Fuzzy Cluster Analysis: Pseudometrics and Fuzzy Clusters

Abstract

Introduction. Clustering problems arise in various spheres of human activity. In cases where there are no initial data sufficient for statistical analysis or information obtained from experts is used, fuzzy models are proposed that take into account different types of uncertainty and more argumentatively reflect real situations that model systems of different purposes. Particular attention is drawn to invariance in problems with different types of data measured in different scales according to the classification of S. Stevens. It is known that when solving cluster analysis problems using the transitive closure operation with respect to the equivalence that is obtained, such connections between objects as similarity and dissimilarity are changed. Therefore, it is necessary to take into account the problem of adequacy when developing models and algorithms for solving problems of fuzzy cluster analysis. The purpose of the paper is an analyzing the problem of adequacy of the results of fuzzy cluster analysis on the introduction of metrics and pseudometrics on fuzzy sets in the presence of several qualitative and quantitative characteristics of objects. Propose an approach that ensures the adequacy of pseudometrics, that is, provides invariance with respect to permissible transformations of the values of fuzzy features, and also ensures the division of objects into equivalence classes without distorting the distance between them. Results. Axiomatic definitions of a fuzzy cluster and a fuzzy α level cluster are proposed, which are introduced as fuzzy sets of elements similar to certain elements of a given set, if the condition is met: the dissimilarity ratio must be an invariant pseudometric. This condition is ensured by the use of the linguistic correlation coefficient when calculating fuzzy relations of similarity and dissimilarity. Based on the definition of a fuzzy cluster of α level and threshold conorm, the distance between fuzzy clusters of α level is determined. Conclusions. The proposed approach can be the basis for the development of algorithms for solving cluster analysis problems. This provides a meaningful interpretation of the obtained clusters, and the possibility of clarifying the results in further studies of their structure. Keywords: fuzzy set, conorm, metric, pseudometric, fuzzy similarity relation, fuzzy cluster.