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Clustering in Banach spaces. (English) Zbl 0786.62059
Novák, Vilém (ed.) et al., Fuzzy approach to reasoning and decision- making. Selected papers of the international symposium held at Bechyně, Czechoslovakia, 25-29 June 1990. Dordrecht, Prague: Kluwer Academic Publishers, Akademia. Theory Decis. Libr., Ser. D. 8, 173-184 (1992).

Summary: We extend the Hard and Fuzzy $c$-Means (HCM/FCM) clustering algorithms to the case where the (dis)similarity measure on pairs of numerical vectors includes two members of the Minkowski or $p$-norm family, viz., the $p=1$ and $p=\infty$ (or “sup”) norms. We note that a basic exchange algorithm can be used to find approximate critical points of the new objective functions. This method broadens the applications horizon of the FCM family by enabling users to match “discontinuous” multidimensional numerical data structures with similarity measures which have nonhyperelliptical topologies.

For example, data drawn from a mixture of uniform distributions have sharp or “boxy” edges; the $\left(p=1$ and $p=\infty \right)$ norms have open and closed sets that match these shapes. We illustrate the technique with a small artificial data set, and compare the results with the $c$-means clustering solution produced using the Euclidean (inner product) norm.

##### MSC:
 62H30 Classification and discrimination; cluster analysis (statistics) 91C20 Clustering (Social and behavioral sciences) 46N30 Applications of functional analysis in probability theory and statistics