Fuzzy Equivalence Relations as Similarity Measure in Agglomerative Clustering Algorithm

This chapter studies hierarchical clustering, particularly agglomerative clustering, where fuzzy equivalence relations are considered as a measure of the similarity of objects and clusters, in which transitivity is defined on the basis of various t-norms: Łukasiewicz, product, and Hamacher’s t-norm. In construction of fuzzy equivalence relation is involved a tool called an additive generator. For objects and clusters, internal and external similarities are introduced using aggregation of corresponding equivalence relations. In the paper, a quality measure of the cluster system is presented using fuzzy equivalence relations. Additionally, the paper provides a comparative analysis between standard clustering methods (KMeans, C-Means) and the proposed method, with fuzzy equivalence relations, where transitivity is defined on the basis of t-norms.