Modern Multidimensional Scaling

1. Front Matter

   Pages i-xxi

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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

      Pages 3-18

   3. Constructing MDS Representations

      Pages 19-35

   4. MDS Models and Measures of Fit

      Pages 37-61

   5. Three Applications of MDS

      Pages 63-85

   6. MDS and Facet Theory

      Pages 87-109

   7. How to Obtain Proximities

      Pages 111-133

3. MDS Models and Solving MDS Problems

   1. Front Matter

      Pages 135-135

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   2. Matrix Algebra for MDS

      Pages 137-168

   3. A Majorization Algorithm for Solving MDS

      Pages 169-197

   4. Metric and Nonmetric MDS

      Pages 199-225

   5. Confirmatory MDS

      Pages 227-246

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

      Pages 247-260

   7. Classical Scaling

      Pages 261-267

   8. Special Solutions, Degeneracies, and Local Minima

      Pages 269-289

4. Unfolding

   1. Front Matter

      Pages 291-291

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   2. Unfolding

      Pages 293-316

   3. Avoiding Trivial Solutions in Unfolding

      Pages 317-334

   4. Special Unfolding Models

      Pages 335-355

Multidimensionalscaling(MDS)isatechniquefortheanalysisofsimilarity 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 forasetofcountries.MDSattemptstomodelsuchdataasdistancesamong pointsinageometricspace.Themainreasonfordoingthisisthatonewants 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 mapthedata,themappingfunction,thealgorithmsusedto?ndanoptimal 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 di?erent purposes for which MDS has been used, to various ways of looking at or “interpreting” an MDS representation, or to di?erences in the data required for the particular models. Inthisbook,wegiveafairlycomprehensivepresentationofMDS.Forthe reader with applied interests only, the ?rst six chapters of Part I should be su?cient. 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
• marketing
• modeling
• multidimensional scaling
• statistics

From the reviews of the second edition:

"[Modern Multidimensional Scaling: Theory and Applications] is without a doubt the most comprehensive and most rigorous book on MDS...The second edition is considerably (140 pages) longer than the first, mostly because of much more material on MDS of rectangluar matrices (also known as unfolding) and MDS of asymmetric matrices is included...this is currently by far the best available book on MDS, and it is quite likely to stay in that position for a long time." Journal of Statistical Software, August 2005

"This is an updated and expanded version of the first edition … . the exercises at the end of each chapter are an attractive feature. I can recommend the book enthusiastically." (W.J. Krzanowski, Short Book Reviews, Vol. 26 (1), 2006)

"The authors provide a comprehensive treatment of multidimensional scaling (MDS), a family of statistical techniques for analyzing similarity or dissimilarity data on a set of objects. … This book may be used as an introduction to MDS for students in psychology, sociology and marketing … . It is also well suited for a variety of advanced courses on MDS topics." (Ivan Krivý, Zentralblatt MATH, Vol. 1085, 2006)

• ZUMA, Mannheim, Germany

  Ingwer Borg

• Econometric Institute, Erasmus University, Rotterdam, The Netherlands

  Patrick J. F. Groenen

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