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| Preface | |
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| Fundamentals of MDS | |
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| The Four Purposes of Multidimensional Scaling | |
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| MDS as an Exploratory Technique | |
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| MDS for Testing Structural Hypotheses | |
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| MDS for Exploring Psychological Structures | |
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| MDS as a Model of Similarity Judgments | |
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| The Different Roots of MDS | |
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| Exercises | |
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| Constructing MDS Representations | |
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| Constructing Ratio MDS Solutions | |
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| Constructing Ordinal MDS Solutions | |
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| Comparing Ordinal and Ratio MDS Solutions | |
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| On Flat and Curved Geometries | |
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| General Properties of Distance Representations | |
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| Exercises | |
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| MDS Models and Measures of Fit | |
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| Basics of MDS Models | |
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| Errors, Loss Functions, and Stress | |
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| Stress Diagrams | |
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| Stress per Point | |
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| Evaluating Stress | |
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| Recovering True Distances by Metric MDS | |
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| Further Variants of MDS Models | |
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| Exercises | |
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| Three Applications of MDS | |
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| The Circular Structure of Color Similarities | |
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| The Regionality of Morse Codes Confusions | |
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| Dimensions of Facial Expressions | |
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| General Principles of Interpreting MDS Solutions | |
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| Exercises | |
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| MDS and Facet Theory | |
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| Facets and Regions in MDS Space | |
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| Regional Laws | |
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| Multiple Facetizations | |
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| Partitioning MDS Spaces Using Facet Diagrams | |
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| Prototypical Roles of Facets | |
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| Criteria for Choosing Regions | |
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| Regions and Theory Construction | |
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| Regions, Clusters, and Factors | |
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| Exercises | |
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| How to Obtain Proximities | |
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| Types of Proximities | |
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| Collecting Direct Proximities | |
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| Deriving Proximities by Aggregating over Other Measures | |
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| Proximities from Converting Other Measures | |
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| Proximities from Co-Occurrence Data | |
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| Choosing a Particular Proximity | |
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| Exercises | |
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| MDS Models and Solving MDS Problems | |
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| Matrix Algebra for MDS | |
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| Elementary Matrix Operations | |
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| Scalar Functions of Vectors and Matrices | |
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| Computing Distances Using Matrix Algebra | |
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| Eigendecompositions | |
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| Singular Value Decompositions | |
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| Some Further Remarks on SVD | |
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| Linear Equation Systems | |
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| Computing the Eigendecomposition | |
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| Configurations that Represent Scalar Products | |
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| Rotations | |
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| Exercises | |
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| A Majorization Algorithm for Solving MDS | |
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| The Stress Function for MDS | |
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| Mathematical Excursus: Differentiation | |
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| Partial Derivatives and Matrix Traces | |
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| Minimizing a Function by Iterative Majorization | |
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| Visualizing the Majorization Algorithm for MDS | |
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| Majorizing Stress | |
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| Exercises | |
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| Metric and Nonmetric MDS | |
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| Allowing for Transformations of the Proximities | |
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| Monotone Regression | |
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| The Geometry of Monotone Regression | |
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| Tied Data in Ordinal MDS | |
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| Rank-Images | |
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| Monotone Splines | |
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| A Priori Transformations Versus Optimal Transformations | |
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| Exercises | |
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| Confirmatory MDS | |
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| Blind Loss Functions | |
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| Theory-Compatible MDS: An Example | |
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| Imposing External Constraints on MDS Representations | |
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| Weakly Constrained MDS | |
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| General Comments on Confirmatory MDS | |
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| Exercises | |
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| MDS Fit Measures, Their Relations, and Some Algorithms | |
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| Normalized Stress and Raw Stress | |
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| Other Fit Measures and Recent Algorithms | |
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| Using Weights in MDS | |
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| Exercises | |
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| Classical Scaling | |
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| Finding Coordinates in Classical Scaling | |
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| A Numerical Example for Classical Scaling | |
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| Choosing a Different Origin | |
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| Advanced Topics | |
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| Exercises | |
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| Special Solutions, Degeneracies, and Local Minima | |
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| A Degenerate Solution in Ordinal MDS | |
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| Avoiding Degenerate Solutions | |
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| Special Solutions: Almost Equal Dissimilarities | |
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| Local Minima | |
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| Unidimensional Scaling | |
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| Full-Dimensional Scaling | |
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| The Tunneling Method for Avoiding Local Minima | |
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| Distance Smoothing for Avoiding Local Minima | |
