Multidimensional Similarity Structure Analysis

Multidimensional Similarity Structure Analysis comprises a class of models that represent similarity among entities (for example, variables, items, objects, persons, etc.) in multidimensional space to permit one to grasp more easily the interrelations and patterns present in the data. The book is oriented to both researchers who have little or no previous exposure to data scaling and have no more than a high school background in mathematics and to investigators who would like to extend their analyses in the direction of hypothesis and theory testing or to more intimately understand these analytic procedures. The book is repleted with examples and illustrations of the various techniques drawn largely, but not restrictively, from the social sciences, with a heavy emphasis on the concrete, geometric or spatial aspect of the data representations.

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.