Modern multidimensional scaling: theory and applications
(Publisher-supplied data) 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...
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Springer
2005
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Springer series in statistics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | (Publisher-supplied data) 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. |
| Beschreibung: | XXI, 614 S. graph. Darst. |
| ISBN: | 0387251502 9780387251509 |
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| 520 | 3 | |a (Publisher-supplied data) 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. | |
| 650 | 4 | |a Psychométrie | |
| 650 | 4 | |a Échelle multidimensionnelle | |
| 650 | 4 | |a Échelle multidimensionnelle - Informatique | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Algorithms | |
| 650 | 4 | |a Data Interpretation, Statistical |x methods | |
| 650 | 4 | |a Models, Statistical | |
| 650 | 4 | |a Multidimensional scaling | |
| 650 | 4 | |a Multidimensional scaling |x Data processing | |
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Datensatz im Suchindex
| _version_ | 1804133626039762944 |
|---|---|
| adam_text | Contents
Preface vii
I Fundamentals of MDS 1
1 The Four Purposes of Multidimensional Scaling 3
1.1 MDS as an Exploratory Technique.............. 4
1.2 MDS for Testing Structural Hypotheses........... 6
1.3 MDS for Exploring Psychological Structures......... 9
1.4 MDS as a Model of Similarity Judgments.......... 11
1.5 The Different Roots of MDS.................. 13
1.6 Exercises ............................ 15
2 Constructing MDS Representations 19
2.1 Constructing Ratio MDS Solutions.............. 19
2.2 Constructing Ordinal MDS Solutions............. 23
2.3 Comparing Ordinal and Ratio MDS Solutions........ 29
2.4 On Flat and Curved Geometries ............... 30
2.5 General Properties of Distance Representations....... 33
2.6 Exercises ............................ 34
3 MDS Models and Measures of Fit 37
3.1 Basics of MDS Models..................... 37
3.2 Errors, Loss Functions, and Stress.............. 41
xvi Contents
3.3 Stress Diagrams......................... 42
3.4 Stress per Point......................... 44
3.5 Evaluating Stress........................ 47
3.6 Recovering True Distances by Metric MDS ......... 55
3.7 Further Variants of MDS Models............... 57
3.8 Exercises ............................ 59
4 Three Applications of MDS 63
4.1 The Circular Structure of Color Similarities......... 63
4.2 The Regionality of Morse Codes Confusions......... 68
4.3 Dimensions of Facial Expressions............... 73
4.4 General Principles of Interpreting MDS Solutions...... 80
4.5 Exercises ............................ 82
5 MDS and Facet Theory 87
5.1 Facets and Regions in MDS Space.............. 87
5.2 Regional Laws ......................... 91
5.3 Multiple Facetizations..................... 93
5.4 Partitioning MDS Spaces Using Facet Diagrams....... 95
5.5 Prototypical Roles of Facets.................. 99
5.6 Criteria for Choosing Regions................. 100
5.7 Regions and Theory Construction............... 102
5.8 Regions, Clusters, and Factors ................ 104
5.9 Exercises ............................ 105
6 How to Obtain Proximities 111
6.1 Types of Proximities...................... Ill
6.2 Collecting Direct Proximities................. 112
6.3 Deriving Proximities by Aggregating over Other Measures . 119
6.4 Proximities from Converting Other Measures........ 125
6.5 Proximities from Co-Occurrence Data............ 126
6.6 Choosing a Particular Proximity............... 128
6.7 Exercises ............................ 130
II MDS Models and Solving MDS Problems 135
7 Matrix Algebra for MDS 137
7.1 Elementary Matrix Operations................ 137
7.2 Scalar Functions of Vectors and Matrices .......... 142
7.3 Computing Distances Using Matrix Algebra......... 144
7.4 Eigendecompositions...................... 146
7.5 Singular Value Decompositions................ 150
7.6 Some Further Remarks on SVD................ 152
7.7 Linear Equation Systems ................... 154
Contents xvii
