The intuitionistic fuzzy set (IFS) theory, originated by Atanassov [K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96], has been used in a wide range of applications, such as logic programming, medical diagnosis, pattern recognition, and decision making, etc. However, so far there has been little investigation of the clustering techniques of IFSs. In this paper, we define the concepts of association matrix and equivalent association matrix, and introduce some methods for calculating the association coefficients of IFSs. Then, we propose a clustering algorithm for IFSs. The algorithm uses the association coefficients of IFSs to construct an association matrix, and utilizes a procedure to transform it into an equivalent association matrix. The -cutting matrix of the equivalent association matrix is used to cluster the given IFSs. Moreover, we extend the algorithm to cluster interval-valued intuitionistic fuzzy sets (IVIFSs), and finally, demonstrate the effectiveness of our clustering algorithm by experimental results.
A geometrical representation of an intuitionistic fuzzy set is a point of departure for our proposal of distances between intuitionistic fuzzy sets. New definitions are introduced and compared with the approach used for fuzzy sets. It is shown that all three parameters describing intuitionistic fuzzy sets should be taken into account while calculating those distances. (C) 2000 Elsevier Science B.V. All rights reserved.
A non-probabilistic-type entropy measure for intuitionistic fuzzy sets is proposed. It is a result of a geometric interpretation of intuitionistic fuzzy sets and uses a ratio of distances between them proposed in Szmidt and Kacprzyk (to appear). It is also shown that the proposed measure can be defined in terms of the ratio of intuitionistic fuzzy cardinalities: of F intersection F and F union F .
Characterization of dissimilarity/divergence between intuitionistic fuzzy sets (IFSs) is important as it has applications in different areas including image segmentation and decision making. This study deals with the problem of comparison of intuitionistic fuzzy sets. An axiomatic definition of divergence measures for IFSs is presented, which are particular cases of dissimilarities between IFSs. The relationships among IF-divergences, IF-dissimilarities, and IF-distances are studied. Finally, we propose a very general framework for comparison of IFSs, where depending on the conditions imposed on a particular function, we can realize measures of distance, dissimilarity, and divergence for IFSs. Some methods for building divergence measures for IFSs are also introduced, as well as some examples of IF-divergences. In particular, we have proved some results that can be used to generate measures of divergence for fuzzy sets as well as for intuitionistic fuzzy sets.
In this work, considering the information carried by the membership degree and the non-membership degree in Atanassov’s intuitionistic fuzzy sets (IFSs) as a vector representation with the two elements, a cosine similarity measure and a weighted cosine similarity measure between IFSs are proposed based on the concept of the cosine similarity measure for fuzzy sets. To demonstrate the efficiency of the proposed cosine similarity measures, the existing similarity measures between IFSs are compared with the cosine similarity measure between IFSs by numerical examples. Finally, the cosine similarity measures are applied to pattern recognition and medical diagnosis.
A measure of knowledge is often viewed as a dual measure of entropy in a fuzzy system; thus, it appears that the less entropy may always accompany the greater amount of knowledge. Actually, this does not reflect the reality in the context of Atanassov's intuitionistic fuzzy sets (A-IFSs). In this paper, we introduce a novel axiomatic framework for measuring the amount of knowledge associated with A-IFSs, as opposed to a measure of fuzzy entropy. We present an axiomatic definition of knowledge measure for A-IFSs first and then develop a new robust model that strictly complies with these axioms. More efforts are made to form the main properties of two types of axioms (respectively, for fuzzy entropy and knowledge measure) into a unified framework, under which the numerical relationship between these two kinds of measures is discussed in considerable detail. This helps to clear up a fundamental misunderstanding aforementioned and ultimately to draw a firm conclusion on this topic. In particular, the developed model, for its excellent performance in experiments as well as ability to capture the unique features of A-IFSs, can be used to tackle some special problems that are difficult to handle by using fuzzy entropy alone, such as making a difference between such special cases in which there are a large number of arguments in favor but an equally large number of arguments in disapproval at the same time.