Multidimensional Scaling

✍ Scribed by Trevor F. Cox, Michael A. A. Cox

• Publisher: Chapman & Hall/CRC
• Year: 2000
• Tongue: English
• Leaves: 295
• Series: MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY 88
• Edition: 2
• Category: Library

✦ Synopsis

Multidimensional scaling covers a variety of statistical techniques in the area of multivariate data analysis. Geared toward dimensional reduction and graphical representation of data, it arose within the field of the behavioral sciences, but now holds techniques widely used in many disciplines. Multidimensional Scaling, Second Edition extends the popular first edition and brings it up to date. It concisely but comprehensively covers the area, summarizing the mathematical ideas behind the various techniques and illustrating the techniques with real-life examples. A computer disk containing programs and data sets accompanies the book.

✦ Table of Contents

Cover......Page 1
Series Title List......Page 2
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 12
1.1 Introduction......Page 13
Ordinal scale......Page 15
Number of ways......Page 16
1.2.2 Multidimensional scaling models......Page 17
Unidimensional scaling......Page 18
Unfolding......Page 19
1.3 Proximities......Page 20
1.3.1 Similarity/ dissimilarity coefficients for mixed data......Page 26
I.3.3 Similarity of species populations......Page 30
I.3.5 The metric nature of dissimilarities......Page 33
1.3.6 Dissimilarity of variables......Page 34
Example......Page 36
Nominal and ordinal data......Page 24
1.4 Matrix results......Page 37
1.4.2 The singular value decomposition......Page 38
An example......Page 39
Generalized SVD......Page 40
1.4.3 The Moore-Penrose inverse......Page 41
2.2 Classical scaling......Page 42
2.2.1 Recovery of coordinates......Page 43
2.2.2 Dissimilarities as Euclidean distances......Page 45
2.2.3 Classical scaling in practice......Page 47
2.2.5 A practical algorithm for classical scaling......Page 49
2.2.6 A grave example......Page 50
2.2.7 Classical scaling and principal components......Page 54
Optimal transformations of the variables......Page 55
2.2.8 The additive constant problem......Page 56
2.4 Metric least squares scaling......Page 60
Least squares scaling of the skulls......Page 62
2.5 Critchley’s intermediate method......Page 63
2.6 Unidimensional scaling......Page 64
2.6.1 A classic example......Page 66
2.7 Grouped dissimilarities......Page 68
2.8 Inverse scaling......Page 69
3.1 Introduction......Page 72
A simple example......Page 73
3.1.1 Rp space and the Minkowski metric......Page 74
3.2 Kruskal’s approach......Page 75
3.2.1 Minimising S with respect to the disparities......Page 76
3.2.2 A configuration with minimum stress......Page 79
3.2.3 Kruskal's iterative technique......Page 80
3.2.4 Nonmetric scaling of breakfast cereals......Page 82
3.2.5 STRESS l/2, monotonicity, ties and missing data......Page 84
3.3 The Guttman approach......Page 86
Differentiability of stress......Page 87
Limits for stress......Page 88
3.4.1 Interpretation of stress......Page 90
3.5 How many dimensions?......Page 99
3.6 Starting configurations......Page 100
3.7 Interesting axes in the configuration......Page 101
4.1 Other formulations of MDS......Page 104
4.2 MDS Diagnostics......Page 105
Robust parameter estimation......Page 107
4.4 Interactive MDS......Page 109
4.5 Dynamic MDS......Page 110
An example......Page 112
4.6 Constrained MDS......Page 114
4.6.1 Spherical MDS......Page 116
4.7 Statistical inference for MDS......Page 118
Asymptotic confidence regions......Page 121
4.8 Asymmetric dissimilarities......Page 127
5.1 Introduction......Page 133
