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Galois closed entity sets and \(k\)-balls of quasi-ultrametric multi-way dissimilarities. (English) Zbl 1133.62335

Summary: Quasi-ultrametric multi-way dissimilarities and their respective sets of \(k\)-balls extend the fundamental bijection in classification between ultrametric pairwise dissimilarities and indexed hierarchies. We show that nonempty Galois closed subsets of a finite entity set coincide with \(k\)-balls of some quasi-ultrametric multi-way dissimilarity. This result relates the order theoretic Galois connection based clustering approach to the dissimilarity based one. Moreover, it provides an effective way to specify easy-to-interpret cluster systems, from complex data sets, as well as to derive informative attribute implications.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
91C20 Clustering in the social and behavioral sciences
06B99 Lattices
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