Generalized Factor Analyses for Contingency Tables

Abstract

Quotient dissimilarities constitute a broad aggregation-invariant fam- ily; among them, f-dissimilarities are Euclidean embeddable (Bavaud 2002). We present a non linear principal components analysis (NPA) applicable to any quo- tient dissimilarity, based upon the spectral decomposition of the central inertia. For f-dissimilarities, the same decomposition yields a non linear correspondence analysis (NCA), permitting to modulate as finely as wished the contributions of positive or negative deviations from independence. The resulting coordinates exactly reproduce the original dissimilarities between rows or between columns; however, Huygens’s weak principle is generally violated, as measured by some quantity we call eccen- tricity.