7.8 Computing the Eigendecomposition ............. 157
7.9 Configurations that Represent Scalar Products....... 160
7.10 Rotations............................ 160
7.11 Exercises ............................ 163
8 A Majorization Algorithm for Solving MDS 169
8.1 The Stress Function for MDS................. 169
8.2 Mathematical Excursus: Differentiation ........... 171
8.3 Partial Derivatives and Matrix Traces............ 176
8.4 Minimizing a Function by Iterative Majorization...... 178
8.5 Visualizing the Majorization Algorithm for MDS...... 184
8.6 Majorizing Stress........................ 185
8.7 Exercises ............................ 194
9 Metric and Nonmetric MDS 199
9.1 Allowing for Transformations of the Proximities....... 199
9.2 Monotone Regression...................... 205
9.3 The Geometry of Monotone Regression............ 209
9.4 Tied Data in Ordinal MDS .................. 211
9.5 Rank-Images.......................... 213
9.6 Monotone Splines........................ 214
9.7 A Priori Transformations Versus Optimal Transformations . 221
9.8 Exercises ............................ 224
10 Confirmatory MDS 227
10.1 Blind Loss Functions...................... 227
10.2 Theory-Compatible MDS: An Example............ 228
10.3 Imposing External Constraints on MDS Representations . . 230
10.4 Weakly Constrained MDS................... 237
10.5 General Comments on Confirmatory MDS.......... 242
10.6 Exercises ............................ 244
11 MDS Fit Measures, Their Relations, and
Some Algorithms 247
11.1 Normalized Stress and Raw Stress.............. 247
11.2 Other Fit Measures and Recent Algorithms......... 250
11.3 Using Weights in MDS..................... 254
11.4 Exercises ............................ 258
12 Classical Scaling 261
12.1 Finding Coordinates in Classical Scaling........... 261
12.2 A Numerical Example for Classical Scaling ......... 263
12.3 Choosing a Different Origin.................. 264
12.4 Advanced Topics........................ 265
12.5 Exercises ............................ 267
xvüi Contents
13 Special Solutions, Degeneracies, and Local Minima 269
13.1 A Degenerate Solution in Ordinal MDS ........... 269
13.2 Avoiding Degenerate Solutions................ 272
13.3 Special Solutions: Almost Equal Dissimilarities....... 274
13.4 Local Minima.......................... 276
13.5 Unidimensional Scaling .................... 278
13.6 Full-Dimensional Scaling.................... 281
13.7 The Tunneling Method for Avoiding Local Minima..... 283
13.8 Distance Smoothing for Avoiding Local Minima....... 284
13.9 Exercises ............................ 288
III Unfolding 291
14 Unfolding 293
14.1 The Ideal-Point Model..................... 293
14.2 A Majorizing Algorithm for Unfolding............ 297
14.3 Unconditional Versus Conditional Unfolding......... 299
14.4 Trivial Unfolding Solutions and r2.............. 301
14.5 Isotonic Regions and Indeterminacies............. 305
14.6 Unfolding Degeneracies in Practice and Metric Unfolding . 308
14.7 Dimensions in Multidimensional Unfolding.......... 312
14.8 Multiple Versus Multidimensional Unfolding......... 313
14.9 Concluding Remarks...................... 314
14.10 Exercises............................ 314
15 Avoiding Trivial Solutions in Unfolding 317
15.1 Adjusting the Unfolding Data................. 317
15.2 Adjusting the Transformation................. 322
15.3 Adjustments to the Loss Function .............. 324
15.4 Summary............................ 330
15.5 Exercises ............................ 331
16 Special Unfolding Models 335
16.1 External Unfolding....................... 335
16.2 The Vector Model of Unfolding................ 336
16.3 Weighted Unfolding ...................... 342
16.4 Value Scales and Distances in Unfolding........... 345
16.5 Exercises ............................ 352
IV MDS Geometry as a Substantive Model 357
17 MDS as a Psychological Model 359
17.1 Physical and Psychological Space...............359
Contents xix
17.2 Minkowski Distances...................... 363
17.3 Identifying the True Minkowski Distance........... 367
17.4 The Psychology of Rectangles................. 372
