Modern multidimensional scaling : theory and applications /

Modern multidimensional scaling : theory and applications /

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Bibliographic Details
Author / Creator:Borg, Ingwer.
Edition:2nd ed.
Imprint:New York : Springer, c2005.
Description:xxi, 614 p. : ill. ; 24 cm.
Language:English
Series:Springer series in statistics
Subject:
Multidimensional scaling.
Multidimensional scaling -- Data processing.
Psychometrics.
Multidimensional scaling.
Multidimensional scaling -- Data processing.
Psychometrics.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/5723832
Hidden Bibliographic Details
Other authors / contributors:Groenen, Patrick J. F.
ISBN:0387251502
Notes:Includes bibliographical references and index.
Table of Contents:
  • Preface
  • I. Fundamentals of MDS
  • 1. The Four Purposes of Multidimensional Scaling
  • 1.1. MDS as an Exploratory Technique
  • 1.2. MDS for Testing Structural Hypotheses
  • 1.3. 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
  • 2.3. 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
  • 3.2. Errors, Loss Functions, and Stress
  • 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
  • 4. Three Applications of MDS
  • 4.1. The Circular Structure of Color Similarities
  • 4.2. The Regionality of Morse Codes Confusions
  • 4.3. Dimensions of Facial Expressions
  • 4.4. General Principles of Interpreting MDS Solutions
  • 4.5. Exercises
  • 5. MDS and Facet Theory
  • 5.1. Facets and Regions in MDS Space
  • 5.2. Regional Laws
  • 5.3. Multiple Facetizations
  • 5.4. Partitioning MDS Spaces Using Facet Diagrams
  • 5.5. Prototypical Roles of Facets
  • 5.6. Criteria for Choosing Regions
  • 5.7. Regions and Theory Construction
  • 5.8. Regions, Clusters, and Factors
  • 5.9. Exercises
  • 6. How to Obtain Proximities
  • 6.1. Types of Proximities
  • 6.2. Collecting Direct Proximities
  • 6.3. Deriving Proximities by Aggregating over Other Measures
  • 6.4. Proximities from Converting Other Measures
  • 6.5. Proximities from Co-Occurrence Data
  • 6.6. Choosing a Particular Proximity
  • 6.7. Exercises
  • II. MDS Models and Solving MDS Problems
  • 7. 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
  • 7.8. Computing the Eigendecomposition
  • 7.9. Configurations that Represent Scalar Products
  • 7.10. Rotations
  • 7.11. Exercises
  • 8. A Majorization Algorithm for Solving MDS
  • 8.1. The Stress Function for MDS
  • 8.2. Mathematical Excursus: Differentiation
  • 8.3. Partial Derivatives and Matrix Traces
  • 8.4. Minimizing a Function by Iterative Majorization
  • 8.5. Visualizing the Majorization Algorithm for MDS
  • 8.6. Majorizing Stress
  • 8.7. Exercises
  • 9. Metric and Nonmetric MDS
  • 9.1. Allowing for Transformations of the Proximities
  • 9.2. Monotone Regression
  • 9.3. The Geometry of Monotone Regression
  • 9.4. Tied Data in Ordinal MDS
  • 9.5. Rank-Images
  • 9.6. Monotone Splines
  • 9.7. A Priori Transformations Versus Optimal Transformations
  • 9.8. Exercises
  • 10. Confirmatory MDS
  • 10.1. Blind Loss Functions
  • 10.2. Theory-Compatible MDS: An Example
  • 10.3. Imposing External Constraints on MDS Representations
  • 10.4. Weakly Constrained MDS
  • 10.5. General Comments on Confirmatory MDS
  • 10.6. Exercises
  • 11. MDS Fit Measures, Their Relations, and Some Algorithms
  • 11.1. Normalized Stress and Raw Stress
  • 11.2. Other Fit Measures and Recent Algorithms
  • 11.3. Using Weights in MDS
  • 11.4. Exercises
  • 12. Classical Scaling
  • 12.1. Finding Coordinates in Classical Scaling
  • 12.2. A Numerical Example for Classical Scaling
  • 12.3. Choosing a Different Origin
  • 12.4. Advanced Topics
  • 12.5. Exercises
  • 13. Special Solutions, Degeneracies, and Local Minima
  • 13.1. A Degenerate Solution in Ordinal MDS
  • 13.2. Avoiding Degenerate Solutions
  • 13.3. Special Solutions: Almost Equal Dissimilarities
  • 13.4. Local Minima
  • 13.5. Unidimensional Scaling
  • 13.6. Full-Dimensional Scaling
  • 13.7. The Tunneling Method for Avoiding Local Minima
  • 13.8. Distance Smoothing for Avoiding Local Minima