5.2 Procrustes analysis......Page 134
Optimal rotation......Page 136
5.2.1 Procrustes analysis in practice......Page 137
5.2.2 The projection case......Page 139
5.3 Historic maps......Page 140
5.4.1 Weighted Procrustes rotation......Page 142
5.4.2 Generalized Procrustes analpsis......Page 145
5.4.3 The coefficient of congruence......Page 147
5.4.4 Oblique Procrustes problem......Page 148
5.4.5 Perturbation analysis......Page 149
6.2 Monkeys......Page 151
6.3 Whisky......Page 153
6.4 Aeroplanes......Page 156
6.5 Yoghurts......Page 158
6.6 Bees......Page 159
7.2 The classic biplot......Page 162
7.2.1 An example......Page 163
7.2.2 Principal component biplots......Page 166
7.3 Another approach......Page 168
7.4 Categorical variables......Page 171
8.1 Introduction......Page 173
8.2 Nonmetric unidimensional unfolding......Page 174
8.3 Nonmetric multidimensional unfolding......Page 177
8.4 Metric multidimensional unfolding......Page 181
8.4.1 The rating of nations......Page 185
9.2 Analysis of two- way contingency tables......Page 188
9.2.1 Distances between rows (columns) in a contingency table......Page 191
9.3 The theory of correspondence analysis......Page 192
9.3.1 The cancer example......Page 194
A single plot......Page 197
9.3.2 Inertia......Page 198
9.4.1 Algorithm for solution......Page 200
9.4.2 An example: the Munsingen data......Page 201
9.4.3 The whisky data......Page 202
9.4.4 The correspondence analysis connection......Page 204
9.4.5 Two-way weighted dissimilarity coefficients......Page 205
9.5 Multiple correspondence analysis......Page 207
9.5.1 A three-way example......Page 209
10.2 The Tucker-Messick model......Page 211
10.3.1 The algorithm for solution......Page 212
10.3.2 Identifying groundwater populations......Page 214
10.3.3 Extended INDSCAL models......Page 216
Carroll-Chang decomposition of W i......Page 217
10.5 PINDIS......Page 218
11.1.1 The theory......Page 222
Level constraints......Page 223
The optimal scaling phase......Page 224
Model estimation phase......Page 225
11.2 SMACOF......Page 226
11.2. I The majorization algorithm......Page 227
11.2.2 The majorizing method for nonmetric MDS......Page 230
11.3 Gifi......Page 231
11.3.1 Homogeneity......Page 232
HOMALS......Page 233
12.1 CANDECOMP, PARAFAC and CANDELINC......Page 238
12.2 DEDICOM and GIPSCAL......Page 240
12.3 The Tucker models......Page 241
12.4 One-mode, n-way models......Page 243
12.5 Two-mode, three-way asymmetric scaling......Page 248
12.6 Three-way unfolding......Page 250
CRC Press Downloads and Updates......Page 0
A.l Computer programs......Page 251
Minimum system requirements......Page 252
DOS Users......Page 253
A.2.3 To run the menu......Page 254
Data manipulation programs......Page 255
The data sets......Page 256
Figures in the text......Page 257
Example 1: Classical scaling of the skull data......Page 259
Example 2: Nonmetric MDS of the Kellog data......Page 260
Example 4: Individual differences scaling of groundwater samples......Page 261
Example 6: Reciprocal Averaging of the Munsingen data......Page 262
A.5.1 Data format......Page 263
Dissimilarities for Individual Differences Scaling......Page 264
Indicator Matrix......Page 265
DAT2UNF......Page 266
LINEAR......Page 267
MDSCAL-T......Page 268
NONLIN......Page 269
RECAVDIS......Page 270
UNFOLDIN......Page 271
VECJOIN......Page 272
VEC-PLOT......Page 273
B......Page 274
C......Page 275
D......Page 277
F......Page 279
G......Page 280
H......Page 282
K......Page 284
L......Page 286
M......Page 287
P......Page 288
R......Page 289
S......Page 290
T......Page 292
V......Page 293
W......Page 294
Z......Page 295