Multidimensional Similarity Structure Analysis

1 - Construction of SSA Representations.- 1.1 Finding a Point Representation for a Set of Distances.- 1.2 Admissible Transformations.- 1.3 Outline of an Ordinal SSA Procedure.- 1.4 Solution Sets in Ordinal SSA.- 1.5 Comparing Ordinal and Ratio SSA Solutions.- 1.6 Isotonic Transformations.- 2 - Ordinal SSA by Iterative Optimization.- 2.1 Simultaneous Point Movements and Force Vectors.- 2.2 Constructing Point Movements.- 2.3 Introducing a Cartesian Coordinate System.- 2.4 Computing Correction Factors.- 2.5 Mathematical Excursus: Vectors.- 2.6 Computing Point Translations.- 2.7 Applying the Formulas.- 3 - Monotone Regression.- 3.1 Shepard Diagrams and Monotone Functions.- 3.2 Disparities as Target Distances.- 3.3 Moving Points Relative to Disparities.- 3.4 Computing Disparities.- 3.5 Tied Data.- 3.6 Missing Data.- 3.7 Computing Target Distances with Tied and Missing Data.- 4 - SSA Models, Measures of Fit, and Their Optimization.- 4.1 Some SSA Models.- 4.2 Errors and Measures of Fit.- 4.3 Minimization of Stress and Alienation.- 4.4 Mathematical Excursus: Differentiation.- 4.5 Determining Point Movements by Differentiation.- 4.6 Some Problems of the Gradient Method.- 5 - Three Applications of SSA.- 5.1 An Ordinal SSA of Some Data on Color Perception.- 5.2 Conditional Error Measures.- 5.3 Overall Criteria for Goodness-of-Fit of an SSA Solution.- 5.4 Some Similarity Data on Morse Codes.- 5.5 An SSA of the Symmetrized Data.- 5.6 A Dimensional Theory for the Perception of Facial Expressions.- 5.7 Ordinal SSA Representations and the Schlosberg Scales.- 5.8 Fitting External Scales, Conditional Error Measures.- 6 - SSA and Facet Theory.- 6.1 Dimensions and Partitions.- 6.2 A Study of Wellbeing: Design.- 6.3 Regional Hypotheses.- 6.4 Simplexes.- 6.5 A Second Study on Wellbeing.- 6.6 Hypotheses on Partial Structures and SSA Analyses.- 6.7 Hypotheses on Global Structure.- 6.8 SSA Analyses of Global Hypotheses.- 6.9 Discussion.- 7 - Degenerate Solutions in Ordinal SSA.- 7.1 Degenerate Ordinal SSA Representations: An Example.- 7.2 Properties of Degenerate Solutions.- 7.3 Avoiding Degeneracies: Metric SSA.- 7.4 Avoiding Degeneracies: Scaling Data Subsets; Increasing Dimensionality; Local Criteria.- 7.5 Avoiding Degeneracies: Reflecting Variables.- 8 - Computer Simulation Studies on SSA.- 8.1 Data, Error, and Distances.- 8.2 Stress for Random Data.- 8.3 Stress for Data with Different Error Components.- 8.4 Empirical and Simulated Stress Functions.- 8.5 Recovering Known Distances under Noise Conditions.- 8.6 Minkowski Distances and Over/Under-Compression.- 8.7 Subsampling.- 8.8 Recovering a Known Monotonic Transformation Function.- 8.9 Recovery for Incomplete Data.- 8.10 Recovery for Degraded Data.- 8.11 Metric Determinacy of Metric and Rank-Linear SSA under Monotone Transformations of the Data.- 9 - Multidimensional Unfolding.- 9.1 Within- and Between-Proximities: Off-Diagonal Corner Matrices.- 9.2 Unconditional Unfolding.- 9.3 Trivial Unfolding Solutions and S2.- 9.4 Conditional Unfolding.- 9.5 Isotonic Regions.- 9.6 Metric Determinacies and Partial Degeneracies.- 9.7 Some Remarks on Metric Conditional Unfolding.- 10 - Generalized and Metric Unfolding.- 10.1 External Unfolding.- 10.2 Weighted Unfolding.- 10.3 The Vector Model of Unfolding.- 10.4 Subjective Value Scales and Distances in Unfolding.- 10.5 Problems in Dimensional Interpretations in Multidimensional Unfolding.