Multidimensional Scaling

Abstract

In Chapter 3, we noted in passing that one of the most useful ways of using principal components analysis was to obtain a low-dimensional “map” of the data that preserved as far as possible the Euclidean distances between the observations in the space of the original q variables. In this chapter, we will make this aspect of principal component analysis more explicit and also introduce a class of other methods, labelled multidimensional scaling, that aim to produce similar maps of data but do not operate directly on the usual multivariate data matrix, X. Instead they are applied to distance matrices (see Chapter 1), which are derived from the matrix X (an example of a distance matrix derived from a small set of multivariate data is shown in Subsection 4.4.2), and also to so-called dissimilarity or similarity matrices that arise directly in a number of ways, in particular from judgements made by human raters about how alike pairs of objects, stimuli, etc., of interest are. An example of a directly observed dissimilarity matrix is shown in Table 4.5, with judgements about political and war leaders that had major roles in World War II being given by a subject after receiving the simple instructions to rate each pair of politicians on a nine-point scale, with 1 indicating two politicians they regard as very similar and 9 indicating two they regard as very dissimilar. (If the nine point-scale had been 1 for very dissimilar and 9 for very similar, then the result would have been a rating of similarity, although similarities are often scaled to lie in a [0; 1] interval. The term proximity is often used to encompass both dissimilarity and similarity ratings.)