17.5 Axiomatic Foundations of Minkowski Spaces......... 377
17.6 Subadditivity and the MBR Metric.............. 381
17.7 Minkowski Spaces, Metric Spaces, and Psychological Models 385
17.8 Exercises ............................ 386
18 Scalar Products and Euclidean Distances 389
18.1 The Scalar Product Function................. 389
18.2 Collecting Scalar Products Empirically............ 392
18.3 Scalar Products and Euclidean Distances: Formal Relations 397
18.4 Scalar Products and Euclidean Distances:
Empirical Relations ...................... 400
18.5 MDS of Scalar Products.................... 403
18.6 Exercises ............................ 408
19 Euclidean Embeddings 411
19.1 Distances and Euclidean Distances.............. 411
19.2 Mapping Dissimilarities into Distances............ 415
19.3 Maximal Dimensionality for Perfect Interval MDS ..... 418
19.4 Mapping Fallible Dissimilarities into Euclidean Distances . 419
19.5 Fitting Dissimilarities into a Euclidean Space........ 424
19.6 Exercises ............................ 425
V MDS and Related Methods 427
20 Procrustes Procedures 429
20.1 The Problem.......................... 429
20.2 Solving the Orthogonal Procrustean Problem........ 430
20.3 Examples for Orthogonal Procrustean Transformations . . . 432
20.4 Procrustean Similarity Transformations........... 434
20.5 An Example of Procrustean Similarity Transformations . . 436
20.6 Configurational Similarity and Correlation Coefficients . . . 437
20.7 Configurational Similarity and Congruence Coefficients . . . 439
20.8 Artificial Target Matrices in Procrustean Analysis ..... 441
20.9 Other Generalizations of Procrustean Analysis ....... 444
20.10 Exercises............................ 445
21 Three-Way Procrustean Models 449
21.1 Generalized Procrustean Analysis............... 449
21.2 Helm s Color Data....................... 451
21.3 Generalized Procrustean Analysis............... 454
21.4 Individual Differences Models: Dimension Weights..... 457
xx Contents
21.5 An Application of the Dimension-Weighting Model..... 462
21.6 Vector Weightings ....................... 465
21.7 Pindis, a Collection of Procrustean Models......... 469
21.8 Exercises ............................ 471
22 Three-Way MDS Models 473
22.1 The Model: Individual Weights on Fixed Dimensions .... 473
22.2 The Generalized Euclidean Model............... 479
22.3 Overview of Three-Way Models in MDS........... 482
22.4 Some Algebra of Dimension-Weighting Models....... 485
22.5 Conditional and Unconditional Approaches......... 489
22.6 On the Dimension-Weighting Models............. 491
22.7 Exercises ............................ 492
23 Modeling Asymmetric Data 495
23.1 Symmetry and Skew-Symmetry................ 495
23.2 A Simple Model for Skew-Symmetric Data.......... 497
23.3 The Gower Model for Skew-Symmetries........... 498
23.4 Modeling Skew-Symmetry by Distances........... 500
23.5 Embedding Skew-Symmetries as Drift Vectors into
MDS Plots ........................... 502
23.6 Analyzing Asymmetry by Unfolding............. 503
23.7 The Slide-Vector Model.................... 506
23.8 The Hill-Climbing Model ................... 509
23.9 The Radius-Distance Model.................. 512
23.10 Using Asymmetry Models .................. 514
23.11 Overview............................ 515
23.12 Exercises............................ 515
24 Methods Related to MDS 519
24.1 Principal Component Analysis ................ 519
24.2 Correspondence Analysis.................... 526
24.3 Exercises ............................ 537
VI Appendices 541
A Computer Programs for MDS 543
A.I Interactive MDS Programs .................. 544
A.2 MDS Programs with High-Resolution Graphics....... 550
A.3 MDS Programs without High-Resolution Graphics..... 562
B Notation 569
References 573
Contents xxi
Author Index 599
Subject Index 605
|
| any_adam_object | 1 |
| author | Borg, Ingwer 1945- Groenen, Patrick J. F. 1964- |
| author_GND | (DE-588)121327191 (DE-588)140945199 |
| author_facet | Borg, Ingwer 1945- Groenen, Patrick J. F. 1964- |
| author_role | aut aut |
| author_sort | Borg, Ingwer 1945- |
| author_variant | i b ib p j f g pjf pjfg |
| building | Verbundindex |
