Modern Multidimensional Scaling

Transcription

1 Ingwer Borg Patrick Groenen Modern Multidimensional Scaling Theory and Applications With 116 Figures Springer

2 Contents Preface vii I Fundamentals of MDS 1 1 The Four Purposes of Multidimensional Scaling MDS as an Exploratory Technique MDS for Testing Structural Hypotheses MDS for Exploring Psychological Structures MDS as a Model of Similarity Judgments The Different Roots of MDS 13 2 Constructing MDS Representations Constructing Ratio MDS Solutions Constructing Ordinal MDS Solutions Comparing Ordinal and Ratio MDS Solutions On Flat and Curved Geometries 26 3 MDS Models and Measures of Fit Basics of MDS Models Errors, Loss Functions, and Stress Stress Diagrams Evaluating Stress Recovering True Distances by Metric MDS 45

3 xiv Contents 3.6 Further Variants of MDS Models 47 4 Three Applications of MDS The Circular Structure of Color Similarities The Regionality of Morse Codes Confusions Dimensions of Facial Expressions General Principles of Interpreting MDS Solutions 68 5 MDS and Facet Theory Facets and Regions in MDS Space Regional Laws Multiple Facetizations Partitioning MDS Spaces Using Facet Diagrams Prototypical Roles of Facets Criteria for Choosing Regions Regions and Theory Construction Regions, Clusters, and Factors 87 6 How to Obtain Proximities Types of Proximities Collecting Direct Proximities Deriving Proximities by Aggregating over Other Measures Proximities from Converting Other Measures Proximities from Co-occurrence Data Choosing a Particular Proximity 104 II MDS Models and Solving MDS Problems Matrix Algebra for MDS Elementary Matrix Operations Scalar Functions of Vectors and Matrices Computing Distances Using Matrix Algebra Eigendecompositions Singular Value Decompositions Some Further Remarks on SVD Linear Equation Systems Computing the Eigendecomposition Configurations that Represent Scalar Products Rotations A Majorization Algorithm for Solving MDS The Stress Function for MDS Mathematical Excursus: Differentiation Partial Derivatives and Matrix Traces 142

4 Contents xv 8.4 Minimizing a Function by Iterative Majorization Majorizing Stress Metric and Nonmetric MDS Allowing for Transformations of the Proximities Monotone Regression The Geometry of Monotone Regression Tied Data in Ordinal MDS Rank-Images Monotone Splines Confirmatory MDS Blind Loss Functions Theory-Compatible MDS: An Example Imposing External Constraints on MDS Representations Weakly Constrained MDS General Comments on Confirmatory MDS MDS Fit Measures, Their Relations, and Some Algorithms Normalized Stress and Raw Stress Other Fit Measures and Recent Algorithms Classical Scaling Finding Coordinates in Classical Scaling A Numerical Example for Classical Scaling Choosing a Different Origin Advanced Topics Special Solutions, Degeneracies, and Local Minima Special Solutions: Almost Equal Dissimilarities A Degenerate Solution in Ordinal MDS Avoiding Degenerate Solutions Local Minima Unidimensional Scaling Full-Dimensional Scaling The Tunneling Method for Avoiding Local Minima 227 III Unfolding Unfolding The Ideal-Point Model A Majorizing Algorithm for Unfolding Unconditional Versus Conditional Unfolding Trivial Unfolding Solutions and a 2 239

5 xvi Contents 14.5 Isotonic Regions and Indeterminacies Unfolding Degeneracies in Practice and Metric Unfolding An Ordinal-Interval Approach to Unfolding Dimensions in Multidimensional Unfolding Multiple Versus Multidimensional Unfolding Special Unfolding Models External Unfolding The Vector Model of Unfolding Weighted Unfolding Value Scales and Distances in Unfolding 263 IV MDS Geometry as a Substantive Model MDS as a Psychological Model Physical and Psychological Space Minkowski Distances Identifying the True Minkowski Distance The Psychology of Rectangles Axiomatic Foundations of Minkowski Spaces Subadditivity and the MBR Metric Minkowski Spaces, Metric Spaces, and Psychological Models Scalar Products and Euclidean Distances The Scalar Product Function Collecting Scalar Products Empirically Scalar Products and Euclidean Distances: Formal Relations Scalar Products and Euclidean Distances: Empirical Relations MDS of Scalar Products Euclidean Embeddings Distances and Euclidean Distances Mapping Proximities into Distances Maximal Dimensionality for Perfect Interval MDS Mapping Fallible Proximities into Euclidean Distances Fitting Proximities into a Euclidean Space 334 V MDS and Related Methods Procrustes Procedures The Problem Solving the Orthogonal Procrustean Problem Examples for Orthogonal Procrustean Transformations Procrustean Similarity Transformations 344

6 Contents xvii 19.5 An Example of Procrustean Similarity Transformations Measuring Configurational Similarity by the Correlation Coefficient Measuring Configurational Similarity by the Congruence Coefficient Artificial Target Matrices in Procrustean Analysis Other Generalizations of Procrustean Analysis Three-Way Procrustean Models Generalized Procrustean Analysis Helm's Color Data Generalized Procrustean Analysis Individual Differences Models: Dimension Weights An Application of the Dimension-Weighting Model Vector Weightings PINDIS, a Collection of Procrustean Models Three-Way MDS Models The Model: Individual Weights on Fixed Dimensions The Generalized Euclidean Model Some Algebra of Dimension-Weighting Models Conditional and Unconditional Approaches On the Dimension-Weighting Models Methods Related to MDS Principal Component Analysis Models for Asymmetric Data Correspondence Analysis 408 VI Appendices 417 A Computer Programs for MDS 419 B Notation 435 References 437 Author Index 457 Subject Index 463