1Multidimensional Dependencies in Classi cation and Ordination

1 ABSTRACT The relations between two distance matrices on the same nite set are analyzed, via metric scaling, by correlating principal axis. Some applications are given and illustrated with examples. 1.1 Introduction Dissimilarities, similarities and distances are fundamental concepts in mul-tidimensional scaling and related topics. Euclidean and Mahalanobis distance also play a basic role in techniques such as regression and discrimination. In many cases, distances are computed by observing variables on individuals, but in general, the dependence between variables is not taken into account. Mahalanobis, and its extension Rao's distance, is an important exception. This distance between two observations x;y, say, (x ? y) 0 ?1 (x ? y) depends on the covariance matrix , hence its computation is not possible when the variables are categorical, binary or mixed. In such situations, distances are obtained from similarities such as Jaccard (binary), matching (categorical) and Gower (mixed) coeecients. Other coeecients are possible, but none of them has the property of including the relationships among variables. Motivated by this problem, CF97b] recently introduced related metric scaling, a new multidimensional scaling method to represent objects when two distances are deened on them. This method is based on the construction of a joint distance that has some compatible properties, especially identifying and discarding redundant information.