2. Linear Multidimensional Scaling
Society for Industrial and Applied Mathematics eBooks · 2006
Chapter 1 gave an optimization strategy based on iterative quadratic assignment (QA) for the linear unidimensional scaling (LUS) task in the L2-norm, with all implementations carried out within a MATLAB computational environment. The central LUS task involves arranging the n objects in a set S = {O1, O2, …, On} along a single dimension, defined by coordinates x1, x2, …, xn, based on an n × n symmetric proximity matrix P = {pij}, whose (exclusively nondiagonal) nonnegative entries are given a dissimilarity interpretation (pij = pji for 1 ≤ i, j ≤ n; pii = 0 for 1 ≤ i ≤ n). The L2 criterion ∑ i<j ( pij − | xj − xi | )2 2.1 is minimized by the choice of the coordinates. The present chapter will give extensions to multidimensional scaling in the city-block metric (see Arabie, 1991, for a review of uses of this metric) for the L2-norm. The computational routines to be discussed and illustrated are again freely available as MATLAB M-files. We also note that most of the references given in Chapter 1 would also be relevant here as background material on the basic LUS task, but that review will not be repeated. Also, we will not discuss (in this chapter) comparisons to other methods (or strategies) for multidimensional scaling in the city-block metric—for the development of some of these alternatives, see Brusco (2001), Brusco and Stahl (2005), Groenen, Heiser, and Meulman (1999), Hubert, Arabie, and Meulman (1997), and Hubert, Arabie, and Hesson-McInnis (1992). In the extensions to city-block multidimensional scaling being pursued, a slight generalization to the basic unidimensional task that incorporates an additional additive constant will prove extremely convenient. So, in Section 2.1 we emphasize the more general leastsquares loss function of the form ∑ i