多维标度理论，在经济金融领域应用广泛，特别在大数据处理方面。
（另外，作者I.Borg P.J.F.Groenen 另一本应用技术参考书：I.Borg Patrick J.F. Groenen P.Mair Applied Multidimensional Scaling Springer 2013，请参考：https://bbs.pinggu.org/thread-2440842-1-1.html）

I.Borg P.J.F.Groenen
Modern Multidimensional Scaling Theory and Applications Second Edition 2005
Preface vii
I Fundamentals of MDS 1
1 The Four Purposes of Multidimensional Scaling 3
1.1 MDS as an Exploratory Technique . . . . . . . . . . . . . . 4
1.2 MDS for Testing Structural Hypotheses . . . . . . . . . . . 6
1.3 MDS for Exploring Psychological Structures . . . . . . . . . 9
1.4 MDS as aModel of Similarity Judgments . . . . . . . . . . 11
1.5 The Different Roots ofMDS . . . . . . . . . . . . . . . . . . 13
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Constructing MDS Representations 19
2.1 Constructing Ratio MDS Solutions . . . . . . . . . . . . . . 19
2.2 Constructing OrdinalMDS Solutions . . . . . . . . . . . . . 23
2.3 Comparing Ordinal and Ratio MDS Solutions . . . . . . . . 29
2.4 On Flat and Curved Geometries . . . . . . . . . . . . . . . 30
2.5 General Properties of Distance Representations . . . . . . . 33
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 MDS Models and Measures of Fit 37
3.1 Basics ofMDSModels . . . . . . . . . . . . . . . . . . . . . 37
3.2 Errors, Loss Functions, and Stress . . . . . . . . . . . . . . 41
xvi Contents
3.3 Stress Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Stress per Point . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.5 Evaluating Stress . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Recovering True Distances byMetricMDS . . . . . . . . . 55
3.7 Further Variants ofMDSModels . . . . . . . . . . . . . . . 57
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Three Applications of MDS 63
4.1 The Circular Structure of Color Similarities . . . . . . . . . 63
4.2 The Regionality of Morse Codes Confusions . . . . . . . . . 68
4.3 Dimensions of Facial Expressions . . . . . . . . . . . . . . . 73
4.4 General Principles of InterpretingMDS Solutions . . . . . . 80
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5 MDS and Facet Theory 87
5.1 Facets and Regions inMDS Space . . . . . . . . . . . . . . 87
5.2 Regional Laws . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Multiple Facetizations . . . . . . . . . . . . . . . . . . . . . 93
5.4 PartitioningMDS Spaces Using Facet Diagrams . . . . . . . 95
5.5 Prototypical Roles of Facets . . . . . . . . . . . . . . . . . . 99
5.6 Criteria for Choosing Regions . . . . . . . . . . . . . . . . . 100
5.7 Regions and Theory Construction . . . . . . . . . . . . . . . 102
5.8 Regions, Clusters, and Factors . . . . . . . . . . . . . . . . 104
5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 How to Obtain Proximities 111
6.1 Types of Proximities . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Collecting Direct Proximities . . . . . . . . . . . . . . . . . 112
6.3 Deriving Proximities by Aggregating over Other Measures . 119
6.4 Proximities fromConverting OtherMeasures . . . . . . . . 125
6.5 Proximities from Co-Occurrence Data . . . . . . . . . . . . 126
6.6 Choosing a Particular Proximity . . . . . . . . . . . . . . . 128
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
II MDS Models and Solving MDS Problems 135
7 Matrix Algebra for MDS 137
7.1 ElementaryMatrix Operations . . . . . . . . . . . . . . . . 137
7.2 Scalar Functions of Vectors andMatrices . . . . . . . . . . 142
7.3 Computing Distances UsingMatrix Algebra . . . . . . . . . 144
7.4 Eigendecompositions . . . . . . . . . . . . . . . . . . . . . . 146
7.5 Singular Value Decompositions . . . . . . . . . . . . . . . . 150
7.6 Some Further Remarks on SVD . . . . . . . . . . . . . . . . 152
7.7 Linear Equation Systems . . . . . . . . . . . . . . . . . . . 154
Contents xvii
7.8 Computing the Eigendecomposition . . . . . . . . . . . . . 157
7.9 Configurations that Represent Scalar Products . . . . . . . 160
7.10 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8 A Majorization Algorithm for Solving MDS 169
8.1 The Stress Function forMDS . . . . . . . . . . . . . . . . . 169
8.2 Mathematical Excursus: Differentiation . . . . . . . . . . . 171
8.3 Partial Derivatives andMatrix Traces . . . . . . . . . . . . 176
8.4 Minimizing a Function by IterativeMajorization . . . . . . 178
