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Quotient dissimilarities, Euclidean embeddability, and Huygens’ weak principle. (English) Zbl 1033.62055

Jajuga, Krzysztof et al., Classification, clustering, and data analysis. Recent advances and applications. Papers presented at the eighth conference of the International Federation of Classification Societies (IFCS), Cracow, Poland, July 16–19, 2002. Berlin: Springer (ISBN 3-540-43691-X/pbk). Studies in Classification, Data Analysis, and Knowledge Organization, 195-202 (2002).
Summary: We introduce a broad class of categorical dissimilarities, the quotient dissimilarities, for which aggregation invariance is automatically satisfied. This class contains the chi-square, ratio, Kullback-Leibler and Hellinger dissimilarities, as well as presumably new “power” and “threshold” dissimilarity families. For a large sub-class of the latter, the product dissimilarities, we show that the Euclidean embeddability property on one hand and the weak Huygens’ principle on the other hand are mutually exclusive, the only exception being provided by the chi-square dissimilarity \(D^\chi\). Various suggestions are presented, aimed at generalizing Factorial Correspondence Analysis beyond the chi-square metric, by nonlinear distortion of departures from independence. In particular, the central inertia appearing in one formulation precisely amounts to the mutual information of information theory.
For the entire collection see [Zbl 1026.00018].

MSC:

62H17 Contingency tables
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H25 Factor analysis and principal components; correspondence analysis
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