Modern Multidimensional Scaling : Theory and Applications

Theory and Applications

By: I. Borg, P. J. F. Groenen

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Hardcover | 1 February 2009 | Edition Number 2

At a Glance

Hardcover
640 Pages

Edition Type
Revised

Dimensions(cm)
23.5 x 16.51 x 3.18

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The book provides a comprehensive treatment of multidimensional scaling (MDS), a family of statistical techniques for analyzing the structure of (dis)similarity data. Such data are widespread, including, for example, intercorrelations of survey items, direct ratings on the similarity on choice objects, or trade indices for a set of countries. MDS represents the data as distances among points in a geometric space of low dimensionality. This map can help to see patterns in the data that are not obvious from the data matrices. MDS is also used as a psychological model for judgments of similarity and preference.

This book may be used as an introduction to MDS for students in psychology, sociology, and marketing. The prerequisite is an elementary background in statistics. The book is also well suited for a variety of advanced courses on MDS topics. All the mathematics required for more advanced topics is developed systematically.

This second edition is not only a complete overhaul of its predecessor, but also adds some 140 pages of new material. Many chapters are revised or have sections reflecting new insights and developments in MDS. There are two new chapters, one on asymmetric models and the other on unfolding. There are also numerous exercises that help the reader to practice what he or she has learned, and to delve deeper into the models and its intricacies. These exercises make it easier to use this edition in a course. All data sets used in the book can be downloaded from the web. The appendix on computer programs has also been updated and enlarged to reflect the state of the art.

Industry Reviews

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 Krivy, Zentralblatt MATH, Vol. 1085, 2006)

