Modern Multidimensional Scaling

1. Front Matter

   Pages i-xvii

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2. Fundamentals of MDS

   1. Front Matter

      Pages 1-1

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   2. The Four Purposes of Multidimensional Scaling

      • Ingwer Borg, Patrick Groenen

      Pages 3-14

   3. Constructing MDS Representations

      • Ingwer Borg, Patrick Groenen

      Pages 15-28

   4. MDS Models and Measures of Fit

      • Ingwer Borg, Patrick Groenen

      Pages 29-48

   5. Three Applications of MDS

      • Ingwer Borg, Patrick Groenen

      Pages 49-69

   6. MDS and Facet Theory

      • Ingwer Borg, Patrick Groenen

      Pages 71-89

   7. How to Obtain Proximities

      • Ingwer Borg, Patrick Groenen

      Pages 91-106

3. MDS Models and Solving MDS Problems

   1. Front Matter

      Pages 107-107

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   2. Matrix Algebra for MDS

      • Ingwer Borg, Patrick Groenen

      Pages 109-134

   3. A Majorization Algorithm for Solving MDS

      • Ingwer Borg, Patrick Groenen

      Pages 135-157

   4. Metric and Nonmetric MDS

      • Ingwer Borg, Patrick Groenen

      Pages 159-180

   5. Confirmatory MDS

      • Ingwer Borg, Patrick Groenen

      Pages 181-197

   6. MDS Fit Measures, Their Relations, and Some Algorithms

      • Ingwer Borg, Patrick Groenen

      Pages 199-205

   7. Classical Scaling

      • Ingwer Borg, Patrick Groenen

      Pages 207-212

   8. Special Solutions, Degeneracies, and Local Minima

      • Ingwer Borg, Patrick Groenen

      Pages 213-228

4. Unfolding

   1. Front Matter

      Pages 229-229

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   2. Unfolding

      • Ingwer Borg, Patrick Groenen

      Pages 231-252

   3. Special Unfolding Models

      • Ingwer Borg, Patrick Groenen

      Pages 253-269

5. MDS Geometry as a Substantive Model

   1. Front Matter

      Pages 271-271

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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.

• algorithms
• best fit
• correlation
• data analysis
• multidimensional scaling
• sociology
• statistics

• Zentrum für Umfragen, Methoden und Analysen, Mannheim, Germany

  Ingwer Borg

• Department of Data Theory, Leiden University, Leiden, The Netherlands

  Patrick Groenen

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