Modern Multidimensional Scaling Theory and Applications

ISBN-10: 0387251502

ISBN-13: 9780387251509

Edition: 2nd 2005 (Revised)

Authors: Ingwer Borg, P. J. F. Groenen

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The book provides a comprehensive treatment of multidimensional scaling (MDS), a family of statistical techniques for analyzing the structure of (dis)similarity data. Such data are widespread, including, for example, intercorrelations of survey items, direct ratings on the similarity on choice objects, or trade indices for a set of countries. MDS represents the data as distances among points in a geometric space of low dimensionality. This map can help to see patterns in the data that are not obvious from the data matrices. MDS is also used as a psychological model for judgments of similarity and preference. This book may be used as an introduction to MDS for students in psychology, sociology, and marketing. The prerequisite is an elementary background in statistics. The book is also well suited for a variety of advanced courses on MDS topics. All the mathematics required for more advanced topics is developed systematically. This second edition is not only a complete overhaul of its predecessor, but also adds some 140 pages of new material. Many chapters are revised or have sections reflecting new insights and developments in MDS. There are two new chapters, one on asymmetric models and the other on unfolding. There are also numerous exercises that help the reader to practice what he or she has learned, and to delve deeper into the models and its intricacies. These exercises make it easier to use this edition in a course. All data sets used in the book can be downloaded from the web. The appendix on computer programs has also been updated and enlarged to reflect the state of the art.
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Book details

List price: $199.00
Edition: 2nd
Copyright year: 2005
Publisher: Springer
Publication date: 8/4/2005
Binding: Hardcover
Pages: 614
Size: 6.50" wide x 9.25" long x 1.25" tall
Weight: 2.244
Language: English

Fundamentals of MDS
The Four Purposes of Multidimensional Scaling
MDS as an Exploratory Technique
MDS for Testing Structural Hypotheses
MDS for Exploring Psychological Structures
MDS as a Model of Similarity Judgments
The Different Roots of MDS
Constructing MDS Representations
Constructing Ratio MDS Solutions
Constructing Ordinal MDS Solutions
Comparing Ordinal and Ratio MDS Solutions
On Flat and Curved Geometries
General Properties of Distance Representations
MDS Models and Measures of Fit
Basics of MDS Models
Errors, Loss Functions, and Stress
Stress Diagrams
Stress per Point
Evaluating Stress
Recovering True Distances by Metric MDS
Further Variants of MDS Models
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
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
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
MDS Models and Solving MDS Problems
Matrix Algebra for MDS
Elementary Matrix Operations
Scalar Functions of Vectors and Matrices
Computing Distances Using Matrix Algebra
Singular Value Decompositions
Some Further Remarks on SVD
Linear Equation Systems
Computing the Eigendecomposition
Configurations that Represent Scalar Products
A Majorization Algorithm for Solving MDS
The Stress Function for MDS
Mathematical Excursus: Differentiation
Partial Derivatives and Matrix Traces
Minimizing a Function by Iterative Majorization
Visualizing the Majorization Algorithm for MDS
Majorizing Stress
Metric and Nonmetric MDS
Allowing for Transformations of the Proximities
Monotone Regression
The Geometry of Monotone Regression
Tied Data in Ordinal MDS
Monotone Splines
A Priori Transformations Versus Optimal Transformations
Confirmatory MDS
Blind Loss Functions
Theory-Compatible MDS: An Example
Imposing External Constraints on MDS Representations
Weakly Constrained MDS
General Comments on Confirmatory MDS
MDS Fit Measures, Their Relations, and Some Algorithms
Normalized Stress and Raw Stress
Other Fit Measures and Recent Algorithms
Using Weights in MDS
Classical Scaling
Finding Coordinates in Classical Scaling
A Numerical Example for Classical Scaling
Choosing a Different Origin
Advanced Topics
Special Solutions, Degeneracies, and Local Minima
A Degenerate Solution in Ordinal MDS
Avoiding Degenerate Solutions
Special Solutions: Almost Equal Dissimilarities
Local Minima
Unidimensional Scaling
Full-Dimensional Scaling
The Tunneling Method for Avoiding Local Minima
Distance Smoothing for Avoiding Local Minima
The Ideal-Point Model
A Majorizing Algorithm for Unfolding
Unconditional Versus Conditional Unfolding
Trivial Unfolding Solutions and [sigma subscript 2]
Isotonic Regions and Indeterminacies
Unfolding Degeneracies in Practice and Metric Unfolding
Dimensions in Multidimensional Unfolding
Multiple Versus Multidimensional Unfolding
Concluding Remarks
Avoiding Trivial Solutions in Unfolding
Adjusting the Unfolding Data
Adjusting the Transformation
Adjustments to the Loss Function
Special Unfolding Models
External Unfolding
The Vector Model of Unfolding
Weighted Unfolding
Value Scales and Distances in Unfolding
MDS Geometry as a Substantive Model
MDS as a Psychological Model
Physical and Psychological Space
Minkowski Distances
Identifying the True Minkowski Distance
The Psychology of Rectangles
Axiomatic Foundations of Minkowski Spaces
Subadditivity and the MBR Metric
Minkowski Spaces, Metric Spaces, and Psychological Models
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
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
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
Three-Way Procrustean Models
Generalized Procrustean Analysis
Helm's Color Data
Generalized Procrustean Analysis
Individual Differences Models: Dimension Weights
An Application of the Dimension-Weighting Model
Vector Weightings
Pindis, a Collection of Procrustean Models
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
Modeling Asymmetric 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
Methods Related to MDS
Principal Component Analysis
Correspondence Analysis
Computer Programs for MDS
Interactive MDS Programs
MDS Programs with High-Resolution Graphics
MDS Programs without High-Resolution Graphics
Author Index
Subject Index
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