Metric on the Space of Systems Behavior Functions Represented by Fuzzy Measures

Abstract

G. Klir proposed to describe the behavior of complex systems using behavior functions (BFs) - invariant constraints on the set of system states. BFs are one of the most productive tools for studying the functioning of systems. To study systems, it is necessary to have a metric for measuring of the difference between two BFs. To describe BFs modern researchers do not use distributions other than probability or possibility. But these distributions can be considered as special cases of Sugeno fuzzy measures, the use of which greatly expands the possibilities in the study of systems. However, metrics to measure the difference between fuzzy measures have not been developed. Therefore, in this article, the authors proposed a new metric and an algorithm for its calculation for the case when BFs are described by Sugeno fuzzy measures. This metric is based on the Cartesian product of fuzzy measures and the use of our proposed concentration function. The metric makes it possible to compare the behavior of systems in the case of describing BFs by Sugeno fuzzy measures with different modalities, as well as to ensure the priority of taking into account the set of the most significant states of the system.