8.5 Visualizing theMajorization AlgorithmforMDS . . . . . . 184
8.6 Majorizing Stress . . . . . . . . . . . . . . . . . . . . . . . . 185
8.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
9 Metric and Nonmetric MDS 199
9.1 Allowing for Transformations of the Proximities . . . . . . . 199
9.2 Monotone Regression . . . . . . . . . . . . . . . . . . . . . . 205
9.3 The Geometry of Monotone Regression . . . . . . . . . . . . 209
9.4 Tied Data in Ordinal MDS . . . . . . . . . . . . . . . . . . 211
9.5 Rank-Images . . . . . . . . . . . . . . . . . . . . . . . . . . 213
9.6 Monotone Splines . . . . . . . . . . . . . . . . . . . . . . . . 214
9.7 A Priori Transformations Versus Optimal Transformations . 221
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10 Confirmatory MDS 227
10.1 Blind Loss Functions . . . . . . . . . . . . . . . . . . . . . . 227
10.2 Theory-CompatibleMDS: An Example . . . . . . . . . . . . 228
10.3 Imposing External Constraints on MDS Representations . . 230
10.4 Weakly ConstrainedMDS . . . . . . . . . . . . . . . . . . . 237
10.5 General Comments on Confirmatory MDS . . . . . . . . . . 242
10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
11 MDS Fit Measures, Their Relations, and
Some Algorithms 247
11.1 Normalized Stress and Raw Stress . . . . . . . . . . . . . . 247
11.2 Other Fit Measures and Recent Algorithms . . . . . . . . . 250
11.3 UsingWeights inMDS . . . . . . . . . . . . . . . . . . . . . 254
11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
12 Classical Scaling 261
12.1 Finding Coordinates in Classical Scaling . . . . . . . . . . . 261
12.2 A Numerical Example for Classical Scaling . . . . . . . . . 263
12.3 Choosing a Different Origin . . . . . . . . . . . . . . . . . . 264
12.4 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . 265
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
xviii Contents
13 Special Solutions, Degeneracies, and Local Minima 269
13.1 A Degenerate Solution in OrdinalMDS . . . . . . . . . . . 269
13.2 Avoiding Degenerate Solutions . . . . . . . . . . . . . . . . 272
13.3 Special Solutions: Almost Equal Dissimilarities . . . . . . . 274
13.4 LocalMinima . . . . . . . . . . . . . . . . . . . . . . . . . . 276
13.5 Unidimensional Scaling . . . . . . . . . . . . . . . . . . . . 278
13.6 Full-Dimensional Scaling . . . . . . . . . . . . . . . . . . . . 281
13.7 The Tunneling Method for Avoiding Local Minima . . . . . 283
13.8 Distance Smoothing for Avoiding LocalMinima . . . . . . . 284
13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
III Unfolding 291
14 Unfolding 293
14.1 The Ideal-PointModel . . . . . . . . . . . . . . . . . . . . . 293
14.2 AMajorizing Algorithmfor Unfolding . . . . . . . . . . . . 297
14.3 Unconditional Versus Conditional Unfolding . . . . . . . . . 299
14.4 Trivial Unfolding Solutions and σ2 . . . . . . . . . . . . . . 301
14.5 Isotonic Regions and Indeterminacies . . . . . . . . . . . . . 305
14.6 Unfolding Degeneracies in Practice and Metric Unfolding . 308
14.7 Dimensions inMultidimensional Unfolding . . . . . . . . . . 312
14.8 Multiple VersusMultidimensional Unfolding . . . . . . . . . 313
14.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 314
14.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
15 Avoiding Trivial Solutions in Unfolding 317
15.1 Adjusting the Unfolding Data . . . . . . . . . . . . . . . . . 317
15.2 Adjusting the Transformation . . . . . . . . . . . . . . . . . 322
15.3 Adjustments to the Loss Function . . . . . . . . . . . . . . 324
15.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
16 Special Unfolding Models 335
16.1 External Unfolding . . . . . . . . . . . . . . . . . . . . . . . 335
16.2 The VectorModel of Unfolding . . . . . . . . . . . . . . . . 336
16.3 Weighted Unfolding . . . . . . . . . . . . . . . . . . . . . . 342
16.4 Value Scales and Distances in Unfolding . . . . . . . . . . . 345
16.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
IV MDS Geometry as a Substantive Model 357
17 MDS as a Psychological Model 359
17.1 Physical and Psychological Space . . . . . . . . . . . . . . . 359
Contents xix
17.2 Minkowski Distances . . . . . . . . . . . . . . . . . . . . . . 363
17.3 Identifying the TrueMinkowski Distance . . . . . . . . . . . 367
17.4 The Psychology of Rectangles . . . . . . . . . . . . . . . . . 372
17.5 Axiomatic Foundations ofMinkowski Spaces . . . . . . . . . 377