- 11 - Generalized SSA Procedures.- 11.1 SSA for a Block-Partitioned Data Matrix.- 11.2 SSA for Replicated Data.- 11.3 A Generalized Loss Function.- 11.4 Degeneration in Unfolding Revisited.- 11.5 Some Illustrations of Unfolding Degeneracies.- 11.6 An Ordinal-Interval Approach to Unfolding.- 12 - Confirmatory SSA (1).- 12.1 Blind Loss Functions.- 12.2 Theory-compatible SSA: An Example.- 12.3 Imposing External Constraints on SSA Representations.- 12.4 A Further Example for Defining External Constraints.- 12.5 Enforcing Order Constraints onto SSA Distances.- 13 - Confirmatory SSA (2).- 13.1 Comparing Fit and Equivalency of Different SSA Representations.- 13.2 Some Forms of Contiguity.- 13.3 A System of Contiguity Forms.- 13.4 Biconditional Structures: Simplex and Circumplex.- 14 - Physical and Psychological Spaces.- 14.1 Physical and Psychological Spaces: An Example.- 14.2 Using Ordinal MDS to Find the True Generalization Function.- 14.3 Minkowski Metrics.- 14.4 Physical Stimulus Space and Different Minkowski Metrics: An Experiment.- 14.5 Identifying the True Minkowski Parameter.- 14.6 Robustness of the Euclidean Metric When Another Minkowski Metric is True.- 14.7 Minkowski Distances and Other Composition Rules.- 15 - SSA as Multidimensional Scaling.- 15.1 Multidimensional Scaling.- 15.2 MDS with the City-Block Metric as a Composition Rule.- 15.3 Choosing between Different Dimension Systems.- 15.4 Some More General Conclusions.- 16 - Scalar Products.- 16.1 Scalar Products.- 16.2 Data Collection Procedures Yielding Scalar Products: A Psychophysical Example.- 16.3 SSA of Scalar Products.- 16.4 Scalar Products and Empirical Data: An Example on Emotions.- 16.5 Scalar Products and Distances: Formal Relations.- 16.6 Scalar Products and Distances: Empirical Relations.- 16.7 SSA Representations of v- and p-Data.- 17 - Matrix Algebra for SSA.- 17.1 Elementary Matrix Operations.- 17.2 Linear Equation Systems and Matrix Inverses.- 17.3 Finding a Configuration that Represents Scalar Products.- 17.4 Rotations to Principal Axes.- 17.5 Eigendecompositions.- 17.6 Computing Eigenvalues.- 18 - Mappings of Data in Distances.- 18.1 Scalar-Product Matrices and Distances: Positive Semi-Definiteness.- 18.2 Distances and Euclidean Distances.- 18.3 Proximities and Distances: An Algebraic View.- 18.4 Interval and Ordinal Proximities and Dimensionality of their SSA Representations.- 18.5 Interval Proximities and Distances: A Statistical View.- 18.6 Interval Proximities and Distances: An Optimization View.- 19 - Procrustes Procedures.- 19.1 The Problem.- 19.2 Differentiation of Matrix Traces and the Linear Procrustes Problem.- 19.3 Mathematical Excursus: Differentiation under Side Constraints.- 19.4 Solving the Orthogonal Procrustean Problem.- 19.5 Examples for Orthogonal Procrustean Transformations.- 19.6 Procrustean Similarity Transformations.- 19.7 An Example for Procrustean Similarity Transformations.- 19.8 Artificial Target Matrices and other Generalizations.- 19.9 Measuring Configurational Similarity by an Index.- 20 - Individual Differences Models.- 20.1 Generalized Procrustean Analysis.- 20.2 Individual Differences Models: Dimensional Weightings.- 20.3 An Application of the Dimensional-Weighting Model.- 20.4 Vector Weightings.- 20.5 PINDIS.- 20.6 Direct Approaches to Dimensional-Weighting Models.- 20.7 INDSCAL.- 20.8 Some Algebraic Properties of Dimensional-Weighting Models.- 20.9 Matrix-Conditional and Unconditional Approaches.- References.- Name Index.