Measure vs Variable Duality in Correspondence Analysis

Henry Rouanet and Brigitte Le Roux

Université René Descartes, Paris, France

Rouanet@math-info.univ-paris5.fr and Lerb@math-info.univ-paris5.fr

Abstract

The formal approach used by Benzécri to develop CA was not an accidental matter of notation, but an integral part of the construction (Benzécri & Coll, 1973). The properties of CA are entirely founded on the underlying mathematical theory, essentially abstract linear algebra , as found in the classical mathematical books by MacLane, Halmos, etc.: finite-dimensional vector space, homomorphism, scalar product, etc. The cornerstone of the formal-geometric approach is the measure vs variable duality, which formalizes the distinction between two sorts of quantities: those for which grouping units entails summing (adding up) values, such as weights, frequencies, etc., we call them measures (like in mathematical measure theory), versus those for which grouping units entails averaging values, such as scores, rates, etc., we call them variables. This duality is reflected in the duality notation (alias transition notation), putting lower indices for measures and upper indices for variables. See Rouanet & Le Roux (1993), Le Roux & Rouanet (2004).

In the paper, we describe measure vs variable duality in CA at the following two crucial stages of geometric modeling: i) Construction of clouds and the chi-square metric. The marginal frequencies of the table firstly provide reference measures over rows and columns. Secondly, they define Euclidean isomorphisms from variable vector spaces to dual measure vector spaces, hence scalar products and Euclidean norms, therefore they determine without arbitrariness the chi-square metric over those spaces. ii) Principal directions of clouds and principal coordinates. The fundamental mathematical result is that the solution of spectral equations is the singular decomposition of two adjoint homomorphisms and/or the associated bilinear form. Applying these results to CA immediately yields the transition equations and the reconstitution formulas.

CA is a sase in point to exemplify the superiority of the formal approach to multivariate statistics over the usual matrix approach.

References

Benzécri J.P. & Coll.(1973). Analyse des Données, Volume 2, Analyse des Correspondances. Paris, Dunod.

Rouanet H. & Le Roux B. (1993). Analyse des Données Multidimensionnelles Paris, Dunod.

Le Roux B., Rouanet H. Geometric Data Analysis (in press). Dordrecht, Kluwer.