17.6 Subadditivity and the MBR Metric . . . . . . . . . . . . . . 381
17.7 Minkowski Spaces, Metric Spaces, and Psychological Models 385
17.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
18 Scalar Products and Euclidean Distances 389
18.1 The Scalar Product Function . . . . . . . . . . . . . . . . . 389
18.2 Collecting Scalar Products Empirically . . . . . . . . . . . . 392
18.3 Scalar Products and Euclidean Distances: Formal Relations 397
18.4 Scalar Products and Euclidean Distances:
Empirical Relations . . . . . . . . . . . . . . . . . . . . . . 400
18.5 MDS of Scalar Products . . . . . . . . . . . . . . . . . . . . 403
18.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
19 Euclidean Embeddings 411
19.1 Distances and Euclidean Distances . . . . . . . . . . . . . . 411
19.2 Mapping Dissimilarities into Distances . . . . . . . . . . . . 415
19.3 Maximal Dimensionality for Perfect IntervalMDS . . . . . 418
19.4 Mapping Fallible Dissimilarities into Euclidean Distances . 419
19.5 Fitting Dissimilarities into a Euclidean Space . . . . . . . . 424
19.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
V MDS and Related Methods 427
20 Procrustes Procedures 429
20.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 429
20.2 Solving the Orthogonal Procrustean Problem . . . . . . . . 430
20.3 Examples for Orthogonal Procrustean Transformations . . . 432
20.4 Procrustean Similarity Transformations . . . . . . . . . . . 434
20.5 An Example of Procrustean Similarity Transformations . . 436
20.6 Configurational Similarity and Correlation Coefficients . . . 437
20.7 Configurational Similarity and Congruence Coefficients . . . 439
20.8 Artificial TargetMatrices in Procrustean Analysis . . . . . 441
20.9 Other Generalizations of Procrustean Analysis . . . . . . . 444
20.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
21 Three-Way Procrustean Models 449
21.1 Generalized Procrustean Analysis . . . . . . . . . . . . . . . 449
21.2 Helm’s Color Data . . . . . . . . . . . . . . . . . . . . . . . 451
21.3 Generalized Procrustean Analysis . . . . . . . . . . . . . . . 454
21.4 Individual DifferencesModels: DimensionWeights . . . . . 457
xx Contents
21.5 An Application of the Dimension-WeightingModel . . . . . 462
21.6 VectorWeightings . . . . . . . . . . . . . . . . . . . . . . . 465
21.7 Pindis, a Collection of ProcrusteanModels . . . . . . . . . 469
21.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
22 Three-Way MDS Models 473
22.1 The Model: Individual Weights on Fixed Dimensions . . . . 473
22.2 The Generalized EuclideanModel . . . . . . . . . . . . . . . 479
22.3 Overview of Three-WayModels inMDS . . . . . . . . . . . 482
22.4 Some Algebra of Dimension-WeightingModels . . . . . . . 485
22.5 Conditional and Unconditional Approaches . . . . . . . . . 489
22.6 On the Dimension-WeightingModels . . . . . . . . . . . . . 491
22.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
23 Modeling Asymmetric Data 495
23.1 Symmetry and Skew-Symmetry . . . . . . . . . . . . . . . . 495
23.2 A Simple Model for Skew-Symmetric Data . . . . . . . . . . 497
23.3 The GowerModel for Skew-Symmetries . . . . . . . . . . . 498
23.4 Modeling Skew-Symmetry by Distances . . . . . . . . . . . 500
23.5 Embedding Skew-Symmetries as Drift Vectors into
MDS Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
23.6 Analyzing Asymmetry by Unfolding . . . . . . . . . . . . . 503
23.7 The Slide-VectorModel . . . . . . . . . . . . . . . . . . . . 506
23.8 The Hill-Climbing Model . . . . . . . . . . . . . . . . . . . 509
23.9 The Radius-DistanceModel . . . . . . . . . . . . . . . . . . 512
23.10 Using AsymmetryModels . . . . . . . . . . . . . . . . . . 514
23.11 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
23.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
24 Methods Related to MDS 519
24.1 Principal Component Analysis . . . . . . . . . . . . . . . . 519
24.2 Correspondence Analysis . . . . . . . . . . . . . . . . . . . . 526
24.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
VI Appendices 541
A Computer Programs for MDS 543
A.1 InteractiveMDS Programs . . . . . . . . . . . . . . . . . . 544
A.2 MDS Programs with High-Resolution Graphics . . . . . . . 550
A.3 MDS Programs without High-Resolution Graphics . . . . . 562
B Notation 569
References 573
Contents xxi
Author Index 599
Subject Index 605