  • 13.9. Exercises
  • III. Unfolding
  • 14. Unfolding
  • 14.1. The Ideal-Point Model
  • 14.2. A Majorizing Algorithm for Unfolding
  • 14.3. Unconditional Versus Conditional Unfolding
  • 14.4. Trivial Unfolding Solutions and [sigma subscript 2]
  • 14.5. Isotonic Regions and Indeterminacies
  • 14.6. Unfolding Degeneracies in Practice and Metric Unfolding
  • 14.7. Dimensions in Multidimensional Unfolding
  • 14.8. Multiple Versus Multidimensional Unfolding
  • 14.9. Concluding Remarks
  • 14.10. Exercises
  • 15. Avoiding Trivial Solutions in Unfolding
  • 15.1. Adjusting the Unfolding Data
  • 15.2. Adjusting the Transformation
  • 15.3. Adjustments to the Loss Function
  • 15.4. Summary
  • 15.5. Exercises
  • 16. Special Unfolding Models
  • 16.1. External Unfolding
  • 16.2. The Vector Model of Unfolding
  • 16.3. Weighted Unfolding
  • 16.4. Value Scales and Distances in Unfolding
  • 16.5. Exercises
  • IV. MDS Geometry as a Substantive Model
  • 17. MDS as a Psychological Model
  • 17.1. Physical and Psychological Space
  • 17.2. Minkowski Distances
  • 17.3. Identifying the True Minkowski Distance
  • 17.4. The Psychology of Rectangles
  • 17.5. Axiomatic Foundations of Minkowski Spaces
  • 17.6. Subadditivity and the MBR Metric
  • 17.7. Minkowski Spaces, Metric Spaces, and Psychological Models
  • 17.8. Exercises
  • 18. Scalar Products and Euclidean Distances
  • 18.1. The Scalar Product Function
  • 18.2. Collecting Scalar Products Empirically
  • 18.3. Scalar Products and Euclidean Distances: Formal Relations
  • 18.4. Scalar Products and Euclidean Distances: Empirical Relations
  • 18.5. MDS of Scalar Products
  • 18.6. Exercises
  • 19. Euclidean Embeddings
  • 19.1. Distances and Euclidean Distances
  • 19.2. Mapping Dissimilarities into Distances
  • 19.3. Maximal Dimensionality for Perfect Interval MDS
  • 19.4. Mapping Fallible Dissimilarities into Euclidean Distances
  • 19.5. Fitting Dissimilarities into a Euclidean Space
  • 19.6. Exercises
  • V. MDS and Related Methods
  • 20. Procrustes Procedures
  • 20.1. The Problem
  • 20.2. Solving the Orthogonal Procrustean Problem
  • 20.3. Examples for Orthogonal Procrustean Transformations
  • 20.4. Procrustean Similarity Transformations
  • 20.5. An Example of Procrustean Similarity Transformations
  • 20.6. Configurational Similarity and Correlation Coefficients
  • 20.7. Configurational Similarity and Congruence Coefficients
  • 20.8. Artificial Target Matrices in Procrustean Analysis
  • 20.9. Other Generalizations of Procrustean Analysis
  • 20.10. Exercises
  • 21. Three-Way Procrustean Models
  • 21.1. Generalized Procrustean Analysis
  • 21.2. Helm's Color Data
  • 21.3. Generalized Procrustean Analysis
  • 21.4. Individual Differences Models: Dimension Weights
  • 21.5. An Application of the Dimension-Weighting Model
  • 21.6. Vector Weightings
  • 21.7. Pindis, a Collection of Procrustean Models
  • 21.8. Exercises
  • 22. Three-Way MDS Models
  • 22.1. The Model: Individual Weights on Fixed Dimensions
  • 22.2. The Generalized Euclidean Model
  • 22.3. Overview of Three-Way Models in MDS
  • 22.4. Some Algebra of Dimension-Weighting Models
  • 22.5. Conditional and Unconditional Approaches
  • 22.6. On the Dimension-Weighting Models
  • 22.7. Exercises
  • 23. Modeling Asymmetric Data
  • 23.1. Symmetry and Skew-Symmetry
  • 23.2. A Simple Model for Skew-Symmetric Data
  • 23.3. The Gower Model for Skew-Symmetries
  • 23.4. Modeling Skew-Symmetry by Distances
  • 23.5. Embedding Skew-Symmetries as Drift Vectors into MDS Plots
  • 23.6. Analyzing Asymmetry by Unfolding
  • 23.7. The Slide-Vector Model
  • 23.8. The Hill-Climbing Model
  • 23.9. The Radius-Distance Model
  • 23.10. Using Asymmetry Models
  • 23.11. Overview
  • 23.12. Exercises
  • 24. Methods Related to MDS
  • 24.1. Principal Component Analysis
  • 24.2. Correspondence Analysis
  • 24.3. Exercises
  • VI. Appendices
  • A. 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
  • B. Notation
  • References
  • Author Index
  • Subject Index
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