Abstract
A fuzzy equivalence relation can be used for clustering. But when using it in applications, we often get a similarity relation rather than an equivalence one because of various reasons. We need to reform it into an equivalence relation close to it to cluster. A commonly used method is transitive closure method, but it usually results in serious distortions about the relation. This paper further studies fuzzy similarity and equivalence relations using fuzzy graphs, and obtains some new results. The defects of transitive closure method are analyzed, and an improved clustering algorithm is given, but it cannot eliminate the inconsistency phenomenon in classification hierarchy structure. To solve this problem, the optimal fuzzy equivalence relation of similarity relation is studied. An optimization model which can derive it exactly is given, but it is too complex for applications. An effective approximation algorithm to get the optimal equivalence relation is thus presented. Several examples and some discussions are also given to illustrate the given methods.
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The author would like to acknowledge the financial support from National Natural Science Foundation of China (Grant No. 12171445)
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Chai, Z. A study of clustering on optimal fuzzy equivalence relations. Soft Comput 27, 1415–1424 (2023). https://doi.org/10.1007/s00500-022-07654-z
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DOI: https://doi.org/10.1007/s00500-022-07654-z