Modern Multidimensional Scaling

Transcription

1 Ingwer Borg Patrick J.F. Groenen Modern Multidimensional Scaling Theory and Applications Second Edition With 176 Illustrations ~ Springer

2 Preface vii I Fundamentals of MDS 1 1 The Four Purposes of Multidimensional Scaling 1.1 MDS as an Exploratory Technique. 1.2 MDS for Testing Structural Hypotheses MDS for Exploring Psychological Structures. 1.4 MDS as a Model of Similarity Judgments 1.5 The Different Roots of MDS. 1.6 Exercises. 2 Constructing MDS Representations 2.1 Constructing Ratio MDS Solutions. 2.2 Constructing Ordinal MDS Solutions Comparing Ordinal and Ratio MDS Solutions 2.4 On Flat and Curved Geometries. 2.5 General Properties of Distance Representations 2.6 Exercises. 3 MDS Models and Measures of Fit 3.1 Basics of MDS Models Errors, Loss Functions, and Stress

3 XVI 3.3 Stress Diagrams. 3.4 Stress per Point. 3.5 Evaluating Stress 3.6 Recovering True Distances by Metric MDS 3.7 Further Variants of MDS Models 3.8 Exercises 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 Exercises 82 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 Exercises 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 Exercises MDS Models and Solving MDS Problems Matrix Algebra for MDS 7.1 Elementary Matrix Operations. 7.2 Scalar Functions of Vectors and Matrices 7.3 Computing Distances Using Matrix Algebra 7.4 Eigendecompositions. 7.5 Singular Value Decompositions 7.6 Some Further Remarks on SVD 7.7 Linear Equation Systems

4 XVll 7.8 Computing the Eigendecomposition Configurations that Represent Scalar Products Rotations Exercises A Majorization Algorithm for Solving MDS The Stress Punction for MDS Mathematical Excursus: Differentiation Partial Derivatives and Matrix Traces Minimizing a Function by Iterative Majorization Visualizing the Majorization Algorithm for MDS Majorizing Stress Exercises Metric and Nonmetric MDS Allowing for Transformations of thc Proximitieti Monotone Regression The Geomctry of Monotone Regression Tied Data in Ordinal MDS Rank-Images., Monotone Splines APriori Transformations Versus Optimal TransfoI'Inations Exercises Confirmatory MDS Blind Loss Punctions Theory-Compatible MDS: An Example Imposing External Constraints on MDS Representations Weakly Constrained MDS General Comments on Confirmatory MDS Exercises MDS Fit Measures, Their Relations, and Some Algorithms Normalized Stress and Raw Stress Other Fit Measures and Recent Algorithms Using Weights in MDS Exercises Classical Scaling Finding Coordinates in Classical Scaling A Numerical Example for Classical Scaling Choosing a Different Origin Advanced Topics Exercises 267

5 xviii 13 Special Solutions, Degeneracies, and Local Minima ADegenerate Solution in Ordinal MDS Avoiding Degenerate Solutions Special Solutions: Almost Equal Dissimilarities Local Minima Unidimensional Scaling Full-Dimensional Scaling The Thnneling Method for Avoiding Local Minima Distance Smoothing for Avoiding Local Minima Exercises 288 III U nfolding Unfolding The Ideal-Point Model A Majorizing Algorithm for Unfolding Unconditional Versus Conditional Unfolding Trivial Unfolding Solutions and Isotonic Regions and Indeterminacies Unfolding Degeneracies in Practice and Metric Unfolding Dimensions in Multidimensional Unfolding Multiple Versus Multidimensional Unfolding Concluding Remarks Exercises Avoiding Trivial Solutions in Unfolding Adjusting the Unfolding Data Adjusting the Transformation Adjustments to the Loss Function Summary Exercises Special Unfolding Models External Unfolding The Vector Model of Unfolding Weighted Unfolding Value Scales and Distances in Unfolding Exercises 352 IV MDS Geometry as a Substantive Model MDS as a Psychological Model 17.1 Physical and Psychologieal Space

6 xix 17.2 Minkowski Distances Identifying the True Minkowski Distance Thc Psychology of Rectangles Axiomatic Foundations of Minkowski Spaces Subadditivity and the MBR Metric Minkowski Spaces, Metric Spaces, and Psychological Models Exercises 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 Exercises Euclidean Embeddings Distances and Euclidean Distances Mapping Dissimilarities into Distances Maximal Dimensionality for Perfect Interval MDS Mapping Fallible Dissimilarities into Euclidean Distances Fitting Dissimilarities into a Euclidean Space Exercises 425 V MDS and Related Methods Procrustes Procedures The Problem Solving the Orthogonal Procrustean Problem Examples for Orthogonal Procrustean Transformations Procrustean Similarity Transformations An Example of Procrustean Similarity Transformations Configurational Similarity and Correlation Coefficients Configurational Similarity and Congruence Coefficients Artificial Target Matrices in Procrustean Analysis Other Generalizations of Procrustean Analysis Exercises Three-Way Procrustean Models Generalized Procrustean Analysis Helm's Color Data Generalized Procrustean Analysis Individual Differences Models: Dimension Weights 457

7 xx 21.5 An Application of the Dimension-Weighting Model 21.6 Vector Weightings PINDIS, a Collection of Procrustean Models 21.8 Exercises Three-Way MDS Models The Model: Individual Weights on Fixed Dimensions The Generalized Euclidean Model Overview of Three-Way Models in MDS Some Algebra of Dimension-Weighting Models Conditional and Unconditional Approaches On the Dimension-Weighting Models Exercises Modeling Asymmetrie Data Symmetry and Skew-Symmetry A Simple Model for Skew-Symmetric Data The Gower Model for Skew-Symmetries Modeling Skew-Symmetry by Distances Embedding Skew-Symmetries as Drift Vectors into MDS Plots Analyzing Asymmetry by Unfolding The Slide-Vector Model The Hill-Climbing Model The Radius-Distance Model Using Asymmetry Models Overview Exercises Methods Related to MDS Principal Component Analysis Correspondence Analysis Exercises 537 VI Appendices 541 A B Computer Programs for MDS A.1 Interactive MDS Programs. A.2 MDS Programs with High-Resolution Graphics.. A.3 MDS Programs without High-Resolution Graphics Notation References 573

8 xxi Author Index 599 Subjeet Index 605