| bvnumber | BV020049873 |
| callnumber-first | B - Philosophy, Psychology, Religion |
| callnumber-label | BF39 |
| callnumber-raw | BF39.2.M85 |
| callnumber-search | BF39.2.M85 |
| callnumber-sort | BF 239.2 M85 |
| callnumber-subject | BF - Psychology |
| classification_rvk | QH 234 SK 840 |
| classification_tum | SOZ 720f PSY 470f |
| ctrlnum | (OCoLC)61260823 (DE-599)BVBBV020049873 |
| dewey-full | 519.535 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.535 |
| dewey-search | 519.535 |
| dewey-sort | 3519.535 |
| dewey-tens | 510 - Mathematics |
| discipline | Soziologie Psychologie Mathematik Wirtschaftswissenschaften |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV020049873 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T20:11:42Z |
| institution | BVB |
| isbn | 0387251502 9780387251509 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013370828 |
| oclc_num | 61260823 |
| open_access_boolean | |
| owner | DE-384 DE-706 DE-473 DE-BY-UBG DE-19 DE-BY-UBM DE-83 DE-N2 DE-M347 DE-188 DE-91 DE-BY-TUM DE-20 |
| owner_facet | DE-384 DE-706 DE-473 DE-BY-UBG DE-19 DE-BY-UBM DE-83 DE-N2 DE-M347 DE-188 DE-91 DE-BY-TUM DE-20 |
| physical | XXI, 614 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spelling | Borg, Ingwer 1945- Verfasser (DE-588)121327191 aut Modern multidimensional scaling theory and applications Ingwer Borg ; Patrick J. F. Groenen 2. ed. New York Springer 2005 XXI, 614 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in statistics (Publisher-supplied data) 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. Psychométrie Échelle multidimensionnelle Échelle multidimensionnelle - Informatique Datenverarbeitung Algorithms Data Interpretation, Statistical methods Models, Statistical Multidimensional scaling Multidimensional scaling Data processing Psychometrics Psychometrics methods Multidimensionale Skalierung (DE-588)4075090-5 gnd rswk-swf Multidimensionale Skalierung (DE-588)4075090-5 s DE-604 Groenen, Patrick J. F. 1964- Verfasser (DE-588)140945199 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013370828&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borg, Ingwer 1945- Groenen, Patrick J. F. 1964- Modern multidimensional scaling theory and applications Psychométrie Échelle multidimensionnelle Échelle multidimensionnelle - Informatique Datenverarbeitung Algorithms Data Interpretation, Statistical methods Models, Statistical Multidimensional scaling Multidimensional scaling Data processing Psychometrics Psychometrics methods Multidimensionale Skalierung (DE-588)4075090-5 gnd |
| subject_GND | (DE-588)4075090-5 |
| title | Modern multidimensional scaling theory and applications |
| title_auth | Modern multidimensional scaling theory and applications |
| title_exact_search | Modern multidimensional scaling theory and applications |
| title_full | Modern multidimensional scaling theory and applications Ingwer Borg ; Patrick J. F. Groenen |
| title_fullStr | Modern multidimensional scaling theory and applications Ingwer Borg ; Patrick J. F. Groenen |
| title_full_unstemmed | Modern multidimensional scaling theory and applications Ingwer Borg ; Patrick J. F. Groenen |
| title_short | Modern multidimensional scaling |
| title_sort | modern multidimensional scaling theory and applications |
| title_sub | theory and applications |
| topic | Psychométrie Échelle multidimensionnelle Échelle multidimensionnelle - Informatique Datenverarbeitung Algorithms Data Interpretation, Statistical methods Models, Statistical Multidimensional scaling Multidimensional scaling Data processing Psychometrics Psychometrics methods Multidimensionale Skalierung (DE-588)4075090-5 gnd |
| topic_facet | Psychométrie Échelle multidimensionnelle Échelle multidimensionnelle - Informatique Datenverarbeitung Algorithms Data Interpretation, Statistical methods Models, Statistical Multidimensional scaling Multidimensional scaling Data processing Psychometrics Psychometrics methods Multidimensionale Skalierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013370828&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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