Bezdek, James C. 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 \((p=1\) and \(p=\infty)\) 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.For the entire collection see [Zbl 0777.00046]. Cited in 1 Document MSC: 62H30 Classification and discrimination; cluster analysis (statistical aspects) 91C20 Clustering in the social and behavioral sciences 46N30 Applications of functional analysis in probability theory and statistics Keywords:fuzzy \(c\)-means clustering algorithms; Banach spaces; \(p\)-norm family; Minkowski norms; hard clustering algorithms; \(c\)-means clustering; exchange algorithm; approximate critical points; similarity measures; nonhyperelliptical topologies; mixture of uniform distributions PDFBibTeX XMLCite \textit{J. C. Bezdek}, in: Fuzzy approach to reasoning and decision-making. Selected papers of the international symposium held at Bechyně, Czechoslovakia, 25-29 June 1990. Dordrecht: Kluwer Academic Publishers. 173--184 (1992; Zbl 0786.62059)