Abstract
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 DX. Various suggestions are presented, aimed at generalizing Factorial Correspondence Analysis beyond the chi-square metric, by non-linear distortion of departures from independence. In particular, the central inertia appearing in one formulation precisely amounts to the mutual information of Information Theory. 1
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BAVAUD, F. (2000): On a class of Aggregation-invariant Dissimilarities obeying the weak Huygens’ principle. In H.A.L. Kiers and al. (Eds.): Data Analysis, Classification and Related Methods. Springer, New York, 131–136.
CRESSIE, N. and READ, T.R.C. (1984): Multinomial goodness-of-fit tests.J.R.Statist.Soc.B, 46,440–464.
CRITCHLEY, F. and FICHET, B. (1994): The partial order by inclusion of the classes and dissimilarity on a finite set, and some of their basic properties. In B. Van Custem (Ed.): Classification and Dissimilarity Analysis. Lecture Notes in Statistics, Springer, New York, 5–65.
ESCOFIER, B. (1978): Analyse factorielle et distances répondant au principe d’équivalence distributionnelle. Revue de Statistique Appliquée, 26, 29–37
GOWER, J.C. (1982): Euclidean distance geometry. The Mathematical Scientist, 7, 1–14
JOLY, S. and LE CALVE, G. (1994): Similarity functions. In B. Van Custem (Ed.): Classification and Dissimilarity Analysis. Lecture Notes in Statistics, Springer, New York, 67–86.
LEBART, L. (1969): L’analyse statistique de la contiguïté. Publications de l’ISUP, XVIII, 81–112
SCHOENBERG, I.J. (1935) Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espaces vectoriels distancés applicables vectorielle-merit sur l’espace de Hilbert”. Annals of Mathematics, 36, 724–732
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© 2002 Springer-Verlag Berlin Heidelberg
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Bavaud, F. (2002). Quotient Dissimilarities, Euclidean Embeddability, and Huygens’ Weak Principle. In: Jajuga, K., Sokołowski, A., Bock, HH. (eds) Classification, Clustering, and Data Analysis. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56181-8_21
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DOI: https://doi.org/10.1007/978-3-642-56181-8_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43691-1
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