Supervised multidimensional scaling for visualization, classification, and bipartite ranking

Least squares multidimensional scaling (MDS) is a classical method for representing an n x n dissimilarity matrix D. One seeks a set of configuration points z(1), ... , Z(n) is an element of R-S such that D is well approximated by the Euclidean distances between the configuration points: D-ij approximate to parallel to Z(i) - Z(j)parallel to(2). Suppose that in addition to D. a vector of associated binary class labels y is an element of {1, 2}(n) corresponding to the n observations is available. We propose an extension to MDS that incorporates this outcome vector. Our proposal, supervised multidimensional scaling (SMDS), seeks a set of configuration points z(1,) ... , z(n) is an element of R-S such that D-ij approximate to parallel to Z(i) - Z(j)parallel to(2), and such that z(is) > z(is) for s = 1, ... , S tends to occur when y(i) > y(j). This results in a new way to visualize the observations. In addition, we show that SMDS leads to a method for the classification of test observations, which can also be interpreted as a solution to the bipartite ranking problem. This method is explored in a simulation study, as well as on a prostate cancer gene expression data set and on a handwritten digits data set. (C) 2010 Elsevier B.V. All rights reserved.