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.

MSC:

 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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 [1] Bavaud F (2002) Quotient dissimilarities, Euclidean embeddability, and Huygens’ weak principle. In: Jaguja K, Sokolowski A, Bock HH (eds) Classification, clustering and data analysis. Springer, New York, pp 194–202 · Zbl 1033.62055 [2] Bavaud F (2006) Spectral clustering and multidimensional scaling: a unified view. In: Batagelj V, Bock HH, Ferligoj A, Ziberna A (eds) Data science and classification. Springer, New York, pp 131–139 [3] Bezdek D (1981) Pattern recognition. Plenum Press, New York · Zbl 0503.68065 [4] Blumenthal LM (1953) Theory and applications of distance geometry. University Press, Oxford · Zbl 0050.38502 [5] Celeux G, Govaert G (1992) A classification EM algorithm and two stochastic versions. Comput Stat Data Anal 14: 315–332 · Zbl 0937.62605 [6] Cover TM, Thomas JA (1991) Elements of information theory. Wiley, New York [7] Gray RM, Neuhoff DL (1998) Quantization. IEEE Trans Inf Theory 44: 2325–2383 · Zbl 1016.94016 [8] McLachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley, New York · Zbl 0882.62012 [9] Miyamoto S, Ichihashi H, Honda K (2008) Algorithms for fuzzy clustering: methods in c-means clustering with applications. Springer, New York · Zbl 1147.68073 [10] Rose K (1998) Deterministic Annealing for clustering, compression, classification, regression, and related optimization problems. Proc IEEE 86: 2210–2239 [11] Rose K, Gurewitz E, Fox GC (1990) Statistical mechanics and phase transitions in clustering. Phys Rev Lett 65: 945–948 [12] Runkler TA (2007) Relational fuzzy clustering. In: Valentede Oliveira J, Pedrycz W (eds) Advances in fuzzy clustering and its applications. Wiley, Chichester, pp 31–52 [13] Schoenberg IJ (1935) Remarks to Maurice Fréchet’s article ”Sur la définition axiomatique d’une classe d’espaces vectoriels distancés applicables vectoriellement sur l’espace de Hilbert”. Ann Math 36: 724–732 · Zbl 0012.30703
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