Modern Multidimensional Scaling

by Borg, Ingwer; Groenen, Patrick J. F.

• ISBN13: 9780387251509
• ISBN10: 0387251502
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-08-18
• Publisher: Springer Verlag

Summary

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.

Author Biography

Ingwer Borg is Scientific Director at the Center for Survey Methodology (ZUMA) in Mannheim, Germany, and Professor of Psychology at the University of Giessen, Germany Patrick J.F. Groenen is Professor in Statistics at the Econometric Institute of the Erasmus University Rotterdam, the Netherlands

Table of Contents

Preface	vii
I Fundamentals of MDS	1	(134)
The Four Purposes of Multidimensional Scaling	3	(16)
MDS as an Exploratory Technique	4	(2)
MDS for Testing Structural Hypotheses	6	(3)
MDS for Exploring Psychological Structures	9	(2)
MDS as a Model of Similarity Judgments	11	(2)
The Different Roots of MDS	13	(2)
Exercises	15	(4)
Constructing MDS Representations	19	(18)
Constructing Ratio MDS Solutions	19	(4)
Constructing Ordinal MDS Solutions	23	(6)
Comparing Ordinal and Ratio MDS Solutions	29	(1)
On Flat and Curved Geometries	30	(3)
General Properties of Distance Representations	33	(1)
Exercises	34	(3)
MDS Models and Measures of Fit	37	(26)
Basics of MDS Models	37	(4)
Errors, Loss Functions, and Stress	41	(1)
Stress Diagrams	42	(2)
Stress per Point	44	(3)
Evaluating Stress	47	(8)
Recovering True Distances by Metric MDS	55	(2)
Further Variants of MDS Models	57	(2)
Exercises	59	(4)
Three Applications of MDS	63	(24)
The Circular Structure of Color Similarities	63	(5)
The Regionality of Morse Codes Confusions	68	(5)
Dimensions of Facial Expressions	73	(7)
General Principles of Interpreting MDS Solutions	80	(2)
Exercises	82	(5)
MDS and Facet Theory	87	(24)
Facets and Regions in MDS Space	87	(4)
Regional Laws	91	(2)
Multiple Facetizations	93	(2)
Partitioning MDS Spaces Using Facet Diagrams	95	(4)
Prototypical Roles of Facets	99	(1)
Criteria for Choosing Regions	100	(2)
Regions and Theory Construction	102	(2)
Regions, Clusters, and Factors	104	(1)
Exercises	105	(6)
How to Obtain Proximities	111	(24)
Types of Proximities	111	(1)
Collecting Direct Proximities	112	(7)
Deriving Proximities by Aggregating over Other Measures	119	(6)
Proximities from Converting Other Measures	125	(1)
Proximities from Co-Occurrence Data	126	(2)
Choosing a Particular Proximity	128	(2)
Exercises	130	(5)
II MDS Models and Solving MDS Problems	135	(156)
Matrix Algebra for MDS	137	(32)
Elementary Matrix Operations	137	(5)
Scalar Functions of Vectors and Matrices	142	(2)
Computing Distances Using Matrix Algebra	144	(2)
Eigendecompositions	146	(4)
Singular Value Decompositions	150	(2)
Some Further Remarks on SVD	152	(2)
Linear Equation Systems	154	(3)
Computing the Eigendecomposition	157	(3)
Configurations that Represent Scalar Products	160	(1)
Rotations	160	(3)
Exercises	163	(6)
A Majorization Algorithm for Solving MDS	169	(30)
The Stress Function for MDS	169	(2)
Mathematical Excursus: Differentiation	171	(5)
Partial Derivatives and Matrix Traces	176	(2)
Minimizing a Function by Iterative Majorization	178	(6)
Visualizing the Majorization Algorithm for MDS	184	(1)
Majorizing Stress	185	(9)
Exercises	194	(5)
Metric and Nonmetric MDS	199	(28)
Allowing for Transformations of the Proximities	199	(6)
Monotone Regression	205	(4)
The Geometry of Monotone Regression	209	(2)
Tied Data in Ordinal MDS	211	(2)
Rank-Images	213	(1)
Monotone Splines	214	(7)
A Priori Transformations Versus Optimal Transformations	221	(3)
Exercises	224	(3)
Confirmatory MDS	227	(20)
Blind Loss Functions	227	(1)
Theory-Compatible MDS: An Example	228	(2)
Imposing External Constraints on MDS Representations	230	(7)
Weakly Constrained MDS	237	(5)
General Comments on Confirmatory MDS	242	(2)
Exercises	244	(3)
MDS Fit Measures, Their Relations, and Some Algorithms	247	(14)
Normalized Stress and Raw Stress	247	(3)
Other Fit Measures and Recent Algorithms	250	(4)
Using Weights in MDS	254	(4)
Exercises	258	(3)
Classical Scaling	261	(8)
Finding Coordinates in Classical Scaling	261	(2)
A Numerical Example for Classical Scaling	263	(1)
Choosing a Different Origin	264	(1)
Advanced Topics	265	(2)
Exercises	267	(2)
Special Solutions, Degeneracies, and Local Minima	269	(22)
A Degenerate Solution in Ordinal MDS	269	(3)
