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Multidimensional scaling (MDS) is a technique for the analysis of similarity or dissimilarity data on a set of objects. Such data may be intercorrelations of test items, ratings of similarity on political candidates, or trade indices for a set of countries. MDS attempts to model such data as distances among points in a geometric space. The main reason for doing this is that one wants a graphical display of the structure of the data, one that is much easier to understand than an array of numbers and, moreover, one that displays the essential information in the data, smoothing out noise. There are numerous varieties of MDS. Some facets for distinguishing among them are the particular type of geometry into which one wants to map the data, the mapping function, the algorithms used to find an optimal data representation, the treatment of statistical error in the models, or the possibility to represent not just one but several similarity matrices at the same time. Other facets relate to the different purposes for which MDS has been used, to various ways of looking at or "interpreting" an MDS representation, or to differences in the data required for the particular models. In this book, we give a fairly comprehensive presentation of MDS. For the reader with applied interests only, the first six chapters of Part I should be sufficient. They explain the basic notions of ordinary MDS, with an emphasis on how MDS can be helpful in answering substantive questions.
I Fundamentals of MDS.- 1 The Four Purposes of Multidimensional Scaling.- 2 Constructing MDS Representations.- 3 MDS Models and Measures of Fit.- 4 Three Applications of MDS.- 5 MDS and Facet Theory.- 6 How to Obtain Proximities.- II MDS Models and Solving MDS Problems.- 7 Matrix Algebra for MDS.- 8 A Majorization Algorithm for Solving MDS.- 9 Metric and Nonmetric MDS.- 10 Confirmatory MDS.- 11 MDS Fit Measures, Their Relations, and Some Algorithms.- 12 Classical Scaling.- 13 Special Solutions, Degeneracies, and Local Minima.- III Unfolding.- 14 Unfolding.- 15 Special Unfolding Models.- IV MDS Geometry as a Substantive Model.- 16 MDS as a Psychological Model.- 17 Scalar Products and Euclidean Distances.- 18 Euclidean Embeddings.- V MDS and Related Methods.- 19 Procrustes Procedures.- 20 Three-Way Procrustean Models.- 21 Three-Way MDS Models.- 22 Methods Related to MDS.- VI Appendices.- A Computer Programs for MDS.- B Notation.- References.- Author Index.