Aggregation invariance in general clustering approaches. (English) Zbl 1306.62137

Summary: General clustering deals with weighted objects and fuzzy memberships. We investigate the group- or object-aggregation-invariance properties possessed by the relevant functionals (effective number of groups or objects, centroids, dispersion, mutual object-group information, etc.). The classical squared Euclidean case can be generalized to non-Euclidean distances, as well as to non-linear transformations of the memberships, yielding the \(c\)-means clustering algorithm as well as two presumably new procedures, the convex and pairwise convex clustering. Cluster stability and aggregation-invariance of the optimal memberships associated to the various clustering schemes are examined as well.


62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H86 Multivariate analysis and fuzziness
82B26 Phase transitions (general) in equilibrium statistical mechanics
94A17 Measures of information, entropy
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
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