The Representation of Symmetric Proximity Data: Dimensions and Classifications

A review is given for the data analysis task of representing a symmetric proximity matrix, defined for some object set, by a sum of matrices each having the restrictive anti-Robinson (AR) form. An emphasis is placed on the inclusion of an optimal monotonic transformation of the given proximity matrix and what each AR component of an additive decomposition might be depicting by imposing further restrictions to obtain approximating matrices that are strongly AR, or that provide unidimensional scales or ultrametrics. Three published data sets are used to illustrate the process of constructing the initial decomposition and then giving a substantive interpretation subsequently for each of the terms in the fitted sum. An extension to circular anti-Robinson (CAR) matrices is also discussed briefly and illustrated, along with further restrictions to circular unidimensional scales and circular strongly AR forms.