Xingxing He

Some new approaches to constructing similarity measures

The purpose of this paper is threefold. Firstly, based on the concept of fuzzy equivalence given by Fodor and Roubens, we propose different methods for the construction of fuzzy equivalencies from biresiduations, automorphisms, fuzzy negations, t-norms and t-conorms. Secondly, we give two manners of generating similarity measures of P.Z. Wang from fuzzy equivalencies. Finally, some properties of the two kinds of similarity measures are discussed and compared.

Robustness of fuzzy connectives and fuzzy reasoning

Abstract In fuzzy control, practical fuzzy reasoning schemes are likely to be perturbed by various types of noise, and thus analysis of the stability and robustness of fuzzy reasoning are important issue. We used a concept similar to the modulus of continuity to characterize the robustness of fuzzy connectives and present robustness results for various fuzzy connectives. We investigated the robustness of fuzzy reasoning from the perspective of perturbation of membership functions. We propose a method for judging the most robust elements of different classes of fuzzy connectives. The results obtained are compared with previous findings in the literature.

Dissimilarity functions and divergence measures between fuzzy sets

In this paper we propose two approaches to constructing divergence measures. The construction is based on the use of dissimilarity functions and fuzzy equivalencies. Firstly, we introduce some ways of generating dissimilarity functions. Then, we present several formulae of divergence measures. Finally, we examine the properties of divergence measures as a whole.

Relations among similarity measure, subsethood measure and fuzzy entropy

In this paper we study the relations among similarity measure, subsethood measure and fuzzy entropy and present several propositions that similarity measure, subsethood measure and fuzzy entropy can be transformed by each other based on their axiomatic definitions. Some new formulae to calculate similarity measure, subsethood measure and fuzzy entropy are proposed.

Properties of Raha's similarity-based approximate reasoning method

A similarity-based approximate reasoning methodology that requires the construction of fuzzy relations was proposed by Raha et al. [26]. This paper investigates properties of four inferred conclusions calculated by Rahas similarity-based approximate reasoning method. The relationships among the four inferred conclusions are examined. The monotonicity and the approximation property for similarity-based approximate reasoning methods are defined. Then the monotonicity and the approximation property of Rahas similarity-based approximate reasoning method are studied.

On the associative property of fuzzy Xor connectives

The connective fuzzy Xor has been studied by Bedregal et al. [B.C. Bedregal, R.H.S. Reiser and G.P. Dimuro, Xor-Implications and E-Implications: Classes of fuzzy implications based on fuzzy Xor, Electronic Notes in Theoretical Computer Science 247 2009 5--18] and two methods for constructing fuzzy Xor are given. As an important property of classical Xor connective, associativity is also defined in the definition of fuzzy Xor. However, we find that some of the fuzzy Xor connectives constructed with the given methods do not satisfy associative property. This paper studies the conditions under which the fuzzy Xor constructed with the methods given in [1] satisfy associative property.

Robustness of fuzzy connectives and fuzzy reasoning with respect to general divergence measures

Abstract This paper discusses the robustness of fuzzy connectives and fuzzy reasoning with respect to general divergence measures. First of all, the concept of DF-metric is proposed. Secondly, several DF-metrics are introduced as well as their properties and some inequalities about them. Then a formula of divergence measure composed by DF-metric is presented. Finally, based on the proposed divergence measures, the concept of perturbations of fuzzy sets is extended. According to the extended concept, the perturbation parameters raised by various fuzzy connectives are studied and the perturbations of fuzzy reasoning are also investigated.

On the TL-transitivity of fuzzy similarity measures

Abstract In fuzzy neural networks, it is usual to perform rule matching by employing fuzzy similarity measures. A fuzzy similarity measure is reserved for a binary fuzzy relation. In fuzzy relational calculus, the notion of T-transitivity is indispensable because it is strongly related to the concept of a metric. In this study, we consider whether the two types of fuzzy similarity measures given by Li et al. (2014) [24] satisfy T L -transitivity. We also investigate the T L -transitivity of fuzzy equivalencies.

Similarity measures of interval-valued fuzzy sets

Interval-valued fuzzy set (briefly, IVFS) is a generalization of fuzzy set that may better model imperfect information. Similarity measure is a useful tool for determining the similarity degree between two fuzzy concepts. In this paper, the definition of similarity measure of IVFSs is presented. Three approaches to constructing similarity measures of IVFSs are proposed. Based on the approaches, three computation formulae for calculating the similarity degree between IVFSs are obtained. The properties of similarity measures of IVFSs are examined as a whole.

Three constructive methods for the definition of interval-valued fuzzy equivalencies

Abstract In this paper, we introduce the concept of an interval-valued fuzzy equivalence as an interval extension of the fuzzy equivalence. The partial orders between fuzzy equivalencies are studied. Three ways of constructing interval-valued fuzzy equivalencies from given fuzzy equivalencies and aggregation functions are provided. Some examples of interval-valued fuzzy equivalencies built by the proposed constructive methods are shown. The properties and structures of interval-valued fuzzy equivalencies are also studied.