MDS Models and Measures of Fit

Abstract

MDS models are defined by specifying how given similarity or dissimilarity data, the proximities p _ij , are mapped into distances of an m-dimensional MDS configuration X. The mapping is specified by a representation function, f : P _ij →d _ij (X), which specifies how the proximities should be related to the distances. In practice, one usually does not attempt to strictly satisfy f. Rather, what is sought is a configuration (in a given dimensionality) whose distances satisfy f as closely as possible. The condition “as closely as” is quantified by a badness-of-fit measure or loss function. The loss function is a mathematical expression that aggregates the representation errors, e _ij = f (p _ij )−d _ij (X), over all pairs (i, j). A normed sum-of-squares of these errors defines Stress, the most common loss function in MDS. How Stress should be evaluated is a major issue in MDS. It is discussed at length in this chapter, and various criteria are presented.