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| Exercises | |
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| Unfolding | |
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| Unfolding | |
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| The Ideal-Point Model | |
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| A Majorizing Algorithm for Unfolding | |
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| Unconditional Versus Conditional Unfolding | |
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| Trivial Unfolding Solutions and [sigma subscript 2] | |
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| Isotonic Regions and Indeterminacies | |
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| Unfolding Degeneracies in Practice and Metric Unfolding | |
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| Dimensions in Multidimensional Unfolding | |
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| Multiple Versus Multidimensional Unfolding | |
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| Concluding Remarks | |
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| Exercises | |
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| Avoiding Trivial Solutions in Unfolding | |
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| Adjusting the Unfolding Data | |
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| Adjusting the Transformation | |
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| Adjustments to the Loss Function | |
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| Summary | |
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| Exercises | |
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| Special Unfolding Models | |
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| External Unfolding | |
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| The Vector Model of Unfolding | |
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| Weighted Unfolding | |
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| Value Scales and Distances in Unfolding | |
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| Exercises | |
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| MDS Geometry as a Substantive Model | |
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| MDS as a Psychological Model | |
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| Physical and Psychological Space | |
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| Minkowski Distances | |
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| Identifying the True Minkowski Distance | |
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| The Psychology of Rectangles | |
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| Axiomatic Foundations of Minkowski Spaces | |
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| Subadditivity and the MBR Metric | |
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| Minkowski Spaces, Metric Spaces, and Psychological Models | |
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| Exercises | |
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| Scalar Products and Euclidean Distances | |
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| The Scalar Product Function | |
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| Collecting Scalar Products Empirically | |
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| Scalar Products and Euclidean Distances: Formal Relations | |
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| Scalar Products and Euclidean Distances: Empirical Relations | |
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| MDS of Scalar Products | |
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| Exercises | |
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| Euclidean Embeddings | |
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| Distances and Euclidean Distances | |
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| Mapping Dissimilarities into Distances | |
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| Maximal Dimensionality for Perfect Interval MDS | |
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| Mapping Fallible Dissimilarities into Euclidean Distances | |
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| Fitting Dissimilarities into a Euclidean Space | |
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| Exercises | |
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| MDS and Related Methods | |
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| Procrustes Procedures | |
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| The Problem | |
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| Solving the Orthogonal Procrustean Problem | |
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| Examples for Orthogonal Procrustean Transformations | |
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| Procrustean Similarity Transformations | |
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| An Example of Procrustean Similarity Transformations | |
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| Configurational Similarity and Correlation Coefficients | |
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| Configurational Similarity and Congruence Coefficients | |
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| Artificial Target Matrices in Procrustean Analysis | |
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| Other Generalizations of Procrustean Analysis | |
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| Exercises | |
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| Three-Way Procrustean Models | |
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| Generalized Procrustean Analysis | |
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| Helm's Color Data | |
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| Generalized Procrustean Analysis | |
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| Individual Differences Models: Dimension Weights | |
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| An Application of the Dimension-Weighting Model | |
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| Vector Weightings | |
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| Pindis, a Collection of Procrustean Models | |
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| Exercises | |
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| Three-Way MDS Models | |
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| The Model: Individual Weights on Fixed Dimensions | |
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| The Generalized Euclidean Model | |
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| Overview of Three-Way Models in MDS | |
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| Some Algebra of Dimension-Weighting Models | |
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| Conditional and Unconditional Approaches | |
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| On the Dimension-Weighting Models | |
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| Exercises | |
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| Modeling Asymmetric Data | |
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| Symmetry and Skew-Symmetry | |
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| A Simple Model for Skew-Symmetric Data | |
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| The Gower Model for Skew-Symmetries | |
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| Modeling Skew-Symmetry by Distances | |
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| Embedding Skew-Symmetries as Drift Vectors into MDS Plots | |
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| Analyzing Asymmetry by Unfolding | |
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| The Slide-Vector Model | |
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| The Hill-Climbing Model | |
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| The Radius-Distance Model | |
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| Using Asymmetry Models | |
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| Overview | |
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| Exercises | |
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| Methods Related to MDS | |
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| Principal Component Analysis | |
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| Correspondence Analysis | |
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| Exercises | |
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| Appendices | |
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| Computer Programs for MDS | |
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| Interactive MDS Programs | |
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| MDS Programs with High-Resolution Graphics | |
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| MDS Programs without High-Resolution Graphics | |
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| Notation | |
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| References | |
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| Author Index | |
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| Subject Index | |