and fuzzy entropies are also given.We next turn to the application part. For one thing, we study similarity-based approximate reasoning method. For another thing, we study robustness of fuzzy reasoning with respect to general divergence measures.The similarity-based approximate reasoning method is a fuzzy reasoning method that different from the Compositional Rule of Inference (CRI, for short) method. In Chapter 6 we define the monotonicity and the approximation property for similarity-based approximate reasoning method. The monotonicity and the approximation property of Raha’s similarity-based approximate reasoning method are studied and compared.In fuzzy control, it is reasonable to require that fuzzy reasoning results should not change much for a slight change in inputs. The robustness of fuzzy reasoning has become a particular focus in fuzzy research. In Chapter 7 we study robustness of fuzzy connectives and fuzzy reasoning respect to Chebyshev distance. We use a concept similar to the modulus of continuity to characterize the robustness of fuzzy connectives. We present robustness results for various fuzzy connectives. We transform the problem of robustness of fuzzy reasoning to the robustness of the corresponding fuzzy connectives. We show that the robustness of fuzzy reasoning is directly linked to the selection of t-norms and fuzzy implications. We also propose a method for judging the most robust elements of different classes of fuzzy connectives.In Chapter 8 we present the concept of DF-metric and introduce some new DF-metrics. We analyze properties of DF-metrics through several inequalities about them. Since DF-metrics are special dissimilarity functions, we use them to construct divergence measures. Based on the proposed divergence measures, we obtain an extension of the concept of perturbations of fuzzy sets. According to the new concept, the perturbation parameters raised by various fuzzy connectives are studied and the perturbations of fuzzy reasoning are also discussed.