.

0 (hodnocen0 x )

BK

2nd ed.

New York : Springer, c2005

xxi, 614 s. : il. ; 24 cm

ISBN 0-387-25150-2 (váz.)

Springer series in statistics

Obsahuje bibliografii na s. [573]-597, bibliografické odkazy a rejstříky

000239871

Contents // 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 a Model of Similarity Judgments... 11 // 1.5 The Different Roots of MDS... 13 // 1.6 Exercises ... 15 // 2 Constructing MDS Representations 19 // 2.1 Constructing Ratio MDS Solutions... 19 // 2.2 Constructing Ordinal MDS 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 of MDS Models... 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 by Metric MDS ... 55 // 3.7 Further Variants of MDS Models... 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 Interpreting MDS Solutions... 80 // 4.5 Exercises ... 82 // 5 MDS and Facet Theory 87 // 5.1 Facets and Regions in MDS Space... 87 // 5.2 Regional Laws ... 91 // 5.3 Multiple Facetizations... 93 // 5.4 Partitioning MDS 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...Ill // 6.2 Collecting Direct Proximities...112 // 6.3 Deriving Proximities by Aggregating over Other Measures . 119 // 6.4 Proximities from Converting Other Measures...125 // 6.5 Proximities from Co-Occurrence Data...126 // 6.6 Choosing a Particular Proximity...128 // 6.7 Exercises ...130 // w // II MDS Models and Solving MDS Problems 135 // 7 Matrix Algebra for MDS 137 // 7.1 Elementary Matrix Operations...137 // 7.2 Scalar Functions of Vectors and Matrices ...142 // 7.3 Computing Distances Using Matrix 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 for MDS... 169 // 8.2 Mathematical Excursus: Differentiation ... 171 // 8.3 Partial Derivatives and Matrix Traces... 176 // 8.4 Minimizing a Function by Iterative Majorization...178 // 8.5 Visualizing the Majorization Algorithm for MDS... 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-Compatible MDS: An Example...228 // 10.3 Imposing External Constraints on MDS Representations . . 230 // 10.4 Weakly Constrained MDS...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 Using Weights in MDS...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 Ordinal MDS...269 // 13.2 Avoiding Degenerate Solutions...272 // 13.3 Special Solutions: Almost Equal Dissimilarities...274 // 13.4 Local Minima...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 Local Minima...284 // 13.9 Exercises ...288 // III Unfolding 291 // 14 Unfolding 293 // 14.1 The Ideal-Point Model...293 // 14.2 A Majorizing Algorithm for Unfolding...297 // 14.3 Unconditional Versus Conditional Unfolding...299 // 14.4 Trivial Unfolding Solutions and a2...301 // 14.5 Isotonic Regions and Indeterminacies...305 // 14.6 Unfolding Degeneracies in Practice and Metric Unfolding . 308 // 14.7 Dimensions in Multidimensional Unfolding...312 // 14.8 Multiple Versus Multidimensional 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 Vector Model 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 // XÍX // 17.2 Minkowski Distances...363 // 17.3 Identifying the True Minkowski Distance...367 // 17.4 The Psychology of Rectangles...372 // 17.5 Axiomatic Foundations of Minkowski 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 Interval MDS ...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 Target Matrices 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 Differences Models: Dimension Weights...457 // XX Contents // 21.5 An Application of the Dimension-Weighting Model...462 // 21.6 Vector Weightings...465 // 21.7 PiNDlS, a Collection of Procrustean Models...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 Euclidean Model...479 // 22.3 Overview of Three-Way Models in MDS...482 // 22.4 Some Algebra of Dimension-Weighting Models ...485 // 22.5 Conditional and Unconditional Approaches...489 // 22.6 On the Dimension-Weighting Models...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 Gower Model 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-Vector Model...506 // 23.8 The Hill-Climbing Model ...509 // 23.9 The Radius-Distance Model...512 // 23.10 Using Asymmetry Models ...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.l Interactive MDS Programs ...544 // A.2 MDS Programs

with High-Resolution Graphics...550 // A.3 MDS Programs without High-Resolution Graphics...562 // ? Notation 569 // References // 573