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Preface	p. vii
Fundamentals of MDS	p. 1
The Four Purposes of Multidimensional Scaling	p. 3
MDS as an Exploratory Technique	p. 4
MDS for Testing Structural Hypotheses	p. 6
MDS for Exploring Psychological Structures	p. 9
MDS as a Model of Similarity Judgments	p. 11
The Different Roots of MDS	p. 13
Exercises	p. 15
Constructing MDS Representations	p. 19
Constructing Ratio MDS Solutions	p. 19
Constructing Ordinal MDS Solutions	p. 23
Comparing Ordinal and Ratio MDS Solutions	p. 29
On Flat and Curved Geometries	p. 30
General Properties of Distance Representations	p. 33
Exercises	p. 34
MDS Models and Measures of Fit	p. 37
Basics of MDS Models	p. 37
Errors, Loss Functions, and Stress	p. 41
Stress Diagrams	p. 42
Stress per Point	p. 44
Evaluating Stress	p. 47
Recovering True Distances by Metric MDS	p. 55
Further Variants of MDS Models	p. 57
Exercises	p. 59
Three Applications of MDS	p. 63
The Circular Structure of Color Similarities	p. 63
The Regionality of Morse Codes Confusions	p. 68
Dimensions of Facial Expressions	p. 73
General Principles of Interpreting MDS Solutions	p. 80
Exercises	p. 82
MDS and Facet Theory	p. 87
Facets and Regions in MDS Space	p. 87
Regional Laws	p. 91
Multiple Facetizations	p. 93
Partitioning MDS Spaces Using Facet Diagrams	p. 95
Prototypical Roles of Facets	p. 99
Criteria for Choosing Regions	p. 100
Regions and Theory Construction	p. 102
Regions, Clusters, and Factors	p. 104
Exercises	p. 105
How to Obtain Proximities	p. 111
Types of Proximities	p. 111
Collecting Direct Proximities	p. 112
Deriving Proximities by Aggregating over Other Measures	p. 119
Proximities from Converting Other Measures	p. 125
Proximities from Co-Occurrence Data	p. 126
Choosing a Particular Proximity	p. 128
Exercises	p. 130
MDS Models and Solving MDS Problems	p. 135
Matrix Algebra for MDS	p. 137
Elementary Matrix Operations	p. 137
Scalar Functions of Vectors and Matrices	p. 142
Computing Distances Using Matrix Algebra	p. 144
Eigendecompositions	p. 146
Singular Value Decompositions	p. 150
Some Further Remarks on SVD	p. 152
Linear Equation Systems	p. 154
Computing the Eigendecomposition	p. 157
Configurations that Represent Scalar Products	p. 160
Rotations	p. 160
Exercises	p. 163
A Majorization Algorithm for Solving MDS	p. 169
The Stress Function for MDS	p. 169
Mathematical Excursus: Differentiation	p. 171
Partial Derivatives and Matrix Traces	p. 176
Minimizing a Function by Iterative Majorization	p. 178
Visualizing the Majorization Algorithm for MDS	p. 184
Majorizing Stress	p. 185
Exercises	p. 194
Metric and Nonmetric MDS	p. 199
Allowing for Transformations of the Proximities	p. 199
Monotone Regression	p. 205
The Geometry of Monotone Regression	p. 209
Tied Data in Ordinal MDS	p. 211
Rank-Images	p. 213
Monotone Splines	p. 214
A Priori Transformations Versus Optimal Transformations	p. 221
Exercises	p. 224
Confirmatory MDS	p. 227
Blind Loss Functions	p. 227
Theory-Compatible MDS: An Example	p. 228
Imposing External Constraints on MDS Representations	p. 230
Weakly Constrained MDS	p. 237
General Comments on Confirmatory MDS	p. 242
Exercises	p. 244
MDS Fit Measures, Their Relations, and Some Algorithms	p. 247
Normalized Stress and Raw Stress	p. 247
Other Fit Measures and Recent Algorithms	p. 250
Using Weights in MDS	p. 254
Exercises	p. 258
Classical Scaling	p. 261
Finding Coordinates in Classical Scaling	p. 261
A Numerical Example for Classical Scaling	p. 263
Choosing a Different Origin	p. 264
Advanced Topics	p. 265
Exercises	p. 267
Special Solutions, Degeneracies, and Local Minima	p. 269
A Degenerate Solution in Ordinal MDS	p. 269
Avoiding Degenerate Solutions	p. 272
Special Solutions: Almost Equal Dissimilarities	p. 274
Local Minima	p. 276
Unidimensional Scaling	p. 278
Full-Dimensional Scaling	p. 281
The Tunneling Method for Avoiding Local Minima	p. 283
Distance Smoothing for Avoiding Local Minima	p. 284
Exercises	p. 288
Unfolding	p. 291
Unfolding	p. 293
The Ideal-Point Model	p. 293
A Majorizing Algorithm for Unfolding	p. 297
Unconditional Versus Conditional Unfolding	p. 299
Trivial Unfolding Solutions and [sigma subscript 2]	p. 301
Isotonic Regions and Indeterminacies	p. 305
Unfolding Degeneracies in Practice and Metric Unfolding	p. 308
Dimensions in Multidimensional Unfolding	p. 312
Multiple Versus Multidimensional Unfolding	p. 313
Concluding Remarks	p. 314
Exercises	p. 314
Avoiding Trivial Solutions in Unfolding	p. 317
Adjusting the Unfolding Data	p. 317
Adjusting the Transformation	p. 322
Adjustments to the Loss Function	p. 324
Summary	p. 330
Exercises	p. 331
Special Unfolding Models	p. 335
External Unfolding	p. 335
The Vector Model of Unfolding	p. 336
Weighted Unfolding	p. 342
Value Scales and Distances in Unfolding	p. 345
Exercises	p. 352
MDS Geometry as a Substantive Model	p. 357
MDS as a Psychological Model	p. 359
Physical and Psychological Space	p. 359
Minkowski Distances	p. 363
Identifying the True Minkowski Distance	p. 367
The Psychology of Rectangles	p. 372
Axiomatic Foundations of Minkowski Spaces	p. 377
Subadditivity and the MBR Metric	p. 381
Minkowski Spaces, Metric Spaces, and Psychological Models	p. 385
Exercises	p. 386
Scalar Products and Euclidean Distances	p. 389
The Scalar Product Function	p. 389
Collecting Scalar Products Empirically	p. 392
Scalar Products and Euclidean Distances: Formal Relations	p. 397
Scalar Products and Euclidean Distances: Empirical Relations	p. 400
MDS of Scalar Products	p. 403
Exercises	p. 408
Euclidean Embeddings	p. 411
Distances and Euclidean Distances	p. 411
Mapping Dissimilarities into Distances	p. 415
Maximal Dimensionality for Perfect Interval MDS	p. 418
Mapping Fallible Dissimilarities into Euclidean Distances	p. 419
Fitting Dissimilarities into a Euclidean Space	p. 424
Exercises	p. 425
MDS and Related Methods	p. 427
Procrustes Procedures	p. 429
The Problem	p. 429
Solving the Orthogonal Procrustean Problem	p. 430
Examples for Orthogonal Procrustean Transformations	p. 432
Procrustean Similarity Transformations	p. 434
An Example of Procrustean Similarity Transformations	p. 436
Configurational Similarity and Correlation Coefficients	p. 437
Configurational Similarity and Congruence Coefficients	p. 439
Artificial Target Matrices in Procrustean Analysis	p. 441
Other Generalizations of Procrustean Analysis	p. 444
Exercises	p. 445
Three-Way Procrustean Models	p. 449
Generalized Procrustean Analysis	p. 449
Helm's Color Data	p. 451
Generalized Procrustean Analysis	p. 454
Individual Differences Models: Dimension Weights	p. 457
An Application of the Dimension-Weighting Model	p. 462
Vector Weightings	p. 465
Pindis, a Collection of Procrustean Models	p. 469
Exercises	p. 471
Three-Way MDS Models	p. 473
The Model: Individual Weights on Fixed Dimensions	p. 473
The Generalized Euclidean Model	p. 479
Overview of Three-Way Models in MDS	p. 482
Some Algebra of Dimension-Weighting Models	p. 485
Conditional and Unconditional Approaches	p. 489
On the Dimension-Weighting Models	p. 491
Exercises	p. 492
Modeling Asymmetric Data	p. 495
Symmetry and Skew-Symmetry	p. 495
A Simple Model for Skew-Symmetric Data	p. 497
The Gower Model for Skew-Symmetries	p. 498
Modeling Skew-Symmetry by Distances	p. 500
Embedding Skew-Symmetries as Drift Vectors into MDS Plots	p. 502
Analyzing Asymmetry by Unfolding	p. 503
The Slide-Vector Model	p. 506
The Hill-Climbing Model	p. 509
The Radius-Distance Model	p. 512
Using Asymmetry Models	p. 514
Overview	p. 515
Exercises	p. 515
Methods Related to MDS	p. 519
Principal Component Analysis	p. 519
Correspondence Analysis	p. 526
Exercises	p. 537
Appendices	p. 541
Computer Programs for MDS	p. 543
Interactive MDS Programs	p. 544
MDS Programs with High-Resolution Graphics	p. 550
MDS Programs without High-Resolution Graphics	p. 562
Notation	p. 569
References	p. 573
Author Index	p. 599
Subject Index	p. 605
Table of Contents provided by Ingram. All Rights Reserved.

Modern Multidimensional Scaling : Theory and Applications

Theory and Applications

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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