Multidimensional Scaling

The term ‘Multidimensional Scaling’ or MDS is used in two essentially different ways in statistics (de Leeuw & Heiser 1980a). MDS in the wide sense refers to any technique that produces a multidimensional geometric representation of data, where quantitative or qualitative relationships in the data are made to correspond with geometric relationships in the representation. MDS in the narrow sense starts with information about some form of dissimilarity between the elements of a set of objects, and it constructs its geometric representation from this information. Thus the data are dissimilarities, which are distance-like quantities (or similarities, which are inversely related to distances). In this chapter we will discuss narrow-sense MDS only, because we do not want to dilute the definition of the technique so as to include almost all of multivariate analysis. We emphasize from the beginning that MDS is a descriptive technique, in which the notion of statistical inference is almost completely absent. There have been some attempts to introduce statistical models and corresponding estimating and testing methods, but they have been largely unsuccessful. We introduce some quick notation. Dissimilarities are written as δi j , and distances are di j (X). Here i and j are the objects we are interested in. The n × p matrix X is the configuration, with coordinates of the objects in Rp. Often, we also have as data weights wi j reflecting the importance or precision of dissimilarity δi j .