Divergence and Similarity Characteristics for Two Fuzzy Measures Based on Associated Probabilities

The article deals with the definitions of the distance, divergence, and similarity characteristics between two finite fuzzy measures, which are generalizations of the same definitions between two finite probability distributions. As is known, a fuzzy measure can be uniquely represented by the so-cal...

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Bibliographic Details
Published in:Axioms Vol. 13; no. 11; p. 776
Main Authors: Sirbiladze, Gia, Midodashvili, Bidzina, Manjafarashvili, Teimuraz
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.11.2024
Subjects:
Analysis
associated probabilities
Clustering
Decision-making
Digital video recorders
distance generator
distances
Distribution (Probability theory)
Divergence
divergence indexes
Entropy
Expected values
fuzzy measure identification
fuzzy measures
Fuzzy sets
Machine learning
Parameter identification
Probability
Probability distribution
Similarity
ISSN:2075-1680, 2075-1680
Online Access:Get full text
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Summary:The article deals with the definitions of the distance, divergence, and similarity characteristics between two finite fuzzy measures, which are generalizations of the same definitions between two finite probability distributions. As is known, a fuzzy measure can be uniquely represented by the so-called its associated probability class (APC). The idea of generalization is that new definitions of distance, divergence, and similarity between fuzzy measures are reduced to the definitions of distance, divergence, and similarity between the APCs of fuzzy measures. These definitions are based on the concept of distance generator. The proof of the correctness of generalizations is provided. Constructed distance, similarity, and divergence relations can be used in such applied problems as: determining the difference between Dempster-Shafer belief structures; Constructions of collaborative filtering similarity relations; non-additive and interactive parameters of machine learning in phase space metrics definition, object clustering, classification and other tasks. In this work, a new concept is used in the fuzzy measure identification problem for a certain multi-attribute decision-making (MADM) environment. For this, a conditional optimization problem with one objective function representing the distance, divergence or similarity index is formulated. Numerical examples are discussed and a comparative analysis of the obtained results is presented.
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ISSN:2075-1680
2075-1680
DOI:10.3390/axioms13110776