Abstract

Multidimensional scaling (MDS) is a versatile technique that can be used to visualize objects based on observed dissimilarities. When dissimilarities are asymmetric, for example when considering confusion rates between stimuli or exchanges between units, making a visualization capturing the asymmetry is not trivial. Moreover, when there are many objects, interpretation of MDS maps may become difficult. Clustering the objects can alleviate such problems and, when object related information is available in the form of external variables, relating this information to the factors underlying the low dimensional representation may further help in interpreting results. In this paper, we propose a least-squares model that partitions the objects and represents the cluster centroids in low dimensional space for the symmetric part, and using bimensions, planes in which the asymmetry between two objects is read as the area of a triangle, for the skew-symmetric part. Furthermore, the coordinates for both representations are constrained to be linear combinations of the external variables. The loss splits exactly into a symmetric and a skew-symmetric term, the loadings of both parts have closed-form solutions at a fixed partition, and the partition is updated with cluster-level statistics. Hence, evaluating the move of an object does not involve large (number of observations by number of observation) matrices and can be done efficiently. The loadings of the symmetric part are determined up to a rotation, which we fix with orthogonal coordinates or orthogonal loadings, as well as with a disjoint version, in which each external variable loads on one dimension. The external variables restrict the solution only when the rank of their cluster means is below the number of clusters minus one, and double centring removes the dominance component of the asymmetry. For symmetric dissimilarities our model reduces to the model of Kiers, Vicari and Vichi (2005). A simulation study and empirical applications illustrate the model.

Kiers, H., Vicari, D., and Vichi, M. (2005). Simultaneous classification and multidimensional scaling with external information. Psychometrika, 70(3):433–460.