Avoiding Degenerate Solutions	272	(2)
Special Solutions: Almost Equal Dissimilarities	274	(2)
Local Minima	276	(2)
Unidimensional Scaling	278	(3)
Full-Dimensional Scaling	281	(2)
The Tunneling Method for Avoiding Local Minima	283	(1)
Distance Smoothing for Avoiding Local Minima	284	(4)
Exercises	288	(3)
III Unfolding	291	(66)
Unfolding	293	(24)
The Ideal-Point Model	293	(4)
A Majorizing Algorithm for Unfolding	297	(2)
Unconditional Versus Conditional Unfolding	299	(2)
Trivial Unfolding Solutions and σ2	301	(4)
Isotonic Regions and Indeterminacies	305	(3)
Unfolding Degeneracies in Practice and Metric Unfolding	308	(4)
Dimensions in Multidimensional Unfolding	312	(1)
Multiple Versus Multidimensional Unfolding	313	(1)
Concluding Remarks	314	(1)
Exercises	314	(3)
Avoiding Trivial Solutions in Unfolding	317	(18)
Adjusting the Unfolding Data	317	(5)
Adjusting the Transformation	322	(2)
Adjustments to the Loss Function	324	(6)
Summary	330	(1)
Exercises	331	(4)
Special Unfolding Models	335	(22)
External Unfolding	335	(1)
The Vector Model of Unfolding	336	(6)
Weighted Unfolding	342	(3)
Value Scales and Distances in Unfolding	345	(7)
Exercises	352	(5)
IV MDS Geometry as a Substantive Model	357	(70)
MDS as a Psychological Model	359	(30)
Physical and Psychological Space	359	(4)
Minkowski Distances	363	(4)
Identifying the True Minkowski Distance	367	(5)
The Psychology of Rectangles	372	(5)
Axiomatic Foundations of Minkowski Spaces	377	(4)
Subadditivity and the MBR Metric	381	(4)
Minkowski Spaces, Metric Spaces, and Psychological Models	385	(1)
Exercises	386	(3)
Scalar Products and Euclidean Distances	389	(22)
The Scalar Product Function	389	(3)
Collecting Scalar Products Empirically	392	(5)
Scalar Products and Euclidean Distances: Formal Relations	397	(3)
Scalar Products and Euclidean Distances: Empirical Relations	400	(3)
MDS of Scalar Products	403	(5)
Exercises	408	(3)
Euclidean Embeddings	411	(16)
Distances and Euclidean Distances	411	(4)
Mapping Dissimilarities into Distances	415	(3)
Maximal Dimensionality for Perfect Interval MDS	418	(1)
Mapping Fallible Dissimilarities into Euclidean Distances	419	(5)
Fitting Dissimilarities into a Euclidean Space	424	(1)
Exercises	425	(2)
V MDS and Related Methods	427	(114)
Procrustes Procedures	429	(20)
The Problem	429	(1)
Solving the Orthogonal Procrustean Problem	430	(2)
Examples for Orthogonal Procrustean Transformations	432	(2)
Procrustean Similarity Transformations	434	(2)
An Example of Procrustean Similarity Transformations	436	(1)
Configurational Similarity and Correlation Coefficients	437	(2)
Configurational Similarity and Congruence Coefficients	439	(2)
Artificial Target Matrices in Procrustean Analysis	441	(3)
Other Generalizations of Procrustean Analysis	444	(1)
Exercises	445	(4)
Three-Way Procrustean Models	449	(24)
Generalized Procrustean Analysis	449	(2)
Helm's Color Data	451	(3)
Generalized Procrustean Analysis	454	(3)
Individual Differences Models: Dimension Weights	457	(5)
An Application of the Dimension-Weighting Model	462	(3)
Vector Weightings	465	(4)
PINDIS, a Collection of Procrustean Models	469	(2)
Exercises	471	(2)
Three-Way MDS Models	473	(22)
The Model: Individual Weights on Fixed Dimensions	473	(6)
The Generalized Euclidean Model	479	(3)
Overview of Three-Way Models in MDS	482	(3)
Some Algebra of Dimension-Weighting Models	485	(4)
Conditional and Unconditional Approaches	489	(2)
On the Dimension-Weighting Models	491	(1)
Exercises	492	(3)
Modeling Asymmetric Data	495	(24)
Symmetry and Skew-Symmetry	495	(2)
A Simple Model for Skew-Symmetric Data	497	(1)
The Gower Model for Skew-Symmetries	498	(2)
Modeling Skew-Symmetry by Distances	500	(2)
Embedding Skew-Symmetries as Drift Vectors into MDS Plots	502	(1)
Analyzing Asymmetry by Unfolding	503	(3)
The Slide-Vector Model	506	(3)
The Hill-Climbing Model	509	(3)
The Radius-Distance Model	512	(2)
Using Asymmetry Models	514	(1)
Overview	515	(1)
Exercises	515	(4)
Methods Related to MDS	519	(22)
Principal Component Analysis	519	(7)
Correspondence Analysis	526	(11)
Exercises	537	(4)
VI Appendices	541	(32)
A Computer Programs for MDS	543	(26)
A.1 Interactive MDS Programs	544	(6)
A.2 MDS Programs with High-Resolution Graphics	550	(12)
A.3 MDS Programs without High-Resolution Graphics	562	(7)
B Notation	569	(4)
References	573	(26)
Author Index	599	(6)
Subject Index	605